Chapter 10

 

Approximate theories for solids with special shapes:

 rods, beams, membranes, plates and shells

 

 

 

10.2 Motion and Deformation of slender rods

 

The figure illustrates the problem to be solved.   We suppose that a long, initially straight rod is subjected to forces and moments that cause it to stretch, bend and twist into a complex three dimensional shape, which we wish to determine.  The initial shape need not necessarily be stress free.  Consequently, we can solve problems involving a rod that is bent and twisted in its unloaded configuration (such as a helical spring) by first mapping it onto an intermediate, straight reference configuration, and then analyzing the deformation of this shape.

 

 

 

 

 

 

10.2.1 Variables characterizing the geometry of the rod’s cross-section

 

The figure illustrates a generic cross-section of the (undeformed) rod.

 

We will characterize the shape of the cross-section as follows:

  1. We introduce three mutually perpendicular, unit basis vectors { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@ , with e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  pointing parallel to the axis of the cylinder, and e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaaaaa@372B@  parallel to the principal moments of area of the (undeformed) cross section. 
  2. We introduce a coordinate system ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@366E@  within the cross section, with origin at the centroid of the cross-section.
  3. The cross-sectional area of the rod is denoted by A= A d A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacqGH9aqpdaWdrbqaaiaadsgaaS qaaiaadgeaaeqaniabgUIiYdGccaWGbbaaaa@371D@
  4. The principal moments of area of the cross-section are defined as

I 1 = A x 2 2 dA I 2 = A x 1 2 dA I 3 = A ( x 1 2 + x 2 2 )dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccq GH9aqpdaWdrbqaaiaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGc caWGKbGaamyqaaWcbaGaamyqaaqab0Gaey4kIipakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadMeadaWgaaWcbaGaaG OmaaqabaGccqGH9aqpdaWdrbqaaiaadIhadaqhaaWcbaGaaGymaaqa aiaaikdaaaGccaWGKbGaamyqaaWcbaGaamyqaaqab0Gaey4kIipaki aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadMeada WgaaWcbaGaaG4maaqabaGccqGH9aqpdaWdrbqaamaabmaabaGaamiE amaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaadIhadaqhaa WcbaGaaGOmaaqaaiaaikdaaaaakiaawIcacaGLPaaacaWGKbGaamyq aaWcbaGaamyqaaqab0Gaey4kIipaaaa@828E@

  1. We define a moment of area tensor H for the cross-section, with components H 11 = I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIeadaWgaaWcbaGaaGymaiaaigdaae qaaOGaeyypa0JaamysamaaBaaaleaacaaIXaaabeaaaaa@35BC@ , H 22 = I 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIeadaWgaaWcbaGaaGOmaiaaikdaae qaaOGaeyypa0JaamysamaaBaaaleaacaaIYaaabeaaaaa@35BF@ , H 33 = I 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIeadaWgaaWcbaGaaG4maiaaiodaae qaaOGaeyypa0JaamysamaaBaaaleaacaaIZaaabeaaaaa@35C2@  and all other components zero.
  2. In calculations to follow, it will be helpful to note that, because of the choice of origin and coordinate system,

A x 1 dA = A x 2 dA = A x 1 x 2 dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaamiEamaaBaaaleaacaaIXa aabeaakiaadsgacaWGbbaaleaacaWGbbaabeqdcqGHRiI8aOGaeyyp a0JaaGPaVpaapefabaGaamiEamaaBaaaleaacaaIYaaabeaakiaads gacaWGbbaaleaacaWGbbaabeqdcqGHRiI8aOGaeyypa0JaaGPaVpaa pefabaGaamiEamaaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcba GaaGOmaaqabaGccaWGKbGaamyqaiabg2da9iaaicdaaSqaaiaadgea aeqaniabgUIiYdaaaa@4D75@

 

Principal moments of area and their directions are listed for a few simple geometries below.  Recall also that area moments of inertia for hollow sections can be calculated by subtraction.

 

Areas and area moments of inertia for simple cross-sections

A=ab I 1 = b 3 a/12 I 2 = a 3 b/12 I 3 =ab( a 2 + b 2 )/12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyqaiabg2da9iaadggacaWGIb aabaGaamysamaaBaaaleaacaaIXaaabeaakiabg2da9iaadkgadaah aaWcbeqaaiaaiodaaaGccaWGHbGaai4laiaaigdacaaIYaaabaGaam ysamaaBaaaleaacaaIYaaabeaakiabg2da9iaadggadaahaaWcbeqa aiaaiodaaaGccaWGIbGaai4laiaaigdacaaIYaaabaGaamysamaaBa aaleaacaaIZaaabeaakiabg2da9iaadggacaWGIbGaaiikaiaadgga daahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGIbWaaWbaaSqabeaaca aIYaaaaOGaaiykaiaac+cacaaIXaGaaGOmaaaaaa@5037@

A=πab I 1 =π b 3 a/4 I 2 =π a 3 b/4 I 3 =πab( a 2 + b 2 )/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyqaiabg2da9iabec8aWjaadg gacaWGIbaabaGaamysamaaBaaaleaacaaIXaaabeaakiabg2da9iab ec8aWjaadkgadaahaaWcbeqaaiaaiodaaaGccaWGHbGaai4laiaais daaeaacaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeqiWdaNa amyyamaaCaaaleqabaGaaG4maaaakiaadkgacaGGVaGaaGinaaqaai aadMeadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcqaHapaCcaWGHbGa amOyaiaacIcacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaam OyamaaCaaaleqabaGaaGOmaaaakiaacMcacaGGVaGaaGinaaaaaa@5500@

A= a 2 3 /4 I 1 = I 2 = a 4 3 /96 I 3 = a 4 3 /48 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyqaiabg2da9iaadggadaahaa WcbeqaaiaaikdaaaGcdaGcaaqaaiaaiodaaSqabaGccaGGVaGaaGin aaqaaiaadMeadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcaWGjbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0JaamyyamaaCaaaleqabaGaaGin aaaakmaakaaabaGaaG4maaWcbeaakiaac+cacaaI5aGaaGOnaaqaai aadMeadaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaWGHbWaaWbaaSqa beaacaaI0aaaaOWaaOaaaeaacaaIZaaaleqaaOGaai4laiaaisdaca aI4aaaaaa@48BD@

 

 

 

10.2.2 Coordinate systems and variables characterizing the deformation of a rod

 

 The orientation of the straight rod is characterized using the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  basis described in the preceding section.

 The position vector of a material particle in the reference configuration is x= x i e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0JaamiEamaaBaaale aacaWGPbaabeaakiaahwgadaWgaaWcbaGaamyAaaqabaaaaa@38F6@ , where x 1 = x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaamiEamaaBaaaleaacaaIYaaabeaakiabg2da9iaaicda aaa@3969@  corresponds to the centroid of the cross section, and x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@  is the height above the base of the cylinder.

 After deformation, the axis of the cylinder lies on a smooth curve.  The point that lies at x= x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0JaamiEamaaBaaale aacaaIZaaabeaakiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@3894@  in the undeformed solid moves to a new position y=r( x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0JaaCOCaiaacIcaca WG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@3912@  after deformation.

 The orientation of the cross-section after deformation will be described by introducing a basis of mutually perpendicular unit vectors { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ , chosen so that m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaiodaaeqaaa aa@34A5@  is parallel to the axis of the deformed rod, and is m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaigdaaeqaaa aa@34A3@  parallel to the line of material points that lay along e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa aa@349B@  in the reference configuration (or, more precisely, parallel to the projection of this line perpendicular to m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaiodaaeqaaa aa@34A5@  ).  Note that the three basis vectors { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  are all functions of x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@ , and if the rod is moving, they are also functions of time.

 The orientation of { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  can be specified by three Euler angles (θ,ϕ,ψ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaeqiUdeNaaiilaiabew9aMj aacYcacqaHipqEcaGGPaaaaa@3ACB@ , which characterize the rigid rotation that maps { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  onto  { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ .   To visualize the significance of the three angles, note that the rotation can be accomplished in three stages (i) rotate the basis vectors through an angle ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@348E@  about the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  axis.   This results in a new set of vectors { e ^ 1 , e ^ 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiqahwgagaqcamaaBaaale aacaaIXaaabeaakiaacYcaceWHLbGbaKaadaWgaaWcbaGaaGOmaaqa baGccaGGSaGaaCyzamaaBaaaleaacaaIZaaabeaaaOGaay5Eaiaaw2 haaaaa@3C17@ ; (ii) Rotate these new vectors through an angle θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@347C@  about the e ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHLbGbaKaadaWgaaWcbaGaaGOmaa qabaaaaa@34AC@  axis.  This rotates the vectors onto a second configuration { e ˜ 1 , e ^ 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiqahwgagaacamaaBaaale aacaaIXaaabeaakiaacYcaceWHLbGbaKaadaWgaaWcbaGaaGOmaaqa baGccaGGSaGaaCyBamaaBaaaleaacaaIZaaabeaaaOGaay5Eaiaaw2 haaaaa@3C1E@ ; (iii) finally, rotate these vectors through angle ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  about the m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaiodaaeqaaa aa@34A5@  direction, to create the { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  vectors.   

 Relationships between the Euler angles and the curve characterizing the axis of the rod will be given shortly: these results will show that the angles ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@348E@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@347C@  can be determined from the shape of the axis.   The angle ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  is an independent degree of freedom, and quantifies the rotation of the rod’s cross-section about its axis.  

 We let s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbaaaa@33BE@  denote the arc length measured along the axis of the rod in the deformed configuration.

 The velocity of the rod is characterized by the velocity vector of its axis, v=dr/dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWGKbGaaCOCaiaac+ cacaWGKbGaamiDaaaa@3706@

 The rate of rotation of the rod is characterized by the angular velocity ω= ω i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8acqGH9aqpcqaHjpWDdaWgaaWcba GaamyAaaqabaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaaaa@37E4@  of the basis vectors { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ .  It will be shown in Sect 10.2.3 that the angular velocity can be related to the velocity v of the bar’s axis and its twist ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  by

ω= m 3 × dv ds + ψ ˙ m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8acqGH9aqpcaWHTbWaaSbaaSqaai aaiodaaeqaaOGaey41aq7aaSaaaeaacaWGKbGaaCODaaqaaiaadsga caWGZbaaaiabgUcaRiqbeI8a5zaacaGaaCyBamaaBaaaleaacaaIZa aabeaaaaa@3F54@

 The acceleration of the rod is characterized by the acceleration vector of its axis, a=dv/dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGKbGaaCODaiaac+ cacaWGKbGaamiDaaaa@36F5@

 The angular acceleration of the rod is characterized by the angular acceleration α= α i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahg7acqGH9aqpcqaHXoqydaWgaaWcba GaamyAaaqabaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaaaa@379E@  of the basis vectors.  It will be shown below that the angular acceleration can be related to the acceleration a and velocity v of the bar’s axis and its twist ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHipqEaaa@3494@  by

α= dω dt = m 3 × da ds 2( dv ds m 3 ) m 3 × dv ds + dψ dt { dv ds ( dv ds m 3 ) m 3 }+ d 2 ψ d t 2 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahg7acqGH9aqpdaWcaaqaaiaadsgaca WHjpaabaGaamizaiaadshaaaGaeyypa0JaaCyBamaaBaaaleaacaaI ZaaabeaakiabgEna0oaalaaabaGaamizaiaahggaaeaacaWGKbGaam 4CaaaacqGHsislcaaIYaWaaeWaaeaadaWcaaqaaiaadsgacaWH2baa baGaamizaiaadohaaaGaeyyXICTaaCyBamaaBaaaleaacaaIZaaabe aaaOGaayjkaiaawMcaaiaah2gadaWgaaWcbaGaaG4maaqabaGccqGH xdaTdaWcaaqaaiaadsgacaWH2baabaGaamizaiaadohaaaGaey4kaS YaaSaaaeaacaWGKbGaeqiYdKhabaGaamizaiaadshaaaWaaiWaaeaa daWcaaqaaiaadsgacaWH2baabaGaamizaiaadohaaaGaeyOeI0Yaae WaaeaadaWcaaqaaiaadsgacaWH2baabaGaamizaiaadohaaaGaeyyX ICTaaCyBamaaBaaaleaacaaIZaaabeaaaOGaayjkaiaawMcaaiaah2 gadaWgaaWcbaGaaG4maaqabaaakiaawUhacaGL9baacqGHRaWkdaWc aaqaaiaadsgadaahaaWcbeqaaiaaikdaaaGccqaHipqEaeaacaWGKb GaamiDamaaCaaaleqabaGaaGOmaaaaaaGccaWHTbWaaSbaaSqaaiaa iodaaeqaaaaa@742B@

 

 

 

10.2.3 Additional deformation measures and useful kinematic relations

 

In this section we introduce some additional measures of the deformation of the rod, as well as several useful relations between the various deformation measures.

 

 The curve corresponding to the axis of the deformed rod is often characterized by its tangent, normal and binormal vectors, together with its curvature, and its torsion.   These are defined as follows.

  1. The tangent vector t= dr ds = dr d x 3 d x 3 ds = m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH0bGaeyypa0ZaaSaaaeaacaWGKb GaaCOCaaqaaiaadsgacaWGZbaaaiabg2da9maalaaabaGaamizaiaa hkhaaeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaaaaGcdaWcaa qaaiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaa dohaaaGaeyypa0JaaCyBamaaBaaaleaacaaIZaaabeaaaaa@4620@
  2. The normal vector and curvature are defined so that κn= dt ds = d m 3 d x 3 d x 3 ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAcaWHUbGaeyypa0ZaaSaaae aacaWGKbGaaCiDaaqaaiaadsgacaWGZbaaaiabg2da9maalaaabaGa amizaiaah2gadaWgaaWcbaGaaG4maaqabaaakeaacaWGKbGaamiEam aaBaaaleaacaaIZaaabeaaaaGcdaWcaaqaaiaadsgacaWG4bWaaSba aSqaaiaaiodaaeqaaaGcbaGaamizaiaadohaaaaaaa@45D7@ , where n is a unit vector
  3. The binormal vector is defined as b=t×n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbGaeyypa0JaaCiDaiabgEna0k aah6gaaaa@38C2@
  4. The triad of unit vectors {t,n,b} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCiDaiaacYcacaWHUbGaai ilaiaahkgacaGG9baaaa@3905@  defines the Frenet Basis for the curve
  5. The torsion of the curve is defined as τ=n db ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHepaDcqGH9aqpcqGHsislcaWHUb GaeyyXIC9aaSaaaeaacaWGKbGaaCOyaaqaaiaadsgacaWGZbaaaaaa @3D84@ .  Note that the torsion is simply a geometric property of the curve MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it is not necessarily related to the rod’s twist.

These variables are not sufficient to completely describe the deformation, however, since the twist of the rod can vary independently of the shape of its axis.

 

 The two bases { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@ , { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  can be related in terms of the Euler angles as follows.

m 1 =( cosψcosθcosϕsinψsinϕ ) e 1 +( cosψcosθsinϕ+sinψcosϕ ) e 2 cosψsinθ e 3 m 2 =( cosψsinϕ+sinψcosθcosϕ ) e 1 +( cosψcosϕsinψcosθsinϕ ) e 2 +sinψsinθ e 3 m 3 =sinθ( cosϕ e 1 +sinϕ e 2 )+cosθ e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaah2gadaWgaaWcbaGaaGymaa qabaGccqGH9aqpdaqadaqaaiGacogacaGGVbGaai4CaiabeI8a5jGa cogacaGGVbGaai4CaiabeI7aXjGacogacaGGVbGaai4Caiabew9aMj abgkHiTiGacohacaGGPbGaaiOBaiabeI8a5jGacohacaGGPbGaaiOB aiabew9aMbGaayjkaiaawMcaaiaahwgadaWgaaWcbaGaaGymaaqaba GccqGHRaWkdaqadaqaaiGacogacaGGVbGaai4CaiabeI8a5jGacoga caGGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9aMjabgU caRiGacohacaGGPbGaaiOBaiabeI8a5jGacogacaGGVbGaai4Caiab ew9aMbGaayjkaiaawMcaaiaahwgadaWgaaWcbaGaaGOmaaqabaGccq GHsislciGGJbGaai4BaiaacohacqaHipqEciGGZbGaaiyAaiaac6ga cqaH4oqCcaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaaCyBamaaBa aaleaacaaIYaaabeaakiabg2da9iabgkHiTmaabmaabaGaci4yaiaa c+gacaGGZbGaeqiYdKNaci4CaiaacMgacaGGUbGaeqy1dyMaey4kaS Iaci4CaiaacMgacaGGUbGaeqiYdKNaci4yaiaac+gacaGGZbGaeqiU deNaci4yaiaac+gacaGGZbGaeqy1dygacaGLOaGaayzkaaGaaCyzam aaBaaaleaacaaIXaaabeaakiabgUcaRmaabmaabaGaci4yaiaac+ga caGGZbGaeqiYdKNaci4yaiaac+gacaGGZbGaeqy1dyMaeyOeI0Iaci 4CaiaacMgacaGGUbGaeqiYdKNaci4yaiaac+gacaGGZbGaeqiUdeNa ci4CaiaacMgacaGGUbGaeqy1dygacaGLOaGaayzkaaGaaCyzamaaBa aaleaacaaIYaaabeaakiabgUcaRiGacohacaGGPbGaaiOBaiabeI8a 5jGacohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaaWcbaGaaG4maa qabaaakeaacaWHTbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaci4C aiaacMgacaGGUbGaeqiUde3aaeWaaeaaciGGJbGaai4Baiaacohacq aHvpGzcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaci4Caiaa cMgacaGGUbGaeqy1dyMaaCyzamaaBaaaleaacaaIYaaabeaaaOGaay jkaiaawMcaaiabgUcaRiGacogacaGGVbGaai4CaiabeI7aXjaahwga daWgaaWcbaGaaG4maaqabaaaaaa@DEDD@

These results can be derived by calculating the effects of the sequence of three rotations.  Note also that since both sets of basis vectors are triads of mutually perpendicular unit vectors, they must be related by

m i =R e i e i = R T m i = m i R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaaCOuaiaahwgadaWgaaWcbaGaamyAaaqabaGccaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaCyzamaaBaaaleaacaWGPbaabeaa kiabg2da9iaahkfadaahaaWcbeqaaiaadsfaaaGccaWHTbWaaSbaaS qaaiaadMgaaeqaaOGaeyypa0JaaCyBamaaBaaaleaacaWGPbaabeaa kiaahkfaaaa@57FA@

where R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbaaaa@33A1@  is a proper orthogonal tensor that can be visualized as a rigid rotation.   The rotation tensor can be expressed in several different forms:

  1. It can be expressed as the sum of three dyadic products R= m i e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaeyypa0JaaCyBamaaBaaale aacaWGPbaabeaakiabgEPielaahwgadaWgaaWcbaGaamyAaaqabaaa aa@3AD2@
  2. It can be expressed as components in either { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  or { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ , which can be written in dyadic notation as R ij ee e i e j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaa0baaSqaaiaadMgacaWGQb aabaGaaCyzaiaahwgaaaGccaWHLbWaaSbaaSqaaiaadMgaaeqaaOGa ey4LIqSaaCyzamaaBaaaleaacaWGQbaabeaaaaa@3DB1@  or R ij mm m i m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaa0baaSqaaiaadMgacaWGQb aabaGaaCyBaiaah2gaaaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaOGa ey4LIqSaaCyBamaaBaaaleaacaWGQbaabeaaaaa@3DD1@ .  Surprisingly, the components both bases are equal, and are given by R ij mm = R ij ee = e i m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaa0baaSqaaiaadMgacaWGQb aabaGaaCyBaiaah2gaaaGccqGH9aqpcaWGsbWaa0baaSqaaiaadMga caWGQbaabaGaaCyzaiaahwgaaaGccqGH9aqpcaWHLbWaaSbaaSqaai aadMgaaeqaaOGaeyyXICTaaCyBamaaBaaaleaacaWGQbaabeaaaaa@44DD@ . The components can be expressed in terms of the Euler angles as a matrix

R ij mm = R ij ee [ cosψcosθcosϕsinψsinϕ ( cosψsinϕ+sinψcosθcosϕ ) sinϕsinθ cosψcosθsinϕ+sinψcosϕ cosψcosϕsinψcosθsinϕ sinθsinϕ cosψsinθ sinψsinθ cosθ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaa0baaSqaaiaadMgacaWGQb aabaGaaCyBaiaah2gaaaGccqGH9aqpcaWGsbWaa0baaSqaaiaadMga caWGQbaabaGaaCyzaiaahwgaaaGccqGHHjIUdaWadaqaauaabeqadm aaaeaaciGGJbGaai4BaiaacohacqaHipqEciGGJbGaai4Baiaacoha cqaH4oqCciGGJbGaai4BaiaacohacqaHvpGzcqGHsislciGGZbGaai yAaiaac6gacqaHipqEciGGZbGaaiyAaiaac6gacqaHvpGzaeaacqGH sisldaqadaqaaiGacogacaGGVbGaai4CaiabeI8a5jGacohacaGGPb GaaiOBaiabew9aMjabgUcaRiGacohacaGGPbGaaiOBaiabeI8a5jGa cogacaGGVbGaai4CaiabeI7aXjGacogacaGGVbGaai4Caiabew9aMb GaayjkaiaawMcaaaqaaiGacohacaGGPbGaaiOBaiabew9aMjGacoha caGGPbGaaiOBaiabeI7aXbqaaiGacogacaGGVbGaai4CaiabeI8a5j GacogacaGGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9a MjabgUcaRiGacohacaGGPbGaaiOBaiabeI8a5jGacogacaGGVbGaai 4Caiabew9aMbqaaiGacogacaGGVbGaai4CaiabeI8a5jGacogacaGG VbGaai4Caiabew9aMjabgkHiTiGacohacaGGPbGaaiOBaiabeI8a5j GacogacaGGVbGaai4CaiabeI7aXjGacohacaGGPbGaaiOBaiabew9a MbqaaiGacohacaGGPbGaaiOBaiabeI7aXjGacohacaGGPbGaaiOBai abew9aMbqaaiabgkHiTiGacogacaGGVbGaai4CaiabeI8a5jGacoha caGGPbGaaiOBaiabeI7aXbqaaiGacohacaGGPbGaaiOBaiabeI8a5j GacohacaGGPbGaaiOBaiabeI7aXbqaaiGacogacaGGVbGaai4Caiab eI7aXbaaaiaawUfacaGLDbaaaaa@CDB0@

 

 In further calculations the variation of basis vectors m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaadMgaaeqaaa aa@34D6@  with distance s along the deformed rod will play a central role.  To visualize this quantity, imagine that the basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaaCyBam aaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaaikda aeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3DD8@  travels up the deformed rod. The basis vectors will then rotate with an angular velocity that depends on the curvature and twist of the deformed rod, suggesting that we can characterize the rate of change of orientation with arc-length by a vector κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH6oaaaa@340C@ , analogous to an angular velocity vector.  The curvature vector can be expressed as components in the basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaaCyBam aaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaaikda aeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3DD8@  as κ= κ i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWH6oGaeyypa0 JaeqOUdS2aaSbaaSqaaiaadMgaaeqaaOGaaCyBamaaBaaaleaacaWG Pbaabeaaaaa@3BF2@ .  This vector has the following properties

1.      The curvature vector is (by definition) related to the rate of change of  m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaadMgaaeqaaa aa@34D6@  with s by d m i ds =κ× m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHTbWaaSbaaS qaaiaadMgaaeqaaaGcbaGaamizaiaadohaaaGaeyypa0JaaCOUdiab gEna0kaah2gadaWgaaWcbaGaamyAaaqabaaaaa@3E2D@ , which can be expanded out to show that

d m 1 ds = κ 2 m 3 + κ 3 m 2 d m 2 ds = κ 1 m 3 κ 3 m 1 d m 3 ds = κ 1 m 2 + κ 2 m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaads gacaWHTbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaadohaaaGa eyypa0JaeyOeI0IaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaaCyBam aaBaaaleaacaaIZaaabeaakiabgUcaRiabeQ7aRnaaBaaaleaacaaI Zaaabeaakiaah2gadaWgaaWcbaGaaGOmaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaadsgacaWHTbWaaSba aSqaaiaaikdaaeqaaaGcbaGaamizaiaadohaaaGaeyypa0JaeqOUdS 2aaSbaaSqaaiaaigdaaeqaaOGaaCyBamaaBaaaleaacaaIZaaabeaa kiabgkHiTiabeQ7aRnaaBaaaleaacaaIZaaabeaakiaah2gadaWgaa WcbaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVpaalaaabaGaamizaiaah2gadaWgaaWcbaGaaG 4maaqabaaakeaacaWGKbGaam4CaaaacqGH9aqpcqGHsislcqaH6oWA daWgaaWcbaGaaGymaaqabaGccaWHTbWaaSbaaSqaaiaaikdaaeqaaO Gaey4kaSIaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaaCyBamaaBaaa leaacaaIXaaabeaaaaa@7B7B@

2.      The components κ 1 , κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAdaWgaaWcbaGaaGymaaqaba GccaGGSaGaeqOUdS2aaSbaaSqaaiaaikdaaeqaaaaa@38B3@  quantify the bending of the rod, and are related the curvature κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAaaa@3478@  and the binormal vector b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbaaaa@33B1@  of the curve traced by the axis of the deformed rod by κ 1 m 1 + κ 2 m 2 =κb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAdaWgaaWcbaGaaGymaaqaba GccaWHTbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqOUdS2aaSba aSqaaiaaikdaaeqaaOGaaCyBamaaBaaaleaacaaIYaaabeaakiabg2 da9iabeQ7aRjaahkgaaaa@4061@ .  You can show this result by comparing the formula for d m 3 /ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHTbWaaSbaaSqaaiaaiodaae qaaOGaai4laiaadsgacaWGZbaaaa@35EE@  with the formula for b.

