Chapter 5

 

Analytical techniques and solutions for linear elastic solids

 

 

 

In the preceding chapter, we solved some simple linear elastic boundary value problems.  The problems were trivial, however, in that the stress, strain and displacement fields were axially or spherically symmetric.  Most problems of practical interest involve fully 3D stress and displamement fields.

 

It is extremely difficult to solve general boundary value problems.  However, some of the best mathematicians over the past 200 years have turned their attention to this matter, and have developed several very elegant techniques.  None of these are completely general, but solutions derived using these techniques have provided invaluable insight into the behavior of deformable solids. 

 

Sadly, these days any fool with a PC and a finite element package can solve virtually any linear elastic boundary value problem, so you will not be able to make a living calculating exact elasticity solutions.  Nevertheless, some exact solutions are of fundamental practical importance. Examples include contact problems, solutions for cracks, stress concentrations, thermal stress problems, and problems involving defects such as dislocations in solids.  It is worth seeing how such solutions were derived.

 

In addition, there are some very powerful theorems in linear elasticity (such as the principle of minimum potential energy, and the reciprocal theorem), which can be used to directly calculate quantities of interest without necessarily having to solve all the governing equations directly.

 

In this chapter, we present a very brief survey of the field of linear elasticity.  Specifically,

1.      We will outline some important general features of solutions to boundary and initial value problems;

2.      We will discuss some solution techniques and present solutions to selected boundary value problems of interest;

3.      We will discuss energy methods for solving problems involving linear elastic solids, including the principle of minimum potential energy, the reciprocal theorem, and the Rayleigh-Ritz method for estimating natural frequencies of vibrating elastic solids.

 

 


5.1 General Principles

 

 

5.1.1 Summary of the governing equations of linear elasticity

 

Static problems. We already listed the governing equations of linear elasticity in our discussion of simple axisymmetric problems. They are repeated here for convenience.  

 

Given:

1.      The shape of the solid in its unloaded condition R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamOuaaaa@31B4@

2.      The initial stress field in the solid (we will take this to be zero)

3.      The elastic constants for the solid C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaam4qamaaBaaaleaacaWGPbGaamOAai aadUgacaWGSbaabeaaaaa@358F@  and its mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@331E@

4.      The thermal expansion coefficients for the solid, and temperature change from the initial configuration ΔT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam ivaaaa@3835@

5.      A body force distribution b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCOyaaaa@31C8@  (per unit mass) acting on the solid

6.      Boundary conditions, specifying displacements u * (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaaCyDamaaCaaaleqabaGaaiOkaaaaki aacIcacaWH4bGaaiykaaaa@351A@  on a portion 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIy7aaSbaaSqaaiaaigdaaeqaaO GaamOuaaaa@340B@  or tractions on a portion 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeyOaIy7aaSbaaSqaaiaaikdaaeqaaO GaamOuaaaa@340C@  of the boundary of R

 

Calculate displacements, strains and stresses satisfying the governing equations of linear elastostatics

ε ij = 1 2 ( u i x j + u j x i ) σ ij = C ijkl ( ε kl α kl ΔT) σ ij x i + ρ 0 b j =0 u i = u i * on 1 R σ ij n i = t j * on 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacqaH1oqzdaWgaaWcbaGaamyAai aadQgaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqa daqaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaO qaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYa aSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaey OaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaa caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaWGdbWa aSbaaSqaaiaadMgacaWGQbGaam4AaiaadYgaaeqaaOGaaiikaiabew 7aLnaaBaaaleaacaWGRbGaamiBaaqabaGccqGHsislcqaHXoqydaWg aaWcbaGaam4AaiaadYgaaeqaaOGaeuiLdqKaamivaiaacMcacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVpaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqa baaaaOGaey4kaSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyam aaBaaaleaacaWGQbaabeaakiabg2da9iaaicdaaeaacaWG1bWaaSba aSqaaiaadMgaaeqaaOGaeyypa0JaamyDamaaDaaaleaacaWGPbaaba GaaiOkaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaab+gacaqGUbGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHciITdaWgaaWcbaGaaGym aaqabaGccaWGsbGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaOGaamOBamaaBaaaleaacaWGPbaabeaakiabg2 da9iaadshadaqhaaWcbaGaamOAaaqaaiaacQcaaaGccaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caqGVbGaaeOBaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaeyOaIy7aaSbaaSqaaiaaikdaaeqaaOGaamOuaaaaaa@EA16@

