Problems for Chapter 2

 

Governing Equations of Solid Mechanics

 

 

 

2.1.  Mathematical Description of Shape Changes in Solids

 

 

2.1.1.      A thin film of material is deformed in simple shear during a plate impact experiment, as shown in the figure.

2.1.1.1.            Write down expressions for the displacement field in the film, in terms of x 1 , x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaa aaa@3ABC@  d and h, expressing your answer as components in the basis shown.

2.1.1.2.            Calculate the Lagrange strain tensor associated with the deformation, expressing your answer as components in the basis shown.

2.1.1.3.            Calculate the infinitesimal strain tensor for the deformation, expressing your answer as components in the basis shown.

2.1.1.4.            Find the principal values of the infinitesimal strain tensor, in terms of d and h

 

 

2.1.2.      Find a displacement field corresponding to a uniform infinitesimal strain field ε ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3428@ .  (Don’t make this hard MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in particular do not use the complicated approach described in Section 2.1.20.  Instead, think about what kind of function, when differentiated, gives a constant). Is the displacement unique? 

 

 

2.1.3.      Find a formula for the displacement field that generates zero infinitesimal strain.   

 

 

2.1.4.      Find a displacement field that corresponds to a uniform Lagrange strain tensor E ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@334B@ .  Is the displacement unique?   Find a formula for the most general displacement field that generates a uniform Lagrange strain.

 

 

2.1.5.      The displacement field in a homogeneous, isotropic circular shaft twisted through angle α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37D8@  at one end is given by

u 1 = x 1 [cos( α x 3 L )1] x 2 sin( α x 3 L ) u 2 = x 1 sin( α x 3 L )+ x 2 [cos( α x 3 L )1] u 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyDamaaBaaaleaacaaIXaaabe aakiabg2da9iaadIhadaWgaaWcbaGaaGymaaqabaGccaGGBbGaci4y aiaac+gacaGGZbWaaeWaaeaadaWcaaqaaiabeg7aHjaadIhadaWgaa WcbaGaaG4maaqabaaakeaacaWGmbaaaaGaayjkaiaawMcaaiabgkHi TiaaigdacaGGDbGaeyOeI0IaamiEamaaBaaaleaacaaIYaaabeaaki GacohacaGGPbGaaiOBamaabmaabaWaaSaaaeaacqaHXoqycaWG4bWa aSbaaSqaaiaaiodaaeqaaaGcbaGaamitaaaaaiaawIcacaGLPaaaae aacaWG1bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaamiEamaaBaaa leaacaaIXaaabeaakiGacohacaGGPbGaaiOBamaabmaabaWaaSaaae aacqaHXoqycaWG4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamitaaaa aiaawIcacaGLPaaacqGHRaWkcaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaai4waiGacogacaGGVbGaai4CamaabmaabaWaaSaaaeaacqaHXoqy caWG4bWaaSbaaSqaaiaaiodaaeqaaaGcbaGaamitaaaaaiaawIcaca GLPaaacqGHsislcaaIXaGaaiyxaaqaaiaadwhadaWgaaWcbaGaaG4m aaqabaGccqGH9aqpcaaIWaaaaaa@6DA2@

2.1.5.1.            Calculate the matrix of components of the deformation gradient tensor

2.1.5.2.            Calculate the matrix of components of the Lagrange strain tensor.  Is the strain tensor a function of x 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaG4maaqabaaaaa@381F@ ?  Why?

2.1.5.3.            Find an expression for the increase in length of a material fiber of initial length dl, which is on the outer surface of the cylinder and initially oriented in the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaG4maaqabaaaaa@3810@  direction.

2.1.5.4.            Show that material fibers initially oriented in the e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGOmaaqabaaaaa@380F@  and e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGOmaaqabaaaaa@380F@  directions do not change their length.

2.1.5.5.            Calculate the principal values and directions of the Lagrange strain tensor at the point x 1 =a, x 2 =0, x 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadIhadaWgaa WcbaGaaGymaaqabaGccqGH9aqpcaWGHbGaaiilaiaaykW7caaMc8Ua aGPaVlaaykW7caWG4bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaG imaiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadIhadaWg aaWcbaGaaG4maaqabaGccqGH9aqpcaaIWaaaaa@50B5@ .  Hence, deduce the orientations of the material fibers that have the greatest and smallest increase in length.

2.1.5.6.            Calculate the components of the infinitesimal strain tensor.  Show that, for small values of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37D8@ , the infinitesimal strain tensor is identical to the Lagrange strain tensor, but for finite rotations the two measures of deformation differ.

2.1.5.7.            Use the infinitesimal strain tensor to obtain estimates for the lengths of material fibers initially oriented with the three basis vectors. Where is the error in this estimate greatest? How large can α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHbaa@37D8@  be before the error in this estimate reaches 10%?

 

 

 

2.1.6.      To measure the in-plane deformation of a sheet of metal during a forming process, your managers place three small hardness indentations on the sheet.  Using a travelling microscope, they determine that the initial lengths of the sides of the triangle formed by the three indents are 1cm, 1cm, 1.414cm, as shown in the picture below.  After deformation, the sides have lengths 1.5cm, 2.0cm and 2.8cm.  Your managers would like to use this information to determine the in MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ plane components of the Lagrange strain tensor.  Unfortunately, being business economics graduates, they are unable to do this.

2.1.6.1.            Explain how the measurements can be used to determine E 11 , E 22 , E 12 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaaGymaiaaigdaae qaaOGaaiilaiaaykW7caaMc8UaamyramaaBaaaleaacaaIYaGaaGOm aaqabaGccaGGSaGaaGPaVlaaykW7caaMc8UaamyramaaBaaaleaaca aIXaGaaGOmaaqabaaaaa@40E9@  and do the calculation.