3.      The curvature vector can also be expressed in terms of the position vector of the rod’s centroid as

κ i m i = dr ds × d 2 r d s 2 + κ 3 dr ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAdaWgaaWcbaGaamyAaaqaba GccaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacaWG KbGaaCOCaaqaaiaadsgacaWGZbaaaiabgEna0oaalaaabaGaamizam aaCaaaleqabaGaaGOmaaaakiaahkhaaeaacaWGKbGaam4CamaaCaaa leqabaGaaGOmaaaaaaGccqGHRaWkcqaH6oWAdaWgaaWcbaGaaG4maa qabaGcdaWcaaqaaiaadsgacaWHYbaabaGaamizaiaadohaaaGaaGPa Vdaa@4D4A@

The component of curvature κ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaaa a@3323@  cannot in general be expressed in terms of r, because the rotation of the rod’s cross-section about its centroid axis may provide an additional, independent contribution to κ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaaa a@3323@ .  For the special case where m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGymaaqabaaaaa@3265@  and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGOmaaqabaaaaa@3266@  are everywhere parallel to the normal vector n and binormal b, respectively, it follows that κ 3 =bdn/ds=ndb/ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaki abg2da9iaahkgacqGHflY1caWGKbGaaCOBaiaac+cacaWGKbGaam4C aiabg2da9iabgkHiTiaah6gacqGHflY1caWGKbGaaGPaVlaahkgaca GGVaGaamizaiaadohaaaa@4703@ .  In this case, κ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaaa a@3323@  is equal to the torsion of the curve. 

 

 The rate of change of  m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaadMgaaeqaaa aa@34D6@  with distance s can also be expressed in terms of the Euler angles.  For example, the derivative of m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaiodaaeqaaa aa@34A5@  can be calculated as follows

m 3 =sinθ( cosϕ e 1 +sinϕ e 2 )+cosθ e 3 d m 3 ds =cosθ dθ ds ( cosϕ e 1 +sinϕ e 2 )+sinθ( sinϕ e 1 +cosϕ e 2 ) dϕ ds sinθ dθ ds e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaah2gadaWgaaWcbaGaaG4maa qabaGccqGH9aqpciGGZbGaaiyAaiaac6gacqaH4oqCdaqadaqaaiGa cogacaGGVbGaai4Caiabew9aMjaahwgadaWgaaWcbaGaaGymaaqaba GccqGHRaWkciGGZbGaaiyAaiaac6gacqaHvpGzcaWHLbWaaSbaaSqa aiaaikdaaeqaaaGccaGLOaGaayzkaaGaey4kaSIaci4yaiaac+gaca GGZbGaeqiUdeNaaCyzamaaBaaaleaacaaIZaaabeaaaOqaaiabgkDi EpaalaaabaGaamizaiaah2gadaWgaaWcbaGaaG4maaqabaaakeaaca WGKbGaam4CaaaacqGH9aqpciGGJbGaai4BaiaacohacqaH4oqCdaWc aaqaaiaadsgacqaH4oqCaeaacaWGKbGaam4CaaaadaqadaqaaiGaco gacaGGVbGaai4Caiabew9aMjaahwgadaWgaaWcbaGaaGymaaqabaGc cqGHRaWkciGGZbGaaiyAaiaac6gacqaHvpGzcaWHLbWaaSbaaSqaai aaikdaaeqaaaGccaGLOaGaayzkaaGaey4kaSIaci4CaiaacMgacaGG UbGaeqiUde3aaeWaaeaacqGHsislciGGZbGaaiyAaiaac6gacqaHvp GzcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaci4yaiaac+ga caGGZbGaeqy1dyMaaCyzamaaBaaaleaacaaIYaaabeaaaOGaayjkai aawMcaamaalaaabaGaamizaiabew9aMbqaaiaadsgacaWGZbaaaiab gkHiTiGacohacaGGPbGaaiOBaiabeI7aXnaalaaabaGaamizaiabeI 7aXbqaaiaadsgacaWGZbaaaiaahwgadaWgaaWcbaGaaG4maaqabaaa aaa@97E4@

Similar results for m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaigdaaeqaaa aa@34A3@  and m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaikdaaeqaaa aa@34A4@  are left as exercises.

 

 The bending curvatures κ 1 , κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAdaWgaaWcbaGaaGymaaqaba GccaGGSaGaeqOUdS2aaSbaaSqaaiaaikdaaeqaaaaa@38B3@  and the twist rate κ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAdaWgaaWcbaGaaG4maaqaba aaaa@3561@  are related to the Euler angles by

κ 1 =sin(ψ) dθ ds cos(ψ)sinθ dϕ ds κ 2 =cos(ψ) dθ ds +sin(ψ)sinθ dϕ ds κ 3 = dψ ds + dϕ ds cosθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH6oWAdaWgaa WcbaGaaGymaaqabaGccqGH9aqpciGGZbGaaiyAaiaac6gacaGGOaGa eqiYdKNaaiykamaalaaabaGaamizaiabeI7aXbqaaiaadsgacaWGZb aaaiabgkHiTiGacogacaGGVbGaai4CaiaacIcacqaHipqEcaGGPaGa ci4CaiaacMgacaGGUbGaeqiUde3aaSaaaeaacaWGKbGaeqy1dygaba GaamizaiaadohaaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7cqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGH9a qpciGGJbGaai4BaiaacohacaGGOaGaeqiYdKNaaiykamaalaaabaGa amizaiabeI7aXbqaaiaadsgacaWGZbaaaiabgUcaRiGacohacaGGPb GaaiOBaiaacIcacqaHipqEcaGGPaGaci4CaiaacMgacaGGUbGaeqiU de3aaSaaaeaacaWGKbGaeqy1dygabaGaamizaiaadohaaaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlabeQ7aRnaaBaaaleaacaaIZaaabeaakiabg2da9maalaaaba GaamizaiabeI8a5bqaaiaadsgacaWGZbaaaiabgUcaRmaalaaabaGa amizaiabew9aMbqaaiaadsgacaWGZbaaaiGacogacaGGVbGaai4Cai abeI7aXbaa@9F7A@

These results can be derived from the two different formulas for d m i /ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCyBamaaBaaaleaacaWGPb aabeaakiaac+cacaWGKbGaam4Caaaa@385D@ , together with the equations relating  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  and { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  in terms of the Euler angles. 

 

 The arc length s along the rod’s centerline is related to the position vector of the rod’s axis by

ds d x 3 = dr d x 3 dr d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGZbaabaGaam izaiaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaeyypa0ZaaOaaaeaa daWcaaqaaiaadsgacaWHYbaabaGaamizaiaadIhadaWgaaWcbaGaaG 4maaqabaaaaOGaeyyXIC9aaSaaaeaacaWGKbGaaCOCaaqaaiaadsga caWG4bWaaSbaaSqaaiaaiodaaeqaaaaaaeqaaaaa@447F@

 

 Some relationships between the time derivatives of these various kinematic quantities are also useful in subsequent calculations.  The rate of change in shape of the rod can be characterized by the velocity of the axis v( s )=dr/dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhadaqadaqaaiaadohaaiaawIcaca GLPaaacqGH9aqpcaWGKbGaaCOCaiaac+cacaWGKbGaamiDaaaa@3987@  and the time rate of change of the cross-sectional rotation dψ/dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacqaHipqEcaGGVaGaamizaiaads haaaa@35D4@ .

 

 The time derivative of the tangent vector is a convenient way to characterize the rate of change of bending of the rod.   This is related to the velocity of the rod’s centerline by

dt dt d m 3 dt = τ ˙ 1 m 1 + τ ˙ 2 m 2 = d ds ( dr dt ) dr ds [ dr ds d ds ( dr dt ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaahshaaeaacaWGKb GaamiDaaaacqGHHjIUdaWcaaqaaiaadsgacaWHTbWaaSbaaSqaaiaa iodaaeqaaaGcbaGaamizaiaadshaaaGaeyypa0JafqiXdqNbaiaada WgaaWcbaGaaGymaaqabaGccaWHTbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIafqiXdqNbaiaadaWgaaWcbaGaaGOmaaqabaGccaWHTbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaWGKbaabaGaamiz aiaadohaaaWaaeWaaeaadaWcaaqaaiaadsgacaWHYbaabaGaamizai aadshaaaaacaGLOaGaayzkaaGaeyOeI0YaaSaaaeaacaWGKbGaaCOC aaqaaiaadsgacaWGZbaaamaadmaabaWaaSaaaeaacaWGKbGaaCOCaa qaaiaadsgacaWGZbaaaiabgwSixpaalaaabaGaamizaaqaaiaadsga caWGZbaaamaabmaabaWaaSaaaeaacaWGKbGaaCOCaaqaaiaadsgaca WG0baaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@646E@

If we express the velocity in components dr/dt= v i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHYbGaai4laiaadsgacaWG0b Gaeyypa0JaamODamaaBaaaleaacaWGPbaabeaakiaah2gadaWgaaWc baGaamyAaaqabaaaaa@3A36@  and recall m 3 =dr/ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGKbGaaCOCaiaac+cacaWGKbGaam4Caaaa@37EF@  we can write this as

τ ˙ 1 m 1 + τ ˙ 2 m 2 =( d v 1 ds v 2 κ 3 + v 3 κ 2 ) m 1 +( d v 2 ds + v 1 κ 3 v 3 κ 1 ) m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbes8a0zaacaWaaSbaaSqaaiaaigdaae qaaOGaaCyBamaaBaaaleaacaaIXaaabeaakiabgUcaRiqbes8a0zaa caWaaSbaaSqaaiaaikdaaeqaaOGaaCyBamaaBaaaleaacaaIYaaabe aakiabg2da9maabmaabaWaaSaaaeaacaWGKbGaamODamaaBaaaleaa caaIXaaabeaaaOqaaiaadsgacaWGZbaaaiabgkHiTiaadAhadaWgaa WcbaGaaGOmaaqabaGccqaH6oWAdaWgaaWcbaGaaG4maaqabaGccqGH RaWkcaWG2bWaaSbaaSqaaiaaiodaaeqaaOGaeqOUdS2aaSbaaSqaai aaikdaaeqaaaGccaGLOaGaayzkaaGaaCyBamaaBaaaleaacaaIXaaa beaakiabgUcaRmaabmaabaWaaSaaaeaacaWGKbGaamODamaaBaaale aacaaIYaaabeaaaOqaaiaadsgacaWGZbaaaiabgUcaRiaadAhadaWg aaWcbaGaaGymaaqabaGccqaH6oWAdaWgaaWcbaGaaG4maaqabaGccq GHsislcaWG2bWaaSbaaSqaaiaaiodaaeqaaOGaeqOUdS2aaSbaaSqa aiaaigdaaeqaaaGccaGLOaGaayzkaaGaaCyBamaaBaaaleaacaaIYa aabeaaaaa@62E8@

It is important to note that the components τ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbes8a0zaacaWaaSbaaSqaaiaadMgaae qaaaaa@3370@  are not equal to the time derivatives of the components of the tangent vector t, because the basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  varies with time.

 

 The time derivatives of the basis vectors can also be quantified by an angular velocity vector ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8aaaa@31DD@ , which satisfies m ˙ i =ω× m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqah2gagaGaamaaBaaaleaacaWGPbaabe aakiabg2da9iaahM8acqGHxdaTcaWHTbWaaSbaaSqaaiaadMgaaeqa aaaa@392D@ .   The components of ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8aaaa@31DD@  are readily shown to be

ω= m 3 × dv ds + ψ ˙ m 3 = τ ˙ 2 m 1 + τ ˙ 1 m 2 + ψ ˙ m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8acqGH9aqpcaWHTbWaaSbaaSqaai aaiodaaeqaaOGaey41aq7aaSaaaeaacaWGKbGaaCODaaqaaiaadsga caWGZbaaaiabgUcaRiqbeI8a5zaacaGaaCyBamaaBaaaleaacaaIZa aabeaakiabg2da9iabgkHiTiqbes8a0zaacaWaaSbaaSqaaiaaikda aeqaaOGaaCyBamaaBaaaleaacaaIXaaabeaakiabgUcaRiqbes8a0z aacaWaaSbaaSqaaiaaigdaaeqaaOGaaCyBamaaBaaaleaacaaIYaaa beaakiabgUcaRiqbeI8a5zaacaGaaCyBamaaBaaaleaacaaIZaaabe aaaaa@5019@

 

 The time derivatives of the remaining basis vectors follow as

d m 1 dt = τ ˙ 1 m 3 + ψ ˙ m 2 d m 2 dt = τ ˙ 2 m 3 ψ ˙ m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHTbWaaSbaaS qaaiaaigdaaeqaaaGcbaGaamizaiaadshaaaGaeyypa0JaeyOeI0Ia fqiXdqNbaiaadaWgaaWcbaGaaGymaaqabaGccaWHTbWaaSbaaSqaai aaiodaaeqaaOGaey4kaSIafqiYdKNbaiaacaWHTbWaaSbaaSqaaiaa ikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWc aaqaaiaadsgacaWHTbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaamizai aadshaaaGaeyypa0JaeyOeI0IafqiXdqNbaiaadaWgaaWcbaGaaGOm aaqabaGccaWHTbWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IafqiYdK NbaiaacaWHTbWaaSbaaSqaaiaaigdaaeqaaaaa@685D@

 

 The time derivative of the arc length of the centerline is related to its velocity as follows

d dt ( ds d x 3 )= m 3 ds d x 3 d ds ( dr dt )= ds d x 3 ( d v 3 ds v 1 κ 2 + v 2 κ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaabmaabaWaaSaaaeaacaWGKbGaam4CaaqaaiaadsgacaWG4bWa aSbaaSqaaiaaiodaaeqaaaaaaOGaayjkaiaawMcaaiabg2da9iaah2 gadaWgaaWcbaGaaG4maaqabaGccqGHflY1daWcaaqaaiaadsgacaWG ZbaabaGaamizaiaadIhadaWgaaWcbaGaaG4maaqabaaaaOWaaSaaae aacaWGKbaabaGaamizaiaadohaaaWaaeWaaeaadaWcaaqaaiaadsga caWHYbaabaGaamizaiaadshaaaaacaGLOaGaayzkaaGaeyypa0ZaaS aaaeaacaWGKbGaam4CaaqaaiaadsgacaWG4bWaaSbaaSqaaiaaioda aeqaaaaakmaabmaabaWaaSaaaeaacaWGKbGaamODamaaBaaaleaaca aIZaaabeaaaOqaaiaadsgacaWGZbaaaiabgkHiTiaadAhadaWgaaWc baGaaGymaaqabaGccqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGHRa WkcaWG2bWaaSbaaSqaaiaaikdaaeqaaOGaeqOUdS2aaSbaaSqaaiaa igdaaeqaaaGccaGLOaGaayzkaaaaaa@6302@

 

 We shall also require the gradient of the angular velocity ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8aaaa@31DD@ , which quantifies the rate of change of bending. We shall give this vector the symbol κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWfGaqaaiaahQ7aaSqabeaacqGHhi s0aaaaaa@35DB@  to denote its physical significance: it can be interpreted (see Appendix E) as the co-rotational time derivative of the curvature vector, as follows

κ = κ i m i = dω ds = dκ dt ω×κ+ d x 3 ds d s ˙ d x 3 κ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaxacabaGaaCOUdaWcbeqaaiabgEGird aakiabg2da9maaxacabaGaeqOUdS2aaSbaaSqaaiaadMgaaeqaaaqa beaacqGHhis0aaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0 ZaaSaaaeaacaWGKbGaaCyYdaqaaiaadsgacaWGZbaaaiabg2da9maa laaabaGaamizaiaahQ7aaeaacaWGKbGaamiDaaaacqGHsislcaWHjp Gaey41aqRaaCOUdiabgUcaRmaalaaabaGaamizaiaadIhadaWgaaWc baGaaG4maaqabaaakeaacaWGKbGaam4CaiaaxcW7aaWaaSaaaeaaca WGKbGabm4CayaacaaabaGaamizaiaadIhadaWgaaWcbaGaaG4maaqa baaaaOGaaCOUdaaa@58A8@

Evaluating the derivatives of ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8aaaa@31DD@  shows that

κ 1 = d τ ˙ 2 ds τ ˙ 1 κ 3 + ψ ˙ κ 2 κ 2 = d τ ˙ 1 ds τ ˙ 2 κ 3 ψ ˙ κ 1 κ 3 = d ψ ˙ ds + τ ˙ 1 κ 1 + τ ˙ 2 κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaxacabaGaeqOUdSgaleqabaGaey4bIe naaOWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaa caWGKbGafqiXdqNbaiaadaWgaaWcbaGaaGOmaaqabaaakeaacaWGKb Gaam4CaaaacqGHsislcuaHepaDgaGaamaaBaaaleaacaaIXaaabeaa kiabeQ7aRnaaBaaaleaacaaIZaaabeaakiabgUcaRiqbeI8a5zaaca GaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8+aaCbiaeaacqaH6oWAaSqabeaacqGHhis0aa GcdaWgaaWcbaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiaadsgacuaH epaDgaGaamaaBaaaleaacaaIXaaabeaaaOqaaiaadsgacaWGZbaaai abgkHiTiqbes8a0zaacaWaaSbaaSqaaiaaikdaaeqaaOGaeqOUdS2a aSbaaSqaaiaaiodaaeqaaOGaeyOeI0IafqiYdKNbaiaacqaH6oWAda WgaaWcbaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VpaaxacabaGaeqOUdSgaleqabaGaey4bIenaaOWaaSbaaSqaaiaaio daaeqaaOGaeyypa0ZaaSaaaeaacaWGKbGafqiYdKNbaiaaaeaacaWG KbGaam4CaaaacqGHRaWkcuaHepaDgaGaamaaBaaaleaacaaIXaaabe aakiabeQ7aRnaaBaaaleaacaaIXaaabeaakiabgUcaRiqbes8a0zaa caWaaSbaaSqaaiaaikdaaeqaaOGaeqOUdS2aaSbaaSqaaiaaikdaae qaaaaa@86C2@

The co-rotational time derivative of curvature must be used to quantify bending rate (instead of the time derivative dκ/dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCOUdiaac+cacaWGKbGaam iDaaaa@378A@  ) to correct for the fact that rigid rotations and pure stretching do not change bending.

 

 Finally, to solve dynamic problems, we will need to be able to describe the linear and angular acceleration of the bar.  The linear acceleration is most conveniently characterized by the acceleration of the centerline of the bar a=dv/ds= d 2 r/d t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacqGH9aqpcaWGKbGaaCODaiaac+ cacaWGKbGaam4Caiabg2da9iaadsgadaahaaWcbeqaaiaaikdaaaGc caWHYbGaai4laiaadsgacaWG0bWaaWbaaSqabeaacaaIYaaaaaaa@3E4F@

 

 The angular acceleration of the bar’s cross-section can be characterized by the angular acceleration α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahg7aaaa@31C5@  of the basis vectors { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ .  A straightforward calculation shows that

α= dω dt = m 3 × da ds 2( dv ds m 3 ) m 3 × dv ds + dψ dt { dv ds ( dv ds m 3 ) m 3 }+ d 2 ψ d t 2 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahg7acqGH9aqpdaWcaaqaaiaadsgaca WHjpaabaGaamizaiaadshaaaGaeyypa0JaaCyBamaaBaaaleaacaaI ZaaabeaakiabgEna0oaalaaabaGaamizaiaahggaaeaacaWGKbGaam 4CaaaacqGHsislcaaIYaWaaeWaaeaadaWcaaqaaiaadsgacaWH2baa baGaamizaiaadohaaaGaeyyXICTaaCyBamaaBaaaleaacaaIZaaabe aaaOGaayjkaiaawMcaaiaah2gadaWgaaWcbaGaaG4maaqabaGccqGH xdaTdaWcaaqaaiaadsgacaWH2baabaGaamizaiaadohaaaGaey4kaS YaaSaaaeaacaWGKbGaeqiYdKhabaGaamizaiaadshaaaWaaiWaaeaa daWcaaqaaiaadsgacaWH2baabaGaamizaiaadohaaaGaeyOeI0Yaae WaaeaadaWcaaqaaiaadsgacaWH2baabaGaamizaiaadohaaaGaeyyX ICTaaCyBamaaBaaaleaacaaIZaaabeaaaOGaayjkaiaawMcaaiaah2 gadaWgaaWcbaGaaG4maaqabaaakiaawUhacaGL9baacqGHRaWkdaWc aaqaaiaadsgadaahaaWcbeqaaiaaikdaaaGccqaHipqEaeaacaWGKb GaamiDamaaCaaaleqabaGaaGOmaaaaaaGccaWHTbWaaSbaaSqaaiaa iodaaeqaaaaa@742B@

 

 The second time derivative of the basis vectors can then be calculated as

d 2 m i d t 2 = d dt ( ω× m i )= dω dt × m i +ω× d m i dt =α× m i +ω×( ω× m i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizamaaCaaaleqabaGaaG Omaaaakiaah2gadaWgaaWcbaGaamyAaaqabaaakeaacaWGKbGaamiD amaaCaaaleqabaGaaGOmaaaaaaGccqGH9aqpdaWcaaqaaiaadsgaae aacaWGKbGaamiDaaaadaqadaqaaiaahM8acqGHxdaTcaWHTbWaaSba aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaaca WGKbGaaCyYdaqaaiaadsgacaWG0baaaiabgEna0kaah2gadaWgaaWc baGaamyAaaqabaGccqGHRaWkcaWHjpGaey41aq7aaSaaaeaacaWGKb GaaCyBamaaBaaaleaacaWGPbaabeaaaOqaaiaadsgacaWG0baaaiab g2da9iaahg7acqGHxdaTcaWHTbWaaSbaaSqaaiaadMgaaeqaaOGaey 4kaSIaaCyYdiabgEna0oaabmaabaGaaCyYdiabgEna0kaah2gadaWg aaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaaa@66CE@

 

 

 

10.2.4 Approximating the displacement, velocity and acceleration in the rod

 

The position vector after deformation of the material point that has coordinates x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaa aa@34DF@  in the undeformed rod can be expressed as

y( x k )=r( x 3 )+ η i ( x k ) m i ( x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaaiikaiaadIhadaWgaaWcba Gaam4AaaqabaGccaGGPaGaeyypa0JaaCOCaiaacIcacaWG4bWaaSba aSqaaiaaiodaaeqaaOGaaiykaiabgUcaRiabeE7aOnaaBaaaleaaca WGPbaabeaakiaacIcacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaaiyk aiaah2gadaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEamaaBaaale aacaaIZaaabeaakiaacMcaaaa@491F@

This is a completely general expression.   We now introduce a series of approximations that are based on the assumptions that

  1. The rod is thin compared with its length;
  2. The radius of curvature of the rod (due to bending) is much larger than the characteristic dimension of its cross section;
  3. The rate of change of twist of the rod dψ/ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaeqiYdKNaai4laiaadsgaca WGZbaaaa@3811@  has the same order of magnitude as the bending curvature of the rod.
  4. The material in the rod experiences small distorsions MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  i.e. the change in length of any infinitesimal material fiber in the rod is much less than its undeformed length.  

 

With this in mind, we assume that  η i ( x j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH3oaAdaWgaaWcbaGaamyAaaqaba GccaGGOaGaamiEamaaBaaaleaacaWGQbaabeaakiaacMcaaaa@3911@  can be approximated by a function of the form

η α = x α + f αβ ( x 3 ) x β η 3 = u 3 ( x β , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH3oaAdaWgaaWcbaGaeqySdegabe aakiabg2da9iaadIhadaWgaaWcbaGaeqySdegabeaakiabgUcaRiaa dAgadaWgaaWcbaGaeqySdeMaeqOSdigabeaakiaacIcacaWG4bWaaS baaSqaaiaaiodaaeqaaOGaaiykaiaadIhadaWgaaWcbaGaeqOSdiga beaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH3oaAdaWgaaWc baGaaG4maaqabaGccqGH9aqpcaWG1bWaaSbaaSqaaiaaiodaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaeqOSdigabeaakiaacYcacaWG4bWa aSbaaSqaaiaaiodaaeqaaOGaaiykaaaa@65ED@

where the Greek indices α,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycaGGSaGaeqOSdigaaa@36B6@  can have values 1 and 2, and f αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaaaaa@371D@  can be regarded as the first term in a Taylor expansion of η α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH3oaAdaWgaaWcbaGaeqySdegabe aaaaa@363D@ .  The definition of m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaaigdaaeqaaa aa@34A3@  requires that f 21 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiaaikdacaaIXa aabeaakiabg2da9iaaicdaaaa@371E@ .  We assume in addition that

d f αβ d x 3 x β 0 d u 3 d x 3 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWGMbWaaSbaaS qaaiabeg7aHjabek7aIbqabaaakeaacaWGKbGaamiEamaaBaaaleaa caaIZaaabeaaaaGccaWG4bWaaSbaaSqaaiabek7aIbqabaGccqGHij YUcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVpaalaaabaGaamizaiaadwhadaWgaaWcbaGaaG4maaqabaaa keaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaaaaGccaaMc8Uaey isISRaaGimaiaaykW7aaa@6428@

for all possible choices of α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@ .  The constants f αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaaaaa@371D@  can be thought of as the components of a homogeneous in-plane deformation applied to the cross section, while the function u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaiodaaeqaaa aa@34A9@  describes the warping of the cross-section.  To decouple the warping from the axial displacement of the rod, we require that

A u 3 ( x α , x 3 )dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8+aa8quaeaacaWG1bWaaSbaaS qaaiaaiodaaeqaaOGaaiikaiaadIhadaWgaaWcbaGaeqySdegabeaa kiaacYcacaWG4bWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaadsgaca WGbbGaeyypa0JaaGimaaWcbaGaamyqaaqab0Gaey4kIipaaaa@4388@

In addition, for small distorsions, the deformation must satisfy f αβ <<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaGccqGH8aapcqGH8aapcaaIXaaaaa@39EA@  and d u 3 /d x γ <<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyDamaaBaaaleaacaaIZa aabeaakiaac+cacaWGKbGaamiEamaaBaaaleaacqaHZoWzaeqaaOGa eyipaWJaeyipaWJaaGymaaaa@3CD5@ , the rod curvatures must satisfy κ β x α <<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAdaWgaaWcbaGaeqOSdigabe aakiaadIhadaWgaaWcbaGaeqySdegabeaakiabgYda8iabgYda8iaa igdaaaa@3BE4@  for all α,β=1,2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycaGGSaGaeqOSdiMaeyypa0 JaaGymaiaacYcacaaIYaaaaa@39E3@ , and the variation of arc-length s along the axis of the deformed rod with x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@  must satisfy ds/d x 3 1<<1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaam4Caiaac+cacaWGKbGaam iEamaaBaaaleaacaaIZaaabeaakiabgkHiTiaaigdacqGH8aapcqGH 8aapcaaIXaaaaa@3C9E@ .

