Chapter 3

 

Constitutive Models MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  Relations between Stress and Strain

 

 

 

 

3.2 Linear elastic material behavior

 

You are probably familiar with the behavior of a linear elastic material from introductory materials courses.

 

3.2.1 Isotropic, linear elastic material behavior

 

If you conduct a uniaxial tensile test on almost any material, and keep the stress levels sufficiently low, you will observe the following behavior:

 The specimen deforms reversibly:  If you remove the loads, the solid returns to its original shape.

 The strain in the specimen depends only on the stress applied to it MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it doesn’t depend on the rate of loading, or the history of loading.

 For most materials, the stress is a linear function of strain, as shown in the picture above.  Because the strains are small, this is true whatever stress measure is adopted (Cauchy stress or nominal stress), and is true whatever strain measure is adopted (Lagrange strain or infinitesimal strain).

 For most, but not all, materials, the material has no characteristic orientation.  Thus, if you cut a tensile specimen out of a block of material, as shown in the figure, the the stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain curve will be independent of the orientation of the specimen relative to the block of material.  Such materials are said to be isotropic.

 If you heat a specimen of the material, increasing its temperature uniformly, it will generally change its shape slightly.  If the material is isotropic (no preferred material orientation) and homogeneous, then the specimen will simply increase in size, without shape change.

 

 

 

3.2.2 Stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E8@ strain relations for isotropic, linear elastic materials. Young’s Modulus, Poissons ratio and the Thermal Expansion Coefficient.

 

Before writing down stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain relations, we need to decide what strain and stress measures we want to use.  Because the model only works for small shape changes

 Deformation is characterized using the infinitesimal strain tensor ε ij =( u i / x j + u j / x i )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaqadaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyA aaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaki abgUcaRiabgkGi2kaadwhadaWgaaWcbaGaamOAaaqabaGccaGGVaGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaai aac+cacaaIYaaaaa@4880@  defined in Section 2.1.7.  This is convenient for calculations, but has the disadvantage that linear elastic constitutive equations can only be used if the solid experiences small rotations, as well as small shape changes. 

 All stress measures are taken to be equal.  We can use the Cauchy stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  as the stress measure.

 

You probably already know the stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain relations for an isotropic, linear elastic solid.  They are repeated below for convenience.

[ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ]= 1 E [ 1 ν ν 0 0 0 ν 1 ν 0 0 0 ν ν 1 0 0 0 0 0 0 2( 1+ν ) 0 0 0 0 0 0 2( 1+ν ) 0 0 0 0 0 0 2( 1+ν ) ][ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ]+αΔT[ 1 1 1 0 0 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabyqaaaaabaGaeqyTdu 2aaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiabew7aLnaaBaaaleaa caaIYaGaaGOmaaqabaaakeaacqaH1oqzdaWgaaWcbaGaaG4maiaaio daaeqaaaGcbaGaaGOmaiabew7aLnaaBaaaleaacaaIYaGaaG4maaqa baaakeaacaaIYaGaeqyTdu2aaSbaaSqaaiaaigdacaaIZaaabeaaaO qaaiaaikdacqaH1oqzdaWgaaWcbaGaaGymaiaaikdaaeqaaaaaaOGa ay5waiaaw2faaiabg2da9maalaaabaGaaGymaaqaaiaadweaaaWaam WaaeaafaqabeGbgaaaaaqaaiaaigdaaeaacqGHsislcqaH9oGBaeaa cqGHsislcqaH9oGBaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacq GHsislcqaH9oGBaeaacaaIXaaabaGaeyOeI0IaeqyVd4gabaGaaGim aaqaaiaaicdaaeaacaaIWaaabaGaeyOeI0IaeqyVd4gabaGaeyOeI0 IaeqyVd4gabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqa aiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaikdadaqadaqaaiaaig dacqGHRaWkcqaH9oGBaiaawIcacaGLPaaaaeaacaaIWaaabaGaaGim aaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYa WaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaaabaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaca aIWaaabaGaaGOmamaabmaabaGaaGymaiabgUcaRiabe27aUbGaayjk aiaawMcaaaaaaiaawUfacaGLDbaadaWadaqaauaabeqageaaaaqaai abeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakeaacqaHdpWCdaWg aaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaio dacaaIZaaabeaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaG4maaqa baaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaeq 4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaaaaaakiaawUfacaGLDbaa cqGHRaWkcqaHXoqycqqHuoarcaWGubWaamWaaeaafaqabeGbbaaaae aacaaIXaaabaGaaGymaaqaaiaaigdaaeaacaaIWaaabaGaaGimaaqa aiaaicdaaaaacaGLBbGaayzxaaaaaa@A385@

Here, E and ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@322F@  are Young’s modulus and Poisson’s ratio, α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3216@  is the coefficient of thermal expansion, and ΔT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadsfaaaa@32A6@  is the increase in temperature of the solid.  The remaining relations can be deduced from the fact that both σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3433@  and ε ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3417@  are symmetric. 

 