 

Dynamic problems Dynamic problems are essentially identical, except that the boundary conditions must be specified as functions of time, and the initial displacement and velocity field must be specified.  In this case the governing equations are

ε ij = 1 2 ( u i x j + u j x i ) σ ij = C ijkl ( ε kl α kl ΔT) σ ij x i + ρ 0 b j = ρ 0 2 u j t 2 u i = u i * (t)on 1 R σ ij n i = t j * (t)on 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGceaqabeaacqaH1oqzdaWgaaWcbaGaamyAai aadQgaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqa daqaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaWGPbaabeaaaO qaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaey4kaSYa aSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadQgaaeqaaaGcbaGaey OaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaaakiaawIcacaGLPaaa caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiab g2da9iaadoeadaWgaaWcbaGaamyAaiaadQgacaWGRbGaamiBaaqaba GccaGGOaGaeqyTdu2aaSbaaSqaaiaadUgacaWGSbaabeaakiabgkHi Tiabeg7aHnaaBaaaleaacaWGRbGaamiBaaqabaGccqqHuoarcaWGub GaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVpaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqa baaaaOGaey4kaSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGaamOyam aaBaaaleaacaWGQbaabeaakiabg2da9iabeg8aYnaaBaaaleaacaaI WaaabeaakmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaam yDamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2kaadshadaahaaWc beqaaiaaikdaaaaaaaGcbaGaamyDamaaBaaaleaacaWGPbaabeaaki abg2da9iaadwhadaqhaaWcbaGaamyAaaqaaiaacQcaaaGccaGGOaGa amiDaiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caqGVbGaaeOBaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaeyOaIy7aaSbaaSqaaiaaig daaeqaaOGaamOuaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBa aaleaacaWGPbGaamOAaaqabaGccaWGUbWaaSbaaSqaaiaadMgaaeqa aOGaeyypa0JaamiDamaaDaaaleaacaWGQbaabaGaaiOkaaaakiaacI cacaWG0bGaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaab+gacaqGUb GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHciITdaWgaaWcbaGa aGOmaaqabaGccaWGsbaaaaa@FA15@

 

 

5.1.2 Alternative form of the governing equations MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E7@  the Navier equation

 

The governing equations can be simplified by eliminating stress and strain from the governing equations, and solving directly for the displacements.  In this case the linear momentum balance equation (in terms of displacement) reduces to

C ijkl x i ( u k x l α kl ΔT )+ ρ 0 b j = ρ 0 2 u j t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKaaalaadoeakm aaBaaajeaWbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGcdaWcaaqa aiabgkGi2cqaaiabgkGi2kaadIhadaWgaaWcbaGaamyAaaqabaaaaO WaaeWaaeaadaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaam4Aaaqa baaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadYgaaeqaaaaakiabgk HiTiabeg7aHnaaBaaajeaWbaGaam4AaiaadYgaaeqaaOGaeuiLdqKa amivaaGaayjkaiaawMcaaKaaalabgUcaRiabeg8aYPWaaSbaaSqaai aaicdaaeqaaKaaalaadkgakmaaBaaajeaWbaGaamOAaaqabaqcaaSa eyypa0JaeqyWdiNcdaWgaaWcbaGaaGimaaqabaGcdaWcaaqaaiabgk Gi2oaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaamOAaaqa baaakeaacqGHciITcaWG0bWaaWbaaSqabeaacaaIYaaaaaaaaaa@64E4@

For the special case of an isotropic solid with shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@322E@  and Poisson ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3230@  and uniform temperature ΔT=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam ivaiabg2da9iaaicdaaaa@39F5@  this equation reduces to