 

The velocity field in the bar can be approximated as

dy dt =v+ f ˙ αβ x β m α + x α m ˙ α + u ˙ 3 m 3 =v+ f ˙ αβ x β m α + x 1 ( τ ˙ 1 m 3 + ψ ˙ m 2 ) x 2 ( τ ˙ 2 m 3 + ψ ˙ m 1 )+ u ˙ 3 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWH5baabaGaam izaiaadshaaaGaeyypa0JaaCODaiabgUcaRiqadAgagaGaamaaBaaa leaacqaHXoqycqaHYoGyaeqaaOGaamiEamaaBaaaleaacqaHYoGyae qaaOGaaCyBamaaBaaaleaacqaHXoqyaeqaaOGaey4kaSIaamiEamaa BaaaleaacqaHXoqyaeqaaOGabCyBayaacaWaaSbaaSqaaiabeg7aHb qabaGccqGHRaWkceWG1bGbaiaadaWgaaWcbaGaaG4maaqabaGccaWH TbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaCODaiabgUcaRiqadA gagaGaamaaBaaaleaacqaHXoqycqaHYoGyaeqaaOGaamiEamaaBaaa leaacqaHYoGyaeqaaOGaaCyBamaaBaaaleaacqaHXoqyaeqaaOGaey 4kaSIaamiEamaaBaaaleaacaaIXaaabeaakmaabmaabaGaeyOeI0Ia fqiXdqNbaiaadaWgaaWcbaGaaGymaaqabaGccaWHTbWaaSbaaSqaai aaiodaaeqaaOGaey4kaSIafqiYdKNbaiaacaWHTbWaaSbaaSqaaiaa ikdaaeqaaaGccaGLOaGaayzkaaGaeyOeI0IaamiEamaaBaaaleaaca aIYaaabeaakmaabmaabaGafqiXdqNbaiaadaWgaaWcbaGaaGOmaaqa baGccaWHTbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIafqiYdKNbai aacaWHTbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaey4k aSIabmyDayaacaWaaSbaaSqaaiaaiodaaeqaaOGaaCyBamaaBaaale aacaaIZaaabeaaaaa@7CC0@

where it has been assumed that u 3 << x α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH8aapcqGH8aapcaWG4bWaaSbaaSqaaiabeg7aHbqabaaaaa@3745@  and f αβ << x α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaeqySdeMaeqOSdi gabeaakiabgYda8iabgYda8iaadIhadaWgaaWcbaGaeqySdegabeaa aaa@39B9@  for all α,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaacYcacqaHYoGyaaa@3478@ .

 

Finally, the acceleration field within the bar will be approximated as

d 2 y d t 2 =a+ x α d 2 m α d t 2 =a+ x α { α× m α +ω×( ω× m α ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizamaaCaaaleqabaGaaG OmaaaakiaahMhaaeaacaWGKbGaamiDamaaCaaaleqabaGaaGOmaaaa aaGccqGH9aqpcaWHHbGaey4kaSIaamiEamaaBaaaleaacqaHXoqyae qaaOWaaSaaaeaacaWGKbWaaWbaaSqabeaacaaIYaaaaOGaaCyBamaa BaaaleaacqaHXoqyaeqaaaGcbaGaamizaiaadshadaahaaWcbeqaai aaikdaaaaaaOGaeyypa0JaaCyyaiabgUcaRiaadIhadaWgaaWcbaGa eqySdegabeaakmaacmaabaGaaCySdiabgEna0kaah2gadaWgaaWcba GaeqySdegabeaakiabgUcaRiaahM8acqGHxdaTdaqadaqaaiaahM8a cqGHxdaTcaWHTbWaaSbaaSqaaiabeg7aHbqabaaakiaawIcacaGLPa aaaiaawUhacaGL9baaaaa@5D7D@

Here, all time derivatives of u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaaaaa@326B@  and f αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaeqySdeMaeqOSdi gabeaaaaa@34DF@  have been neglected.  This is not so much because they are small, but because they represent a crude approximation to the distortion of the cross-section.  The time derivatives of these quantities are associated with short wavelength oscillations in the bar, which cannot be modeled accurately using the assumed displacement field. 

 

 

 

10.2.5 Approximating the deformation gradient

 

Based on the assumptions listed in Section 10.2.3, the deformation gradient in the rod can be approximated by

F ds d x 3 (1 κ 2 x 1 + κ 1 x 2 ) m 3 e 3 + ds d x 3 x 1 κ 3 m 2 e 3 ds d x 3 x 2 κ 3 m 1 e 3 +( δ αβ + f αβ ) m α e β + d u 3 d x β m 3 e β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahAeacaaMc8UaaGPaVlabgI Ki7oaalaaabaGaamizaiaadohaaeaacaWGKbGaamiEamaaBaaaleaa caaIZaaabeaaaaGccaGGOaGaaGymaiabgkHiTiabeQ7aRnaaBaaale aacaaIYaaabeaakiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWk cqaH6oWAdaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaik daaeqaaOGaaiykaiaah2gadaWgaaWcbaGaaG4maaqabaGccqGHxkcX caWHLbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSYaaSaaaeaacaWGKb Gaam4CaaqaaiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaaakiaa dIhadaWgaaWcbaGaaGymaaqabaGccqaH6oWAdaWgaaWcbaGaaG4maa qabaGccaWHTbWaaSbaaSqaaiaaikdaaeqaaOGaey4LIqSaaCyzamaa BaaaleaacaaIZaaabeaakiaaykW7cqGHsisldaWcaaqaaiaadsgaca WGZbaabaGaamizaiaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaamiE amaaBaaaleaacaaIYaaabeaakiabeQ7aRnaaBaaaleaacaaIZaaabe aakiaah2gadaWgaaWcbaGaaGymaaqabaGccqGHxkcXcaWHLbWaaSba aSqaaiaaiodaaeqaaaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqGHRaWkcaGGOaGaeqiTdq2aaSbaaSqaaiabeg7a Hjabek7aIbqabaGccqGHRaWkcaWGMbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaGccaGGPaGaaCyBamaaBaaaleaacqaHXoqyaeqaaOGaey4L IqSaaCyzamaaBaaaleaacqaHYoGyaeqaaOGaey4kaSYaaSaaaeaaca WGKbGaamyDamaaBaaaleaacaaIZaaabeaaaOqaaiaadsgacaWG4bWa aSbaaSqaaiabek7aIbqabaaaaOGaaCyBamaaBaaaleaacaaIZaaabe aakiabgEPielaahwgadaWgaaWcbaGaeqOSdigabeaaaaaa@B7A0@

The first line of this expression quantifies the effects of axial stretching, bending and twisting of the rod.  The second line approximates the distorsion of its cross-section.

 

The deformation gradient can also be decomposed as

F=RG=HR MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiaahEeacq GH9aqpcaWHibGaaCOuaaaa@38F8@

where R is the rigid rotation satisfying m i =R e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaaCOuaiaahwgadaWgaaWcbaGaamyAaaqabaaaaa@38C9@ , and G and H are deformation gradient like tensors that describe the change in shape of the rod.  These tensors are most conveniently expressed as components in  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  and { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ , respectively MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  we can represent this in diadic notation as G= G ij ee e i e j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHhbGaeyypa0Jaam4ramaaDaaale aacaWGPbGaamOAaaqaaiaahwgacaWHLbaaaOGaaCyzamaaBaaaleaa caWGPbaabeaakiabgEPielaahwgadaWgaaWcbaGaamOAaaqabaaaaa@3F7C@  or H= H ij mm m i m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHibGaeyypa0JaamisamaaDaaale aacaWGPbGaamOAaaqaaiaah2gacaWHTbaaaOGaaCyBamaaBaaaleaa caWGPbaabeaakiabgEPielaah2gadaWgaaWcbaGaamOAaaqabaaaaa@3F9E@ .  The components can be expressed in matrix form as

G ij ee = H ij mm [ 1+ f 11 f 12 κ 3 x 2 ds d x 3 0 1+ f 22 κ 3 x 1 ds d x 3 d u 3 d x 1 d u 3 d x 2 ( 1+ κ 1 x 2 κ 2 x 1 ) ds d x 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGhbWaa0baaSqaaiaadMgacaWGQb aabaGaaCyzaiaahwgaaaGccqGH9aqpcaWGibWaa0baaSqaaiaadMga caWGQbaabaGaaCyBaiaah2gaaaGccqGHHjIUdaWadaqaauaabeqadm aaaeaacaaIXaGaey4kaSIaamOzamaaBaaaleaacaaIXaGaaGymaaqa baaakeaacaWGMbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiabgk HiTiabeQ7aRnaaBaaaleaacaaIZaaabeaakiaadIhadaWgaaWcbaGa aGOmaaqabaGcdaWcaaqaaiaadsgacaWGZbaabaGaamizaiaadIhada WgaaWcbaGaaG4maaqabaaaaaGcbaGaaGimaaqaaiaaigdacqGHRaWk caWGMbWaaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabeQ7aRnaaBa aaleaacaaIZaaabeaakiaadIhadaWgaaWcbaGaaGymaaqabaGcdaWc aaqaaiaadsgacaWGZbaabaGaamizaiaadIhadaWgaaWcbaGaaG4maa qabaaaaaGcbaWaaSaaaeaacaWGKbGaamyDamaaBaaaleaacaaIZaaa beaaaOqaaiaadsgacaWG4bWaaSbaaSqaaiaaigdaaeqaaaaaaOqaam aalaaabaGaamizaiaadwhadaWgaaWcbaGaaG4maaqabaaakeaacaWG KbGaamiEamaaBaaaleaacaaIYaaabeaaaaaakeaadaqadaqaaiaaig dacqGHRaWkcqaH6oWAdaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSba aSqaaiaaikdaaeqaaOGaeyOeI0IaeqOUdS2aaSbaaSqaaiaaikdaae qaaOGaamiEamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaamaa laaabaGaamizaiaadohaaeaacaWGKbGaamiEamaaBaaaleaacaaIZa aabeaaaaaaaaGccaGLBbGaayzxaaaaaa@7E07@

 

 

Derivation: The deformation gradient is, by definition, the derivative of the position vector of material particles with respect to their position in the reference configuration, i.e.

F= dy dx = dr d x 3 e 3 + d η i d x j m i e j + η i ( x j ) d m i d x 3 e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0ZaaSaaaeaacaWGKb GaaCyEaaqaaiaadsgacaWH4baaaiabg2da9maalaaabaGaamizaiaa hkhaaeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGHxk cXcaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSYaaSaaaeaacaWG KbGaeq4TdG2aaSbaaSqaaiaadMgaaeqaaaGcbaGaamizaiaadIhada WgaaWcbaGaamOAaaqabaaaaOGaaCyBamaaBaaaleaacaWGPbaabeaa kiabgEPielaahwgadaWgaaWcbaGaamOAaaqabaGccqGHRaWkcqaH3o aAdaWgaaWcbaGaamyAaaqabaGccaGGOaGaamiEamaaBaaaleaacaWG QbaabeaakiaacMcadaWcaaqaaiaadsgacaWHTbWaaSbaaSqaaiaadM gaaeqaaaGcbaGaamizaiaadIhadaWgaaWcbaGaaG4maaqabaaaaOGa ey4LIqSaaCyzamaaBaaaleaacaaIZaaabeaaaaa@6122@

To reduce this to the expression given,

1.      Note that  dr d x 3 = dr ds ds d x 3 = m 3 ds d x 3 d m i d x 3 = d m i ds ds d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHYbaabaGaam izaiaadIhadaWgaaWcbaGaaG4maaqabaaaaOGaeyypa0ZaaSaaaeaa caWGKbGaaCOCaaqaaiaadsgacaWGZbaaamaalaaabaGaamizaiaado haaeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGH9aqp caWHTbWaaSbaaSqaaiaaiodaaeqaaOWaaSaaaeaacaWGKbGaam4Caa qaaiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaaakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VpaalaaabaGaamizaiaah2gadaWgaaWcbaGaamyAaaqabaaakeaaca WGKbGaamiEamaaBaaaleaacaaIZaaabeaaaaGccqGH9aqpdaWcaaqa aiaadsgacaWHTbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaamizaiaado haaaWaaSaaaeaacaWGKbGaam4CaaqaaiaadsgacaWG4bWaaSbaaSqa aiaaiodaaeqaaaaaaaa@755D@

2.      Recall that d m 1 ds = κ 2 m 3 + κ 3 m 2 d m 2 ds = κ 1 m 3 κ 3 m 1 d m 3 ds = κ 1 m 2 + κ 2 m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaads gacaWHTbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaamizaiaadohaaaGa eyypa0JaeyOeI0IaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaaCyBam aaBaaaleaacaaIZaaabeaakiabgUcaRiabeQ7aRnaaBaaaleaacaaI Zaaabeaakiaah2gadaWgaaWcbaGaaGOmaaqabaGccaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaadsgacaWHTbWaaSba aSqaaiaaikdaaeqaaaGcbaGaamizaiaadohaaaGaeyypa0JaeqOUdS 2aaSbaaSqaaiaaigdaaeqaaOGaaCyBamaaBaaaleaacaaIZaaabeaa kiabgkHiTiabeQ7aRnaaBaaaleaacaaIZaaabeaakiaah2gadaWgaa WcbaGaaGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVpaalaaabaGaamizaiaah2gadaWgaaWcbaGaaG 4maaqabaaakeaacaWGKbGaam4CaaaacqGH9aqpcqGHsislcqaH6oWA daWgaaWcbaGaaGymaaqabaGccaWHTbWaaSbaaSqaaiaaikdaaeqaaO Gaey4kaSIaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaaCyBamaaBaaa leaacaaIXaaabeaaaaa@7B7B@

3.      Substitute η α = x α + f αβ ( x 3 ) x β , η 3 = u 3 ( x β , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH3oaAdaWgaaWcbaGaeqySdegabe aakiabg2da9iaadIhadaWgaaWcbaGaeqySdegabeaakiabgUcaRiaa dAgadaWgaaWcbaGaeqySdeMaeqOSdigabeaakiaacIcacaWG4bWaaS baaSqaaiaaiodaaeqaaOGaaiykaiaadIhadaWgaaWcbaGaeqOSdiga beaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8Uaeq4TdG2aaSbaaS qaaiaaiodaaeqaaOGaeyypa0JaamyDamaaBaaaleaacaaIZaaabeaa kiaacIcacaWG4bWaaSbaaSqaaiabek7aIbqabaGccaGGSaGaamiEam aaBaaaleaacaaIZaaabeaakiaacMcaaaa@58BA@  and neglect the derivatives of f and u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaiodaaeqaaa aa@34A9@  with respect to x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@

 

The decomposition F=RG MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiaahEeaaa a@3646@  follows trivially by substituting m i =R e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaaCOuaiaahwgadaWgaaWcbaGaamyAaaqabaaaaa@38C9@  into the dyadic representation of F and rearranging the result.  A similar approach gives F=HR MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCisaiaahkfaaa a@3647@ .

 

 

 

 

10.2.6 Other strain measures

 

It is straightforward to compute additional strain measures from the deformation gradient.  Only a partial list will be given here.

 

  1. The determinant of the deformation gradient follows as

J=det(F)=det(G)=det(H)=(1+ f 11 )(1+ f 22 )( 1+ κ 1 x 2 κ 2 x 1 ) ds d x 3 ds d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca GG0bGaaiikaiaahAeacaGGPaGaeyypa0JaciizaiaacwgacaGG0bGa aiikaiaahEeacaGGPaGaeyypa0JaciizaiaacwgacaGG0bGaaiikai aahIeacaGGPaGaeyypa0JaaiikaiaaigdacqGHRaWkcaWGMbWaaSba aSqaaiaaigdacaaIXaaabeaakiaacMcacaGGOaGaaGymaiabgUcaRi aadAgadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaaiykamaabmaabaGa aGymaiabgUcaRiabeQ7aRnaaBaaaleaacaaIXaaabeaakiaadIhada WgaaWcbaGaaGOmaaqabaGccqGHsislcqaH6oWAdaWgaaWcbaGaaGOm aaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaa WaaSaaaeaacaWGKbGaam4CaaqaaiaadsgacaWG4bWaaSbaaSqaaiaa iodaaeqaaaaakiabgIKi7oaalaaabaGaamizaiaadohaaeaacaWGKb GaamiEamaaBaaaleaacaaIZaaabeaaaaaaaa@6A17@

 

  1. The components of the left and right Cauchy-Green tensors can be computed from B=F F T =H H T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHcbGaeyypa0JaaCOraiaahAeada ahaaWcbeqaaiaadsfaaaGccqGH9aqpcaWHibGaaCisamaaCaaaleqa baGaamivaaaaaaa@3AF3@  and C= F T F= G T G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbGaeyypa0JaaCOramaaCaaale qabaGaamivaaaakiaahAeacqGH9aqpcaWHhbWaaWbaaSqabeaacaWG ubaaaOGaaC4raaaa@3AFC@ , where G and H were defined in 10.2.4.   C and B are most conveniently expressed as components in { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  and { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ , respectively MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  we can represent this in diadic notation as C= C ij ee e i e j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbGaeyypa0Jaam4qamaaDaaale aacaWGPbGaamOAaaqaaiaahwgacaWHLbaaaOGaaCyzamaaBaaaleaa caWGPbaabeaakiabgEPielaahwgadaWgaaWcbaGaamOAaaqabaaaaa@3F74@  or B= B ij mm m i m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHcbGaeyypa0JaamOqamaaDaaale aacaWGPbGaamOAaaqaaiaah2gacaWHTbaaaOGaaCyBamaaBaaaleaa caWGPbaabeaakiabgEPielaah2gadaWgaaWcbaGaamOAaaqabaaaaa@3F92@ . For small distorsions, the result can be approximated by

C ij ee = B ij mm [ 1+2 f 11 f 12 ( d u 3 d x 1 κ 3 x 2 ) 1+2 f 22 ( d u 3 d x 2 + κ 3 x 1 ) sym ( ds d x 3 ) 2 +2 κ 1 x 2 2 κ 2 x 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb aabaGaaCyzaiaahwgaaaGccqGH9aqpcaWGcbWaa0baaSqaaiaadMga caWGQbaabaGaaCyBaiaah2gaaaGccqGHHjIUdaWadaqaauaabeqadm aaaeaacaaIXaGaey4kaSIaaGOmaiaadAgadaWgaaWcbaGaaGymaiaa igdaaeqaaaGcbaGaamOzamaaBaaaleaacaaIXaGaaGOmaaqabaaake aadaqadaqaamaalaaabaGaamizaiaadwhadaWgaaWcbaGaaG4maaqa baaakeaacaWGKbGaamiEamaaBaaaleaacaaIXaaabeaaaaGccqGHsi slcqaH6oWAdaWgaaWcbaGaaG4maaqabaGccaWG4bWaaSbaaSqaaiaa ikdaaeqaaaGccaGLOaGaayzkaaaabaGaeSO7I0eabaGaaGymaiabgU caRiaaikdacaWGMbWaaSbaaSqaaiaaikdacaaIYaaabeaaaOqaamaa bmaabaWaaSaaaeaacaWGKbGaamyDamaaBaaaleaacaaIZaaabeaaaO qaaiaadsgacaWG4bWaaSbaaSqaaiaaikdaaeqaaaaakiabgUcaRiab eQ7aRnaaBaaaleaacaaIZaaabeaakiaadIhadaWgaaWcbaGaaGymaa qabaaakiaawIcacaGLPaaaaeaacaWGZbGaamyEaiaad2gaaeaacqWI VlctaeaadaqadaqaamaalaaabaGaamizaiaadohaaeaacaWGKbGaam iEamaaBaaaleaacaaIZaaabeaaaaaakiaawIcacaGLPaaadaahaaWc beqaaiaaikdaaaGccqGHRaWkcaaIYaGaeqOUdS2aaSbaaSqaaiaaig daaeqaaOGaamiEamaaBaaaleaacaaIYaaabeaakiabgkHiTiaaikda cqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaig daaeqaaaaaaOGaay5waiaaw2faaaaa@818E@

 

  1. The Lagrange strain is defined as E=(CI)/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHfbGaeyypa0Jaaiikaiaahoeacq GHsislcaWHjbGaaiykaiaac+cacaaIYaaaaa@39ED@ .  Its components follow trivially from the preceding formula.  Note that the matrix of components for E resembles the formula for the infinitesimal strain components in a straight bar subjected to axial stretching, bending and twist deformation.   However, if the bent rod does not lie in one plane, the twisting measure κ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaaa a@3323@  includes contributions from both the rotation of the rod’s cross section about its axis, and also from the bending of the rod.