2.1.6.2.            Is it possible to determine the deformation gradient from the measurements provided?  Why?  If not, what additional measurements would be required to determine the deformation gradient?

 

 

 

2.1.7.      To track the deformation in a slowly moving glacier, three survey stations are installed in the shape of an equilateral triangle, spaced 100m apart, as shown in the picture.  After a suitable period of time, the spacing between the three stations is measured again, and found to be 90m, 110m and 120m, as shown in the figure.  Assuming that the deformation of the glacier is homogeneous over the region spanned by the survey stations, please compute the components of the Lagrange strain tensor associated with this deformation, expressing your answer as components in the basis shown.

 

 

2.1.8.      Compose a limerick that will help you to remember the distinction between engineering shear strains and the formal (mathematical) definition of shear strain.

 

 

2.1.9.      A rigid body motion is a nonzero displacement field that does not distort any infinitesimal volume element within a solid.  Thus, a rigid body displacement induces no strain, and hence no stress, in the solid.  The deformation corresponding to a 3D rigid rotation about an axis through the origin is

y=Rxor y i = R ij x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0JaaCOuaiabgwSixl aahIhacaaMc8UaaGPaVlaaykW7caaMc8Uaae4BaiaabkhacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG5bWaaSbaaSqaaiaadM gaaeqaaOGaeyypa0JaamOuamaaBaaaleaacaWGPbGaamOAaaqabaGc caWG4bWaaSbaaSqaaiaadQgaaeqaaaaa@5272@

where R must satisfy R R T = R T R=I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaeyyXICTaaCOuamaaCaaale qabaGaamivaaaakiabg2da9iaahkfadaahaaWcbeqaaiaadsfaaaGc cqGHflY1caWHsbGaeyypa0JaaCysaaaa@3FC4@ , det(R)>0.

2.1.9.1.            Show that the Lagrange strain associated with this deformation is zero.

2.1.9.2.            As a specific example, consider the deformation

[ y 1 y 2 y 3 ]=[ cosθ sinθ 0 sinθ cosθ 0 0 0 1 ][ x 1 x 2 x 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmqaaaqaaiaadMhada WgaaWcbaGaaGymaaqabaaakeaacaWG5bWaaSbaaSqaaiaaikdaaeqa aaGcbaGaamyEamaaBaaaleaacaaIZaaabeaaaaaakiaawUfacaGLDb aacqGH9aqpdaWadaqaauaabeqadmaaaeaaciGGJbGaai4Baiaacoha cqaH4oqCaeaacqGHsislciGGZbGaaiyAaiaac6gacqaH4oqCaeaaca aIWaaabaGaci4CaiaacMgacaGGUbGaeqiUdehabaGaci4yaiaac+ga caGGZbGaeqiUdehabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaG ymaaaaaiaawUfacaGLDbaadaWadaqaauaabeqadeaaaeaacaWG4bWa aSbaaSqaaiaaigdaaeqaaaGcbaGaamiEamaaBaaaleaacaaIYaaabe aaaOqaaiaadIhadaWgaaWcbaGaaG4maaqabaaaaaGccaGLBbGaayzx aaaaaa@59E3@

This is the displacement field caused by rotating a solid through an angle θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@322D@  about the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@324E@  axis.  Find the deformation gradient for this displacement field, and show that the deformation gradient tensor is orthogonal, as predicted above.  Show also that the infinitesimal strain tensor for this displacement field is not generally zero, but is of order θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXnaaCaaaleqabaGaaGOmaaaaaa a@3316@  if θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@322D@  is small.

2.1.9.3.            If the displacements are small, we can find a simpler representation for a rigid body displacement.  Consider a deformation of the form

y i = ijk ω j x k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaykW7caWG5bWaaSbaaSqaaiaadMgaae qaaOGaeyypa0JaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4Aaaqa baGccqaHjpWDdaWgaaWcbaGaamOAaaqabaGccaWG4bWaaSbaaSqaai aadUgaaeqaaaaa@3EBC@

Here ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8aaaa@31CD@  is a vector with magnitude <<1, which represents an infinitesimal rotation about an axis parallel to ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8aaaa@31CD@ .  Show that the infinitesimal strain tensor associated with this displacement is always zero.   Show further that the Lagrange strain associated with this displacement field is

E ij = 1 2 ( δ ij ω k ω k ω i ω j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaaiab es7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccqaHjpWDdaWgaaWcba Gaam4AaaqabaGccqaHjpWDdaWgaaWcbaGaam4AaaqabaGccqGHsisl cqaHjpWDdaWgaaWcbaGaamyAaaqabaGccqaHjpWDdaWgaaWcbaGaam OAaaqabaaakiaawIcacaGLPaaaaaa@47D8@

This is not, in general, zero.  It is small if all ω k <<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGRbaabeaaki aaykW7cqGH8aapcqGH8aapcaaIXaaaaa@37B8@ .