The inverse relationship can be expressed as

[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ]= E (1+ν)(12ν) [ 1ν ν ν 0 0 0 ν 1ν ν 0 0 0 ν ν 1ν 0 0 0 0 0 0 ( 12ν ) 2 0 0 0 0 0 0 ( 12ν ) 2 0 0 0 0 0 0 ( 12ν ) 2 ][ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ] EαΔT 12ν [ 1 1 1 0 0 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabyqaaaaabaGaeq4Wdm 3aaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiabeo8aZnaaBaaaleaa caaIYaGaaGOmaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaG4maiaaio daaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIZaaabeaaaOqa aiabeo8aZnaaBaaaleaacaaIXaGaaG4maaqabaaakeaacqaHdpWCda WgaaWcbaGaaGymaiaaikdaaeqaaaaaaOGaay5waiaaw2faaiabg2da 9maalaaabaGaamyraaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Maai ykaiaacIcacaaIXaGaeyOeI0IaaGOmaiabe27aUjaacMcaaaWaamWa aeaafaqabeGbgaaaaaqaaiaaigdacqGHsislcqaH9oGBaeaacqaH9o GBaeaacqaH9oGBaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacqaH 9oGBaeaacaaIXaGaeyOeI0IaeqyVd4gabaGaeqyVd4gabaGaaGimaa qaaiaaicdaaeaacaaIWaaabaGaeqyVd4gabaGaeqyVd4gabaGaaGym aiabgkHiTiabe27aUbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaamaalaaabaWaaeWaaeaacaaI XaGaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMcaaaqaaiaaikdaaa aabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaWaaSaaaeaadaqadaqaaiaaigdacqGHsislcaaIYa GaeqyVd4gacaGLOaGaayzkaaaabaGaaGOmaaaaaeaacaaIWaaabaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaada WcaaqaamaabmaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIca caGLPaaaaeaacaaIYaaaaaaaaiaawUfacaGLDbaadaWadaqaauaabe qageaaaaqaaiabew7aLnaaBaaaleaacaaIXaGaaGymaaqabaaakeaa cqaH1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeqyTdu2aaS baaSqaaiaaiodacaaIZaaabeaaaOqaaiaaikdacqaH1oqzdaWgaaWc baGaaGOmaiaaiodaaeqaaaGcbaGaaGOmaiabew7aLnaaBaaaleaaca aIXaGaaG4maaqabaaakeaacaaIYaGaeqyTdu2aaSbaaSqaaiaaigda caaIYaaabeaaaaaakiaawUfacaGLDbaacqGHsisldaWcaaqaaiaadw eacqaHXoqycqqHuoarcaWGubaabaGaaGymaiabgkHiTiaaikdacqaH 9oGBaaWaamWaaeaafaqabeGbbaaaaeaacaaIXaaabaGaaGymaaqaai aaigdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaaaacaGLBbGaayzx aaaaaa@B6D4@

 

HEALTH WARNING: Note the factor of 2 in the strain vector.  Most texts, and most FEM codes use this factor of two, but not all.  In addition, shear strains and stresses are often listed in a different order in the strain and stress vectors.  For isotropic materials this makes no difference, but you need to be careful when listing material constants for anisotropic materials (see below).

 

We can write this expression in a much more convenient form using index notation.  Verify for yourself that the matrix expression above is equivalent to

ε ij = 1+ν E σ ij ν E σ kk δ ij +αΔT δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaaigdacqGHRaWkcqaH9oGBaeaacaWG fbaaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislda Wcaaqaaiabe27aUbqaaiaadweaaaGaeq4Wdm3aaSbaaSqaaiaadUga caWGRbaabeaakiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccq GHRaWkcqaHXoqycqqHuoarcaWGubGaeqiTdq2aaSbaaSqaaiaadMga caWGQbaabeaaaaa@50AA@

 

The inverse relation is

σ ij = E 1+ν { ε ij + ν 12ν ε kk δ ij } EαΔT 12ν δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaey4kaSIaeqyV d4gaamaacmaabaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaaki abgUcaRmaalaaabaGaeqyVd4gabaGaaGymaiabgkHiTiaaikdacqaH 9oGBaaGaeqyTdu2aaSbaaSqaaiaadUgacaWGRbaabeaakiabes7aKn aaBaaaleaacaWGPbGaamOAaaqabaaakiaawUhacaGL9baacqGHsisl daWcaaqaaiaadweacqaHXoqycqqHuoarcaWGubaabaGaaGymaiabgk HiTiaaikdacqaH9oGBaaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaa beaaaaa@5B07@

 

The stress-strain relations are often expressed using the elastic modulus tensor C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3778@  or the elastic compliance tensor S ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3788@  as

σ ij = C ijkl ( ε kl αΔT δ kl ) ε ij = S ijkl σ kl +αΔT δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGPbGaamOAaiaadUga caWGSbaabeaakmaabmaabaGaeqyTdu2aaSbaaSqaaiaadUgacaWGSb aabeaakiabgkHiTiabeg7aHjabfs5aejaadsfacqaH0oazdaWgaaWc baGaam4AaiaadYgaaeqaaaGccaGLOaGaayzkaaGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaeqyTdu2aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2 da9iaadofadaWgaaWcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaGc cqaHdpWCdaWgaaWcbaGaam4AaiaadYgaaeqaaOGaey4kaSIaeqySde MaeuiLdqKaamivaiabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaaa aa@7284@

In terms of elastic constants, C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3778@  and S ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGtbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3788@  are

C ijkl = E 2( 1+ν ) ( δ il δ jk + δ ik δ jl )+ Eν ( 1+ν )( 12ν ) δ ij δ kl S ijkl = 1+ν 2E ( δ il δ jk + δ ik δ jl ) ν E δ ij δ kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadoeadaWgaaWcbaGaamyAai aadQgacaWGRbGaamiBaaqabaGccqGH9aqpdaWcaaqaaiaadweaaeaa caaIYaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaa aaamaabmaabaGaeqiTdq2aaSbaaSqaaiaadMgacaWGSbaabeaakiab es7aKnaaBaaaleaacaWGQbGaam4AaaqabaGccqGHRaWkcqaH0oazda WgaaWcbaGaamyAaiaadUgaaeqaaOGaeqiTdq2aaSbaaSqaaiaadQga caWGSbaabeaaaOGaayjkaiaawMcaaiabgUcaRmaalaaabaGaamyrai abe27aUbqaamaabmaabaGaaGymaiabgUcaRiabe27aUbGaayjkaiaa wMcaamaabmaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaiaawIcaca GLPaaaaaGaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiabes7a KnaaBaaaleaacaWGRbGaamiBaaqabaaakeaacaWGtbWaaSbaaSqaai aadMgacaWGQbGaam4AaiaadYgaaeqaaOGaeyypa0ZaaSaaaeaacaaI XaGaey4kaSIaeqyVd4gabaGaaGOmaiaadweaaaWaaeWaaeaacqaH0o azdaWgaaWcbaGaamyAaiaadYgaaeqaaOGaeqiTdq2aaSbaaSqaaiaa dQgacaWGRbaabeaakiabgUcaRiabes7aKnaaBaaaleaacaWGPbGaam 4AaaqabaGccqaH0oazdaWgaaWcbaGaamOAaiaadYgaaeqaaaGccaGL OaGaayzkaaGaeyOeI0YaaSaaaeaacqaH9oGBaeaacaWGfbaaaiabes 7aKnaaBaaaleaacaWGPbGaamOAaaqabaGccqaH0oazdaWgaaWcbaGa am4AaiaadYgaaeqaaaaaaa@8CBA@

 

 

 

3.2.3 Reduced stress-strain equations for plane deformation of isotropic solids

 

For plane strain or plane stress deformations, some strain or stress components are always zero (by definition) so the stress-strain laws can be simplified. 