 

  1. The rate of deformation tensor D=sym( F ˙ F 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHebGaeyypa0Jaae4CaiaabMhaca qGTbGaaiikaiqahAeagaGaaiaahAeadaahaaWcbeqaaiabgkHiTiaa igdaaaGccaGGPaaaaa@3C5A@  will also be required. It is simplest to calculate the velocity gradient L= F ˙ F 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHmbGaeyypa0JabCOrayaacaGaaC OramaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@381D@  by differentiating the expression given for the velocity vector in the preceding section.

dy dt = dr dt + f ˙ αβ x β m α + x α ω× m α + u ˙ 3 m 3 F ˙ = d dt ( ds d x 3 dr ds ) e 3 +[ f ˙ αβ m α + u ˙ 3 x β m 3 ] e β + f ˙ αβ x β ds d x 3 d m α ds e 3 +( ω× m α ) e α + ds d x 3 d ds ( ω× m α ) e 3 + u ˙ 3 ds d x 3 d m 3 ds e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaalaaabaGaamizaiaahMhaae aacaWGKbGaamiDaaaacqGH9aqpdaWcaaqaaiaadsgacaWHYbaabaGa amizaiaadshaaaGaey4kaSIabmOzayaacaWaaSbaaSqaaiabeg7aHj abek7aIbqabaGccaWG4bWaaSbaaSqaaiabek7aIbqabaGccaWHTbWa aSbaaSqaaiabeg7aHbqabaGccqGHRaWkcaWG4bWaaSbaaSqaaiabeg 7aHbqabaGccaWHjpGaey41aqRaaCyBamaaBaaaleaacqaHXoqyaeqa aOGaey4kaSIabmyDayaacaWaaSbaaSqaaiaaiodaaeqaaOGaaCyBam aaBaaaleaacaaIZaaabeaaaOqaaiabgkDiElqahAeagaGaaiabg2da 9maalaaabaGaamizaaqaaiaadsgacaWG0baaamaabmaabaWaaSaaae aacaWGKbGaam4CaaqaaiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqa aaaakmaalaaabaGaamizaiaahkhaaeaacaWGKbGaam4CaaaaaiaawI cacaGLPaaacqGHxkcXcaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaey4k aSYaamWaaeaaceWGMbGbaiaadaWgaaWcbaGaeqySdeMaeqOSdigabe aakiaah2gadaWgaaWcbaGaeqySdegabeaakiabgUcaRmaalaaabaGa eyOaIyRabmyDayaacaWaaSbaaSqaaiaaiodaaeqaaaGcbaGaeyOaIy RaamiEamaaBaaaleaacqaHYoGyaeqaaaaakiaah2gadaWgaaWcbaGa aG4maaqabaaakiaawUfacaGLDbaacqGHxkcXcaWHLbWaaSbaaSqaai abek7aIbqabaGccqGHRaWkceWGMbGbaiaadaWgaaWcbaGaeqySdeMa eqOSdigabeaakiaadIhadaWgaaWcbaGaeqOSdigabeaakmaalaaaba GaamizaiaadohaaeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaa aaGcdaWcaaqaaiaadsgacaWHTbWaaSbaaSqaaiabeg7aHbqabaaake aacaWGKbGaam4CaaaacqGHxkcXcaWHLbWaaSbaaSqaaiaaiodaaeqa aaGcbaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlabgUcaRmaabmaabaGaaCyYdiabgEna0kaa h2gadaWgaaWcbaGaeqySdegabeaaaOGaayjkaiaawMcaaiabgEPiel aahwgadaWgaaWcbaGaeqySdegabeaakiabgUcaRmaalaaabaGaamiz aiaadohaaeaacaWGKbGaamiEamaaBaaaleaacaaIZaaabeaaaaGcda WcaaqaaiaadsgaaeaacaWGKbGaam4CaaaadaqadaqaaiaahM8acqGH xdaTcaWHTbWaaSbaaSqaaiabeg7aHbqabaaakiaawIcacaGLPaaacq GHxkcXcaWHLbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIabmyDayaa caWaaSbaaSqaaiaaiodaaeqaaOWaaSaaaeaacaWGKbGaam4Caaqaai aadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaaakmaalaaabaGaamiz aiaah2gadaWgaaWcbaGaaG4maaqabaaakeaacaWGKbGaam4Caaaacq GHxkcXcaWHLbWaaSbaaSqaaiaaiodaaeqaaaaaaa@DB57@

Substitute dr/ds= m 3 ,d m i /ds=κ× m i ,dω/ds= κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaaCOCaiaac+cacaWGKbGaam 4Caiabg2da9iaah2gadaWgaaWcbaGaaG4maaqabaGccaGGSaGaaGPa VlaaykW7caaMc8Uaamizaiaah2gadaWgaaWcbaGaamyAaaqabaGcca GGVaGaamizaiaadohacqGH9aqpcaWH6oGaey41aqRaaCyBamaaBaaa leaacaWGPbaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaamizaiaahM8acaGGVaGaamizaiaadohacqGH 9aqpdaWfGaqaaiaahQ7aaSqabeaacqGHhis0aaaaaa@5FFC@ , and note that

F 1 d x 3 ds (1+ κ 2 x 1 κ 1 x 2 ) e 3 m 3 ds d x 3 x 1 κ 3 e 2 m 3 + ds d x 3 x 2 κ 3 e 1 m 3 +( δ αβ f αβ ) e α m β d u 3 d x β e 3 m β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahAeadaahaaWcbeqaaiabgk HiTiaaigdaaaGccaaMc8UaaGPaVlabgIKi7oaalaaabaGaamizaiaa dIhadaWgaaWcbaGaaG4maaqabaaakeaacaWGKbGaam4CaaaacaGGOa GaaGymaiabgUcaRiabeQ7aRnaaBaaaleaacaaIYaaabeaakiaadIha daWgaaWcbaGaaGymaaqabaGccqGHsislcqaH6oWAdaWgaaWcbaGaaG ymaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiaahwga daWgaaWcbaGaaG4maaqabaGccqGHxkcXcaWHTbWaaSbaaSqaaiaaio daaeqaaOGaeyOeI0YaaSaaaeaacaWGKbGaam4CaaqaaiaadsgacaWG 4bWaaSbaaSqaaiaaiodaaeqaaaaakiaadIhadaWgaaWcbaGaaGymaa qabaGccqaH6oWAdaWgaaWcbaGaaG4maaqabaGccaWHLbWaaSbaaSqa aiaaikdaaeqaaOGaey4LIqSaaCyBamaaBaaaleaacaaIZaaabeaaki aaykW7cqGHRaWkdaWcaaqaaiaadsgacaWGZbaabaGaamizaiaadIha daWgaaWcbaGaaG4maaqabaaaaOGaamiEamaaBaaaleaacaaIYaaabe aakiabeQ7aRnaaBaaaleaacaaIZaaabeaakiaahwgadaWgaaWcbaGa aGymaaqabaGccqGHxkcXcaWHTbWaaSbaaSqaaiaaiodaaeqaaaGcba GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWk caGGOaGaeqiTdq2aaSbaaSqaaiabeg7aHjabek7aIbqabaGccqGHsi slcaWGMbWaaSbaaSqaaiabeg7aHjabek7aIbqabaGccaGGPaGaaCyz amaaBaaaleaacqaHXoqyaeqaaOGaey4LIqSaaCyBamaaBaaaleaacq aHYoGyaeqaaOGaeyOeI0YaaSaaaeaacaWGKbGaamyDamaaBaaaleaa caaIZaaabeaaaOqaaiaadsgacaWG4bWaaSbaaSqaaiabek7aIbqaba aaaOGaaCyzamaaBaaaleaacaaIZaaabeaakiabgEPielaah2gadaWg aaWcbaGaeqOSdigabeaaaaaa@B995@

A tedious set of matrix multiplications shows that the components of D in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  are

D ij mm [ f ˙ 11 1 2 f ˙ 12 1 2 ( d x 3 ds d u ˙ 3 d x 1 κ 3 x 2 + κ 2 u ˙ 3 ) f ˙ 22 1 2 ( d x 3 ds d u ˙ 3 d x 2 + κ 3 x 1 κ 1 u ˙ 3 ) sym { d x 3 ds d s ˙ d x 3 +( κ 1 x 2 κ 2 x 1 ) } ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGebWaa0baaSqaaiaadMgacaWGQb aabaGaaCyBaiaah2gaaaGccqGHHjIUdaWadaqaauaabeqadmaaaeaa ceWGMbGbaiaadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaWaaSaaae aacaaIXaaabaGaaGOmaaaaceWGMbGbaiaadaWgaaWcbaGaaGymaiaa ikdaaeqaaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaam aalaaabaGaamizaiaadIhadaWgaaWcbaGaaG4maaqabaaakeaacaWG KbGaam4CaaaadaWcaaqaaiaadsgaceWG1bGbaiaadaWgaaWcbaGaaG 4maaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIXaaabeaaaaGc cqGHsisldaWfGaqaaiabeQ7aRnaaBaaaleaacaaIZaaabeaaaeqaba Gaey4bIenaaOGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRiab eQ7aRnaaBaaaleaacaaIYaaabeaakiqadwhagaGaamaaBaaaleaaca aIZaaabeaaaOGaayjkaiaawMcaaaqaaiabl6UinbqaaiqadAgagaGa amaaBaaaleaacaaIYaGaaGOmaaqabaaakeaadaWcaaqaaiaaigdaae aacaaIYaaaamaabmaabaWaaSaaaeaacaWGKbGaamiEamaaBaaaleaa caaIZaaabeaaaOqaaiaadsgacaWGZbaaamaalaaabaGaamizaiqadw hagaGaamaaBaaaleaacaaIZaaabeaaaOqaaiaadsgacaWG4bWaaSba aSqaaiaaikdaaeqaaaaakiabgUcaRmaaxacabaGaeqOUdS2aaSbaaS qaaiaaiodaaeqaaaqabeaacqGHhis0aaGccaWG4bWaaSbaaSqaaiaa igdaaeqaaOGaeyOeI0IaeqOUdS2aaSbaaSqaaiaaigdaaeqaaOGabm yDayaacaWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaaabaGa am4CaiaadMhacaWGTbaabaGaeS47IWeabaWaaiWaaeaadaWcaaqaai aadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaadoha aaWaaSaaaeaacaWGKbGabm4CayaacaaabaGaamizaiaadIhadaWgaa WcbaGaaG4maaqabaaaaOGaey4kaSYaaeWaaeaadaWfGaqaaiabeQ7a RnaaBaaaleaacaaIXaaabeaaaeqabaGaey4bIenaaOGaamiEamaaBa aaleaacaaIYaaabeaakiabgkHiTmaaxacabaGaeqOUdS2aaSbaaSqa aiaaikdaaeqaaaqabeaacqGHhis0aaGccaWG4bWaaSbaaSqaaiaaig daaeqaaaGccaGLOaGaayzkaaaacaGL7bGaayzFaaaaaaGaay5waiaa w2faaaaa@9C21@

to within second order terms in curvature, f ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35BA@  and d u 3 /d x α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamyDamaaBaaaleaacaaIZa aabeaakiaac+cacaWGKbGaamiEamaaBaaaleaacqaHXoqyaeqaaaaa @3A00@ .

 

 

 

10.2.7 Kinematics of rods that are bent and twisted in the unstressed state

 

It is straightforward to generalize the results in sections 10.2.3-10.2.5 to calculate strain measures for rods that are not straight in their initial configuration.  In this case we must start by describing the geometry of the undeformed rod.  To this end

  1. We denote the distance measured along the axis of the initial, unstressed, twisted rod by s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGZbGbaebaaaa@33D6@
  2. At each point s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGZbGbaebaaaa@33D6@  on the initial rod, we introduce a set of three mutually perpendicular unit vectors { m ¯ 1 , m ¯ 2 , m ¯ 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGabCyBayaaraWaaSbaaSqaai aaigdaaeqaaOGaaiilaiqah2gagaqeamaaBaaaleaacaaIYaaabeaa kiaacYcaceWHTbGbaebadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3C26@ , where m ¯ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHTbGbaebadaWgaaWcbaGaaG4maa qabaaaaa@34BD@  is chosen to be tangent to the axis of the undeformed rod; while m ¯ 1 , m ¯ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHTbGbaebadaWgaaWcbaGaaGymaa qabaGccaGGSaGabCyBayaaraWaaSbaaSqaaiaaikdaaeqaaaaa@376B@  are parallel to the principal moments of inertia of the cross-section.
  3.  We also introduce an arbitrary Cartesian basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@  where the unit vectors denote three fixed directions in space.
  4. The basis vectors { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@  and { m ¯ 1 , m ¯ 2 , m ¯ 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGabCyBayaaraWaaSbaaSqaai aaigdaaeqaaOGaaiilaiqah2gagaqeamaaBaaaleaacaaIYaaabeaa kiaacYcaceWHTbGbaebadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3C26@  together define a set of three Euler angles ( ϕ ¯ (s), θ ¯ (s), ψ ¯ (s) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaaiqbew9aMzaaraGaaiikai aadohacaGGPaGaaiilaiqbeI7aXzaaraGaaiikaiaadohacaGGPaGa aiilaiqbeI8a5zaaraGaaiikaiaadohacaGGPaaacaGLOaGaayzkaa aaaa@4236@ , which completely describe the shape of the undeformed rod.
  5. We define a rotation tensor R ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHsbGbaebaaaa@33B9@  satisfying m ¯ i = R ¯ e i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHTbGbaebadaWgaaWcbaGaamyAaa qabaGccqGH9aqpceWHsbGbaebacaWHLbWaaSbaaSqaaiaadMgaaeqa aaaa@38F9@  that characterizes the orientation of { m ¯ 1 , m ¯ 2 , m ¯ 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGabCyBayaaraWaaSbaaSqaai aaigdaaeqaaOGaaiilaiqah2gagaqeamaaBaaaleaacaaIYaaabeaa kiaacYcaceWHTbGbaebadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3C26@  with respect to { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BC6@ .  The components of R ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHsbGbaebaaaa@33B9@  can be found using the formulas in Section 10.2.3.
  6. We define three curvature components κ ¯ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaH6oWAgaqeamaaBaaaleaacaWGPb aabeaaaaa@35AA@  that characterize the bending and twisting of the initial rod, as follows

κ ¯ 1 =sin( ψ ¯ ) d θ ¯ d s ¯ cos( ψ ¯ )sin θ ¯ d ϕ ¯ d s ¯ κ ¯ 2 =cos( ψ ¯ ) d θ ¯ d s ¯ +sin( ψ ¯ )sin θ ¯ d ϕ ¯ d s ¯ κ ¯ 3 = d ψ ¯ d s ¯ + d ϕ ¯ d s ¯ cos θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH6oWAgaqeam aaBaaaleaacaaIXaaabeaakiabg2da9iGacohacaGGPbGaaiOBaiaa cIcacuaHipqEgaqeaiaacMcadaWcaaqaaiaadsgacuaH4oqCgaqeaa qaaiaadsgaceWGZbGbaebaaaGaeyOeI0Iaci4yaiaac+gacaGGZbGa aiikaiqbeI8a5zaaraGaaiykaiGacohacaGGPbGaaiOBaiqbeI7aXz aaraWaaSaaaeaacaWGKbGafqy1dyMbaebaaeaacaWGKbGabm4Cayaa raaaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UafqOUdSMbaebadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpciGG JbGaai4BaiaacohacaGGOaGafqiYdKNbaebacaGGPaWaaSaaaeaaca WGKbGafqiUdeNbaebaaeaacaWGKbGabm4CayaaraaaaiabgUcaRiGa cohacaGGPbGaaiOBaiaacIcacuaHipqEgaqeaiaacMcaciGGZbGaai yAaiaac6gacuaH4oqCgaqeamaalaaabaGaamizaiqbew9aMzaaraaa baGaamizaiqadohagaqeaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UafqOUdSMbaebadaWg aaWcbaGaaG4maaqabaGccqGH9aqpdaWcaaqaaiaadsgacuaHipqEga qeaaqaaiaadsgaceWGZbGbaebaaaGaey4kaSYaaSaaaeaacaWGKbGa fqy1dyMbaebaaeaacaWGKbGabm4CayaaraaaaiGacogacaGGVbGaai 4CaiqbeI7aXzaaraaaaa@A18A@

 

The deformed shape of the rod is characterized exactly as described in Section 10.2.1, except that the axial distance x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@  is replaced by the arc-length s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGZbGbaebaaaa@33D6@  of the undeformed rod.

 

Assuming small distorsions, the deformation gradient can be expressed in dyadic notation as F= F ij m i m ¯ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaamOramaaBaaale aacaWGPbGaamOAaaqabaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaOGa ey4LIqSabCyBayaaraWaaSbaaSqaaiaadQgaaeqaaaaa@3DC5@ , where the coefficients F ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@359A@  are given below.  The deformation gradient can also be decomposed into two successive rotations and a small distorsion

F=R G ¯ R ¯ T =HR R ¯ T =R R ¯ T G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiqahEeaga qeaiqahkfagaqeamaaCaaaleqabaGaamivaaaakiabg2da9iaahIea caWHsbGabCOuayaaraWaaWbaaSqabeaacaWGubaaaOGaeyypa0JaaC OuaiqahkfagaqeamaaCaaaleqabaGaamivaaaakiaahEeaaaa@41CA@

where the rotation tensors R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbaaaa@33A1@  and R ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHsbGbaebaaaa@33B9@  satisfy m ¯ i = R ¯ e i m i =R e i m i =R R ¯ T m ¯ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHTbGbaebadaWgaaWcbaGaamyAaa qabaGccqGH9aqpceWHsbGbaebacaWHLbWaaSbaaSqaaiaadMgaaeqa aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2 gadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWHsbGaaCyzamaaBaaa leaacaWGPbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaWGPbaabeaakiabg2da 9iaahkfaceWHsbGbaebadaahaaWcbeqaaiaadsfaaaGcceWHTbGbae badaWgaaWcbaGaamyAaaqabaaaaa@5E5B@ , and the tensors G ¯ ,H,G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHhbGbaebacaGGSaGaaCisaiaacY cacaWHhbaaaa@36AF@  can be expressed in component form as   G ¯ = G ¯ ij ee e i e j , H ij mm m i m j , G ij m ¯ m ¯ m ¯ i m ¯ j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHhbGbaebacqGH9aqpceWGhbGbae badaqhaaWcbaGaamyAaiaadQgaaeaacaWHLbGaaCyzaaaakiaahwga daWgaaWcbaGaamyAaaqabaGccqGHxkcXcaWHLbWaaSbaaSqaaiaadQ gaaeqaaOGaaiilaiaaykW7caaMc8UaamisamaaDaaaleaacaWGPbGa amOAaaqaaiaah2gacaWHTbaaaOGaaCyBamaaBaaaleaacaWGPbaabe aakiabgEPielaah2gadaWgaaWcbaGaamOAaaqabaGccaGGSaGaaGPa VlaaykW7caaMc8UaaGPaVlaadEeadaqhaaWcbaGaamyAaiaadQgaae aaceWHTbGbaebaceWHTbGbaebaaaGcceWHTbGbaebadaWgaaWcbaGa amyAaaqabaGccqGHxkcXceWHTbGbaebadaWgaaWcbaGaamOAaaqaba aaaa@60C3@ .  Their components are given by

F ij = G ¯ ij ee = H ij mm = G ij m ¯ m ¯ [ 1+ f 11 f 12 κ 3 x 2 ds d s ¯ + κ ¯ 3 x 2 0 1+ f 22 κ 3 x 1 ds d s ¯ κ ¯ 3 x 1 d u 3 d x 1 d u 3 d x 2 ( 1+ κ 1 x 2 κ 2 x 1 ) ds d s ¯ κ ¯ 1 x 2 + κ ¯ 2 x 1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iqadEeagaqeamaaDaaaleaacaWGPbGaamOAaaqa aiaahwgacaWHLbaaaOGaeyypa0JaamisamaaDaaaleaacaWGPbGaam OAaaqaaiaah2gacaWHTbaaaOGaeyypa0Jaam4ramaaDaaaleaacaWG PbGaamOAaaqaaiqah2gagaqeaiqah2gagaqeaaaakiabggMi6oaadm aabaqbaeqabmWaaaqaaiaaigdacqGHRaWkcaWGMbWaaSbaaSqaaiaa igdacaaIXaaabeaaaOqaaiaadAgadaWgaaWcbaGaaGymaiaaikdaae qaaaGcbaGaeyOeI0IaeqOUdS2aaSbaaSqaaiaaiodaaeqaaOGaamiE amaaBaaaleaacaaIYaaabeaakmaalaaabaGaamizaiaadohaaeaaca WGKbGabm4CayaaraaaaiabgUcaRiqbeQ7aRzaaraWaaSbaaSqaaiaa iodaaeqaaOGaamiEamaaBaaaleaacaaIYaaabeaaaOqaaiaaicdaae aacaaIXaGaey4kaSIaamOzamaaBaaaleaacaaIYaGaaGOmaaqabaaa keaacqaH6oWAdaWgaaWcbaGaaG4maaqabaGccaWG4bWaaSbaaSqaai aaigdaaeqaaOWaaSaaaeaacaWGKbGaam4CaaqaaiaadsgaceWGZbGb aebaaaGaeyOeI0IafqOUdSMbaebadaWgaaWcbaGaaG4maaqabaGcca WG4bWaaSbaaSqaaiaaigdaaeqaaaGcbaWaaSaaaeaacaWGKbGaamyD amaaBaaaleaacaaIZaaabeaaaOqaaiaadsgacaWG4bWaaSbaaSqaai aaigdaaeqaaaaaaOqaamaalaaabaGaamizaiaadwhadaWgaaWcbaGa aG4maaqabaaakeaacaWGKbGaamiEamaaBaaaleaacaaIYaaabeaaaa aakeaadaqadaqaaiaaigdacqGHRaWkcqaH6oWAdaWgaaWcbaGaaGym aaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaeqOUdS 2aaSbaaSqaaiaaikdaaeqaaOGaamiEamaaBaaaleaacaaIXaaabeaa aOGaayjkaiaawMcaamaalaaabaGaamizaiaadohaaeaacaWGKbGabm 4CayaaraaaaiabgkHiTiqbeQ7aRzaaraWaaSbaaSqaaiaaigdaaeqa aOGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRiqbeQ7aRzaara WaaSbaaSqaaiaaikdaaeqaaOGaamiEamaaBaaaleaacaaIXaaabeaa aaaakiaawUfacaGLDbaaaaa@9BAD@

 

The deformation gradient can be written down immediately, by mapping the initial rod onto a fictitious intermediate configuration in which the rod is straight, chosen as follows:

1.      The straight rod has axis parallel to the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  direction

2.      The point at arc-length s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGZbGbaebaaaa@33D6@  in the unstressed rod has coordinates x= s ¯ e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0Jabm4CayaaraGaaC yzamaaBaaaleaacaaIZaaabeaaaaa@37B4@  in the intermediate configuration.

3.      The principal axes of the cross section are parallel to ( e 1 , e 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiykaaaa @388E@  in the intermediate configuration

4.      The cross-section of the rod has the same shape in the intermediate configuration as in the undeformed configuration.

 

The deformed state can be reached in two steps (i) Deform the rod from the unstressed configuration to the intermediate configuration, with a deformation gradient F ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHgbGbaebaaaa@33AD@ .  The components of F ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWHgbGbaebaaaa@33AD@  can be calculated as the inverse of the deformation gradient that maps the intermediate straight rod onto the undeformed shape.  (ii) Deform the rod from the straight configuration to the deformed configuration, with a deformation gradient F ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahAeagaqcaaaa@3167@ .  The total deformation gradient follows as F= F ^ F ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAeacqGH9aqpceWHgbGbaKaaceWHgb Gbaebaaaa@3423@

 

 

 

10.2.8 Representation of forces and moments in slender rods

 

The figure shows a generic cross-section of the rod, in the deformed configuration. To define measures of internal and external force acting on the rod, we define the following variables

*      A basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=fYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaaCyBam aaBaaaleaacaaIXaaabeaakiaacYcacaWHTbWaaSbaaSqaaiaaikda aeqaaOGaaiilaiaah2gadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3DD8@  with unit vectors chosen following the scheme described in 10.2.2.  We define the following vector components in this basis:

*      The body force acting on the rod b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaa aa@34C7@ .  For simplicity, we shall assume that the body force is uniform within the cross section (but b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaadMgaaeqaaa aa@34C7@  may vary along the length of the rod).

*      The tractions acting on the exterior surface of the rod t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG0bWaaSbaaSqaaiaadMgaaeqaaa aa@34D9@

*      The Cauchy stress within the rod σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@ .

 

External forces and moments acting on the rod are characterized by

  1. The force per unit length acting on the rod, p= p i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHWbGaeyypa0JaamiCamaaBaaale aacaWGPbaabeaakiaah2gadaWgaaWcbaGaamyAaaqabaaaaa@38EE@ .  The force components can be calculated from the tractions and body force acting on the rod as f i ( x 3 )=A b i + C t i ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaeyypa0Ja amyqaiaadkgadaWgaaWcbaGaamyAaaqabaGccqGHRaWkdaWdrbqaai aadshadaWgaaWcbaGaamyAaaqabaGccaWGKbGaam4CaaWcbaGaam4q aaqab0Gaey4kIipaaaa@43E7@
  2. The moment per unit length acting on the rod, q= q i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHXbGaeyypa0JaamyCamaaBaaale aacaWGPbaabeaakiaah2gadaWgaaWcbaGaamyAaaqabaaaaa@38F0@ .  The moment components can  be calculated from the tractions acting on the exterior surface of the rod as as

q 1 = C x 2 t 3 ds q 2 = C x 1 t 3 ds q 3 = C ( x 1 t 2 x 2 t 1 )ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGXbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Zaa8quaeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamiD amaaBaaaleaacaaIZaaabeaakiaadsgacaWGZbaaleaacaWGdbaabe qdcqGHRiI8aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGXbWaaSbaaSqaaiaaik daaeqaaOGaeyypa0JaeyOeI0Yaa8quaeaacaWG4bWaaSbaaSqaaiaa igdaaeqaaOGaamiDamaaBaaaleaacaaIZaaabeaakiaadsgacaWGZb aaleaacaWGdbaabeqdcqGHRiI8aOGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadghadaWgaa WcbaGaaG4maaqabaGccqGH9aqpdaWdrbqaaiaacIcacaWG4bWaaSba aSqaaiaaigdaaeqaaOGaamiDamaaBaaaleaacaaIYaaabeaakiabgk HiTiaadIhadaWgaaWcbaGaaGOmaaqabaGccaWG0bWaaSbaaSqaaiaa igdaaeqaaOGaaiykaiaadsgacaWGZbaaleaacaWGdbaabeqdcqGHRi I8aaaa@7D7E@

  1. The resultant force acting on each end of the rod.  Each force can be expressed as components as P= P i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaaCiuaiabg2da9iaadcfadaWgaaWcba GaamyAaaqabaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaaaa@37EA@ .  The components are related to the tractions acting on the end of the rod by P i = A t i dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamiuamaaBaaaleaacaWGPbaabeaaki abg2da9maapefabaGaamiDamaaBaaaleaacaWGPbaabeaakiaadsga caWGbbaaleaacaWGbbaabeqdcqGHRiI8aaaa@3BDD@ , where the area integral is taken over the cross section at the appropriate end of the rod.
  2. The resultant moment acting on each end of the rod.  Each moment can be expressed as components as Q= Q i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaaCyuaiabg2da9iaadgfadaWgaaWcba GaamyAaaqabaGccaWHTbWaaSbaaSqaaiaadMgaaeqaaaaa@37EC@ .  The components are related to the tractions acting on the end of the rod by

Q 1 = A x 2 t 3 dA Q 2 = A x 1 t 3 dA Q 3 = A ( x 1 t 2 x 2 t 1 )dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGrbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Zaa8quaeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaamiD amaaBaaaleaacaaIZaaabeaakiaadsgacaWGbbGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGrbWaaSbaaSqa aiaaikdaaeqaaaqaaiaadgeaaeqaniabgUIiYdGccqGH9aqpcqGHsi sldaWdrbqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccaWG0bWaaSba aSqaaiaaiodaaeqaaOGaamizaiaadgeacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVdWcbaGaamyqaaqab0Gaey4k IipakiaadgfadaWgaaWcbaGaaG4maaqabaGccqGH9aqpdaWdrbqaai aacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaamiDamaaBaaaleaa caaIYaaabeaakiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaGcca WG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaadsgacaWGbbaaleaa caWGbbaabeqdcqGHRiI8aaaa@74B6@

 

 

Internal forces and moments in the rod are characterized by the following quantities:

  1. The variation of internal shear stress in the cross section σ α3 ( x β , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaeqySdeMaaG 4maaqabaGccaGGOaGaamiEamaaBaaaleaacqaHYoGyaeqaaOGaaiil aiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@3DE8@
  2. The average in-plane stress components

S αβ = A σ αβ dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaGccqGH9aqpdaWdrbqaaiabeo8aZnaaBaaaleaacqaHXoqy cqaHYoGyaeqaaOGaamizaiaadgeaaSqaaiaadgeaaeqaniabgUIiYd GccaaMc8oaaa@43A7@

  1. Three components of a vector bending moment, defined as

M 1 ( x 3 )= A σ 33 x 2 dA M 2 ( x 3 )= A σ 33 x 1 dA M 3 ( x 3 )= A ( σ 23 x 1 σ 13 x 2 ) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGnbWaaSbaaSqaaiaaigdaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaeyypa0Za a8quaeaacqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGaamiEam aaBaaaleaacaaIYaaabeaakiaadsgacaWGbbaaleaacaWGbbaabeqd cqGHRiI8aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caWGnbWaaSbaaSqaaiaaikda aeqaaOGaaiikaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaey ypa0JaeyOeI0Yaa8quaeaacqaHdpWCdaWgaaWcbaGaaG4maiaaioda aeqaaOGaamiEamaaBaaaleaacaaIXaaabeaakiaadsgacaWGbbaale aacaWGbbaabeqdcqGHRiI8aOGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaad2eadaWgaaWcba GaaG4maaqabaGccaGGOaGaamiEamaaBaaaleaacaaIZaaabeaakiaa cMcacqGH9aqpdaWdrbqaamaabmaabaGaeq4Wdm3aaSbaaSqaaiaaik dacaaIZaaabeaakiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHsisl cqaHdpWCdaWgaaWcbaGaaGymaiaaiodaaeqaaOGaamiEamaaBaaale aacaaIYaaabeaaaOGaayjkaiaawMcaaaWcbaGaamyqaaqab0Gaey4k IipakiaadsgacaWGbbaaaa@8CA7@

  1. The axial force on the cross-section T 3 ( x 3 )= A σ 33 dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaaiodaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaeyypa0Za a8quaeaacqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGaamizai aadgeaaSqaaiaadgeaaeqaniabgUIiYdGccaaMc8oaaa@42A8@
  2. Two additional generalized forces T 1 ( x 3 ), T 2 ( x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaaigdaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaaG4maaqabaGccaGGPaGaaiilaiaa dsfadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEamaaBaaaleaaca aIZaaabeaakiaacMcaaaa@3D9D@ , which represent the transverse shear forces acting on the rod’s cross section.  Unlike the axial force, however, these forces cannot be directly related to the deformation of the rod.  Instead, they are calculated from the bending moments, using the equilibrium equations listed in the next section.  