 

 

2.1.10.  The formula for the deformation due to a rotation through an angle θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@322E@  about an axis parallel to a unit vector n that passes through the origin is

y i =[ cosθ δ ij +(1cosθ) n i n j +sinθ ikj n k ] x j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0ZaamWaaeaaciGGJbGaai4BaiaacohacqaH4oqCcqaH0oaz daWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaaiikaiaaigdacq GHsislciGGJbGaai4BaiaacohacqaH4oqCcaGGPaGaamOBamaaBaaa leaacaWGPbaabeaakiaad6gadaWgaaWcbaGaamOAaaqabaGccqGHRa WkciGGZbGaaiyAaiaac6gacqaH4oqCcqGHiiIZdaWgaaWcbaGaamyA aiaadUgacaWGQbaabeaakiaad6gadaWgaaWcbaGaam4Aaaqabaaaki aawUfacaGLDbaacaWG4bWaaSbaaSqaaiaadQgaaeqaaaaa@5AE3@

2.1.10.1.        Calculate the components of corresponding deformation gradient

2.1.10.2.        Verify that the deformation gradient satisfies F ik F jk = F ki F kj = δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadUgaae qaaOGaamOramaaBaaaleaacaWGQbGaam4AaaqabaGccqGH9aqpcaWG gbWaaSbaaSqaaiaadUgacaWGPbaabeaakiaadAeadaWgaaWcbaGaam 4AaiaadQgaaeqaaOGaeyypa0JaeqiTdq2aaSbaaSqaaiaadMgacaWG Qbaabeaaaaa@41B0@

2.1.10.3.        Find the components of the inverse of the deformation gradient

2.1.10.4.        Verify that both the Lagrange strain tensor and the Eulerian strain tensor are zero for this deformation.  What does this tell you about the distorsion of the material?

2.1.10.5.        Calculate the Jacobian of the deformation gradient.  What does this tell you about volume changes associated with the deformation?

 

 

2.1.11.  In Section 2.1.6 it was stated that the Eulerian strain tensor E ij * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaqhaaWcbaGaamyAaiaadQgaae aacaGGQaaaaaaa@33FA@  can be used to relate the length of a material fiber in a deformable solid before and after deformation, using the formula

l 2 l 0 2 2 l 2 = E ij * n i n j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadYgadaahaaWcbeqaai aaikdaaaGccqGHsislcaWGSbWaaSbaaSqaaiaaicdaaeqaaOWaaWba aSqabeaacaaIYaaaaaGcbaGaaGOmaiaadYgadaahaaWcbeqaaiaaik daaaaaaOGaeyypa0JaamyramaaDaaaleaacaWGPbGaamOAaaqaaiaa cQcaaaGccaWGUbWaaSbaaSqaaiaadMgaaeqaaOGaamOBamaaBaaale aacaWGQbaabeaaaaa@43D1@

where n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@3285@  are the components of a unit vector parallel to the material fiber after deformation.

Derive this result.

 

 


2.1.12.  Suppose that you have measured the Lagrange strain tensor for a deformation, and wish to calculate the Eulerian strain tensor.  On purely physical grounds, do you think this is possible, without calculating the deformation gradient?  If so, find a formula relating Lagrange strain E ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@334B@  to Eulerian strain E ij * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaqhaaWcbaGaamyAaiaadQgaae aacaGGQaaaaaaa@33FA@ .

 

 

2.1.13.  Repeat problem 2.1.6, but instead of calculating the Lagrange strain tensor, find the components of the Eulerian strain tensor E ij * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaqhaaWcbaGaamyAaiaadQgaae aacaGGQaaaaaaa@33FA@  (you can do this directly, or use the results of problem 2.1.12, or both)

 

 

2.1.14.  Repeat problem 2.1.7, but instead of calculating the Lagrange strain tensor, find the components of the Eulerian strain tensor E ij * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaqhaaWcbaGaamyAaiaadQgaae aacaGGQaaaaaaa@33FA@  (you can do this directly, or use the results for problem 2.1.12, or both)

 

 

2.1.15.  The Lagrange strain tensor can be used to calculate the change in angle between any two material fibers in a solid as the solid is deformed.  In this problem you will calculate the formula that can be used to do this.  To this end, consider two infinitesimal material fibers in the undeformed solid, which are characterized by vectors with components d x i (1) = l 1 m i (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaadMgaae aacaGGOaGaaGymaiaacMcaaaGccqGH9aqpcaWGSbWaaSbaaSqaaiaa igdaaeqaaOGaamyBamaaDaaaleaacaWGPbaabaGaaiikaiaaigdaca GGPaaaaaaa@3CA0@  and d x i (2) = l 2 m i (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG4bWaa0baaSqaaiaadMgaae aacaGGOaGaaGOmaiaacMcaaaGccqGH9aqpcaWGSbWaaSbaaSqaaiaa ikdaaeqaaOGaamyBamaaDaaaleaacaWGPbaabaGaaiikaiaaikdaca GGPaaaaaaa@3CA3@ , where m (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaahaaWcbeqaaiaacIcacaaIXa Gaaiykaaaaaaa@33AF@  and  m (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaahaaWcbeqaaiaacIcacaaIYa Gaaiykaaaaaaa@33B0@  are two unit vectors.  Recall that the angle θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXnaaBaaaleaacaaIWaaabeaaaa a@3314@  between  m (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaahaaWcbeqaaiaacIcacaaIXa Gaaiykaaaaaaa@33AF@  and m (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaahaaWcbeqaaiaacIcacaaIYa Gaaiykaaaaaaa@33B0@  before deformation can be calculated from cos θ 0 = m i (1) m i (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacogacaGGVbGaai4CaiabeI7aXnaaBa aaleaacaaIWaaabeaakiabg2da9iaad2gadaqhaaWcbaGaamyAaaqa aiaacIcacaaIXaGaaiykaaaakiaad2gadaqhaaWcbaGaamyAaaqaai aacIcacaaIYaGaaiykaaaaaaa@3F44@ .  Let d y i (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaa0baaSqaaiaadMgaae aacaGGOaGaaGymaiaacMcaaaaaaa@358E@  and d y i (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaa0baaSqaaiaadMgaae aacaGGOaGaaGOmaiaacMcaaaaaaa@358F@  represent the two material fibers after deformation.  Show that the angle between d y i (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaa0baaSqaaiaadMgaae aacaGGOaGaaGymaiaacMcaaaaaaa@358E@  and d y i (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaa0baaSqaaiaadMgaae aacaGGOaGaaGOmaiaacMcaaaaaaa@358F@  can be calculated from the formula