1 12ν 2 u k x k x i + 2 u i x k x k + ρ 0 b i μ = ρ 0 μ 2 u i t 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaaiaaykW6daWcaaqa aiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaam 4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGccqGHRaWkcaaMcS +aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSba aSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGRb aabeaakiabgkGi2kaadIhadaWgaaWcbaGaam4AaaqabaaaaOGaey4k aSIaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOWaaSaaaeaacaWGIbWaaS baaSqaaiaadMgaaeqaaaGcbaGaeqiVd0gaaiabg2da9maalaaabaGa eqyWdi3aaSbaaSqaaiaaicdaaeqaaaGcbaGaeqiVd0gaamaalaaaba GaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamyDamaaBaaaleaacaWG PbaabeaaaOqaaiabgkGi2kaadshadaahaaWcbeqaaiaaikdaaaaaaa aa@69BE@

These are known as the Navier (or Cauchy-Navier) equations of elasticity.

 

The boundary conditions remain as given in the preceding section.

 

 

5.1.3 Superposition and linearity of solutions

 

The governing equations of elasticity are linear.  This has two important consequences:

1.      The stresses, strains and displacements in a solid are directly proportional to the loads (or displacements) applied to the solid. 

2.      If you can find two sets of displacements, strains and stresses that satisfy the governing equations, you can add them to create more solutions.

 

These principles can be illustrated clearly using some of the simple solutions derived in Section 4.1.  For example, examine the solution to the pressurized sphere (Sect 4.1.4).    As an example, the radial stress induced by pressure p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@  on the interior, and zero pressure on the exterior surface is

σ rr = ( p a a 3 ) ( b 3 a 3 ) ( 1 b 3 r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0ZaaSaaaeaadaqadaqaaiaadchadaWgaaWcbaGa amyyaaqabaGccaWGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaay zkaaaabaWaaeWaaeaacaWGIbWaaWbaaSqabeaacaaIZaaaaOGaeyOe I0IaamyyamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaaaaada qadaqaaiaaigdacqGHsisldaWcaaqaaiaadkgadaahaaWcbeqaaiaa iodaaaaakeaacaWGYbWaaWbaaSqabeaacaaIYaaaaaaaaOGaayjkai aawMcaaiaaykW7caaMc8oaaa@4D7E@     

The radial stress induced by pressure p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadkgaaeqaaa aa@34CE@  on the exterior surface, with zero pressure on the interior surface is

σ rr = p b b 3 ( b 3 a 3 ) ( 1 a 3 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacaWGWbWaaSbaaSqaaiaa dkgaaeqaaOGaamOyamaaCaaaleqabaGaaG4maaaaaOqaamaabmaaba GaamOyamaaCaaaleqabaGaaG4maaaakiabgkHiTiaadggadaahaaWc beqaaiaaiodaaaaakiaawIcacaGLPaaaaaWaaeWaaeaacaaIXaGaey OeI0YaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIZaaaaaGcbaGaamOC amaaCaaaleqabaGaaG4maaaaaaaakiaawIcacaGLPaaacaaMc8oaaa@4B59@

Note that in both cases the stress is directly proportional to the pressure.  In addition, to find the radial stress by combined pressures p a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadggaaeqaaa aa@34CD@  on the interior and p b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbWaaSbaaSqaaiaadkgaaeqaaa aa@34CE@  on the exterior surface, you can just add these two solutions.

 

Further examples of superposition and linearity will be given in subsequent sections.

 

 

 

5.1.4 Uniqueness and existence of solutions to the linear elasticity equations

 

The following results are useful:

1.      If only displacements are prescribed on the boundary of the solid, the governing equations of linear elasticity always have a solution, and the solution is unique.

2.      If mixed boundary conditions are specified, a static solution exists and is unique if the displacements constrain rigid motions. A dynamic solution always exists and is unique, provided the velocity field and displacement field at time t=0 are known.

3.      If only tractions are prescribed on the boundary, a static solution exists only if the tractions are in equilibrium.  In this case, the stresses and strains are unique, but the displacements are not.  A dynamic solution always exists and is unique, again, providing initial conditions are known.