 

 

 For a plane strain deformation ε 33 = ε 23 = ε 13 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpcqaH1oqzdaWgaaWcbaGaaGOmaiaaiodaaeqaaOGa eyypa0JaeqyTdu2aaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9i aaicdaaaa@3E36@ .  The stress strain laws are therefore

[ ε 11 ε 22 2 ε 12 ]= (1+ν) E [ 1ν ν 0 ν 1ν 0 0 0 2 ][ σ 11 σ 22 σ 12 ]+( 1+ν )αΔT[ 1 1 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmqaaaqaaiabew7aLn aaBaaaleaacaaIXaGaaGymaaqabaaakeaacqaH1oqzdaWgaaWcbaGa aGOmaiaaikdaaeqaaaGcbaGaaGOmaiabew7aLnaaBaaaleaacaaIXa GaaGOmaaqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaacaGG OaGaaGymaiabgUcaRiabe27aUjaacMcaaeaacaWGfbaaamaadmaaba qbaeqabmWaaaqaaiaaigdacqGHsislcqaH9oGBaeaacqGHsislcqaH 9oGBaeaacaaIWaaabaGaeyOeI0IaeqyVd4gabaGaaGymaiabgkHiTi abe27aUbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaikdaaaaa caGLBbGaayzxaaWaamWaaeaafaqabeWabaaabaGaeq4Wdm3aaSbaaS qaaiaaigdacaaIXaaabeaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGa aGOmaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaikdaaeqaaa aaaOGaay5waiaaw2faaiabgUcaRmaabmaabaGaaGymaiabgUcaRiab e27aUbGaayjkaiaawMcaaiabeg7aHjabfs5aejaadsfadaWadaqaau aabeqadeaaaeaacaaIXaaabaGaaGymaaqaaiaaicdaaaaacaGLBbGa ayzxaaaaaa@6F86@

[ σ 11 σ 22 σ 12 ]= E (1+ν)(12ν) [ 1ν ν 0 ν 1ν 0 0 0 12ν 2 ][ ε 11 ε 22 2 ε 12 ] EαΔT 12ν [ 1 1 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmqaaaqaaiabeo8aZn aaBaaaleaacaaIXaGaaGymaaqabaaakeaacqaHdpWCdaWgaaWcbaGa aGOmaiaaikdaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYa aabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiaadweaaeaa caGGOaGaaGymaiabgUcaRiabe27aUjaacMcacaGGOaGaaGymaiabgk HiTiaaikdacqaH9oGBcaGGPaaaamaadmaabaqbaeqabmWaaaqaaiaa igdacqGHsislcqaH9oGBaeaacqaH9oGBaeaacaaIWaaabaGaeqyVd4 gabaGaaGymaiabgkHiTiabe27aUbqaaiaaicdaaeaacaaIWaaabaGa aGimaaqaamaalaaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaeaaca aIYaaaaaaaaiaawUfacaGLDbaadaWadaqaauaabeqadeaaaeaacqaH 1oqzdaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeqyTdu2aaSbaaS qaaiaaikdacaaIYaaabeaaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGa aGymaiaaikdaaeqaaaaaaOGaay5waiaaw2faaiabgkHiTmaalaaaba Gaamyraiabeg7aHjabfs5aejaadsfaaeaacaaIXaGaeyOeI0IaaGOm aiabe27aUbaadaWadaqaauaabeqadeaaaeaacaaIXaaabaGaaGymaa qaaiaaicdaaaaacaGLBbGaayzxaaaaaa@7770@

σ 33 = Eν( ε 11 + ε 22 ) ( 12ν )( 1+ν ) + EαΔT 12ν , σ 13 = σ 23 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpdaWcaaqaaiaadweacqaH9oGBdaqadaqaaiabew7a LnaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcqaH1oqzdaWgaa WcbaGaaGOmaiaaikdaaeqaaaGccaGLOaGaayzkaaaabaWaaeWaaeaa caaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMcaamaabmaaba GaaGymaiabgUcaRiabe27aUbGaayjkaiaawMcaaaaacqGHRaWkdaWc aaqaaiaadweacqaHXoqycqqHuoarcaWGubaabaGaaGymaiabgkHiTi aaikdacqaH9oGBaaGaaiilaiaaykW7caaMc8Uaeq4Wdm3aaSbaaSqa aiaaigdacaaIZaaabeaakiabg2da9iabeo8aZnaaBaaaleaacaaIYa GaaG4maaqabaGccqGH9aqpcaaIWaaaaa@622E@

In index notation

ε αβ = 1+ν E { σ αβ ν σ γγ δ αβ }+( 1+ν )αΔT δ αβ σ αβ = E 1+ν { ε αβ + ν 12ν ε γγ δ αβ } EαΔT 12ν δ αβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacqaHXoqycqaHYo GyaeqaaOGaeyypa0ZaaSaaaeaacaaIXaGaey4kaSIaeqyVd4gabaGa amyraaaadaGadaqaaiabeo8aZnaaBaaaleaacqaHXoqycqaHYoGyae qaaOGaeyOeI0IaeqyVd4Maeq4Wdm3aaSbaaSqaaiabeo7aNjabeo7a NbqabaGccqaH0oazdaWgaaWcbaGaeqySdeMaeqOSdigabeaaaOGaay 5Eaiaaw2haaiabgUcaRmaabmaabaGaaGymaiabgUcaRiabe27aUbGa ayjkaiaawMcaaiabeg7aHjabfs5aejaadsfacqaH0oazdaWgaaWcba GaeqySdeMaeqOSdigabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo 8aZnaaBaaaleaacqaHXoqycqaHYoGyaeqaaOGaeyypa0ZaaSaaaeaa caWGfbaabaGaaGymaiabgUcaRiabe27aUbaadaGadaqaaiabew7aLn aaBaaaleaacqaHXoqycqaHYoGyaeqaaOGaey4kaSYaaSaaaeaacqaH 9oGBaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaacqaH1oqzdaWgaa WcbaGaeq4SdCMaeq4SdCgabeaakiabes7aKnaaBaaaleaacqaHXoqy cqaHYoGyaeqaaaGccaGL7bGaayzFaaGaeyOeI0YaaSaaaeaacaWGfb GaeqySdeMaeuiLdqKaamivaaqaaiaaigdacqGHsislcaaIYaGaeqyV d4gaaiabes7aKnaaBaaaleaacqaHXoqycqaHYoGyaeqaaaaa@A201@

where Greek subscripts α,β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaacYcacqaHYoGyaaa@3458@  can have values 1 or 2.