cos θ 1 = 2 E ij m i (1) m j (2) +cos θ 0 1+2 E ij m i (1) m j (1) 1+2 E ij m i (2) m j (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGacogacaGGVbGaai4CaiabeI7aXnaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGOmaiaadweadaWg aaWcbaGaamyAaiaadQgaaeqaaOGaamyBamaaDaaaleaacaWGPbaaba GaaiikaiaaigdacaGGPaaaaOGaamyBamaaDaaaleaacaWGQbaabaGa aiikaiaaikdacaGGPaaaaOGaey4kaSIaci4yaiaac+gacaGGZbGaeq iUde3aaSbaaSqaaiaaicdaaeqaaaGcbaWaaOaaaeaacaaIXaGaey4k aSIaaGOmaiaadweadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamyBam aaDaaaleaacaWGPbaabaGaaiikaiaaigdacaGGPaaaaOGaamyBamaa DaaaleaacaWGQbaabaGaaiikaiaaigdacaGGPaaaaaqabaGcdaGcaa qaaiaaigdacqGHRaWkcaaIYaGaamyramaaBaaaleaacaWGPbGaamOA aaqabaGccaWGTbWaa0baaSqaaiaadMgaaeaacaGGOaGaaGOmaiaacM caaaGccaWGTbWaa0baaSqaaiaadQgaaeaacaGGOaGaaGOmaiaacMca aaaabeaaaaaaaa@6486@

 

 

 

 

2.1.16.  Suppose that a solid is subjected to a sequence of two homogeneous deformations (i) a rigid rotation R, followed by (ii) an arbitrary homogeneous deformation F.  Taking the original configuration as reference, find formulas for the following deformation measures for the final configuration of the solid, in terms of F and R:

2.1.16.1.        The deformation gradient

2.1.16.2.        The Left and Right Cauchy-Green deformation tensors

2.1.16.3.        The Lagrange strain

2.1.16.4.        The Eulerian strain.

 

 

2.1.17.  Repeat problem 2.1.16, but this time assume that the sequence of the two deformations is reversed, i.e. the solid is first subjected to an arbitrary homogeneous deformation F, and is subsequently subjected to a rigid rotation R.

 

 

2.1.18.  A spherical shell (see the figure) is made from an incompressible material.  In its undeformed state, the inner and outer radii of the shell are A,B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeacaGGSaGaamOqaaaa@32B5@ .  After deformation, the new values are a,b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacaGGSaGaamOyaaaa@32F5@ .  The deformation in the shell can be described (in Cartesian components) by the equation

y i = ( R 3 + a 3 A 3 ) 1/3 x i R R= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaqadaqaaiaadkfadaahaaWcbeqaaiaaiodaaaGccqGHRaWk caWGHbWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamyqamaaCaaale qabaGaaG4maaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGymaiaa c+cacaaIZaaaaOWaaSaaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaa GcbaGaamOuaaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadkfacqGH9aqp daGcaaqaaiaadIhadaWgaaWcbaGaam4AaaqabaGccaWG4bWaaSbaaS qaaiaadUgaaeqaaaqabaaaaa@66C6@

2.1.18.1.         Calculate the components of the deformation gradient tensor

2.1.18.2.        Verify that the deformation is volume preserving

2.1.18.3.        Find the deformed length of an infinitesimal radial line that has initial length l 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYgadaWgaaWcbaGaaGimaaqabaaaaa@324F@ , expressed as a function of R

2.1.18.4.        Find the deformed length of an infinitesimal circumferential line that has initial length l 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYgadaWgaaWcbaGaaGimaaqabaaaaa@324F@ , expressed as a function of R

2.1.18.5.        Using the results of 2.1.18.3, 2.1.18.4, find the principal stretches for the deformation.

2.1.18.6.        Find the inverse of the deformation gradient, expressed as a function of y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaaaaa@3290@ .  It is best to do this by working out a formula that enables you to calculate x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaaaaa@328F@  in terms of y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaaaaa@3290@  and r= y i y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacqGH9aqpdaGcaaqaaiaadMhada WgaaWcbaGaamyAaaqabaGccaWG5bWaaSbaaSqaaiaadMgaaeqaaaqa baaaaa@36BF@  and differentiate the result rather than to attempt to invert the result of 10.1.

 

 

2.1.19.  Suppose that the spherical shell described in Problem 2.1.18 is continuously expanding (visualize a balloon being inflated).  The rate of expansion can be characterized by the velocity v a =da/dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyyaaqabaGccq GH9aqpcaWGKbGaamyyaiaac+cacaWGKbGaamiDaaaa@37F9@  of the surface that lies at R=A in the undeformed cylinder.

2.1.19.1.        Calculate the velocity field v i =d y i /dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWGKbGaamyEamaaBaaaleaacaWGPbaabeaakiaac+cacaWG KbGaamiDaaaa@393D@  in the sphere as a function of x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaaaaa@328F@

2.1.19.2.        Calculate the velocity field as a function of y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaaaaa@3290@  (there is a long, obvious way to do this and a quick, subtle way)

2.1.19.3.        Calculate the time derivative of the deformation gradient tensor calculated in 2.1.18.1.

2.1.19.4.        Calculate the components of the velocity gradient L ij = v i y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGmbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9maalaaabaGaeyOaIyRaamODamaaBaaaleaacaWG PbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaaaa aa@3DC4@  by differentiating the result of 2.1.19.1

2.1.19.5.        Calculate the components of the velocity gradient using the results of 2.1.19.3 and 2.1.18.6

2.1.19.6.        Calculate the stretch rate tensor D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@334A@ .  Verify that the result represents a volume preserving stretch rate field.