 

The forces T i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGubWaaSbaaSqaaiaadMgaaeqaaa aa@34B9@  and moments M i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGnbWaaSbaaSqaaiaadMgaaeqaaa aa@34B2@  define components of a vector force and moment

  1. T= T i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHubGaeyypa0JaamivamaaBaaale aacaWGPbaabeaakiaah2gadaWgaaWcbaGaamyAaaqabaaaaa@38B6@  is the resultant force acting on an internal cross-section of the rod;
  2. M= M i m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHnbGaeyypa0JaamytamaaBaaale aacaWGPbaabeaakiaah2gadaWgaaWcbaGaamyAaaqabaaaaa@38A8@  is the resultant moment (about the centroid of the cross section) acting on the cross-section.

 

 

 

 

10.2.9 Equations of motion and boundary conditions

 

The internal forces and moments must satisfy the equations of motion

  σ α3 x α σ 31 κ 2 + σ 32 κ 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kabeo8aZnaaBa aaleaacqaHXoqycaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWc baGaeqySdegabeaaaaGccqGHsislcqaHdpWCdaWgaaWcbaGaaG4mai aaigdaaeqaaOGaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaey4kaSIa eq4Wdm3aaSbaaSqaaiaaiodacaaIYaaabeaakiabeQ7aRnaaBaaale aacaaIXaaabeaakiabg2da9iaaicdaaaa@4C82@         dT ds +p=ρAa dM ds + m 3 ×T+q=ρΗα+ω×ρΗω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgacaWHubaabaGaam izaiaadohaaaGaey4kaSIaaCiCaiabg2da9iabeg8aYjaadgeacaWH HbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaadsgacaWHnbaaba GaamizaiaadohaaaGaey4kaSIaaCyBamaaBaaaleaacaaIZaaabeaa kiabgEna0kaahsfacqGHRaWkcaWHXbGaeyypa0JaeqyWdiNaaC4Ldi aahg7acqGHRaWkcaWHjpGaey41aqRaeqyWdiNaaC4LdiaahM8aaaa@74A0@

Here, σ α3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaHXoqycaaIZa aabeaaaaa@34D3@ , T and M are the internal forces and moments in the rod; p,q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahchacaGGSaGaaCyCaaaa@332B@  are the external force and couple per unit length; ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@  is the mass density of the rod; A is its cross-sectional area, H is the area moment of inertia tensor defined in Sect 10.2.1, while a,ω,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahggacaGGSaGaaCyYdiaacYcacaWHXo aaaa@3564@  are the acceleration, angular velocity and angular acceleration of the rod’s centerline. The two equations of motion for T and M clearly represent linear and angular moment balance for an infinitesimal segment of the rod.

 

The equations of motion for T and M are often expressed as components in the { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  basis, as

d T 1 ds κ 3 T 2 + κ 2 T 3 + p 1 =ρA a 1 d T 2 ds + κ 3 T 2 κ 1 T 3 + p 2 =ρA a 2 d T 3 ds + κ 1 T 2 κ 2 T 1 + p 3 =ρA a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaiaadsfadaWgaaWcba GaaGymaaqabaaakeaacaWGKbGaam4CaaaacqGHsislcqaH6oWAdaWg aaWcbaGaaG4maaqabaGccaWGubWaaSbaaSqaaiaaikdaaeqaaOGaey 4kaSIaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaamivamaaBaaaleaa caaIZaaabeaakiabgUcaRiaadchadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcqaHbpGCcaWGbbGaamyyamaaBaaaleaacaaIXaaabeaakiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaS aaaeaacaWGKbGaamivamaaBaaaleaacaaIYaaabeaaaOqaaiaadsga caWGZbaaaiabgUcaRiabeQ7aRnaaBaaaleaacaaIZaaabeaakiaads fadaWgaaWcbaGaaGOmaaqabaGccqGHsislcqaH6oWAdaWgaaWcbaGa aGymaaqabaGccaWGubWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaam iCamaaBaaaleaacaaIYaaabeaakiabg2da9iabeg8aYjaadgeacaWG HbWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7daWcaaqaaiaadsgacaWGubWaaSba aSqaaiaaiodaaeqaaaGcbaGaamizaiaadohaaaGaey4kaSIaeqOUdS 2aaSbaaSqaaiaaigdaaeqaaOGaamivamaaBaaaleaacaaIYaaabeaa kiabgkHiTiabeQ7aRnaaBaaaleaacaaIYaaabeaakiaadsfadaWgaa WcbaGaaGymaaqabaGccqGHRaWkcaWGWbWaaSbaaSqaaiaaiodaaeqa aOGaeyypa0JaeqyWdiNaamyqaiaadggadaWgaaWcbaGaaG4maaqaba aaaa@8FAC@

d M 1 ds κ 3 M 2 + κ 2 M 3 T 2 + q 1 =ρ I 1 α 1 +ρ( I 3 I 2 ) ω 2 ω 3 d M 2 ds + κ 3 M 1 κ 1 M 3 + T 1 + q 2 =ρ I 2 α 2 ρ( I 3 I 1 ) ω 1 ω 3 d M 3 ds κ 2 M 1 + κ 1 M 2 + q 3 = I 3 α 3 +( I 2 I 1 ) ω 2 ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaWGKbGaamytamaaBa aaleaacaaIXaaabeaaaOqaaiaadsgacaWGZbaaaiabgkHiTiabeQ7a RnaaBaaaleaacaaIZaaabeaakiaad2eadaWgaaWcbaGaaGOmaaqaba GccqGHRaWkcqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccaWGnbWaaSba aSqaaiaaiodaaeqaaOGaeyOeI0IaamivamaaBaaaleaacaaIYaaabe aakiabgUcaRiaadghadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH bpGCcaWGjbWaaSbaaSqaaiaaigdaaeqaaOGaeqySde2aaSbaaSqaai aaigdaaeqaaOGaey4kaSIaeqyWdiNaaiikaiaadMeadaWgaaWcbaGa aG4maaqabaGccqGHsislcaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaai ykaiabeM8a3naaBaaaleaacaaIYaaabeaakiabeM8a3naaBaaaleaa caaIZaaabeaaaOqaamaalaaabaGaamizaiaad2eadaWgaaWcbaGaaG OmaaqabaaakeaacaWGKbGaam4CaaaacqGHRaWkcqaH6oWAdaWgaaWc baGaaG4maaqabaGccaWGnbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0 IaeqOUdS2aaSbaaSqaaiaaigdaaeqaaOGaamytamaaBaaaleaacaaI ZaaabeaakiabgUcaRiaadsfadaWgaaWcbaGaaGymaaqabaGccqGHRa WkcaWGXbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaeqyWdiNaamys amaaBaaaleaacaaIYaaabeaakiabeg7aHnaaBaaaleaacaaIYaaabe aakiabgkHiTiabeg8aYjaacIcacaWGjbWaaSbaaSqaaiaaiodaaeqa aOGaeyOeI0IaamysamaaBaaaleaacaaIXaaabeaakiaacMcacqaHjp WDdaWgaaWcbaGaaGymaaqabaGccqaHjpWDdaWgaaWcbaGaaG4maaqa baaakeaadaWcaaqaaiaadsgacaWGnbWaaSbaaSqaaiaaiodaaeqaaa GcbaGaamizaiaadohaaaGaeyOeI0IaeqOUdS2aaSbaaSqaaiaaikda aeqaaOGaamytamaaBaaaleaacaaIXaaabeaakiabgUcaRiabeQ7aRn aaBaaaleaacaaIXaaabeaakiaad2eadaWgaaWcbaGaaGOmaaqabaGc cqGHRaWkcaWGXbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0Jaamysam aaBaaaleaacaaIZaaabeaakiabeg7aHnaaBaaaleaacaaIZaaabeaa kiabgUcaRiaacIcacaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0 IaamysamaaBaaaleaacaaIXaaabeaakiaacMcacqaHjpWDdaWgaaWc baGaaGOmaaqabaGccqaHjpWDdaWgaaWcbaGaaGymaaqabaaaaaa@A781@

Note that:

  1. If the system is in static equilibrium, the right hand sides of all the equations of motion are zero.
  2. In addition, in many dynamic problems, the right hand sides of the angular momentum balance equations may be taken to be approximately zero, since the area moments of inertia are small.   For example, the rotational inertia may be ignored when modeling the vibration of a beam.  The rotational inertia terms can be important if the rod is rotating rapidly: examples include a spinning shaft, or a rotating propeller.

 

Boundary Conditions: The internal stresses, forces and moments must satisfy the following boundary conditions

  1. S αβ = C ( x β t α )dξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaGccqGH9aqpdaWdrbqaamaabmaabaGaamiEamaaBaaaleaa cqaHYoGyaeqaaOGaamiDamaaBaaaleaacqaHXoqyaeqaaaGccaGLOa GaayzkaaGaamizaiabe67a4bWcbaGaam4qaaqab0Gaey4kIipaaaa@4503@
  2. σ 3α n α = t 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaeqySde gabeaakiaad6gadaWgaaWcbaGaeqySdegabeaakiabg2da9iaadsha daWgaaWcbaGaaG4maaqabaaaaa@3A8D@  on C
  3. The ends of the rod may be subjected to a prescribed displacement.   Alternatively, the transverse or axial tractions may be prescribed on the ends of the bar: in this case the internal forces must satisfy T=P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpcaWHqbaaaa@3344@  for s=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcaWGmbaaaa@3357@  and T=P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpcqGHsislcaWHqbaaaa@3431@  for s=0.
  4. The ends of the rod may be subjected to a prescribed rotation.  Alternatively, if the ends are free to rotate, the internal moments must satisfy M=Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2eacqGH9aqpcaWHrbaaaa@333E@  for s=L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcaWGmbaaaa@3357@  and M=Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2eacqGH9aqpcqGHsislcaWHrbaaaa@342B@  for s=0.

 

Derivation: Measures of internal force and the equilibrium equations emerge naturally from the principle of virtual work, which states that the Cauchy stress distribution must satisfy

V 0 J σ ij δ D ij d V 0 + V 0 ρ d 2 y i d t 2 δ v i d V 0 V 0 b i δ v i d V 0 S 2 t i δ v i dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaamOsaiabeo8aZnaaBaaale aacaWGPbGaamOAaaqabaGccqaH0oazcaWGebWaaSbaaSqaaiaadMga caWGQbaabeaakiaadsgacaWGwbWaaSbaaSqaaiaaicdaaeqaaaqaai aadAfadaWgaaadbaGaaGimaaqabaaaleqaniabgUIiYdGccqGHRaWk daWdrbqaaiabeg8aYnaalaaabaGaamizamaaCaaaleqabaGaaGOmaa aakiaadMhadaWgaaWcbaGaamyAaaqabaaakeaacaWGKbGaamiDamaa CaaaleqabaGaaGOmaaaaaaGccqaH0oazcaWG2bWaaSbaaSqaaiaadM gaaeqaaOGaamizaiaadAfadaWgaaWcbaGaaGimaaqabaaabaGaamOv amaaBaaameaacaaIWaaabeaaaSqab0Gaey4kIipakiabgkHiTmaape fabaGaamOyamaaBaaaleaacaWGPbaabeaakiabes7aKjaadAhadaWg aaWcbaGaamyAaaqabaGccaWGKbGaamOvamaaBaaaleaacaaIWaaabe aaaeaacaWGwbWaaSbaaWqaaiaaicdaaeqaaaWcbeqdcqGHRiI8aOGa eyOeI0Yaa8quaeaacaWG0bWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdq MaamODamaaBaaaleaacaWGPbaabeaakiaadsgacaWGbbaaleaacaWG tbWaaSbaaWqaaiaaikdaaeqaaaWcbeqdcqGHRiI8aOGaeyypa0JaaG imaaaa@7050@

for all virtual velocity fields δ v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWG2bWaaSbaaSqaaiaadM gaaeqaaaaa@3680@  and compatible stretch rates δ D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGebWaaSbaaSqaaiaadM gacaWGQbaabeaaaaa@373D@ .  The virtual velocity field and virtual stretch rates in the bar must have the same general form as the actual velocity and stretch rates, as outlined in Section 10.2.4 and 10.2.5.  The virtual velocity and stretch rate can therefore be characterized by δ f ˙ αβ ,δ u 3 ( x α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazceWGMbGbaiaadaWgaaWcba GaeqySdeMaeqOSdigabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7cqaH0oazcaWG1bWaaSbaaSqaaiaaiodaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaeqySdegabeaakiaacMcaaaa@4A84@  and compatible sets of δv,δω,δ s ˙ ,δ κ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWH2bGaaiilaiabes7aKj aahM8acaGGSaGaeqiTdqMabm4CayaacaGaaiilaiabes7aKnaaxaca baGaeqOUdSgaleqabaGaey4bIenaaOWaaSbaaSqaaiaadMgaaeqaaa aa@4364@ .  This has two consequences:

 The virtual work principle can be expressed in terms of the generalized deformation measures and forces defined in the preceding sections as

0 L 0 { ds d x 3 S αβ δ f ˙ αβ + A ( σ α3 dδ u ˙ 3 d x α +( σ 31 κ 2 σ 32 κ 1 )δ u ˙ 3 b 3 δ u ˙ 3 )dA }d x 3 + 0 L 0 { dδ s ˙ d x 3 T 3 +δ κ i M i ds d x 3 }d x 3 + 0 L ρA a i δ v i ds+ 0 L ρ I 1 ( α 1 + ω 2 ω 3 )δ ω 1 ds+ 0 L ρ I 2 ( α 2 ω 1 ω 3 )δ ω 2 ds+ 0 L ρ( I 3 α 3 +( I 1 I 2 ) ω 1 ω 2 )δ ω 3 ds 0 L ( C ( x β t α δ f ˙ αβ + t 3 δ u 3 )dξ ) ds 0 L ( p i δ v i + q i δ ω i ) ds [ P i δ v i + Q i δ ω i ] x 3 =0 [ P i δ v i + Q i δ ω i ] x 3 =L =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaadaWdXbqaamaacmaabaWaaSaaae aacaWGKbGaam4CaaqaaiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqa aaaakiaadofadaWgaaWcbaGaeqySdeMaeqOSdigabeaakiabes7aKj qadAgagaGaamaaBaaaleaacqaHXoqycqaHYoGyaeqaaOGaey4kaSYa a8quaeaadaqadaqaaiabeo8aZnaaBaaaleaacqaHXoqycaaIZaaabe aakmaalaaabaGaamizaiabes7aKjqadwhagaGaamaaBaaaleaacaaI ZaaabeaaaOqaaiaadsgacaWG4bWaaSbaaSqaaiabeg7aHbqabaaaaO Gaey4kaSIaaiikaiabeo8aZnaaBaaaleaacaaIZaGaaGymaaqabaGc cqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGHsislcqaHdpWCdaWgaa WcbaGaaG4maiaaikdaaeqaaOGaeqOUdS2aaSbaaSqaaiaaigdaaeqa aOGaaiykaiabes7aKjqadwhagaGaamaaBaaaleaacaaIZaaabeaaki abgkHiTiaadkgadaWgaaWcbaGaaG4maaqabaGccqaH0oazceWG1bGb aiaadaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPaaacaWGKbGaam yqaaWcbaGaamyqaaqab0Gaey4kIipaaOGaay5Eaiaaw2haaiaadsga caWG4bWaaSbaaSqaaiaaiodaaeqaaaqaaiaaicdaaeaacaWGmbWaaS baaWqaaiaaicdaaeqaaaqdcqGHRiI8aOGaey4kaSYaa8qCaeaadaGa daqaamaalaaabaGaamizaiabes7aKjqadohagaGaaaqaaiaadsgaca WG4bWaaSbaaSqaaiaaiodaaeqaaaaakiaadsfadaWgaaWcbaGaaG4m aaqabaGccqGHRaWkcqaH0oazdaWfGaqaaiabeQ7aRbWcbeqaaiabgE GirdaakmaaBaaaleaacaWGPbaabeaakiaad2eadaWgaaWcbaGaamyA aaqabaGcdaWcaaqaaiaadsgacaWGZbaabaGaamizaiaadIhadaWgaa WcbaGaaG4maaqabaaaaaGccaGL7bGaayzFaaGaamizaiaadIhadaWg aaWcbaGaaG4maaqabaaabaGaaGimaaqaaiaadYeadaWgaaadbaGaaG imaaqabaaaniabgUIiYdaakeaacqGHRaWkdaWdXbqaaiabeg8aYjaa dgeacaWGHbWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdqMaamODamaaBa aaleaacaWGPbaabeaaaeaacaaIWaaabaGaamitaaqdcqGHRiI8aOGa amizaiaadohacqGHRaWkdaWdXbqaaiabeg8aYjaadMeadaWgaaWcba GaaGymaaqabaGcdaqadaqaaiabeg7aHnaaBaaaleaacaaIXaaabeaa kiabgUcaRiabeM8a3naaBaaaleaacaaIYaaabeaakiabeM8a3naaBa aaleaacaaIZaaabeaaaOGaayjkaiaawMcaaiabes7aKjabeM8a3naa BaaaleaacaaIXaaabeaaaeaacaaIWaaabaGaamitaaqdcqGHRiI8aO GaamizaiaadohacqGHRaWkdaWdXbqaaiabeg8aYjaadMeadaWgaaWc baGaaGOmaaqabaGcdaqadaqaaiabeg7aHnaaBaaaleaacaaIYaaabe aakiabgkHiTiabeM8a3naaBaaaleaacaaIXaaabeaakiabeM8a3naa BaaaleaacaaIZaaabeaaaOGaayjkaiaawMcaaiabes7aKjabeM8a3n aaBaaaleaacaaIYaaabeaaaeaacaaIWaaabaGaamitaaqdcqGHRiI8 aOGaamizaiaadohacqGHRaWkdaWdXbqaaiabeg8aYnaabmaabaGaam ysamaaBaaaleaacaaIZaaabeaakiabeg7aHnaaBaaaleaacaaIZaaa beaakiabgUcaRiaacIcacaWGjbWaaSbaaSqaaiaaigdaaeqaaOGaey OeI0IaamysamaaBaaaleaacaaIYaaabeaakiaacMcacqaHjpWDdaWg aaWcbaGaaGymaaqabaGccqaHjpWDdaWgaaWcbaGaaGOmaaqabaaaki aawIcacaGLPaaacqaH0oazcqaHjpWDdaWgaaWcbaGaaG4maaqabaaa baGaaGimaaqaaiaadYeaa0Gaey4kIipakiaadsgacaWGZbaabaGaey OeI0Yaa8qCaeaadaqadaqaamaapefabaWaaeWaaeaacaWG4bWaaSba aSqaaiabek7aIbqabaGccaWG0bWaaSbaaSqaaiabeg7aHbqabaGccq aH0oazceWGMbGbaiaadaWgaaWcbaGaeqySdeMaeqOSdigabeaakiab gUcaRiaadshadaWgaaWcbaGaaG4maaqabaGccqaH0oazcaWG1bWaaS baaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaGaamizaiabe67a4bWc baGaam4qaaqab0Gaey4kIipaaOGaayjkaiaawMcaaaWcbaGaaGimaa qaaiaadYeaa0Gaey4kIipakiaadsgacaWGZbGaeyOeI0Yaa8qCaeaa daqadaqaaiaadchadaWgaaWcbaGaamyAaaqabaGccqaH0oazcaWG2b WaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamyCamaaBaaaleaacaWG Pbaabeaakiabes7aKjabeM8a3naaBaaaleaacaWGPbaabeaaaOGaay jkaiaawMcaaaWcbaGaaGimaaqaaiaadYeaa0Gaey4kIipakiaadsga caWGZbGaaGPaVlabgkHiTmaadmaabaGaamiuamaaBaaaleaacaWGPb aabeaakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqabaGccqGHRaWk caWGrbWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdqMaeqyYdC3aaSbaaS qaaiaadMgaaeqaaaGccaGLBbGaayzxaaWaaSbaaSqaaiaadIhadaWg aaadbaGaaG4maaqabaWccqGH9aqpcaaIWaaabeaakiabgkHiTmaadm aabaGaamiuamaaBaaaleaacaWGPbaabeaakiabes7aKjaadAhadaWg aaWcbaGaamyAaaqabaGccqGHRaWkcaWGrbWaaSbaaSqaaiaadMgaae qaaOGaeqiTdqMaeqyYdC3aaSbaaSqaaiaadMgaaeqaaaGccaGLBbGa ayzxaaWaaSbaaSqaaiaadIhadaWgaaadbaGaaG4maaqabaWccqGH9a qpcaWGmbaabeaakiabg2da9iaaicdaaaaa@5BDE@

 If the virtual work equation is satisfied for all δ f ˙ αβ ,δ u 3 ( x α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazceWGMbGbaiaadaWgaaWcba GaeqySdeMaeqOSdigabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7cqaH0oazcaWG1bWaaSbaaSqaaiaaiodaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaeqySdegabeaakiaacMcaaaa@4A84@  and compatible sets of δv,δω,δ s ˙ ,δ κ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWH2bGaaiilaiabes7aKj aahM8acaGGSaGaeqiTdqMabm4CayaacaGaaiilaiabes7aKnaaxaca baGaeqOUdSgaleqabaGaey4bIenaaOWaaSbaaSqaaiaadMgaaeqaaa aa@4364@ , then the internal forces and moments must satisfy the equilibrium equations and boundary conditions listed above.

 

It is straightforward to derive the first result.  The Jacobian is approximated as Jds/d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOsaiabgIKi7kaadsgacaWGZbGaai 4laiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaaaa@39E5@ ; the components of  δ D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGebWaaSbaaSqaaiaadM gacaWGQbaabeaaaaa@373D@  follow from the formulas given in Section 10.2.6, and the velocity field is approximated using the formula in 10.2.5.  Substituting the definitions given in Section 10.2.7 for generalized internal and external forces immediately gives the required result.  The algebra involved is lengthy and tedious and is left as an exercise.

 

The equilibrium equations and boundary conditions are obtained by substituting various choices of δ f ˙ αβ ,δ u 3 ( x α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazceWGMbGbaiaadaWgaaWcba GaeqySdeMaeqOSdigabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7cqaH0oazcaWG1bWaaSbaaSqaaiaaiodaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaeqySdegabeaakiaacMcaaaa@4A84@  and compatible sets of δv,δω,δ ψ ˙ ,δ s ˙ ,δ κ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWH2bGaaiilaiabes7aKj aahM8acaGGSaGaeqiTdqMafqiYdKNbaiaacaGGSaGaeqiTdqMabm4C ayaacaGaaiilaiabes7aKnaaxacabaGaeqOUdSgaleqabaGaey4bIe naaOWaaSbaaSqaaiaadMgaaeqaaaaa@4790@  into the virtual work equation.