 

 For a plane stress deformation σ 33 = σ 23 = σ 13 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGOmaiaaiodaaeqaaOGa eyypa0Jaeq4Wdm3aaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da9i aaicdaaaa@3E8A@

 

[ ε 11 ε 22 2 ε 12 ]= 1 E [ 1 ν 0 ν 1 0 0 0 2(1+ν) ][ σ 11 σ 22 σ 12 ]+αΔT[ 1 1 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmqaaaqaaiabew7aLn aaBaaaleaacaaIXaGaaGymaaqabaaakeaacqaH1oqzdaWgaaWcbaGa aGOmaiaaikdaaeqaaaGcbaGaaGOmaiabew7aLnaaBaaaleaacaaIXa GaaGOmaaqabaaaaaGccaGLBbGaayzxaaGaeyypa0ZaaSaaaeaacaaI XaaabaGaamyraaaadaWadaqaauaabeqadmaaaeaacaaIXaaabaGaey OeI0IaeqyVd4gabaGaaGimaaqaaiabgkHiTiabe27aUbqaaiaaigda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYaGaaiikaiaaig dacqGHRaWkcqaH9oGBcaGGPaaaaaGaay5waiaaw2faamaadmaabaqb aeqabmqaaaqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaake aacqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaeq4Wdm3a aSbaaSqaaiaaigdacaaIYaaabeaaaaaakiaawUfacaGLDbaacqGHRa WkcqaHXoqycqqHuoarcaWGubWaamWaaeaafaqabeWabaaabaGaaGym aaqaaiaaigdaaeaacaaIWaaaaaGaay5waiaaw2faaaaa@6619@

[ σ 11 σ 22 σ 12 ]= E (1 ν 2 ) [ 1 ν 0 ν 1 0 0 0 (1ν)/2 ][ ε 11 ε 22 2 ε 12 ] EαΔT ( 1ν ) [ 1 1 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaadmaabaqbaeqabmqaaaqaaiabeo8aZn aaBaaaleaacaaIXaGaaGymaaqabaaakeaacqaHdpWCdaWgaaWcbaGa aGOmaiaaikdaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYa aabeaaaaaakiaawUfacaGLDbaacqGH9aqpdaWcaaqaaiaadweaaeaa caGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaaki aacMcaaaWaamWaaeaafaqabeWadaaabaGaaGymaaqaaiabe27aUbqa aiaaicdaaeaacqaH9oGBaeaacaaIXaaabaGaaGimaaqaaiaaicdaae aacaaIWaaabaGaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaai4l aiaaikdaaaaacaGLBbGaayzxaaWaamWaaeaafaqabeWabaaabaGaeq yTdu2aaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiabew7aLnaaBaaa leaacaaIYaGaaGOmaaqabaaakeaacaaIYaGaeqyTdu2aaSbaaSqaai aaigdacaaIYaaabeaaaaaakiaawUfacaGLDbaacqGHsisldaWcaaqa aiaadweacqaHXoqycqqHuoarcaWGubaabaWaaeWaaeaacaaIXaGaey OeI0IaeqyVd4gacaGLOaGaayzkaaaaamaadmaabaqbaeqabmqaaaqa aiaaigdaaeaacaaIXaaabaGaaGimaaaaaiaawUfacaGLDbaaaaa@6FBC@

ε 33 = ν E ( σ 11 + σ 22 )+αΔT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaG4maiaaio daaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacqaH9oGBaeaacaWGfbaa amaabmaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiabgU caRiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakiaawIcacaGL PaaacqGHRaWkcqaHXoqycqqHuoarcaWGubaaaa@48AD@

ε αβ = 1+ν E ( σ αβ ν 1+ν σ γγ δ αβ )+αΔT δ αβ σ αβ = E 1+ν { ε αβ + ν 1ν ε γγ δ αβ } EαΔT 1ν δ αβ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacqaHXoqycqaHYo GyaeqaaOGaeyypa0ZaaSaaaeaacaaIXaGaey4kaSIaeqyVd4gabaGa amyraaaadaqadaqaaiabeo8aZnaaBaaaleaacqaHXoqycqaHYoGyae qaaOGaeyOeI0YaaSaaaeaacqaH9oGBaeaacaaIXaGaey4kaSIaeqyV d4gaaiabeo8aZnaaBaaaleaacqaHZoWzcqaHZoWzaeqaaOGaeqiTdq 2aaSbaaSqaaiabeg7aHjabek7aIbqabaaakiaawIcacaGLPaaacqGH RaWkcqaHXoqycqqHuoarcaWGubGaeqiTdq2aaSbaaSqaaiabeg7aHj abek7aIbqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm 3aaSbaaSqaaiabeg7aHjabek7aIbqabaGccqGH9aqpdaWcaaqaaiaa dweaaeaacaaIXaGaey4kaSIaeqyVd4gaamaacmaabaGaeqyTdu2aaS baaSqaaiabeg7aHjabek7aIbqabaGccqGHRaWkdaWcaaqaaiabe27a UbqaaiaaigdacqGHsislcqaH9oGBaaGaeqyTdu2aaSbaaSqaaiabeo 7aNjabeo7aNbqabaGccqaH0oazdaWgaaWcbaGaeqySdeMaeqOSdiga beaaaOGaay5Eaiaaw2haaiabgkHiTmaalaaabaGaamyraiabeg7aHj abfs5aejaadsfaaeaacaaIXaGaeyOeI0IaeqyVd4gaaiabes7aKnaa BaaaleaacqaHXoqycqaHYoGyaeqaaaaa@9FF3@

 

 

 

3.2.4 Representative values for density, and elastic constants of isotropic solids

 

Most of the data in the table below were taken from the excellent introductory text `Engineering Materials,’ by M.F. Ashby and D.R.H. Jones, Pergamon Press.  The remainder are from random web pages…

 

Note the units MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  values of E are given in GN/ m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiaad6 eacaGGVaGaamyBamaaCaaaleqabaGaaGOmaaaaaaa@3A24@ ; the G stands for Giga, and is short for 10 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dadaahaaWcbeqaaiaaiMdaaaaaaa@385C@ .  The units for density are in Mg m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiaadE gacaWGTbWaaWbaaSqabeaacqGHsislcaaIZaaaaaaa@3A7E@  - that’s Mega grams.  One mega gram is 1000 kg.