 

 

2.1.20.  Repeat Problem 2.1.18.1, 2.1.18.6 and all of 2.1.19, but this time solve the problem using spherical-polar coordinates, using the various formulas for vector and tensor operations given in Appendix E.   In this case, you may assume that a point with position x=R e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaeyypa0JaamOuaiaahwgada WgaaWcbaGaamOuaaqabaaaaa@3795@  in the undeformed solid has position vector

y= ( R 3 + a 3 A 3 ) 1/3 e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahMhacqGH9aqpdaqadaqaaiaadkfada ahaaWcbeqaaiaaiodaaaGccqGHRaWkcaWGHbWaaWbaaSqabeaacaaI ZaaaaOGaeyOeI0IaamyqamaaCaaaleqabaGaaG4maaaaaOGaayjkai aawMcaamaaCaaaleqabaGaaGymaiaac+cacaaIZaaaaOGaaCyzamaa BaaaleaacaWGsbaabeaaaaa@3F8A@

after deformation.

 

 

 

 

2.1.21.  An initially straight beam is bent into a circle with radius R as shown in the figure.  Material fibers that are perpendicular to the axis of the undeformed beam are assumed to remain perpendicular to the axis after deformation, and the beam’s thickness and the length of its axis are assumed to be unchanged.   Under these conditions the deformation can be described as

y 1 =( R x 2 )sin( x 1 /R) y 2 =R(R x 2 )cos( x 1 /R) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpdaqadaqaaiaadkfacqGHsislcaWG4bWaaSbaaSqaaiaaikda aeqaaaGccaGLOaGaayzkaaGaci4CaiaacMgacaGGUbGaaiikaiaadI hadaWgaaWcbaGaaGymaaqabaGccaGGVaGaamOuaiaacMcacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadMhadaWgaaWcbaGaaGOm aaqabaGccqGH9aqpcaWGsbGaeyOeI0IaaiikaiaadkfacqGHsislca WG4bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiGacogacaGGVbGaai4C aiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaai4laiaadkfaca GGPaaaaa@6756@

where, as usual x is the position of a material particle in the undeformed beam, and y is the position of the same particle after deformation.

2.1.21.1.        Calculate the deformation gradient field in the beam, expressing your answer as a function of x 1 , x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamiEamaaBaaaleaacaaIYaaabeaaaaa@34FB@ , and as components in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3978@  shown.

2.1.21.2.        Calculate the Lagrange strain field in the beam.

2.1.21.3.        Calculate the infinitesimal strain field in the beam.

2.1.21.4.        Compare the values of Lagrange strain and infinitesimal strain for two points that lie at ( x 1 =0, x 2 =h) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaeyypa0JaaGimaiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqa aOGaeyypa0JaamiAaiaacMcaaaa@3A11@  and ( x 1 =L, x 2 =0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaeyypa0JaamitaiaacYcacaWG4bWaaSbaaSqaaiaaikdaaeqa aOGaeyypa0JaaGimaiaacMcaaaa@39F5@ .   Explain briefly the physical origin of the difference between the two strain measures at each point.   Recommend maximum allowable values of h/R and L/R for use of the infinitesimal strain measure in modeling beam deflections.

2.1.21.5.        Calculate the deformed length of an infinitesimal material fiber that has length l 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYgadaWgaaWcbaGaaGimaaqabaaaaa@324F@  and orientation e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGymaaqabaaaaa@324D@  in the undeformed beam.  Express your answer as a function of x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaaaaa@325D@ .

2.1.21.6.        Calculate the change in length of an infinitesimal material fiber that has length l 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYgadaWgaaWcbaGaaGimaaqabaaaaa@324F@  and orientation e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaGOmaaqabaaaaa@324E@  in the undeformed beam.

2.1.21.7.        Show that the two material fibers described in 2.1.21.5 and 2.1.21.6 remain mutually perpendicular after deformation.   Is this true for all material fibers that are mutually perpendicular in the undeformed solid?

2.1.21.8.        Find the components in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3978@  of the Left and Right stretch tensors U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwfaaaa@3156@  and V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAfaaaa@3157@  as well as the rotation tensor R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkfaaaa@3153@  for this deformation.  You should be able to write down U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwfaaaa@3156@  and R by inspection, without needing to wade through the laborious general process outlined in Section 2.1.13.  The results can then be used to calculate V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAfaaaa@3157@ .

2.1.21.9.        Find the principal directions of U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwfaaaa@3156@  as well as the principal stretches.  You should be able to write these down using your physical intuition without doing any tedious calculations.  

2.1.21.10.     Let { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3990@  be a basis in which m 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGymaaqabaaaaa@3255@  is parallel to the axis of the deformed beam, as shown in the figure.   Write down the components of each of the unit vectors m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaamyAaaqabaaaaa@3288@  in the basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3978@ .  Hence, compute the transformation matrix Q ij = m i e j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgfadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaaCyBamaaBaaaleaacaWGPbaabeaakiabgwSixlaa hwgadaWgaaWcbaGaamOAaaqabaaaaa@3AD4@  that is used to transform tensor components from { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3978@  to { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3990@ .

2.1.21.11.    Find the components of the deformation gradient tensor, Lagrange strain tensor, as well as U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwfaaaa@3156@    V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAfaaaa@3157@  and R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkfaaaa@3153@  in the basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3990@ .

2.1.21.12.    Find the principal directions of V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAfaaaa@3157@  expressed as components in the basis { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHTbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaah2gadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyB amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3990@ .  Again, you should be able to simply write down this result.