  1. Choosing δ u 3 ( x α )=δv=δ ψ ˙ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWG1bWaaSbaaSqaaiaaio daaeqaaOGaaiikaiaadIhadaWgaaWcbaGaeqySdegabeaakiaacMca cqGH9aqpcqaH0oazcaWH2bGaeyypa0JaeqiTdqMafqiYdKNbaiaacq GH9aqpcaaIWaaaaa@446F@  reduces the virtual work equation to

0 L { A σ αβ δ f ˙ αβ dA }d x 3 0 L ( C ( x β t α δ f ˙ αβ )dξ ) ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdXbqaamaacmaabaWaa8quaeaacq aHdpWCdaWgaaWcbaGaeqySdeMaeqOSdigabeaakiabes7aKjqadAga gaGaamaaBaaaleaacqaHXoqycqaHYoGyaeqaaaqaaiaadgeaaeqani abgUIiYdGccaWGKbGaamyqaaGaay5Eaiaaw2haaiaadsgacaWG4bWa aSbaaSqaaiaaiodaaeqaaaqaaiaaicdaaeaacaWGmbaaniabgUIiYd GccaaMc8UaeyOeI0Yaa8qCaeaadaqadaqaamaapefabaWaaeWaaeaa caWG4bWaaSbaaSqaaiabek7aIbqabaGccaWG0bWaaSbaaSqaaiabeg 7aHbqabaGccqaH0oazceWGMbGbaiaadaWgaaWcbaGaeqySdeMaeqOS digabeaaaOGaayjkaiaawMcaaiaadsgacqaH+oaEaSqaaiaadoeaae qaniabgUIiYdaakiaawIcacaGLPaaaaSqaaiaaicdaaeaacaWGmbaa niabgUIiYdGccaWGKbGaam4Caaaa@6897@

The condition S αβ = C ( x β t α )dξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaGccqGH9aqpdaWdrbqaamaabmaabaGaamiEamaaBaaaleaa cqaHYoGyaeqaaOGaamiDamaaBaaaleaacqaHXoqyaeqaaaGccaGLOa GaayzkaaGaamizaiabe67a4bWcbaGaam4qaaqab0Gaey4kIipaaaa@4503@  follows immediately. 

  1. Choosing δ f αβ =δv=δ ψ ˙ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGMbWaaSbaaSqaaiabeg 7aHjabek7aIbqabaGccqGH9aqpcqaH0oazcaWH2bGaeyypa0JaeqiT dqMafqiYdKNbaiaacqGH9aqpcaaIWaaaaa@42B8@  reduces the virtual work equation to

0 L 0 { A ( σ α3 dδ u ˙ 3 d x α +( σ 31 κ 2 σ 32 κ 1 )δ u ˙ 3 b 3 δ u ˙ 3 )dA }d x 3 0 L 0 ( C ( t 3 δ u ˙ 3 )dξ ) ds d x 3 d x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaa8qCaeaadaGadaqaamaapefabaWaae WaaeaacqaHdpWCdaWgaaWcbaGaeqySdeMaaG4maaqabaGcdaWcaaqa aiaadsgacqaH0oazceWG1bGbaiaadaWgaaWcbaGaaG4maaqabaaake aacaWGKbGaamiEamaaBaaaleaacqaHXoqyaeqaaaaakiabgUcaRiaa cIcacqaHdpWCdaWgaaWcbaGaaG4maiaaigdaaeqaaOGaeqOUdS2aaS baaSqaaiaaikdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaioda caaIYaaabeaakiabeQ7aRnaaBaaaleaacaaIXaaabeaakiaacMcacq aH0oazceWG1bGbaiaadaWgaaWcbaGaaG4maaqabaGccqGHsislcaWG IbWaaSbaaSqaaiaaiodaaeqaaOGaeqiTdqMabmyDayaacaWaaSbaaS qaaiaaiodaaeqaaaGccaGLOaGaayzkaaGaamizaiaadgeaaSqaaiaa dgeaaeqaniabgUIiYdaakiaawUhacaGL9baacaWGKbGaamiEamaaBa aaleaacaaIZaaabeaaaeaacaaIWaaabaGaamitamaaBaaameaacaaI Waaabeaaa0Gaey4kIipakiabgkHiTmaapehabaWaaeWaaeaadaWdrb qaamaabmaabaGaamiDamaaBaaaleaacaaIZaaabeaakiabes7aKjqa dwhagaGaamaaBaaaleaacaaIZaaabeaaaOGaayjkaiaawMcaaiaads gacqaH+oaEaSqaaiaadoeaaeqaniabgUIiYdaakiaawIcacaGLPaaa aSqaaiaaicdaaeaacaWGmbWaaSbaaWqaaiaaicdaaeqaaaqdcqGHRi I8aOWaaSaaaeaacaWGKbGaam4CaaqaaiaadsgacaWG4bWaaSbaaSqa aiaaiodaaeqaaaaakiaadsgacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@83EC@

Recall that (by definition) δ u ˙ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqadwhagaGaamaaBaaaleaaca aIZaaabeaaaaa@3419@  must be chosen to satisfy

A δ u ˙ 3 dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaa8quaeaacqaH0oazceWG1bGbaiaada WgaaWcbaGaaG4maaqabaGccaWGKbGaamyqaaWcbaGaamyqaaqab0Ga ey4kIipakiabg2da9iaaicdaaaa@3C25@

Since the body force is uniform, the term involving b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaaG4maaqabaaaaa@3258@  is zero.  The first integral can then be integrated by parts as follows

A σ α3 δ u 3 x α dA = A ( x α ( σ α3 δ u 3 ) σ α3 x α δ u 3 )dA = C σ α3 δ u 3 n α A δ u 3 σ α3 x α dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiabeo8aZnaaBaaaleaacq aHXoqycaaIZaaabeaakmaalaaabaGaeyOaIyRaeqiTdqMaamyDamaa BaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaeq ySdegabeaaaaGccaWGKbGaamyqaaWcbaGaamyqaaqab0Gaey4kIipa kiabg2da9maapefabaWaaeWaaeaadaWcaaqaaiabgkGi2cqaaiabgk Gi2kaadIhadaWgaaWcbaGaeqySdegabeaaaaGcdaqadaqaaiabeo8a ZnaaBaaaleaacqaHXoqycaaIZaaabeaakiabes7aKjaadwhadaWgaa WcbaGaaG4maaqabaaakiaawIcacaGLPaaacqGHsisldaWcaaqaaiab gkGi2kabeo8aZnaaBaaaleaacqaHXoqycaaIZaaabeaaaOqaaiabgk Gi2kaadIhadaWgaaWcbaGaeqySdegabeaaaaGccqaH0oazcaWG1bWa aSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaGaamizaiaadgeaaS qaaiaadgeaaeqaniabgUIiYdGccqGH9aqpdaWdrbqaaiabeo8aZnaa BaaaleaacqaHXoqycaaIZaaabeaakiabes7aKjaadwhadaWgaaWcba GaaG4maaqabaGccaWGUbWaaSbaaSqaaiabeg7aHbqabaaabaGaam4q aaqab0Gaey4kIipakiabgkHiTmaapefabaGaeqiTdqMaamyDamaaBa aaleaacaaIZaaabeaakmaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaSqa aiabeg7aHjaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacq aHXoqyaeqaaaaakiaadsgacaWGbbaaleaacaWGbbaabeqdcqGHRiI8 aaaa@8C19@

Choosing δ u 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadwhadaWgaaWcbaGaaG4maa qabaGccqGH9aqpcaaIWaaaaa@35DA@  on the boundary yields the equilibrium equation σ α3 / x α σ 31 κ 2 + σ 32 κ 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaeq ySdeMaaG4maaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacqaH XoqyaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaiodacaaIXaaabe aakiabeQ7aRnaaBaaaleaacaaIYaaabeaakiabgUcaRiabeo8aZnaa BaaaleaacaaIZaGaaGOmaaqabaGccqaH6oWAdaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaaIWaaaaa@4D25@ ; choosing any other δ u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadwhadaWgaaWcbaGaaG4maa qabaaaaa@3410@  gives the boundary condition σ 3α n α = t 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaeqySde gabeaakiaad6gadaWgaaWcbaGaeqySdegabeaakiabg2da9iaadsha daWgaaWcbaGaaG4maaqabaaaaa@3A8D@ .

  1. Choosing δ u 3 ( x α )= f αβ =δ r ˙ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWG1bWaaSbaaSqaaiaaio daaeqaaOGaaiikaiaadIhadaWgaaWcbaGaeqySdegabeaakiaacMca cqGH9aqpcaWGMbWaaSbaaSqaaiabeg7aHjabek7aIbqabaGccqGH9a qpcqaH0oazceWHYbGbaiaacqGH9aqpcaaIWaaaaa@4559@ , using δ κ 1 =δ ψ ˙ κ 2 δ κ 2 =δ ψ ˙ κ 1 δ κ 3 =dδ ψ ˙ /ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKnaaxacabaGaeqOUdSgaleqaba Gaey4bIenaaOWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaeqiTdqMa fqiYdKNbaiaacqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH0oazdaWfGaqaaiabeQ7a RbWcbeqaaiabgEGirdaakmaaBaaaleaacaaIYaaabeaakiabg2da9i abgkHiTiabes7aKjqbeI8a5zaacaGaeqOUdS2aaSbaaSqaaiaaigda aeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH0oazdaWfGa qaaiabeQ7aRbWcbeqaaiabgEGirdaakmaaBaaaleaacaaIZaaabeaa kiabg2da9iaadsgacqaH0oazcuaHipqEgaGaaiaac+cacaWGKbGaam 4Caaaa@6B1E@  as well as δ ω 3 =δ ψ ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjabeM8a3naaBaaaleaacaaIZa aabeaakiabg2da9iabes7aKjqbeI8a5zaacaaaaa@396F@  yields

0 L { ( + κ 2 M 1 κ 1 M 2 )δ ψ ˙ + dδ ψ ˙ ds M 3 }d x 3 + 0 L ρ( I 3 α 3 +( I 1 I 2 ) ω 1 ω 2 )δ ψ ˙ ds 0 L ( q 3 δ ψ ˙ ) ds [ Q 3 δ ψ ˙ ] x 3 =0 [ Q 3 δ ψ ˙ ] x 3 =L =0 0 L { ( d M 3 ds + κ 2 M 1 κ 1 M 2 q 3 ρ( I 3 α 3 +( I 1 I 2 ) ω 1 ω 2 ) )δ ψ ˙ }d x 3 + [ ( M 3 Q 3 )δ ψ ˙ ] x 3 =L [ ( M 3 + Q 3 )δ ψ ˙ ] x 3 =0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaapehabaWaaiWaaeaadaqada qaaiabgUcaRiabeQ7aRnaaBaaaleaacaaIYaaabeaakiaad2eadaWg aaWcbaGaaGymaaqabaGccqGHsislcqaH6oWAdaWgaaWcbaGaaGymaa qabaGccaWGnbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGa eqiTdqMafqiYdKNbaiaacqGHRaWkdaWcaaqaaiaadsgacqaH0oazcu aHipqEgaGaaaqaaiaadsgacaWGZbaaaiaad2eadaWgaaWcbaGaaG4m aaqabaaakiaawUhacaGL9baacaWGKbGaamiEamaaBaaaleaacaaIZa aabeaakiabgUcaRmaapehabaGaeqyWdi3aaeWaaeaacaWGjbWaaSba aSqaaiaaiodaaeqaaOGaeqySde2aaSbaaSqaaiaaiodaaeqaaOGaey 4kaSIaaiikaiaadMeadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWG jbWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiabeM8a3naaBaaaleaaca aIXaaabeaakiabeM8a3naaBaaaleaacaaIYaaabeaaaOGaayjkaiaa wMcaaiabes7aKjqbeI8a5zaacaaaleaacaaIWaaabaGaamitaaqdcq GHRiI8aOGaamizaiaadohaaSqaaiaaicdaaeaacaWGmbaaniabgUIi YdaakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyOeI0Yaa8qC aeaadaqadaqaaiaadghadaWgaaWcbaGaaG4maaqabaGccqaH0oazcu aHipqEgaGaaaGaayjkaiaawMcaaaWcbaGaaGimaaqaaiaadYeaa0Ga ey4kIipakiaadsgacaWGZbGaeyOeI0YaamWaaeaacaWGrbWaaSbaaS qaaiaaiodaaeqaaOGaeqiTdqMafqiYdKNbaiaaaiaawUfacaGLDbaa daWgaaWcbaGaamiEamaaBaaameaacaaIZaaabeaaliabg2da9iaaic daaeqaaOGaeyOeI0YaamWaaeaacaWGrbWaaSbaaSqaaiaaiodaaeqa aOGaeqiTdqMafqiYdKNbaiaaaiaawUfacaGLDbaadaWgaaWcbaGaam iEamaaBaaameaacaaIZaaabeaaliabg2da9iaadYeaaeqaaOGaeyyp a0JaaGPaVlaaicdaaeaacqGHshI3daWdXbqaamaacmaabaWaaeWaae aacqGHsisldaWcaaqaaiaadsgacaWGnbWaaSbaaSqaaiaaiodaaeqa aaGcbaGaamizaiaadohaaaGaey4kaSIaeqOUdS2aaSbaaSqaaiaaik daaeqaaOGaamytamaaBaaaleaacaaIXaaabeaakiabgkHiTiabeQ7a RnaaBaaaleaacaaIXaaabeaakiaad2eadaWgaaWcbaGaaGOmaaqaba GccqGHsislcaWGXbWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaeqyW di3aaeWaaeaacaWGjbWaaSbaaSqaaiaaiodaaeqaaOGaeqySde2aaS baaSqaaiaaiodaaeqaaOGaey4kaSIaaiikaiaadMeadaWgaaWcbaGa aGymaaqabaGccqGHsislcaWGjbWaaSbaaSqaaiaaikdaaeqaaOGaai ykaiabeM8a3naaBaaaleaacaaIXaaabeaakiabeM8a3naaBaaaleaa caaIYaaabeaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaiabes7aKj qbeI8a5zaacaaacaGL7bGaayzFaaGaamizaiaadIhadaWgaaWcbaGa aG4maaqabaaabaGaaGimaaqaaiaadYeaa0Gaey4kIipaaOqaaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7cqGHRaWkdaWadaqaaiaacIcacaWGnbWaaSbaaSqaaiaaio daaeqaaOGaeyOeI0IaamyuamaaBaaaleaacaaIZaaabeaakiaacMca cqaH0oazcuaHipqEgaGaaaGaay5waiaaw2faamaaBaaaleaacaWG4b WaaSbaaWqaaiaaiodaaeqaaSGaeyypa0JaamitaaqabaGccqGHsisl daWadaqaaiaacIcacaWGnbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaS IaamyuamaaBaaaleaacaaIZaaabeaakiaacMcacqaH0oazcuaHipqE gaGaaaGaay5waiaaw2faamaaBaaaleaacaWG4bWaaSbaaWqaaiaaio daaeqaaSGaeyypa0JaaGimaaqabaGccqGH9aqpcaaIWaaaaaa@130A@

where we have integrated by parts to obtain the second line.   Choosing δ ψ ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqbeI8a5zaacaaaaa@3404@  to vanish on the ends of the rod yields the equation of motion d M 3 /ds κ 2 M 1 + κ 1 M 2 + q 3 =ρ( I 3 α 3 +( I 1 I 2 ) ω 1 ω 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbGaamytamaaBaaaleaacaaIZa aabeaakiaac+cacaWGKbGaam4CaiabgkHiTiabeQ7aRnaaBaaaleaa caaIYaaabeaakiaad2eadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcq aH6oWAdaWgaaWcbaGaaGymaaqabaGccaWGnbWaaSbaaSqaaiaaikda aeqaaOGaey4kaSIaamyCamaaBaaaleaacaaIZaaabeaakiabg2da9i abeg8aYnaabmaabaGaamysamaaBaaaleaacaaIZaaabeaakiabeg7a HnaaBaaaleaacaaIZaaabeaakiabgUcaRiaacIcacaWGjbWaaSbaaS qaaiaaigdaaeqaaOGaeyOeI0IaamysamaaBaaaleaacaaIYaaabeaa kiaacMcacqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqaHjpWDdaWgaa WcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaaa@5A36@ . Any other choice of  δ ψ ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqbeI8a5zaacaaaaa@3404@  yields the boundary conditions M 3 =± Q 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcqGHXcqScaWGrbWaaSbaaSqaaiaaiodaaeqaaaaa@3700@  on the ends of the rod.

  1. Choosing δ u 3 ( x α )= f αβ =δ ψ ˙ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWG1bWaaSbaaSqaaiaaio daaeqaaOGaaiikaiaadIhadaWgaaWcbaGaeqySdegabeaakiaacMca cqGH9aqpcaWGMbWaaSbaaSqaaiabeg7aHjabek7aIbqabaGccqGH9a qpcqaH0oazcuaHipqEgaGaaiabg2da9iaaicdaaaa@462C@  and substituting δ ω 1 =δ τ ˙ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjabeM8a3naaBaaaleaacaaIXa aabeaakiabg2da9iabgkHiTiabes7aKjqbes8a0zaacaWaaSbaaSqa aiaaikdaaeqaaaaa@3B39@ , δ ω 2 =δ τ ˙ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjabeM8a3naaBaaaleaacaaIYa aabeaakiabg2da9iabes7aKjqbes8a0zaacaWaaSbaaSqaaiaaigda aeqaaaaa@3A4C@ , where δ τ ˙ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqbes8a0zaacaWaaSbaaSqaai abeg7aHbqabaaaaa@35C6@  are the components of a virtual rate of change of the tangent vector t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahshaaaa@3185@  reduces the virtual work equation to

0 L 0 { dδ s ˙ d x 3 T 3 +δ κ i M i ds d x 3 }d x 3 0 L ( p i δ v i q 2 d τ ˙ 1 + q 1 δ τ ˙ 2 ) ds + 0 L ρA a i δ v i ds+ 0 L ρ I 1 ( α 1 + ω 2 ω 3 )δ τ ˙ 2 ds 0 L ρ I 2 ( α 2 ω 1 ω 3 )δ τ ˙ 1 ds [ P i δ v i Q 2 δ τ ˙ 1 + Q 1 δ τ ˙ 2 ] x 3 =0 [ P i δ v i Q 2 δ τ ˙ 1 + Q 1 δ τ ˙ 2 ] x 3 =L =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaadaWdXbqaamaacmaabaWaaSaaae aacaWGKbGaeqiTdqMabm4CayaacaaabaGaamizaiaadIhadaWgaaWc baGaaG4maaqabaaaaOGaamivamaaBaaaleaacaaIZaaabeaakiabgU caRiabes7aKnaaxacabaGaeqOUdSgaleqabaGaey4bIenaaOWaaSba aSqaaiaadMgaaeqaaOGaamytamaaBaaaleaacaWGPbaabeaakmaala aabaGaamizaiaadohaaeaacaWGKbGaamiEamaaBaaaleaacaaIZaaa beaaaaaakiaawUhacaGL9baacaWGKbGaamiEamaaBaaaleaacaaIZa aabeaaaeaacaaIWaaabaGaamitamaaBaaameaacaaIWaaabeaaa0Ga ey4kIipakiabgkHiTmaapehabaWaaeWaaeaacaWGWbWaaSbaaSqaai aadMgaaeqaaOGaeqiTdqMaamODamaaBaaaleaacaWGPbaabeaakiab gkHiTiaadghadaWgaaWcbaGaaGOmaaqabaGccaWGKbGafqiXdqNbai aadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGXbWaaSbaaSqaaiaa igdaaeqaaOGaeqiTdqMafqiXdqNbaiaadaWgaaWcbaGaaGOmaaqaba aakiaawIcacaGLPaaaaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdGc caWGKbGaam4CaaqaaiabgUcaRmaapehabaGaeqyWdiNaamyqaiaadg gadaWgaaWcbaGaamyAaaqabaGccqaH0oazcaWG2bWaaSbaaSqaaiaa dMgaaeqaaaqaaiaaicdaaeaacaWGmbaaniabgUIiYdGccaWGKbGaam 4CaiabgUcaRmaapehabaGaeqyWdiNaamysamaaBaaaleaacaaIXaaa beaakmaabmaabaGaeqySde2aaSbaaSqaaiaaigdaaeqaaOGaey4kaS IaeqyYdC3aaSbaaSqaaiaaikdaaeqaaOGaeqyYdC3aaSbaaSqaaiaa iodaaeqaaaGccaGLOaGaayzkaaGaeqiTdqMafqiXdqNbaiaadaWgaa WcbaGaaGOmaaqabaaabaGaaGimaaqaaiaadYeaa0Gaey4kIipakiaa dsgacaWGZbGaeyOeI0Yaa8qCaeaacqaHbpGCcaWGjbWaaSbaaSqaai aaikdaaeqaaOWaaeWaaeaacqaHXoqydaWgaaWcbaGaaGOmaaqabaGc cqGHsislcqaHjpWDdaWgaaWcbaGaaGymaaqabaGccqaHjpWDdaWgaa WcbaGaaG4maaqabaaakiaawIcacaGLPaaacqaH0oazcuaHepaDgaGa amaaBaaaleaacaaIXaaabeaaaeaacaaIWaaabaGaamitaaqdcqGHRi I8aOGaamizaiaadohacaaMc8UaaGPaVlaaykW7caaMc8oabaGaeyOe I0YaamWaaeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdqMaam ODamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadgfadaWgaaWcbaGa aGOmaaqabaGccqaH0oazcuaHepaDgaGaamaaBaaaleaacaaIXaaabe aakiabgUcaRiaadgfadaWgaaWcbaGaaGymaaqabaGccqaH0oazcuaH epaDgaGaamaaBaaaleaacaaIYaaabeaaaOGaay5waiaaw2faamaaBa aaleaacaWG4bWaaSbaaWqaaiaaiodaaeqaaSGaeyypa0JaaGimaaqa baGccqGHsisldaWadaqaaiaadcfadaWgaaWcbaGaamyAaaqabaGccq aH0oazcaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0Iaamyuamaa BaaaleaacaaIYaaabeaakiabes7aKjqbes8a0zaacaWaaSbaaSqaai aaigdaaeqaaOGaey4kaSIaamyuamaaBaaaleaacaaIXaaabeaakiab es7aKjqbes8a0zaacaWaaSbaaSqaaiaaikdaaeqaaaGccaGLBbGaay zxaaWaaSbaaSqaaiaadIhadaWgaaadbaGaaG4maaqabaWccqGH9aqp caWGmbaabeaakiabg2da9iaaicdaaaaa@EC7D@

To proceed, it is necessary to express δ κ ˙ i ,δ τ ˙ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqbeQ7aRzaacaWaaSbaaSqaai aadMgaaeqaaOGaaiilaiabes7aKjqbes8a0zaacaWaaSbaaSqaaiaa dMgaaeqaaaaa@3A49@  and δ s ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqadohagaGaaaaa@332E@  in terms of the virtual velocity components δ v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadAhadaWgaaWcbaGaamyAaa qabaaaaa@3442@ . The algebra and the resulting equilibrium equations are greatly simplified if the tangent vector t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahshaaaa@3185@  is regarded as an independent kinematic variable.   The relationship between t and dr/ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWHYbGaai4laiaadsgacaWGZb aaaa@3500@  must be enforced by a vector valued Lagrange multiplier T = T 1 m 1 + T 2 m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahsfagaqbaiabg2da9iaadsfadaWgaa WcbaGaaGymaaqabaGccaWHTbWaaSbaaSqaaiaaigdaaeqaaOGaey4k aSIaamivamaaBaaaleaacaaIYaaabeaakiaah2gadaWgaaWcbaGaaG Omaaqabaaaaa@3AB3@ , which must satisfy

0 L ( dr ds t )δ T ˙ ds + 0 L ( dδv ds δ t ˙ ) T ds =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapehabaWaaeWaaeaadaWcaaqaaiaads gacaWHYbaabaGaamizaiaadohaaaGaeyOeI0IaaCiDaaGaayjkaiaa wMcaaiabgwSixlabes7aKjqahsfagaGagaqbaiaadsgacaWGZbaale aacaaIWaaabaGaamitaaqdcqGHRiI8aOGaey4kaSYaa8qCaeaadaqa daqaamaalaaabaGaamizaiabes7aKjaahAhaaeaacaWGKbGaam4Caa aacqGHsislcqaH0oazceWH0bGbaiaaaiaawIcacaGLPaaacqGHflY1 ceWHubGbauaacaWGKbGaam4CaaWcbaGaaGimaaqaaiaadYeaa0Gaey 4kIipakiabg2da9iaaicdaaaa@58E1@

for all variations δ t ˙ ,δ r ˙ ,δ T ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqahshagaGaaiaacYcacqaH0o azceWHYbGbaiaacaGGSaGaeqiTdqMabCivayaacyaafaaaaa@39D2@ . The second integral can be expressed in component form as

0 L ( dδ v α ds T α +( δ v 2 κ 3 +δ v 3 κ 2 ) T 1 +( δ v 1 κ 3 δ v 3 κ 1 ) T 2 δ τ ˙ α T α ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapehabaWaaeWaaeaadaWcaaqaaiaads gacqaH0oazcaWG2bWaaSbaaSqaaiabeg7aHbqabaaakeaacaWGKbGa am4CaaaacaWGubWaaSbaaSqaaiabeg7aHbqabaGccqGHRaWkdaqada qaaiabgkHiTiabes7aKjaadAhadaWgaaWcbaGaaGOmaaqabaGccqaH 6oWAdaWgaaWcbaGaaG4maaqabaGccqGHRaWkcqaH0oazcaWG2bWaaS baaSqaaiaaiodaaeqaaOGaeqOUdS2aaSbaaSqaaiaaikdaaeqaaaGc caGLOaGaayzkaaGaamivamaaBaaaleaacaaIXaaabeaakiabgUcaRm aabmaabaGaeqiTdqMaamODamaaBaaaleaacaaIXaaabeaakiabeQ7a RnaaBaaaleaacaaIZaaabeaakiabgkHiTiabes7aKjaadAhadaWgaa WcbaGaaG4maaqabaGccqaH6oWAdaWgaaWcbaGaaGymaaqabaaakiaa wIcacaGLPaaacaWGubWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0Iaeq iTdqMafqiXdqNbaiaadaWgaaWcbaGaeqySdegabeaakiaadsfadaWg aaWcbaGaeqySdegabeaaaOGaayjkaiaawMcaaaWcbaGaaGimaaqaai aadYeaa0Gaey4kIipakiabg2da9iaaicdaaaa@6EB0@