 

 

 

 5.1.5 Saint-Venant’s principle

 

Saint-Venant’s principle is often invoked to justify approximate solutions to boundary value problems in linear elasticity.  For example, when we solve problems involving bending or axial deformation of slender beams and rods in elementary strength of materials courses, we only specify the resultant forces acting on the ends of a rod, or the magnitudes of point forces acting on a beam, we don’t specify the distribution of traction in detail.  We rely on Saint Venant’s principle to justify this approach. In this context, the principle states the following.

 

The stresses, strains and displacements far from the ends of a rod or beam subjected to end loading depend only on the resultant forces and moments acting on its ends, and do not depend on how the tractions themselves are distributed.

 

Although SVP is widely used, it turns out to be remarkably difficult to prove mathematically.   The difficulty is partly that it is not easy to state the principle itself precisely enough to apply any mathematical machinery to it.  A rigorous statement is given by Sternberg (Q. J. Appl. Mech 11 p. 393 1954), among several other versions.  Here, we will just illustrate the most common applications of the principle through specific examples.

 

One version of SVP can be stated as follows.

 

Suppose that we calculate the stress, strain and displacement induced in a solid by two different traction distributions t (1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahshadaahaaWcbeqaaiaacIcacaaIXa Gaaiykaaaaaaa@33B6@  and t (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahshadaahaaWcbeqaaiaacIcacaaIYa Gaaiykaaaaaaa@33B7@  that act on some small region of a solid with characteristic size a.  If the tractions exert the same resultant force and moment, then the stresses, strains and displacements induced by the two traction distributions at a distance r from the loaded region are identical for large r/a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacaGGVaGaamyyaaaa@3308@ . 

 

In practice `large’ usually means r/a>3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaai4laiaadggacqGH+aGpca aIZaaaaa@372B@ .

 

This principle can be illustrated using a simple example.

 

Consider a large solid with a flat surface, as shown in the picture.  It is possible to calculate formulas for the stresses and displacements induced by various pressure distributions acting on the flat surface MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the procedure to do this will be outlined later.  For now, we will compare the stresses induced by

1.      A uniform pressure p(r)=P/( π a 2 )ra MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacaGGOaGaamOCaiaacMcacqGH9a qpcaWGqbGaai4lamaabmaabaGaeqiWdaNaamyyamaaCaaaleqabaGa aGOmaaaaaOGaayjkaiaawMcaaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caWGYbGaeyizImQaamyyaaaa@52FB@

2.      A parabolic pressure  p(r)= 3P 2π a 2 (1 r 2 / a 2 ) 1/2 ra MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadchacaGGOa GaamOCaiaacMcacqGH9aqpdaWcaaqaaiaaiodacaWGqbaabaGaaGOm aiabec8aWjaaykW6caWGHbWaaWbaaSqabeaacaaIYaaaaaaakiaacI cacaaIXaGaeyOeI0IaamOCamaaCaaaleqabaGaaGOmaaaakiaac+ca caWGHbWaaWbaaSqabeaacaaIYaaaaOGaaiykamaaCaaaleqabaGaaG ymaiaac+cacaaIYaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caWGYbGaeyizImQaamyyaaaa@55EE@

You can verify for yourself that both pressure distributions exert a resultant force P acting in the vertical direction on the surface, and exert zero moment about the origin.  The variation of stress down the axis of symmetry ( r= x 1 2 + x 2 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacqGH9aqpdaGcaaqaaiaadIhada qhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkcaWG4bWaa0baaSqa aiaaikdaaeaacaaIYaaaaaqabaGccqGH9aqpcaaIWaaaaa@3A7E@  ), expressed in cylindrical-polar coordinates, can be derived as

 

 