 

 


2.1.22.  A sheet of material is subjected to a two dimensional homogeneous deformation of the form

y 1 = A 11 x 1 + A 12 x 2 y 2 = A 21 x 1 + A 22 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG5bWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaamyqamaaBaaaleaacaaIXaGaaGymaaqabaGccaWG4bWa aSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyqamaaBaaaleaacaaIXa GaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaamyEamaaBaaaleaacaaIYaaabeaakiabg2da 9iaadgeadaWgaaWcbaGaaGOmaiaaigdaaeqaaOGaamiEamaaBaaale aacaaIXaaabeaakiabgUcaRiaadgeadaWgaaWcbaGaaGOmaiaaikda aeqaaOGaamiEamaaBaaaleaacaaIYaaabeaaaaa@5E75@

where A ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@3595@  are constants.

Suppose that a circle of unit radius is drawn on the undeformed sheet.   This circle is distorted to a smooth curve on the deformed sheet.  Show that the distorted circle is an ellipse, with semi-axes that are parallel to the principal directions of the left stretch tensor V, and that the lengths of the semi-axes of the ellipse are equal to the principal stretches for the deformation.  There are many different ways to approach this calculation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  some are very involved.  The simplest way is probably to assume that the principal directions of V subtend an angle θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGimaaqaba aaaa@3562@  to the { e 1 , e 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiyFaaaa @3935@  basis as shown in the figure, write the polar decomposition A=VR MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHbbGaeyypa0JaaCOvaiabgwSixl aahkfaaaa@389A@  in terms of principal stretches λ 1 , λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH7oaBdaWgaaWcbaGaaGymaaqaba GccaGGSaGaeq4UdW2aaSbaaSqaaiaaikdaaeqaaaaa@38B7@  and θ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGimaaqaba aaaa@3562@ , and then show that y=VRx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH5bGaeyypa0JaaCOvaiabgwSixl aahkfacqGHflY1caWH4baaaa@3C1D@  (where x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4baaaa@33C7@  is on the unit circle) describes an ellipse.

 

 

2.1.23.  A solid is subjected to a rigid rotation so that a unit vector a in the undeformed solid is rotated to a new orientation b.  Find a rotation tensor R that is consistent with this deformation, in terms of the components of a and b.   Is the rotation tensor unique?  If not, find the most general formula for the rotation tensor.

 

 

2.1.24.  In a plate impact experiment, a thin film of material with thickness h is subjected to a homogeneous shear deformation by displacing the upper surface of the film horizontally with a speed v.

2.1.24.1.        Write down the velocity field in the film

2.1.24.2.        Calculate the velocity gradient, the stretch rate and the spin rate

2.1.24.3.        Calculate the instantaneous angular velocity of a material fiber parallel to the e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa aa@349C@  direction in the film

2.1.24.4.        Calculate the instantaneous angular velocity of a material fiber parallel to ( e 1 + e 2 )/ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaaCyzamaaBaaaleaacaaIXa aabeaakiabgUcaRiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGPaGa ai4lamaakaaabaGaaGOmaaWcbeaaaaa@3A4A@

2.1.24.5.        Calculate the stretch rates for the material fibers in 22.3 and 22.4

2.1.24.6.        What is the direction of the material fiber with the greatest angular velocity?  What is the direction of the material fiber with the greatest stretch rate?

 

 

2.1.25.  The velocity field v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2baaaa@33C5@  due to a rigid rotation about an axis through the origin can be characterized by a skew tensor W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHxbaaaa@33A6@  or an angular velocity vector ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHjpaaaa@341B@  defined so that

v=Wxv=ω×x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH2bGaeyypa0JaaC4vaiabgwSixl aahIhacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaahAhacq GH9aqpcaWHjpGaey41aqRaaCiEaaaa@5502@

Find a formula relating the components of W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHxbaaaa@33A6@  and ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHjpaaaa@341B@ . (One way to approach this problem is to calculate a formula for W by taking the time derivative of Rodriguez formula MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  see Sect 2.1.1).

 

 

2.1.26.  A single crystal deforms by shearing on a single active slip system as illustrated in the figure.  The crystal is loaded so that the slip direction s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHZbaaaa@33C2@  and normal to the slip plane m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbaaaa@33BC@  maintain a constant direction during the deformation

2.1.26.1.          Show that the deformation gradient can be expressed in terms of the components of the slip direction s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHZbaaaa@33C2@  and the normal to the slip plane m as F ij = δ ij +γ s i m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGc cqGHRaWkcqaHZoWzcaWGZbWaaSbaaSqaaiaadMgaaeqaaOGaamyBam aaBaaaleaacaWGQbaabeaaaaa@4114@  where γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHZoWzaaa@346D@  denotes the shear, as illustrated in the figure.

2.1.26.2.        Suppose shearing proceeds at some rate γ ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaGaaaaa@3476@ .  At the instant when γ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHZoWzcqGH9aqpcaaIWaaaaa@362D@ , calculate (i) the velocity gradient tensor; (ii) the stretch rate tensor and (iii) the spin tensor associated with the deformation.

 

 

2.1.27.  The properties of many rubbers and foams are specified by functions of the following invariants of the left Cauchy-Green deformation tensor B ij = F ik F jk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGcbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadAeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGa amOramaaBaaaleaacaWGQbGaam4Aaaqabaaaaa@3C5A@ .

I 1 =trace(B)= B kk I 2 = 1 2 ( I 1 2 BB )= 1 2 ( I 1 2 B ik B ki ) I 3 =detB= J 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadMeadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcaqG0bGaaeOCaiaabggacaqGJbGaaeyzaiaacIca caWHcbGaaiykaiabg2da9iaadkeadaWgaaWcbaGaam4AaiaadUgaae qaaaGcbaGaamysamaaBaaaleaacaaIYaaabeaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaaaWaaeWaaeaacaWGjbWaa0baaSqaaiaaig daaeaacaaIYaaaaOGaeyOeI0IaaCOqaiabgwSixlabgwSixlaahkea aiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaam aabmaabaGaamysamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgkHi TiaadkeadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaamOqamaaBaaale aacaWGRbGaamyAaaqabaaakiaawIcacaGLPaaaaeaacaWGjbWaaSba aSqaaiaaiodaaeqaaOGaeyypa0JaciizaiaacwgacaGG0bGaaCOqai abg2da9iaadQeadaahaaWcbeqaaiaaikdaaaaaaaa@65F1@

Invariants of a tensor are defined in Appendix B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  they are functions of the components of a tensor that are independent of the choice of basis.