This equation can simply be added to the virtual work equation to ensure that δ τ ˙ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqbes8a0zaacaWaaSbaaSqaai abeg7aHbqabaaaaa@35C6@  and δ v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadAhadaWgaaWcbaGaamyAaa qabaaaaa@3442@  are consistent.   Finally, recall that the curvature rates and stretch rate are related to δ v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadAhadaWgaaWcbaGaamyAaa qabaaaaa@3442@   δ τ ˙ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjqbes8a0zaacaWaaSbaaSqaai abeg7aHbqabaaaaa@35C6@  by

dδ s ˙ d x 3 = ds d x 3 ( dδ v 3 ds δ v 1 κ 2 +δ v 2 κ 1 ) δ κ 1 = dδ τ ˙ 2 ds δ τ ˙ 1 κ 3 δ κ 2 = dδ τ ˙ 1 ds δ τ ˙ 2 κ 3 δ κ 3 =δ τ ˙ 1 κ 1 +δ τ ˙ 2 κ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaWGKbGaeqiTdqMabm 4CayaacaaabaGaamizaiaadIhadaWgaaWcbaGaaG4maaqabaaaaOGa eyypa0ZaaSaaaeaacaWGKbGaam4CaaqaaiaadsgacaWG4bWaaSbaaS qaaiaaiodaaeqaaaaakmaabmaabaWaaSaaaeaacaWGKbGaeqiTdqMa amODamaaBaaaleaacaaIZaaabeaaaOqaaiaadsgacaWGZbaaaiabgk HiTiabes7aKjaadAhadaWgaaWcbaGaaGymaaqabaGccqaH6oWAdaWg aaWcbaGaaGOmaaqabaGccqGHRaWkcqaH0oazcaWG2bWaaSbaaSqaai aaikdaaeqaaOGaeqOUdS2aaSbaaSqaaiaaigdaaeqaaaGccaGLOaGa ayzkaaGaaGPaVdqaaiabes7aKnaaxacabaGaeqOUdS2aaSbaaSqaai aaigdaaeqaaaqabeaacqGHhis0aaGccqGH9aqpcqGHsisldaWcaaqa aiaadsgacqaH0oazcuaHepaDgaGaamaaBaaaleaacaaIYaaabeaaaO qaaiaadsgacaWGZbaaaiabgkHiTiabes7aKjqbes8a0zaacaWaaSba aSqaaiaaigdaaeqaaOGaeqOUdS2aaSbaaSqaaiaaiodaaeqaaOGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqiTdq2aaCbiaeaa cqaH6oWAdaWgaaWcbaGaaGOmaaqabaaabeqaaiabgEGirdaakiabg2 da9maalaaabaGaamizaiabes7aKjqbes8a0zaacaWaaSbaaSqaaiaa igdaaeqaaaGcbaGaamizaiaadohaaaGaeyOeI0IaeqiTdqMafqiXdq NbaiaadaWgaaWcbaGaaGOmaaqabaGccqaH6oWAdaWgaaWcbaGaaG4m aaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabes7aKnaaxa cabaGaeqOUdS2aaSbaaSqaaiaaiodaaeqaaaqabeaacqGHhis0aaGc cqGH9aqpcqaH0oazcuaHepaDgaGaamaaBaaaleaacaaIXaaabeaaki abeQ7aRnaaBaaaleaacaaIXaaabeaakiabgUcaRiabes7aKjqbes8a 0zaacaWaaSbaaSqaaiaaikdaaeqaaOGaeqOUdS2aaSbaaSqaaiaaik daaeqaaaaaaa@A902@

Substituting these results into the augmented virtual work equation gives

0 L ( ( dδ v α ds T α +δ v 1 κ 3 T 2 δ v 2 κ 3 T 1 +δ v 3 ( κ 1 T 2 + κ 2 T 1 ) )δ τ ˙ α T α ) ds + 0 L { ( dδ v 3 ds +( δ v 1 κ 2 +δ v 2 κ 1 ) ) T 3 } ds + 0 L ρA a i δ v i ds+ 0 L ρ I 1 ( α 1 + ω 2 ω 3 )δ τ ˙ 2 ds 0 L ρ I 2 ( α 2 ω 1 ω 3 )δ τ ˙ 1 ds + 0 L { ( dδ τ ˙ 2 ds κ 3 δ τ ˙ 1 ) M 1 +( dδ τ ˙ 1 ds κ 3 δ τ ˙ 2 ) M 2 +( κ 1 δ τ ˙ 1 + κ 2 δ τ ˙ 2 ) M 3 } ds 0 L ( p i δ v i + q 2 δ τ ˙ 1 q 1 δ τ ˙ 2 ) ds [ P i δ v i + Q 2 δ τ ˙ 1 Q 1 δ τ ˙ 2 ] x 3 =0 [ P i δ v i + Q 2 δ τ ˙ 1 Q 1 δ τ ˙ 2 ] x 3 =L =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xq qrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaadaWdXbqaamaabmaabaWaaeWaae aadaWcaaqaaiaadsgacqaH0oazcaWG2bWaaSbaaSqaaiabeg7aHbqa baaakeaacaWGKbGaam4CaaaacaWGubWaaSbaaSqaaiabeg7aHbqaba GccqGHRaWkcqaH0oazcaWG2bWaaSbaaSqaaiaaigdaaeqaaOGaeqOU dS2aaSbaaSqaaiaaiodaaeqaaOGaamivamaaBaaaleaacaaIYaaabe aakiabgkHiTiabes7aKjaadAhadaWgaaWcbaGaaGOmaaqabaGccqaH 6oWAdaWgaaWcbaGaaG4maaqabaGccaWGubWaaSbaaSqaaiaaigdaae qaaOGaey4kaSIaeqiTdqMaamODamaaBaaaleaacaaIZaaabeaakiaa cIcacqGHsislcqaH6oWAdaWgaaWcbaGaaGymaaqabaGccaWGubWaaS baaSqaaiaaikdaaeqaaOGaey4kaSIaeqOUdS2aaSbaaSqaaiaaikda aeqaaOGaamivamaaBaaaleaacaaIXaaabeaakiaacMcaaiaawIcaca GLPaaacqGHsislcqaH0oazcuaHepaDgaGaamaaBaaaleaacqaHXoqy aeqaaOGaamivamaaBaaaleaacqaHXoqyaeqaaaGccaGLOaGaayzkaa aaleaacaaIWaaabaGaamitaaqdcqGHRiI8aOGaamizaiaadohaaeaa cqGHRaWkdaWdXbqaamaacmaabaWaaeWaaeaadaWcaaqaaiaadsgacq aH0oazcaWG2bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamizaiaadoha aaGaey4kaSYaaeWaaeaacqGHsislcqaH0oazcaWG2bWaaSbaaSqaai aaigdaaeqaaOGaeqOUdS2aaSbaaSqaaiaaikdaaeqaaOGaey4kaSIa eqiTdqMaamODamaaBaaaleaacaaIYaaabeaakiabeQ7aRnaaBaaale aacaaIXaaabeaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaiaadsfa daWgaaWcbaGaaG4maaqabaaakiaawUhacaGL9baaaSqaaiaaicdaae aacaWGmbaaniabgUIiYdGccaWGKbGaam4CaaqaaiabgUcaRmaapeha baGaeqyWdiNaamyqaiaadggadaWgaaWcbaGaamyAaaqabaGccqaH0o azcaWG2bWaaSbaaSqaaiaadMgaaeqaaaqaaiaaicdaaeaacaWGmbaa niabgUIiYdGccaWGKbGaam4CaiabgUcaRmaapehabaGaeqyWdiNaam ysamaaBaaaleaacaaIXaaabeaakmaabmaabaGaeqySde2aaSbaaSqa aiaaigdaaeqaaOGaey4kaSIaeqyYdC3aaSbaaSqaaiaaikdaaeqaaO GaeqyYdC3aaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaGaeqiT dqMafqiXdqNbaiaadaWgaaWcbaGaaGOmaaqabaaabaGaaGimaaqaai aadYeaa0Gaey4kIipakiaadsgacaWGZbGaeyOeI0Yaa8qCaeaacqaH bpGCcaWGjbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacqaHXoqyda WgaaWcbaGaaGOmaaqabaGccqGHsislcqaHjpWDdaWgaaWcbaGaaGym aaqabaGccqaHjpWDdaWgaaWcbaGaaG4maaqabaaakiaawIcacaGLPa aacqaH0oazcuaHepaDgaGaamaaBaaaleaacaaIXaaabeaaaeaacaaI WaaabaGaamitaaqdcqGHRiI8aOGaamizaiaadohacaaMc8UaaGPaVd qaaiabgUcaRmaapehabaWaaiWaaeaadaqadaqaaiabgkHiTmaalaaa baGaamizaiabes7aKjqbes8a0zaacaWaaSbaaSqaaiaaikdaaeqaaa GcbaGaamizaiaadohaaaGaeyOeI0IaeqOUdS2aaSbaaSqaaiaaioda aeqaaOGaeqiTdqMafqiXdqNbaiaadaWgaaWcbaGaaGymaaqabaaaki aawIcacaGLPaaacaWGnbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSYa aeWaaeaadaWcaaqaaiaadsgacqaH0oazcuaHepaDgaGaamaaBaaale aacaaIXaaabeaaaOqaaiaadsgacaWGZbaaaiabgkHiTiabeQ7aRnaa BaaaleaacaaIZaaabeaakiabes7aKjqbes8a0zaacaWaaSbaaSqaai aaikdaaeqaaaGccaGLOaGaayzkaaGaamytamaaBaaaleaacaaIYaaa beaakiabgUcaRmaabmaabaGaeqOUdS2aaSbaaSqaaiaaigdaaeqaaO GaeqiTdqMafqiXdqNbaiaadaWgaaWcbaGaaGymaaqabaGccqGHRaWk cqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqaH0oazcuaHepaDgaGaam aaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiaad2eadaWgaaWc baGaaG4maaqabaaakiaawUhacaGL9baaaSqaaiaaicdaaeaadaWgaa adbaGaamitaaqabaaaniabgUIiYdGccaWGKbGaam4CaaqaaiabgkHi TmaapehabaWaaeWaaeaacaWGWbWaaSbaaSqaaiaadMgaaeqaaOGaeq iTdqMaamODamaaBaaaleaacaWGPbaabeaakiabgUcaRiaadghadaWg aaWcbaGaaGOmaaqabaGccqaH0oazcuaHepaDgaGaamaaBaaaleaaca aIXaaabeaakiabgkHiTiaadghadaWgaaWcbaGaaGymaaqabaGccqaH 0oazcuaHepaDgaGaamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawM caaaWcbaGaaGimaaqaaiaadYeaa0Gaey4kIipakiaadsgacaWGZbGa aGPaVlaaykW7caaMc8UaaGPaVlabgkHiTmaadmaabaGaamiuamaaBa aaleaacaWGPbaabeaakiabes7aKjaadAhadaWgaaWcbaGaamyAaaqa baGccqGHRaWkcaWGrbWaaSbaaSqaaiaaikdaaeqaaOGaeqiTdqMafq iXdqNbaiaadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGrbWaaSba aSqaaiaaigdaaeqaaOGaeqiTdqMafqiXdqNbaiaadaWgaaWcbaGaaG OmaaqabaaakiaawUfacaGLDbaadaWgaaWcbaGaamiEamaaBaaameaa caaIZaaabeaaliabg2da9iaaicdaaeqaaOGaeyOeI0YaamWaaeaaca WGqbWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdqMaamODamaaBaaaleaa caWGPbaabeaakiabgUcaRiaadgfadaWgaaWcbaGaaGOmaaqabaGccq aH0oazcuaHepaDgaGaamaaBaaaleaacaaIXaaabeaakiabgkHiTiaa dgfadaWgaaWcbaGaaGymaaqabaGccqaH0oazcuaHepaDgaGaamaaBa aaleaacaaIYaaabeaaaOGaay5waiaaw2faamaaBaaaleaacaWG4bWa aSbaaWqaaiaaiodaaeqaaSGaeyypa0JaamitaaqabaGccqGH9aqpca aIWaaaaaa@74C3@

This equation must be satisfied for all admissible δ v i ,δ τ ˙ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadAhadaWgaaWcbaGaamyAaa qabaGccaGGSaGaeqiTdqMafqiXdqNbaiaadaWgaaWcbaGaeqySdega beaaaaa@3A3A@ . Considering each component in turn, and integrating by parts appropriately and using I 3 = I 1 + I 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamysamaa BaaaleaacaaIYaaabeaaaaa@37A6@  gives the last five equations of motion, as well as the boundary conditions T=±PM=±Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahsfacqGH9aqpcqGHXcqScaWHqbGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaCytaiabg2da9iab gglaXkaahgfaaaa@4318@  on s=0 and s=L

 

 

 

10.2.10 Constitutive equations relating forces to deformation measures in elastic rods

 

Constitutive equations must relate the deformation measures defined in Section 10.2.3 to the forces defined in 10.2.8.  In this section we list the relationships between these quantities for an isotropic, elastic rod subjected to small distorsions.  For simplicity, the sides of the rod are assumed to be traction free.

 

The results depend on the geometry of the rod’s cross-section, which is characterized as follows. 

  1. Introduce a Cartesian coordinate system ( x 1 , x 2 , x 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamiE amaaBaaaleaacaaIZaaabeaakiaacMcaaaa@390E@  as follows:  the origin for this coordinate system is at the centroid of the cross-section, the basis vectors e 1 , e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaGcca GGSaGaaCyzamaaBaaaleaacaaIYaaabeaaaaa@34ED@  are parallel to the principal axes of inertia for the cross-section, and e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@325F@  is parallel to the rod’s axis.
  2. We denote the cross-sectional area of the rod by A, and the curve bounding the cross-section by C, and let I i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaamyAaaqabaaaaa@3270@  denote the three principal moments of area of the cross-section (see Sect 10.2.1)
  3. We introduce a warping function w( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhacaGGOaGaamiEamaaBaaaleaaca aIXaaabeaakiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiyk aaaa@376A@  to describe the out-of-plane displacement component u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaaaaa@326B@  in the cross-section of the rod.  The warping function is related to the out-of-plane displacement u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaaaaa@326B@  by

u 3 ( x 1 , x 2 , x 3 )=( κ 3 ( x 3 ) κ ¯ 3 ( x 3 ) )w( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGcca GGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaG4maaqaba GccaGGPaGaeyypa0ZaaeWaaeaacqaH6oWAdaWgaaWcbaGaaG4maaqa baGccaGGOaGaamiEamaaBaaaleaacaaIZaaabeaakiaacMcacqGHsi slcuaH6oWAgaqeamaaBaaaleaacaaIZaaabeaakiaacIcacaWG4bWa aSbaaSqaaiaaiodaaeqaaOGaaiykaaGaayjkaiaawMcaaiaadEhaca GGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG4bWaaSba aSqaaiaaikdaaeqaaOGaaiykaaaa@514D@

The warping function depends only on the geometry of the cross-section, and satisfies the following governing equations and boundary conditions

2 w x 1 2 + 2 w x 2 2 w x 1 κ 2 + w x 2 κ 1 =0in A w x α n α = x 2 n 1 x 1 n 2 = x α x α s on C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaam4DaaqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaaqa aiaaikdaaaaaaOGaey4kaSYaaSaaaeaacqGHciITdaahaaWcbeqaai aaikdaaaGccaWG3baabaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaa baGaaGOmaaaaaaGccqGHsisldaWcaaqaaiabgkGi2kaadEhaaeaacq GHciITcaWG4bWaaSbaaSqaaiabggdaXaqabaaaaOGaeqOUdS2aaSba aSqaaiaaikdaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcaWG3baaba GaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGccqaH6oWAdaWg aaWcbaGaaGymaaqabaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caqGPbGaaeOBaiaabccacaqGbbGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7daWcaaqaaiabgkGi2kaadEhaaeaacqGHciITcaWG4b WaaSbaaSqaaiabeg7aHbqabaaaaOGaamOBamaaBaaaleaacqaHXoqy aeqaaOGaeyypa0JaamiEamaaBaaaleaacaaIYaaabeaakiaad6gada WgaaWcbaGaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaaigda aeqaaOGaamOBamaaBaaaleaacaaIYaaabeaakiabg2da9iaadIhada WgaaWcbaGaeqySdegabeaakmaalaaabaGaeyOaIyRaamiEamaaBaaa leaacqaHXoqyaeqaaaGcbaGaeyOaIyRaam4CaaaacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaab+gacaqGUbGa aeiiaiaaboeaaaa@ABF1@

You can easily show that this choice of u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaaaaa@326B@  will automatically satisfy the shear stress equilibrium equation σ α3 / x α σ 31 κ 2 + σ 32 κ 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqaHdpWCdaWgaaWcbaGaeq ySdeMaaG4maaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacqaH XoqyaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaiodacaaIXaaabe aakiabeQ7aRnaaBaaaleaacaaIYaaabeaakiabgUcaRiabeo8aZnaa BaaaleaacaaIZaGaaGOmaaqabaGccqaH6oWAdaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaaIWaaaaa@4D25@  as well as the boundary condition σ 3α n α =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaeqySde gabeaakiaad6gadaWgaaWcbaGaeqySdegabeaakiabg2da9iaaicda aaa@3965@  on C.

  1. Finally we define a modified polar moment of inertia for the cross section as

J 3 = I 3 A ( x 2 w x 1 x 1 w x 2 ) dA MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQeadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaWGjbWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0Yaa8quaeaa daqadaqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGcdaWcaaqaaiabgk Gi2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaa kiabgkHiTiaadIhadaWgaaWcbaGaaGymaaqabaGcdaWcaaqaaiabgk Gi2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqaaiaaikdaaeqaaaaa aOGaayjkaiaawMcaaaWcbaGaamyqaaqab0Gaey4kIipakiaadsgaca WGbbaaaa@4CA6@

 

Calculating the warping function is a nuisance, because it requires the solution to a PDE.  In desperation, you can take w=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this will overestimate the torsional stiffness of the rod, but in many practical applications the error is not significant.   For a better approximation, warping functions can be estimated by neglecting the terms involving κ β w/ x α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH6oWAdaWgaaWcbaGaeqOSdigabe aakiabgkGi2kaadEhacaGGVaGaeyOaIyRaamiEamaaBaaaleaacqaH Xoqyaeqaaaaa@3D92@  in the governing equation.  A few such approximate warping functions and modified polar moments of area are listed in the table below. 

 

Warping functions and modified polar moments of area for simple cross-sections

w= x 1 x 2 n=0 4 (1) n a k n 3 cosh k n b sin k n x 1 sinh k n x 2 J 3 =16 a 3 b{ 1 3 64 π 3 a b n=0 tanh k n b (2n+1) 5 } k n = (2n+1)π 2a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4Daiabg2da9iaadIhadaWgaa WcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOe I0YaaabCaeaadaWcaaqaaiaaisdacaGGOaGaeyOeI0IaaGymaiaacM cadaahaaWcbeqaaiaad6gaaaaakeaacaWGHbGaam4AamaaDaaaleaa caWGUbaabaGaaG4maaaakiGacogacaGGVbGaai4CaiaacIgacaWGRb WaaSbaaSqaaiaad6gaaeqaaOGaamOyaaaaciGGZbGaaiyAaiaac6ga caWGRbWaaSbaaSqaaiaad6gaaeqaaOGaamiEamaaBaaaleaacaaIXa aabeaakiGacohacaGGPbGaaiOBaiaacIgacaWGRbWaaSbaaSqaaiaa d6gaaeqaaOGaamiEamaaBaaaleaacaaIYaaabeaaaeaacaWGUbGaey ypa0JaaGimaaqaaiabg6HiLcqdcqGHris5aaGcbaGaamOsamaaBaaa leaacaaIZaaabeaakiabg2da9iaaigdacaaI2aGaamyyamaaCaaale qabaGaaG4maaaakiaadkgadaGadaqaamaalaaabaGaaGymaaqaaiaa iodaaaGaeyOeI0YaaSaaaeaacaaI2aGaaGinaaqaaiabec8aWnaaCa aaleqabaGaaG4maaaaaaGcdaWcaaqaaiaadggaaeaacaWGIbaaamaa qahabaWaaSaaaeaaciGG0bGaaiyyaiaac6gacaGGObGaam4AamaaBa aaleaacaWGUbaabeaakiaadkgaaeaacaGGOaGaaGOmaiaad6gacqGH RaWkcaaIXaGaaiykamaaCaaaleqabaGaaGynaaaaaaaabaGaamOBai abg2da9iaaicdaaeaacqGHEisPa0GaeyyeIuoaaOGaay5Eaiaaw2ha aaqaaiaadUgadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWcaaqaai aacIcacaaIYaGaamOBaiabgUcaRiaaigdacaGGPaGaeqiWdahabaGa aGOmaiaadggaaaaaaaa@8B93@

w= x 1 x 2 ( a 2 b 2 ) ( a 2 + b 2 ) J 3 = π a 3 b 3 ( a 2 + b 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4Daiabg2da9iabgkHiTmaala aabaGaamiEamaaBaaaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGOaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgk HiTiaadkgadaahaaWcbeqaaiaaikdaaaGccaGGPaaabaWaaeWaaeaa caWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyamaaCaaale qabaGaaGOmaaaaaOGaayjkaiaawMcaaaaaaeaacaWGkbWaaSbaaSqa aiaaiodaaeqaaOGaeyypa0ZaaSaaaeaacqaHapaCcaWGHbWaaWbaaS qabeaacaaIZaaaaOGaamOyamaaCaaaleqabaGaaG4maaaaaOqaamaa bmaabaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkgada ahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaaaaaaa@51EA@

w= x 2 ( x 2 2 3 x 1 2 ) a 3 J 3 = a 4 3 80 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaam4Daiabg2da9maalaaabaGaam iEamaaBaaaleaacaaIYaaabeaakiaacIcacaWG4bWaa0baaSqaaiaa ikdaaeaacaaIYaaaaOGaeyOeI0IaaG4maiaadIhadaqhaaWcbaGaaG ymaaqaaiaaikdaaaGccaGGPaaabaGaamyyamaakaaabaGaaG4maaWc beaaaaaakeaacaWGkbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaS aaaeaacaWGHbWaaWbaaSqabeaacaaI0aaaaOWaaOaaaeaacaaIZaaa leqaaaGcbaGaaGioaiaaicdaaaaaaaa@45C3@

 

 

The force-deformation relations for the rod are

T 3 = EA 2 ( ( ds d s ¯ ) 2 1 )EA( ds d s ¯ 1 ) M α =E I α ( κ α κ ¯ α ) M 3 =μ J 3 ( κ 3 κ ¯ 3 ) σ 13 =μ( κ 3 κ ¯ 3 )( w x 1 x 2 ) σ 23 =μ( κ 3 κ ¯ 3 )( w x 2 + x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamivamaaBaaaleaacaaIZaaabe aakiabg2da9maalaaabaGaamyraiaadgeaaeaacaaIYaaaamaabmaa baWaaeWaaeaadaWcaaqaaiaadsgacaWGZbaabaGaamizaiqadohaga qeaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHsisl caaIXaaacaGLOaGaayzkaaGaaGPaVlabgIKi7kaadweacaWGbbWaae WaaeaadaWcaaqaaiaadsgacaWGZbaabaGaamizaiqadohagaqeaaaa cqGHsislcaaIXaaacaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamytamaaBaaaleaacqaHXoqyaeqaaO Gaeyypa0JaamyraiaadMeadaWgaaWcbaGaeqySdegabeaakiaacIca cqaH6oWAdaWgaaWcbaGaeqySdegabeaakiabgkHiTiqbeQ7aRzaara WaaSbaaSqaaiabeg7aHbqabaGccaGGPaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWGnbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaeqiVd0Maam OsamaaBaaaleaacaaIZaaabeaakiaacIcacqaH6oWAdaWgaaWcbaGa aG4maaqabaGccqGHsislcuaH6oWAgaqeamaaBaaaleaacaaIZaaabe aakiaacMcaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaiodaaeqaaOGa eyypa0JaaGPaVlaaykW7cqaH8oqBcaGGOaGaeqOUdS2aaSbaaSqaai aaiodaaeqaaOGaeyOeI0IafqOUdSMbaebadaWgaaWcbaGaaG4maaqa baGccaGGPaWaaeWaaeaadaWcaaqaaiabgkGi2kaadEhaaeaacqGHci ITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabgkHiTiaadIhadaWg aaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWg aaWcbaGaaGOmaiaaiodaaeqaaOGaeyypa0JaeqiVd0MaaiikaiabeQ 7aRnaaBaaaleaacaaIZaaabeaakiabgkHiTiqbeQ7aRzaaraWaaSba aSqaaiaaiodaaeqaaOGaaiykaiaaykW7daqadaqaamaalaaabaGaey OaIyRaam4DaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaa aOGaey4kaSIaamiEamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawM caaaaaaa@E227@

The two shear force components T α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaeqySdegabeaaaa a@332C@  cannot be related to the deformation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  they are Lagrange multipliers that enforce the condition that the rod does not experience transverse shear, as discussed in the preceding section.