 

Material

Mass density

ρ/Mg m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdiNaaG PaVlaaykW7caGGVaGaaGPaVlaaykW7caaMc8UaamytaiaadEgacaWG TbWaaWbaaSqabeaacqGHsislcaaIZaaaaaaa@44A8@

Youngs Modulus

E/GN m 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiaayk W7caaMc8UaaGPaVlaac+cacaaMc8UaaGPaVlaaykW7caWGhbGaamOt aiaad2gadaahaaWcbeqaaiabgkHiTiaaikdaaaaaaa@451D@

Poisson Ratio
ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4gaaa@37AF@

Expansion coeft K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadUeadaahaaWcbeqaaiabgkHiTiaaig daaaaaaa@330D@

Tungsten Carbide

14 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  17

450 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@ 650

0.22

5× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiwdacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@368D@

Silicon Carbide

2.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  3.2

450

0.22

4× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaisdacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@368C@

Tungsten

13.4

410

0.30

4× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaisdacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@368C@

Alumina

3.9

390

0.25

7× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiEdacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@368F@

Titanium Carbide

4.9

380

0.19

13× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaIZaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3746@

Silicon Nitride

3.2

320 - 270

0.22

3× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiodacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@368B@

Nickel

8.9

215

0.31

14× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaI0aGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3747@

CFRP

1.5-1.6

70 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  200

0.20

2× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@368A@

Iron

7.9

196

0.30

13× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaIZaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3746@

Low alloy steels

7.8

200 - 210

0.30

15× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaI1aGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3748@

Stainless steel

7.5-7.7

190 - 200

0.30

11× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaIXaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3744@

Mild steel

7.8

196

0.30

15× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaI1aGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3748@

Copper

8.9

124

0.34

16× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaI2aGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3749@

Titanium

4.5

116

0.30

9× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiMdacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@3691@

Silicon

2.5-3.2

107

0.22

5× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiwdacqGHxdaTcaaIXaGaaGimamaaCa aaleqabaGaeyOeI0IaaGOnaaaaaaa@368D@

Silica glass

2.6

94

0.16

0.5× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaicdacaGGUaGaaGynaiabgEna0kaaig dacaaIWaWaaWbaaSqabeaacqGHsislcaaI2aaaaaaa@37F9@

Aluminum & alloys

2.6-2.9

69-79

0.35

22× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaaIYaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3746@

Concrete

2.4-2.5

45-50

0.3

10× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaIWaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3743@

GFRP

1.4-2.2

7-45

 

10× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaaIWaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3743@

Wood, parallel grain

0.4-0.8

9-16

0.2

40× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaisdacaaIWaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3746@

Polyimides

1.4

3-5

0.1-0.45

40× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaisdacaaIWaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3746@

Nylon

1.1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  1.2

2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  4

0.25

81× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiIdacaaIXaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@374B@

PMMA

1.2

3.4

0.35-0.4

50× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiwdacaaIWaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3747@

Polycarbonate

1.2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  1.3

2.6

0.36

65× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiAdacaaI1aGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@374D@

Natural Rubbers

0.83-0.91

0.01-0.1

0.49

200× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaaIWaGaaGimaiabgEna0kaaig dacaaIWaWaaWbaaSqabeaacqGHsislcaaI2aaaaaaa@37FE@

PVC

1.3-1.6

0.003-0.01

0.41

70× 10 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiEdacaaIWaGaey41aqRaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiAdaaaaaaa@3749@

 

 

 

3.2.5 Other Elastic Constants MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E7@  bulk, shear and Lame modulus.

 

Young’s modulus and Poisson’s ratio are the most common properties used to characterize elastic solids, but other measures are also used.  For example, we define the shear modulusbulk modulus and Lame modulus of an elastic solid as follows:

Bulk Modulus K= E 3( 12ν ) Shear Modulus μ= E 2( 1+ν ) Lame Modulus λ= νE ( 1+ν )( 12ν ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaeOqaiaabwhacaqGSbGaae4Aai aabccacaqGnbGaae4BaiaabsgacaqG1bGaaeiBaiaabwhacaqGZbGa aeiiaiaadUeacqGH9aqpdaWcaaqaaiaadweaaeaacaaIZaWaaeWaae aacaaIXaGaeyOeI0IaaGOmaiabe27aUbGaayjkaiaawMcaaaaaaeaa caqGtbGaaeiAaiaabwgacaqGHbGaaeOCaiaabccacaqGnbGaae4Bai aabsgacaqG1bGaaeiBaiaabwhacaqGZbGaaeiiaiabeY7aTjaab2da daWcaaqaaiaadweaaeaacaaIYaWaaeWaaeaacaaIXaGaey4kaSIaeq yVd4gacaGLOaGaayzkaaaaaaqaaiaabYeacaqGHbGaaeyBaiaabwga caqGGaGaaeytaiaab+gacaqGKbGaaeyDaiaabYgacaqG1bGaae4Cai aabccacqaH7oaBcqGH9aqpdaWcaaqaaiabe27aUjaadweaaeaadaqa daqaaiaaigdacqGHRaWkcqaH9oGBaiaawIcacaGLPaaadaqadaqaai aaigdacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaaaaaaa@752D@

 

A nice table relating all the possible combinations of moduli to all other possible combinations is given below.  Enjoy!