Case 1: Uniform pressure

σ zz = P π a 2 ( 1 z 3 ( a 2 + z 2 ) 3/2 ) σ rr = σ θθ = P π a 2 ( 1+2ν 2 (1+ν)z ( a 2 + z 2 ) 1/2 + z 3 2 ( a 2 + z 2 ) 3/2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaWG6bGaamOEaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaa dcfaaeaacqaHapaCcaWGHbWaaWbaaSqabeaacaaIYaaaaaaakmaabm aabaGaaGymaiabgkHiTmaalaaabaGaamOEamaaDaaaleaaaeaacaaI ZaaaaaGcbaGaaiikaiaadggadaahaaWcbeqaaiaaikdaaaGccqGHRa WkcaWG6bWaaWbaaSqabeaacaaIYaaaaOGaaiykamaaCaaaleqabaGa aG4maiaac+cacaaIYaaaaaaaaOGaayjkaiaawMcaaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8a ZnaaBaaaleaacaWGYbGaamOCaaqabaGccqGH9aqpcqaHdpWCdaWgaa WcbaGaeqiUdeNaeqiUdehabeaakiabg2da9iabgkHiTmaalaaabaGa amiuaaqaaiabec8aWjaadggadaahaaWcbeqaaiaaikdaaaaaaOWaae WaaeaadaWcaaqaaiaaigdacqGHRaWkcaaIYaGaeqyVd4gabaGaaGOm aaaacqGHsisldaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Maai ykaiaadQhaaeaacaGGOaGaamyyamaaCaaaleqabaGaaGOmaaaakiab gUcaRiaadQhadaahaaWcbeqaaiaaikdaaaGccaGGPaWaaWbaaSqabe aacaaIXaGaai4laiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaWG6bWa aWbaaSqabeaacaaIZaaaaaGcbaGaaGOmaiaacIcacaWGHbWaaWbaaS qabeaacaaIYaaaaOGaey4kaSIaamOEamaaCaaaleqabaGaaGOmaaaa kiaacMcadaahaaWcbeqaaiaaiodacaGGVaGaaGOmaaaaaaaakiaawI cacaGLPaaaaaa@8D2D@

Case 2: Parabolic pressure

σ zz = 3P 2π a 2 a 2 ( a 2 + z 2 ) σ rr = σ θθ = 3P 2π a 2 { (1+ν)( 1 z a tan -1 a z ) 1 2 a 2 a 2 + z 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa aaleaacaWG6bGaamOEaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaa iodacaWGqbaabaGaaGOmaiabec8aWjaadggadaahaaWcbeqaaiaaik daaaaaaOWaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaWa aeWaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOEam aaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaaaacaaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaamOCaiaadkhaaeqaaOGa eyypa0Jaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqabaGccqGH9a qpcqGHsisldaWcaaqaaiaaiodacaWGqbaabaGaaGOmaiabec8aWjaa dggadaahaaWcbeqaaiaaikdaaaaaaOWaaiWaaeaacaGGOaGaaGymai abgUcaRiabe27aUjaacMcadaqadaqaaiaaigdacqGHsisldaWcaaqa aiaadQhaaeaacaWGHbaaaiaabshacaqGHbGaaeOBamaaCaaaleqaba GaaeylaiaabgdaaaGcdaWcaaqaaiaadggaaeaacaWG6baaaaGaayjk aiaawMcaaiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaWaaSaaae aacaWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaGaamyyamaaCaaaleqa baGaaGOmaaaakiabgUcaRiaadQhadaahaaWcbeqaaiaaikdaaaaaaa GccaGL7bGaayzFaaaaaa@892C@

Now, to demonstrate SVP, we want to show that the stresses are equal for large z/a.  We can do this graphically MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the figures below compare the variation of vertical and radial stress down the axis of symmetry with z/a.

                      

 

The stresses induced by the two different pressures are clearly indistinguishable for z/a>3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6bGaai4laiaadggacqGH+aGpca aIZaaaaa@3733@ .  This example helps to quantify what we mean by a `large’ distance.

 

The second commonly used application of SVP is a rather vague statement that

 

 A localized geometrical feature with characteristic size R in a large solid only influences the stress in a region with size approximately 3R surrounding the feature.

 

This is more a rule of thumb than a precise mathematical statement.  It can be illustrated by looking at specific solutions.  For example, the figure below shows the Mises stress contours surrounding a circular hole in a thin rectangular plate that is subjected to extensional loading (calculated using the finite element method). Far from the hole, the stress is uniform.  The contours deviate from the uniform solution in a region that is about three times the hole radius.