2.1.27.1.        Verify that I 1 , I 2 , I 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadMeadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamysamaa BaaaleaacaaIZaaabeaaaaa@395C@  are invariants.  The simplest way to do this is to show that I 1 , I 2 , I 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadMeadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamysamaa BaaaleaacaaIZaaabeaaaaa@395C@  are unchanged during a change of basis. 

2.1.27.2.        In order to calculate stress-strain relations for these materials, it is necessary to evaluate derivatives of the invariants.  Show that

I 1 F ij =2 F ij , I 2 F ij =2( I 1 F ij B ik F kj ), I 3 F ij =2 I 3 F ji 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGjbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOaIyRaamOr amaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaeyypa0JaaGOmaiaadA eadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiilaiaaykW7caaMc8+a aSaaaeaacqGHciITcaWGjbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaey OaIyRaamOramaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaeyypa0Ja aGOmamaabmaabaGaamysamaaBaaaleaacaaIXaaabeaakiaadAeada WgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0IaamOqamaaBaaaleaa caWGPbGaam4AaaqabaGccaWGgbWaaSbaaSqaaiaadUgacaWGQbaabe aaaOGaayjkaiaawMcaaiaacYcacaaMc8UaaGPaVpaalaaabaGaeyOa IyRaamysamaaBaaaleaacaaIZaaabeaaaOqaaiabgkGi2kaadAeada WgaaWcbaGaamyAaiaadQgaaeqaaaaakiabg2da9iaaikdacaWGjbWa aSbaaSqaaiaaiodaaeqaaOGaamOramaaDaaaleaacaWGQbGaamyAaa qaaiabgkHiTiaaigdaaaaaaa@7027@

 

 

 

2.1.28.  The infinitesimal strain field in a long cylinder containing a hole at its center is given by

ε 31 =b x 2 / r 2 ε 32 =b x 1 / r 2 r= x 1 2 + x 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIZaGaaGymaa qabaGccqGH9aqpcqGHsislcaWGIbGaamiEamaaBaaaleaacaaIYaaa beaakiaac+cacaWGYbWaaWbaaSqabeaacaaIYaaaaOGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa Vlabew7aLnaaBaaaleaacaaIZaGaaGOmaaqabaGccqGH9aqpcaWGIb GaamiEamaaBaaaleaacaaIXaaabeaakiaac+cacaWGYbWaaWbaaSqa beaacaaIYaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadkha cqGH9aqpdaGcaaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaa GccqGHRaWkcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaqabaaa aa@70BA@

2.1.28.1.        Show that the strain field satisfies the equations of compatibility.

2.1.28.2.        Show that the strain field is consistent with a displacement field of the form u 3 =θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaG4maaqabaGccq GH9aqpcqaH4oqCaaa@3521@ , where θ=2b tan 1 x 2 / x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaaikdacaWGIbGaci iDaiaacggacaGGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamiE amaaBaaaleaacaaIYaaabeaakiaac+cacaWG4bWaaSbaaSqaaiaaig daaeqaaaaa@3E0D@ .   Note that although the strain field is compatible, the displacement field is multiple valued MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  i.e. the displacements are not equal at θ=2π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaaikdacqaHapaCaa a@35AD@  and θ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=wi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjabg2da9iaaicdaaaa@33EE@ , which supposedly represent the same point in the solid.  Surprisingly, displacement fields like this do exist in solids MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  they are caused by dislocations in a crystal.  These are discussed in more detail in Sections 5.3.4

 

 

 

2.1.29.  The figure shows a test designed to measure the response of a polymer to large shear strains.  The sample is a hollow cylinder with internal radius a 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaaicdaaeqaaa aa@338B@  and external radius a 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaa aa@338C@ .  The inside diameter is bonded to a fixed rigid cylinder.  The external diameter is bonded inside a rigid tube, which is rotated through an angle α(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycaGGOaGaamiDaiaacMcaaa a@35B0@ .  Assume that the specimen deforms as indicated in the figure, i.e. (a) cylindrical sections remain cylindrical; (b) no point in the specimen moves in the axial or radial directions; (c) that a  cylindrical element of material at radius R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbaaaa@3296@  rotates through angle ϕ(R,t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzcaGGOaGaamOuaiaacYcaca WG0bGaaiykaaaa@3750@  about the axis of the specimen. Take the undeformed configuration as reference. Let (R,Θ,Z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkfacaGGSaGaeuiMdeLaai ilaiaadQfacaGGPaaaaa@36B3@  denote the cylindrical-polar coordinates of a material point in the reference configuration, and let { e R , e Θ , e Z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOuaa qabaGccaGGSaGaaCyzamaaBaaaleaacqqHyoquaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOwaaqabaGccaGG9baaaa@3AC6@  be cylindrical-polar basis vectors at (R,Θ,Z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkfacaGGSaGaeuiMdeLaai ilaiaadQfacaGGPaaaaa@36B3@ .  Let (r,θ,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkhacaGGSaGaeqiUdeNaai ilaiaadQhacaGGPaaaaa@3732@  denote the coordinates of this point in the deformed configuration, and let { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOCaa qabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOEaaqabaGccaGG9baaaa@3B47@  by cylindrical-polar basis vectors located at (r,θ,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkhacaGGSaGaeqiUdeNaai ilaiaadQhacaGGPaaaaa@3742@