 

 

Lame

Modulus

λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSbaa@37ED@

Shear

Modulus

μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTbaa@37EF@

Young’s

Modulus

E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweaaaa@3703@

Poisson’s

Ratio

ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUbaa@37F1@

Bulk

Modulus

K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUeaaaa@3709@

λ,μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSjaacY cacqaH8oqBaaa@3A53@

 

 

μ(3λ+2μ) λ+μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeqiVd0Maaiikaiaaiodacq aH7oaBcqGHRaWkcaaIYaGaeqiVd0MaaiykaaqaaiabeU7aSjabgUca RiabeY7aTbaaaaa@3D97@

λ 2(λ+μ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeq4UdWgabaGaaGOmaiaacI cacqaH7oaBcqGHRaWkcqaH8oqBcaGGPaaaaaaa@388C@

3λ+2μ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiabeU7aSjabgUcaRi aaikdacqaH8oqBaeaacaaIZaaaaaaa@36F9@

λ,E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSjaacY cacaWGfbaaaa@3967@

 

Irrational

 

Irrational

Irrational

λ,ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSjaacY cacqaH9oGBaaa@3A55@

 

λ(12ν) 2ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeq4UdWMaaiikaiaaigdacq GHsislcaaIYaGaeqyVd4MaaiykaaqaaiaaikdacqaH9oGBaaaaaa@3A14@

λ(1+ν)(12ν) ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeq4UdWMaaiikaiaaigdacq GHRaWkcqaH9oGBcaGGPaGaaiikaiaaigdacqGHsislcaaIYaGaeqyV d4Maaiykaaqaaiabe27aUbaaaaa@3E06@

 

λ(1+ν) 3ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeq4UdWMaaiikaiaaigdacq GHRaWkcqaH9oGBcaGGPaaabaGaaG4maiabe27aUbaaaaa@394E@

λ,K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeU7aSjaacY cacaaMc8Uaam4saaaa@3AF8@

 

3(Kλ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaacIcacaWGlbGaey OeI0Iaeq4UdWMaaiykaaqaaiaaikdaaaaaaa@36BA@

9K(Kλ) 3Kλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGyoaiaadUeacaGGOaGaam 4saiabgkHiTiabeU7aSjaacMcaaeaacaaIZaGaam4saiabgkHiTiab eU7aSbaaaaa@3B02@

λ 3Kλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeq4UdWgabaGaaG4maiaadU eacqGHsislcqaH7oaBaaaaaa@3659@

 

μ,E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjaacY cacaaMc8UaaGPaVlaadweaaaa@3C7F@

μ(2μE) E3μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeqiVd0Maaiikaiaaikdacq aH8oqBcqGHsislcaWGfbGaaiykaaqaaiaadweacqGHsislcaaIZaGa eqiVd0gaaaaa@3BD9@

 

 

E2μ 2μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamyraiabgkHiTiaaikdacq aH8oqBaeaacaaIYaGaeqiVd0gaaaaa@3712@

μE 3(3μE) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeqiVd0Maamyraaqaaiaaio dacaGGOaGaaG4maiabeY7aTjabgkHiTiaadweacaGGPaaaaaaa@3937@

μ,ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjaacY cacaaMc8UaaGPaVlabe27aUbaa@3D6D@

2μν 12ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGOmaiabeY7aTjabe27aUb qaaiaaigdacqGHsislcaaIYaGaeqyVd4gaaaaa@38BD@

 

2μ(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacqaH8oqBcaGGOaGaaGymaiabgU caRiabe27aUjaacMcaaaa@3787@

 

2μ(1+ν) 3(12ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGOmaiabeY7aTjaacIcaca aIXaGaey4kaSIaeqyVd4MaaiykaaqaaiaaiodacaGGOaGaaGymaiab gkHiTiaaikdacqaH9oGBcaGGPaaaaaaa@3DC9@

μ,K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTjaacY cacaaMc8Uaam4saaaa@3AFA@

3K2μ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaadUeacqGHsislca aIYaGaeqiVd0gabaGaaG4maaaaaaa@3620@

 

9Kμ 3K+μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGyoaiaadUeacqaH8oqBae aacaaIZaGaam4saiabgUcaRiabeY7aTbaaaaa@37E5@

3K2μ 2(3K+μ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaadUeacqGHsislca aIYaGaeqiVd0gabaGaaGOmaiaacIcacaaIZaGaam4saiabgUcaRiab eY7aTjaacMcaaaaaaa@3B9D@

 

E,ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweacaGGSa GaaGPaVlaaykW7cqaH9oGBaaa@3C81@

νE (1+ν)(12ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeqyVd4MaaGPaVlaadweaae aacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcacaGGOaGaaGymaiab gkHiTiaaikdacqaH9oGBcaGGPaaaaaaa@3EA7@

E 2(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamyraaqaaiaaikdacaGGOa GaaGymaiabgUcaRiabe27aUjaacMcaaaaaaa@36AB@

 

 

E 3(12ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamyraaqaaiaaiodacaGGOa GaaGymaiabgkHiTiaaikdacqaH9oGBcaGGPaaaaaaa@3773@

E,K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweacaGGSa GaaGPaVlaadUeaaaa@3A0E@

3K(3KE) 9KE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaadUeacaGGOaGaaG 4maiaadUeacqGHsislcaWGfbGaaiykaaqaaiaaiMdacaWGlbGaeyOe I0Iaamyraaaaaaa@39EB@

3EK 9KE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaadweacaWGlbaaba GaaGyoaiaadUeacqGHsislcaWGfbaaaaaa@3618@

 

3KE 6K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaadUeacqGHsislca WGfbaabaGaaGOnaiaadUeaaaaaaa@354B@

 

ν,K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUjaacY cacaaMc8UaaGPaVlaadUeaaaa@3C87@

3Kν (1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaadUeacqaH9oGBae aacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaaaaaa@386A@

3K(12ν) 2(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaG4maiaadUeacaGGOaGaaG ymaiabgkHiTiaaikdacqaH9oGBcaGGPaaabaGaaGOmaiaacIcacaaI XaGaey4kaSIaeqyVd4Maaiykaaaaaaa@3CE3@

3K(12ν) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiodacaWGlbGaaiikaiaaigdacqGHsi slcaaIYaGaeqyVd4Maaiykaaaa@3769@

 

 

 

 

 

3.2.6 Physical Interpretation of elastic constants for isotropic solids

 

It is important to have a feel for the physical significance of the two elastic constants E and ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@322F@ .