 

Derivation: These results can be derived as follows:

  1. The elastic constitutive equations for materials subjected to small distorsions, but arbitrary rotations, are listed in Section 3.3.   They have the form

Σ ij = E ( 1+ν ) { E ij + ν 12ν ( E kk ) δ ij } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo6atnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaadaqadaqaaiaaigdacqGH RaWkcqaH9oGBaiaawIcacaGLPaaaaaWaaiWaaeaacaWGfbWaaSbaaS qaaiaadMgacaWGQbaabeaakiabgUcaRmaalaaabaGaeqyVd4gabaGa aGymaiabgkHiTiaaikdacqaH9oGBaaGaaiikaiaadweadaWgaaWcba Gaam4AaiaadUgaaeqaaOGaaiykaiabes7aKnaaBaaaleaacaWGPbGa amOAaaqabaaakiaawUhacaGL9baaaaa@4E81@

where Σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo6atnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3415@  are the components of the material stress tensor, and E ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@335B@  are the components of the Lagrange strain tensor. The components of E ij ee =( F ki ee F kj ee δ ij )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaqhaaWcbaGaamyAaiaadQgaae aacaWHLbGaaCyzaaaakiabg2da9iaacIcacaWGgbWaa0baaSqaaiaa dUgacaWGPbaabaGaaCyzaiaahwgaaaGccaWGgbWaa0baaSqaaiaadU gacaWGQbaabaGaaCyzaiaahwgaaaGccqGHsislcqaH0oazdaWgaaWc baGaamyAaiaadQgaaeqaaOGaaiykaiaac+cacaaIYaaaaa@472E@  in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3988@  can be found using the formulas for F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAeaaaa@3157@  given in Section 10.2.7, and when substituted into the constitutive laws give expressions for the components of material stress Σ ij ee MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo6atnaaDaaaleaacaWGPbGaamOAaa qaaiaahwgacaWHLbaaaaaa@35F2@  in terms of the deformation measures f αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaeqySdeMaeqOSdi gabeaaaaa@34DF@ , u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaaaaa@326B@ , κ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaWGPbaabeaaaa a@3354@  and ds/d s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGZbGaai4laiaadsgaceWGZb Gbaebaaaa@3515@ .

  1. The Cauchy stress is related to the material stress by σ=FΣ F T /J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaho8acqGH9aqpcaWHgbGaaC4OdiaahA eadaahaaWcbeqaaiaadsfaaaGccaGGVaGaamOsaaaa@383C@ .  For small distorsions, but arbitrary rotations, we may approximate this by σRΣ R T = Σ ij ee R e i R T e j = Σ ij ee m i m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaho8acqGHijYUcaWHsbGaaC4Odiaahk fadaahaaWcbeqaaiaadsfaaaGccqGH9aqpcqqHJoWudaqhaaWcbaGa amyAaiaadQgaaeaacaWHLbGaaCyzaaaakiaahkfacaWHLbWaaSbaaS qaaiaadMgaaeqaaOGaey4LIqSaaCOuamaaCaaaleqabaGaamivaaaa kiaahwgadaWgaaWcbaGaamOAaaqabaGccqGH9aqpcqqHJoWudaqhaa WcbaGaamyAaiaadQgaaeaacaWHLbGaaCyzaaaakiaah2gadaWgaaWc baGaamyAaaqabaGccqGHxkcXcaWHTbWaaSbaaSqaaiaadQgaaeqaaa aa@5399@ , so the components of the material stress tensor in { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3988@  can be used as an approximation to the components of the Cauchy stress tensor in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@39A0@ .  
  2. Since we have assumed that the tractions on the sides of the rod vanish, the in-plane stress components must satisfy A σ αβ dA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiabeo8aZnaaBaaaleaacq aHXoqycqaHYoGyaeqaaOGaamizaiaadgeaaSqaaiaadgeaaeqaniab gUIiYdGccaaMc8Uaeyypa0JaaGimaaaa@4013@ .  Substituting the formulas for stresses from (2) and noting that A x α dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaamiEamaaBaaaleaacqaHXo qyaeqaaaqaaiaadgeaaeqaniabgUIiYdGccaWGKbGaamyqaiabg2da 9iaaicdaaaa@39CE@  (since that the origin for the coordinate system coincides with the centroid of the cross section) shows that f 11 = f 22 =ν( (ds/d s ¯ ) 2 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaaGymaiaaigdaae qaaOGaeyypa0JaamOzamaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH 9aqpcqGHsislcqaH9oGBdaqadaqaaiaacIcacaWGKbGaam4Caiaac+ cacaWGKbGabm4CayaaraGaaiykamaaCaaaleqabaGaaGOmaaaakiab gkHiTiaaigdaaiaawIcacaGLPaaaaaa@4473@ , f 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaaGymaiaaikdaae qaaOGaeyypa0JaaGimaaaa@34E0@ , and

σ 33 = E 2 ( ( ds d s ¯ ) 2 1+2( κ 1 κ ¯ 1 ) x 2 2( κ 2 κ ¯ 2 ) x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIYaaaamaabmaabaWa aeWaaeaadaWcaaqaaiaadsgacaWGZbaabaGaamizaiqadohagaqeaa aaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI XaGaey4kaSIaaGOmaiaacIcacqaH6oWAdaWgaaWcbaGaaGymaaqaba GccqGHsislcuaH6oWAgaqeamaaBaaaleaacaaIXaaabeaakiaacMca caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaaGOmaiaacIcacq aH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGHsislcuaH6oWAgaqeamaa BaaaleaacaaIYaaabeaakiaacMcacaWG4bWaaSbaaSqaaiaaigdaae qaaaGccaGLOaGaayzkaaaaaa@569C@

  1. Substituting the formula for σ 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maa qabaaaaa@33F1@  into the definitions of T 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaG4maaqabaaaaa@324A@ , M 1 , M 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamytamaaBaaaleaacaaIYaaabeaaaaa@34B5@  given in Section 10.2.8 and noting that A x 1 x 2 dA=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaamiEamaaBaaaleaacaaIXa aabeaakiaadIhadaWgaaWcbaGaaGOmaaqabaaabaGaamyqaaqab0Ga ey4kIipakiaadsgacaWGbbGaeyypa0JaaGimaaaa@3AD9@  (because the basis vectors coincide with the principal axes of inertia) yields

T 3 = EA 2 ( ( ds d s ¯ ) 2 1 )EA( ds d s ¯ 1 ) M α =E I α ( κ α κ ¯ α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsfadaWgaaWcbaGaaG4maaqabaGccq GH9aqpdaWcaaqaaiaadweacaWGbbaabaGaaGOmaaaadaqadaqaamaa bmaabaWaaSaaaeaacaWGKbGaam4CaaqaaiaadsgaceWGZbGbaebaaa aacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGym aaGaayjkaiaawMcaaiaaykW7cqGHijYUcaWGfbGaamyqamaabmaaba WaaSaaaeaacaWGKbGaam4CaaqaaiaadsgaceWGZbGbaebaaaGaeyOe I0IaaGymaaGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaad2eadaWgaaWcbaGaeqySdegabeaakiabg2 da9iaadweacaWGjbWaaSbaaSqaaiabeg7aHbqabaGccaGGOaGaeqOU dS2aaSbaaSqaaiabeg7aHbqabaGccqGHsislcuaH6oWAgaqeamaaBa aaleaacqaHXoqyaeqaaOGaaiykaaaa@72DF@

  1. Recall that the shear stress components σ α3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacqaHXoqycaaIZa aabeaaaaa@34D3@  must satisfy the equilibrium equation and boundary condition

σ α3 x α + σ 31 κ 2 σ 32 κ 1 =0in A σ αβ n α =0on C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiabeg7aHjaaiodaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaa cqaHXoqyaeqaaaaakiabgUcaRiabeo8aZnaaBaaaleaacaaIZaGaaG ymaaqabaGccqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGHsislcqaH dpWCdaWgaaWcbaGaaG4maiaaikdaaeqaaOGaeqOUdS2aaSbaaSqaai aaigdaaeqaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaeyAaiaab6ga caqGGaGaaeyqaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaeqySdeMa eqOSdigabeaakiaad6gadaWgaaWcbaGaeqySdegabeaakiabg2da9i aaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua ae4Baiaab6gacaqGGaGaae4qaaaa@77E6@

Substituting the shear stress components from step (2) into this equilibrium equation and setting u 3 =( κ 3 κ ¯ 3 )w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcaGGOaGaeqOUdS2aaSbaaSqaaiaaiodaaeqaaOGaeyOeI0Ia fqOUdSMbaebadaWgaaWcbaGaaG4maaqabaGccaGGPaGaam4Daaaa@3C1F@  gives the governing equation for w

2 w x 1 2 + 2 w x 2 2 w x 1 κ 2 + w x 2 κ 1 =0in A w x α n α = x 2 n 1 x 1 n 2 = x α x α s on C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaam4DaaqaaiabgkGi2kaadIhadaqhaaWcbaGaaGymaaqa aiaaikdaaaaaaOGaey4kaSYaaSaaaeaacqGHciITdaahaaWcbeqaai aaikdaaaGccaWG3baabaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaa baGaaGOmaaaaaaGccqGHsisldaWcaaqaaiabgkGi2kaadEhaaeaacq GHciITcaWG4bWaaSbaaSqaaiabggdaXaqabaaaaOGaeqOUdS2aaSba aSqaaiaaikdaaeqaaOGaey4kaSYaaSaaaeaacqGHciITcaWG3baaba GaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGccqaH6oWAdaWg aaWcbaGaaGymaaqabaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caqGPbGaaeOBaiaabccacaqGbbGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7daWcaaqaaiabgkGi2kaadEhaaeaacqGHciITcaWG4b WaaSbaaSqaaiabeg7aHbqabaaaaOGaamOBamaaBaaaleaacqaHXoqy aeqaaOGaeyypa0JaamiEamaaBaaaleaacaaIYaaabeaakiaad6gada WgaaWcbaGaaGymaaqabaGccqGHsislcaWG4bWaaSbaaSqaaiaaigda aeqaaOGaamOBamaaBaaaleaacaaIYaaabeaakiabg2da9iaadIhada WgaaWcbaGaeqySdegabeaakmaalaaabaGaeyOaIyRaamiEamaaBaaa leaacqaHXoqyaeqaaaGcbaGaeyOaIyRaam4CaaaacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaab+gacaqGUbGa aeiiaiaaboeaaaa@ABF1@

  1. The shear stresses now follow as

σ 13 =μ( κ 3 κ ¯ 3 )( w x 1 x 2 ) σ 23 =μ( κ 3 κ ¯ 3 )( w x 2 + x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaio daaeqaaOGaeyypa0JaaGPaVlaaykW7cqaH8oqBcaGGOaGaeqOUdS2a aSbaaSqaaiaaiodaaeqaaOGaeyOeI0IafqOUdSMbaebadaWgaaWcba GaaG4maaqabaGccaGGPaWaaeWaaeaadaWcaaqaaiabgkGi2kaadEha aeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabgkHiTi aadIhadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cq aHdpWCdaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaeyypa0JaeqiVd0Ma aiikaiabeQ7aRnaaBaaaleaacaaIZaaabeaakiabgkHiTiqbeQ7aRz aaraWaaSbaaSqaaiaaiodaaeqaaOGaaiykaiaaykW7daqadaqaamaa laaabaGaeyOaIyRaam4DaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaG OmaaqabaaaaOGaey4kaSIaamiEamaaBaaaleaacaaIXaaabeaaaOGa ayjkaiaawMcaaaaa@7540@

Substituting these results into the equation defining M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaG4maaqabaaaaa@3243@  in Section 10.2.6 gives the last equation

M 3 =μ J 3 ( κ 3 κ ¯ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcqaH8oqBcaWGkbWaaSbaaSqaaiaaiodaaeqaaOGaaiikaiab eQ7aRnaaBaaaleaacaaIZaaabeaakiabgkHiTiqbeQ7aRzaaraWaaS baaSqaaiaaiodaaeqaaOGaaiykaaaa@3E73@

 

 

 

 

10.2.11 Strain energy of an elastic rod

 

The total strain energy of an elastic rod can be computed from its curvatures as

Φ= 1 2 0 L { EA ( ds d s ¯ 1 ) 2 +E I 1 ( κ 1 κ ¯ 1 ) 2 +E I 2 ( κ 2 κ ¯ 2 ) 2 +μ J 3 ( κ 3 κ ¯ 3 ) 2 }ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfA6agjabg2da9maalaaabaGaaGymaa qaaiaaikdaaaWaa8qCaeaadaGadaqaaiaadweacaWGbbWaaeWaaeaa daWcaaqaaiaadsgacaWGZbaabaGaamizaiqadohagaqeaaaacqGHsi slcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaamyraiaadMeadaWgaaWcbaGaaGymaaqabaGccaGGOaGaeqOUdS 2aaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IafqOUdSMbaebadaWgaaWc baGaaGymaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaS IaamyraiaadMeadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaeqOUdS2a aSbaaSqaaiaaikdaaeqaaOGaeyOeI0IafqOUdSMbaebadaWgaaWcba GaaGOmaaqabaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIa eqiVd0MaamOsamaaBaaaleaacaaIZaaabeaakiaacIcacqaH6oWAda WgaaWcbaGaaG4maaqabaGccqGHsislcuaH6oWAgaqeamaaBaaaleaa caaIZaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaakiaawUhaca GL9baacaWGKbGaam4CaaWcbaGaaGimaaqaaiaadYeaa0Gaey4kIipa aaa@6B2F@

 

Derivation:  The derivation is similar to the procedure used to compute elastic moment-curvature relations.

  1. The strain energy density in the rod can be computed from the Lagrange strain E ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@335B@  and the Material Stress Σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfo6atnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3415@  as U= Σ ij E ij /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpcqqHJoWudaWgaaWcba GaamyAaiaadQgaaeqaaOGaamyramaaBaaaleaacaWGPbGaamOAaaqa baGccaGGVaGaaGOmaaaa@3A4B@ .  The material stress can be related to the Lagrange strain using the formulas in Section 10.2.10, while the Lagrange strain can be expressed in terms of of the deformation measures f αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaeqySdeMaeqOSdi gabeaaaaa@34DF@ , u 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaaaaa@326B@ , κ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRnaaBaaaleaacaWGPbaabeaaaa a@3354@  and ds/d s ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWGZbGaai4laiaadsgaceWGZb Gbaebaaaa@3515@  using the formulas for the deformation gradient listed in Sections 10.2.7. 
  2. The results can be simplified by recalling that f 11 = f 22 =ν( (ds/d s ¯ ) 2 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaaGymaiaaigdaae qaaOGaeyypa0JaamOzamaaBaaaleaacaaIYaGaaGOmaaqabaGccqGH 9aqpcqGHsislcqaH9oGBdaqadaqaaiaacIcacaWGKbGaam4Caiaac+ cacaWGKbGabm4CayaaraGaaiykamaaCaaaleqabaGaaGOmaaaakiab gkHiTiaaigdaaiaawIcacaGLPaaaaaa@4473@ , f 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgadaWgaaWcbaGaaGymaiaaikdaae qaaOGaeyypa0JaaGimaaaa@34E0@ , which shows that the strain energy density can be approximated as

U= E 2 ( ( ds d s ¯ 1 )+( κ 1 κ ¯ 1 ) x 2 ( κ 2 κ ¯ 2 ) x 1 ) 2 + μ 2 ( κ 3 κ ¯ 3 ) 2 { ( w x 1 x 2 ) 2 + ( w x 2 + x 1 ) 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpdaWcaaqaaiaadweaae aacaaIYaaaamaabmaabaWaaeWaaeaadaWcaaqaaiaadsgacaWGZbaa baGaamizaiqadohagaqeaaaacqGHsislcaaIXaaacaGLOaGaayzkaa Gaey4kaSIaaiikaiabeQ7aRnaaBaaaleaacaaIXaaabeaakiabgkHi TiqbeQ7aRzaaraWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiaadIhada WgaaWcbaGaaGOmaaqabaGccqGHsislcaGGOaGaeqOUdS2aaSbaaSqa aiaaikdaaeqaaOGaeyOeI0IafqOUdSMbaebadaWgaaWcbaGaaGOmaa qabaGccaGGPaGaamiEamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaykW7daWcaaqaai abeY7aTbqaaiaaikdaaaGaaiikaiabeQ7aRnaaBaaaleaacaaIZaaa beaakiabgkHiTiqbeQ7aRzaaraWaaSbaaSqaaiaaiodaaeqaaOGaai ykamaaCaaaleqabaGaaGOmaaaakmaacmaabaWaaeWaaeaadaWcaaqa aiabgkGi2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaae qaaaaakiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaaakiaawIca caGLPaaadaahaaWcbeqaaiaaikdaaaGccaaMc8Uaey4kaSYaaeWaae aadaWcaaqaaiabgkGi2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqa aiaaikdaaeqaaaaakiabgUcaRiaadIhadaWgaaWcbaGaaGymaaqaba aakiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaaakiaawUhacaGL 9baaaaa@7AE4@

where w is the warping function defined in Section 10.2.9. The two terms in this expression represent the strain energy density due to stretching and bending the rod, and twisting the rod, respectively.

  1. The total strain energy follows by integrating U over the volume of the rod.  Using the measures of cross-sectional geometry listed in Section 10.2.1, it is straightforward to show that

V E 2 ( ( ds d s ¯ 1 )+( κ 1 κ ¯ 1 ) x 2 ( κ 2 κ ¯ 2 ) x 1 ) 2 dV= E 2 0 L { A ( ds d s ¯ 1 ) 2 + I 1 ( κ 1 κ ¯ 1 ) 2 + I 2 ( κ 2 κ ¯ 2 ) 2 }ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaSaaaeaacaWGfbaabaGaaG OmaaaadaqadaqaamaabmaabaWaaSaaaeaacaWGKbGaam4Caaqaaiaa dsgaceWGZbGbaebaaaGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabgU caRiaacIcacqaH6oWAdaWgaaWcbaGaaGymaaqabaGccqGHsislcuaH 6oWAgaqeamaaBaaaleaacaaIXaaabeaakiaacMcacaWG4bWaaSbaaS qaaiaaikdaaeqaaOGaeyOeI0IaaiikaiabeQ7aRnaaBaaaleaacaaI YaaabeaakiabgkHiTiqbeQ7aRzaaraWaaSbaaSqaaiaaikdaaeqaaO GaaiykaiaadIhadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaa daahaaWcbeqaaiaaikdaaaaabaGaamOvaaqab0Gaey4kIipakiaads gacaWGwbGaeyypa0ZaaSaaaeaacaWGfbaabaGaaGOmaaaadaWdXbqa amaacmaabaGaamyqamaabmaabaWaaSaaaeaacaWGKbGaam4Caaqaai aadsgaceWGZbGbaebaaaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiabgUcaRiaadMeadaWgaaWcbaGaaGymaa qabaGccaGGOaGaeqOUdS2aaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Ia fqOUdSMbaebadaWgaaWcbaGaaGymaaqabaGccaGGPaWaaWbaaSqabe aacaaIYaaaaOGaey4kaSIaamysamaaBaaaleaacaaIYaaabeaakiaa cIcacqaH6oWAdaWgaaWcbaGaaGOmaaqabaGccqGHsislcuaH6oWAga qeamaaBaaaleaacaaIYaaabeaakiaacMcadaahaaWcbeqaaiaaikda aaaakiaawUhacaGL9baacaWGKbGaam4CaaWcbaGaaGimaaqaaiaadY eaa0Gaey4kIipaaaa@7F72@

  1. Some additional algebra is required to calculate the energy associated with twisting the rod. Begin by noting that

A { ( w x 1 x 2 ) 2 + ( w x 2 + x 1 ) 2 } dA=J+ A ( w x α w x α x 2 w x 1 + x 1 w x 2 )dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaiWaaeaadaqadaqaamaala aabaGaeyOaIyRaam4DaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGym aaqabaaaaOGaeyOeI0IaamiEamaaBaaaleaacaaIYaaabeaaaOGaay jkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiaaykW7cqGHRaWkdaqa daqaamaalaaabaGaeyOaIyRaam4DaaqaaiabgkGi2kaadIhadaWgaa WcbaGaaGOmaaqabaaaaOGaey4kaSIaamiEamaaBaaaleaacaaIXaaa beaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5Eai aaw2haaaWcbaGaamyqaaqab0Gaey4kIipakiaadsgacaWGbbGaeyyp a0JaamOsaiabgUcaRmaapefabaWaaeWaaeaadaWcaaqaaiabgkGi2k aadEhaaeaacqGHciITcaWG4bWaaSbaaSqaaiabeg7aHbqabaaaaOWa aSaaaeaacqGHciITcaWG3baabaGaeyOaIyRaamiEamaaBaaaleaacq aHXoqyaeqaaaaakiabgkHiTiaadIhadaWgaaWcbaGaaGOmaaqabaGc daWcaaqaaiabgkGi2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqaai aaigdaaeqaaaaakiabgUcaRiaadIhadaWgaaWcbaGaaGymaaqabaGc daWcaaqaaiabgkGi2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqaai aaikdaaeqaaaaaaOGaayjkaiaawMcaaiaadsgacaWGbbaaleaacaWG bbaabeqdcqGHRiI8aaaa@77AC@

We need to show that the integral on the right hand side of this expression is zero.

  1. To this end, note that

A w x α w x α dA= A x α ( w w x α ) dA= C w w x α n α ds = C w( x 2 n 1 x 1 n 2 )ds MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaSaaaeaacqGHciITcaWG3b aabaGaeyOaIyRaamiEamaaBaaaleaacqaHXoqyaeqaaaaaaeaacaWG bbaabeqdcqGHRiI8aOWaaSaaaeaacqGHciITcaWG3baabaGaeyOaIy RaamiEamaaBaaaleaacqaHXoqyaeqaaaaakiaadsgacaWGbbGaeyyp a0Zaa8quaeaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhadaWgaa WcbaGaeqySdegabeaaaaGcdaqadaqaaiaadEhadaWcaaqaaiabgkGi 2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqaaiabeg7aHbqabaaaaa GccaGLOaGaayzkaaaaleaacaWGbbaabeqdcqGHRiI8aOGaamizaiaa dgeacqGH9aqpdaWdrbqaaiaadEhadaWcaaqaaiabgkGi2kaadEhaae aacqGHciITcaWG4bWaaSbaaSqaaiabeg7aHbqabaaaaOGaamOBamaa BaaaleaacqaHXoqyaeqaaOGaamizaiaadohaaSqaaiaadoeaaeqani abgUIiYdGccqGH9aqpdaWdrbqaaiaadEhadaqadaqaaiaadIhadaWg aaWcbaGaaGOmaaqabaGccaWGUbWaaSbaaSqaaiaaigdaaeqaaOGaey OeI0IaamiEamaaBaaaleaacaaIXaaabeaakiaad6gadaWgaaWcbaGa aGOmaaqabaaakiaawIcacaGLPaaacaWGKbGaam4CaaWcbaGaam4qaa qab0Gaey4kIipaaaa@78BD@

where we have recalled that the warping function w satisfies 2 w/ x α x α =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaaki aadEhacaGGVaGaeyOaIyRaamiEamaaBaaaleaacqaHXoqyaeqaaOGa eyOaIyRaamiEamaaBaaaleaacqaHXoqyaeqaaOGaeyypa0JaaGimaa aa@3EC0@  in A as well as ( w/ x α ) n α = x 2 n 1 x 1 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaeyOaIyRaam4Daiaac+cacq GHciITcaWG4bWaaSbaaSqaaiabeg7aHbqabaaakiaawIcacaGLPaaa caWGUbWaaSbaaSqaaiabeg7aHbqabaGccqGH9aqpcaWG4bWaaSbaaS qaaiaaikdaaeqaaOGaamOBamaaBaaaleaacaaIXaaabeaakiabgkHi TiaadIhadaWgaaWcbaGaaGymaaqabaGccaWGUbWaaSbaaSqaaiaaik daaeqaaaaa@45B5@  on C, and have used the divergence theorem.

  1. Secondly, note that

A ( x 2 w x 1 + x 1 w x 2 )dA = A ( x 1 ( x 2 w )+ x 2 ( w x 1 ) )dA = C ( ( x 2 w ) n 1 +( w x 1 ) n 2 )dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaWaaeWaaeaacqGHsislcaWG4b WaaSbaaSqaaiaaikdaaeqaaOWaaSaaaeaacqGHciITcaWG3baabaGa eyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaGccqGHRaWkcaWG4b WaaSbaaSqaaiaaigdaaeqaaOWaaSaaaeaacqGHciITcaWG3baabaGa eyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPa aacaWGKbGaamyqaaWcbaGaamyqaaqab0Gaey4kIipakiabg2da9maa pefabaWaaeWaaeaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhada WgaaWcbaGaaGymaaqabaaaaOWaaeWaaeaacqGHsislcaWG4bWaaSba aSqaaiaaikdaaeqaaOGaam4DaaGaayjkaiaawMcaaiabgUcaRmaala aabaGaeyOaIylabaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaa aaGcdaqadaqaaiaadEhacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGcca GLOaGaayzkaaaacaGLOaGaayzkaaGaamizaiaadgeaaSqaaiaadgea aeqaniabgUIiYdGccqGH9aqpdaWdrbqaamaabmaabaWaaeWaaeaacq GHsislcaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaam4DaaGaayjkaiaa wMcaaiaad6gadaWgaaWcbaGaaGymaaqabaGccqGHRaWkdaqadaqaai aadEhacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGa amOBamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiaadsgaca WGbbaaleaacaWGdbaabeqdcqGHRiI8aaaa@7985@

The sum of (5) and (6) is zero.  Using this result and (4) gives the expression for the strain energy of the rod.