 

2.1.29.1.        Write down expressions for  (r,θ,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkhacaGGSaGaeqiUdeNaai ilaiaadQhacaGGPaaaaa@3742@  in terms of (R,Θ,Z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkfacaGGSaGaeuiMdeLaai ilaiaadQfacaGGPaaaaa@36C3@  (this constitutes the deformation mapping)

2.1.29.2.        Let P denote the material point at at (R,Θ,Z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkfacaGGSaGaeuiMdeLaai ilaiaadQfacaGGPaaaaa@36C3@  in the reference configuration. Write down the reference position vector X of P, expressing your answer as components in the basis { e R , e Θ , e Z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOuaa qabaGccaGGSaGaaCyzamaaBaaaleaacqqHyoquaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOwaaqabaGccaGG9baaaa@3AC8@ .

2.1.29.3.        Write down the deformed position vector x of P, expressing your answer in terms of (R,Θ,Z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkfacaGGSaGaeuiMdeLaai ilaiaadQfacaGGPaaaaa@36C3@  and basis vectors { e R , e Θ , e Z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOuaa qabaGccaGGSaGaaCyzamaaBaaaleaacqqHyoquaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOwaaqabaGccaGG9baaaa@3AC8@ .

2.1.29.4.        Find the components of the deformation gradient tensor F in { e R , e Θ , e Z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOuaa qabaGccaGGSaGaaCyzamaaBaaaleaacqqHyoquaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOwaaqabaGccaGG9baaaa@3AC8@ .  (Recall that the gradient operator in cylindrical-polar coordinates is ( e R R + e Θ 1 R Θ + e Z Z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaey4bIeTaeyyyIORaaiikaiaahwgada WgaaWcbaGaamOuaaqabaGcdaWcaaqaaiabgkGi2cqaaiabgkGi2kaa dkfaaaGaey4kaSIaaCyzamaaBaaaleaacqqHyoquaeqaaOWaaSaaae aacaaIXaaabaGaamOuaaaadaWcaaqaaiabgkGi2cqaaiabgkGi2kab fI5arbaacqGHRaWkcaWHLbWaaSbaaSqaaiaadQfaaeqaaOWaaSaaae aacqGHciITaeaacqGHciITcaWGAbaaaiaacMcaaaa@4B37@ ; recall also that e R Θ = e Θ ; e Θ Θ = e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaahwgadaWgaa WcbaGaamOuaaqabaaakeaacqGHciITcqqHyoquaaGaeyypa0JaaCyz amaaBaaaleaacqqHyoquaeqaaOGaai4oaiaaykW7caaMc8UaaGPaVp aalaaabaGaeyOaIyRaaCyzamaaBaaaleaacqqHyoquaeqaaaGcbaGa eyOaIyRaeuiMdefaaiabg2da9iabgkHiTiaahwgadaWgaaWcbaGaam Ouaaqabaaaaa@4CE7@  )

2.1.29.5.        Show that the deformation gradient can be decomposed into a sequence F=RS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaaCOuaiabgwSixl aahofaaaa@3795@  of a simple shear S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbaaaa@329B@  followed by a rigid rotation through angle ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@3387@  about the e Z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaadQfaaeqaaa aa@33B8@  direction R.  In this case the simple shear deformation will have the form

S= e R e R + e Θ e Θ + e Z e Z +k e Θ e R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbGaeyypa0JaaCyzamaaBaaale aacaWGsbaabeaakiaahwgadaWgaaWcbaGaamOuaaqabaGccqGHRaWk caWHLbWaaSbaaSqaaiabfI5arbqabaGccaWHLbWaaSbaaSqaaiabfI 5arbqabaGccqGHRaWkcaWHLbWaaSbaaSqaaiaadQfaaeqaaOGaaCyz amaaBaaaleaacaWGAbaabeaakiabgUcaRiaadUgacaWHLbWaaSbaaS qaaiabfI5arbqabaGccaWHLbWaaSbaaSqaaiaadkfaaeqaaaaa@49FC@

where k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbaaaa@32AF@  is to be determined.

2.1.29.6.        Find the components of F in { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOCaa qabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOEaaqabaGccaGG9baaaa@3B47@ .

2.1.29.7.        Verify that the deformation is volume preserving (i.e. check the value of J=det(F))

2.1.29.8.        Find the components of the right Cauchy-Green deformation tensors in { e R , e Θ , e Z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOuaa qabaGccaGGSaGaaCyzamaaBaaaleaacqqHyoquaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOwaaqabaGccaGG9baaaa@3AC8@  

2.1.29.9.        Find the components of the left Cauchy-Green deformation tensor in { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOCaa qabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOEaaqabaGccaGG9baaaa@3B47@

2.1.29.10.    Find F 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCOramaaCaaaleqabaGaeyOeI0IaaG ymaaaaaaa@3381@  in { e R , e Θ , e Z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOuaa qabaGccaGGSaGaaCyzamaaBaaaleaacqqHyoquaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOwaaqabaGccaGG9baaaa@3AC8@

2.1.29.11.    Find the principal values of the stretch tensor U

2.1.29.12.    Write down the velocity field v in terms of (r,θ,z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaiikaiaadkhacaGGSaGaeqiUdeNaai ilaiaadQhacaGGPaaaaa@3742@  in the basis { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOCaa qabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOEaaqabaGccaGG9baaaa@3B47@

2.1.29.13.    Calculate the spatial velocity gradient L in the basis { e r , e θ , e z } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaai4EaiaahwgadaWgaaWcbaGaamOCaa qabaGccaGGSaGaaCyzamaaBaaaleaacqaH4oqCaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaamOEaaqabaGccaGG9baaaa@3B47@