 

  Young’s modulus E is the slope of the stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain curve in uniaxial tension.  It has dimensions of stress ( N/ m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6eacaGGVaGaamyBamaaCaaaleqaba GaaGOmaaaaaaa@33C8@  ) and is usually large MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for steel, E=210× 10 9 N/m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacqGH9aqpcaaIYaGaaGymaiaaic dacqGHxdaTcaaIXaGaaGimamaaCaaaleqabaGaaGyoaaaakiaaykW7 caqGobGaae4laiaab2gadaahaaWcbeqaaiaabkdaaaaaaa@3DCE@ . You can think of E as a measure of the stiffness of the solid. The larger the value of E, the stiffer the solid.  For a stable material, E>0.

 

 Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUbaa@37F1@  is the ratio of lateral to longitudinal strain in uniaxial tensile stress. It is dimensionless and typically ranges from 0.2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ 0.49, and is around 0.3 for most metals.  For a stable material, 1<ν<0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaaigdacqGH8aapcqaH9oGBcq GH8aapcaaIWaGaaiOlaiaaiwdaaaa@37FA@ . It is a measure of the compressibility of the solid.  If ν=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUjabg2da9iaaicdacaGGUaGaaG ynaaaa@3550@ , the solid is incompressible MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  its volume remains constant, no matter how it is deformed.  If ν=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUjabg2da9iaaicdaaaa@33EF@ , then stretching a specimen causes no lateral contraction.  Some bizarre materials have ν<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUjabgYda8iaaicdaaaa@33ED@  --  if you stretch a round bar of such a material, the bar increases in diameter!!

 

 Thermal expansion coefficient quantifies the change in volume of a material if it is heated in the absence of stress.  It has dimensions of (degrees Kelvin)-1 and is usually very small.  For steel, α610× 10 6 K -1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjabgIKi7kaaiAdacqGHsislca aIXaGaaGimaiabgEna0kaaigdacaaIWaWaaWbaaSqabeaacqGHsisl caaI2aaaaOGaaGPaVlaabUeadaahaaWcbeqaaiaab2cacaqGXaaaaa aa@4033@

 

 The bulk modulus quantifies the resistance of the solid to volume changes.  It has a large value (usually bigger than E).

 

 The shear modulus quantifies its resistance to volume preserving shear deformations.  Its value is usually somewhat smaller than E

 

 

3.2.7 Strain Energy Density for Isotropic Solids

 

Note the following observations

 If you deform a block of material, you do work on it (or, in some cases, it may do work on you…) 

 In an elastic material, the work done during loading is stored as recoverable strain energy in the solid.  If you unload the material, the specimen does work on you, and when it reaches its initial configuration you come out even.

 The work done to deform a specimen depends only on the state of strain at the end of the test.  It is independent of the history of loading. 

 

Based on these observations, we define the strain energy density of a solid as the work done per unit volume to deform a material from a stress free reference state to a loaded state.

 

To write down an expression for the strain energy density, it is convenient to separate the strain into two parts

ε ij = ε ij e + ε ij T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWG LbaaaOGaey4kaSIaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaam ivaaaaaaa@3F38@

where, for an isotropic solid,

ε ij T =αΔT δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadsfaaaGccqGH9aqpcqaHXoqycqqHuoarcaWGubGaeqiTdq2a aSbaaSqaaiaadMgacaWGQbaabeaaaaa@3D8D@

represents the strain due to thermal expansion (known as thermal strain), and

ε ij e = 1+ν E σ ij ν E σ kk δ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadwgaaaGccqGH9aqpdaWcaaqaaiaaigdacqGHRaWkcqaH9oGB aeaacaWGfbaaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccq GHsisldaWcaaqaaiabe27aUbqaaiaadweaaaGaeq4Wdm3aaSbaaSqa aiaadUgacaWGRbaabeaakiabes7aKnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@491D@

is the strain due to mechanical loading (known as elastic strain).

 

Work is done on the specimen only during mechanical loading.  It is straightforward to show that the strain energy density is

U= 1 2 σ ij ε ij e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpdaWcaaqaaiaaigdaae aacaaIYaaaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccqaH 1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWGLbaaaaaa@3C3F@

You can also re-write this as

U= 1+ν 2E σ ij σ ij ν 2E σ kk σ jj U= E 2( 1+ν ) ε ij e ε ij e + Eν 2( 1+ν )( 12ν ) ε jj e ε kk e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyvaiabg2da9maalaaabaGaaG ymaiabgUcaRiabe27aUbqaaiaaikdacaWGfbaaaiabeo8aZnaaBaaa leaacaWGPbGaamOAaaqabaGccqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyOeI0YaaSaaaeaacqaH9oGBaeaacaaIYaGaamyraaaa cqaHdpWCdaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeq4Wdm3aaSbaaS qaaiaadQgacaWGQbaabeaaaOqaaiaadwfacqGH9aqpdaWcaaqaaiaa dweaaeaacaaIYaWaaeWaaeaacaaIXaGaey4kaSIaeqyVd4gacaGLOa Gaayzkaaaaaiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaadwga aaGccqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWGLbaaaOGaey 4kaSYaaSaaaeaacaWGfbGaeqyVd4gabaGaaGOmamaabmaabaGaaGym aiabgUcaRiabe27aUbGaayjkaiaawMcaamaabmaabaGaaGymaiabgk HiTiaaikdacqaH9oGBaiaawIcacaGLPaaaaaGaeqyTdu2aa0baaSqa aiaadQgacaWGQbaabaGaamyzaaaakiabew7aLnaaDaaaleaacaWGRb Gaam4Aaaqaaiaadwgaaaaaaaa@7465@

Observe that

ε ij e = U σ ij σ ij = U ε ij e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaDaaaleaacaWGPbGaamOAaa qaaiaadwgaaaGccqGH9aqpdaWcaaqaaiabgkGi2kaadwfaaeaacqGH ciITcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaaaakiaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq 4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9maalaaabaGa eyOaIyRaamyvaaqaaiabgkGi2kabew7aLnaaDaaaleaacaWGPbGaam OAaaqaaiaadwgaaaaaaaaa@6506@

 

 

 

 

3.2.8 Stress-strain relation for a general anisotropic linear elastic material MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E7@  the elastic stiffness and compliance tensors

 

The simple isotropic model described in the preceding section is unable to describe the response of some materials accurately, even though the material may deform elastically.  This is because some materials do have a characteristic orientation.  For example, in a block of wood, the grain is oriented in a particular direction in the specimen.  The block will be stiffer if it is loaded parallel to the grain than if it is loaded perpendicular to the grain.  The same observation applies to fiber reinforced composite materials. Generally, single crystal specimens of a material will also be anisotropic MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this is important when modeling stress effects in small structures such as microelectronic circuits. Even polycrystalline metals may be anisotropic, because a preferred texture may form in the specimen during manufacture.

 

A more general stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain relation is needed to describe anisotropic solids. 

 

The most general linear stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFuacaaa@37E7@ strain relation has the form

σ ij = C ijkl ( ε kl α kl ΔT ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadMgacaWGQbGaam4Aaiaa dYgaaeqaaOWaaeWaaeaacqaH1oqzdaWgaaWcbaGaam4AaiaadYgaae qaaOGaeyOeI0IaeqySde2aaSbaaSqaaiaadUgacaWGSbaabeaakiab fs5aejaadsfaaiaawIcacaGLPaaaaaa@4638@

Here, C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaaaaa@3529@  is a fourth order tensor (horrors!), known as the elastic stiffness tensor, and α kl = α lk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqydaWgaaWcbaGaam4AaiaadY gaaeqaaOGaeyypa0JaeqySde2aaSbaaSqaaiaadYgacaWGRbaabeaa aaa@3B3E@  is the thermal expansion coefficient tensor. The stress strain relation is invertible:

ε ij = S ijkl σ kl + α ij ΔT MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcaWGtbWaaSbaaSqaaiaadMgacaWGQbGaam4Aaiaa dYgaaeqaaOGaeq4Wdm3aaSbaaSqaaiaadUgacaWGSbaabeaakiabgU caRiabeg7aHnaaBaaaleaacaWGPbGaamOAaaqabaGccqqHuoarcaWG ubaaaa@44B0@

where S ijkl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadofadaWgaa WcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaaaaa@3AFB@  is known as the elastic compliance tensor

 

At first sight it appears that the stiffness tensor has 81 components.  Imagine having to measure and keep track of 81 material properties!  Fortunately, C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaaaaa@3529@  must have the following symmetries

C ijkl = C klij = C jikl = C ijlk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadUgacaWG SbGaamyAaiaadQgaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGQb GaamyAaiaadUgacaWGSbaabeaakiabg2da9iaadoeadaWgaaWcbaGa amyAaiaadQgacaWGSbGaam4Aaaqabaaaaa@466F@

This reduces the number of material constants to 21. The compliance tensor has the same symmetries as C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaa WcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaaaaa@3AEB@ .

 

To see the origin of the symmetries of C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaa WcbaGaamyAaiaadQgacaWGRbGaamiBaaqabaaaaa@3AEB@ , note that

 The stress tensor is symmetric, which is only possible if C ijkl = C jikl . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadQgacaWG PbGaam4AaiaadYgaaeqaaOGaaiOlaaaa@3BA7@

 If a strain energy density exists for the material, the elastic stiffness tensor must satisfy C ijkl = C klij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadUgacaWG SbGaamyAaiaadQgaaeqaaaaa@3AEB@

 The previous two symmetries imply C ijkl = C ijlk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadMgacaWG QbGaamiBaiaadUgaaeqaaaaa@3AEB@ , since C ijkl = C jikl = C klji MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadQgacaWG PbGaam4AaiaadYgaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGRb GaamiBaiaadQgacaWGPbaabeaaaaa@40AD@  and . C ijkl = C lkij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadYgacaWG RbGaamyAaiaadQgaaeqaaaaa@3AEB@

 

To see that C ijkl = C klij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadUgacaWG SbGaamyAaiaadQgaaeqaaaaa@3AEB@ , note that by definition

C ijkl = σ ij ε kl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2kabeo8aZnaa BaaaleaacaWGPbGaamOAaaqabaaakeaacqGHciITcqaH1oqzdaWgaa WcbaGaam4AaiaadYgaaeqaaaaaaaa@408F@

and recall further that the stress is the derivative of the strain energy density with respect to strain

σ ij = U ε ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpdaWcaaqaaiabgkGi2kaadwfaaeaacqGHciITcqaH 1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaaaaaaa@3CA9@

Combining these,

C ijkl = 2 U ε ij ε kl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpdaWcaaqaaiabgkGi2oaaCaaaleqa baGaaGOmaaaakiaadwfaaeaacqGHciITcqaH1oqzdaWgaaWcbaGaam yAaiaadQgaaeqaaOGaeyOaIyRaeqyTdu2aaSbaaSqaaiaadUgacaWG Sbaabeaaaaaaaa@43A6@

Now, note that

2 U ε ij ε kl = 2 U ε kl ε ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaamyvaaqaaiabgkGi2kabew7aLnaaBaaaleaacaWGPbGa amOAaaqabaGccqGHciITcqaH1oqzdaWgaaWcbaGaam4AaiaadYgaae qaaaaakiabg2da9maalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaa aOGaamyvaaqaaiabgkGi2kabew7aLnaaBaaaleaacaWGRbGaamiBaa qabaGccqGHciITcqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa aaa@4C71@

so that

C ijkl = C klij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadUgacaWG SbGaamyAaiaadQgaaeqaaaaa@3ADB@

These symmetries allow us to write the stress-strain relations in a more compact matrix form as

σ=C(εαΔT) σ=[ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ]C=[ c 11 c 12 c 13 c 14 c 15 c 16 c 12 c 22 c 23 c 24 c 25 c 26 c 13 c 23 c 33 c 34 c 35 c 36 c 14 c 24 c 34 c 44 c 45 c 46 c 15 c 25 c 35 c 45 c 55 c 56 c 16 c 26 c 36 c 46 c 56 c 66 ]ε=[ ε 11 ε 22 ε 33 2 ε 23 2 ε 13 2 ε 12 ]α=[ α 11 α 22 α 33 2 α 23 2 α 13 2 α 12 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaaho8acqGH9aqpcaWHdbGaai ikaiaahw7acqGHsislcaWHXoGaeuiLdqKaamivaiaacMcaaeaacaWH dpGaeyypa0JaaGPaVpaadmaabaqbaeqabyqaaaaabaGaeq4Wdm3aaS baaSqaaiaaigdacaaIXaaabeaaaOqaaiabeo8aZnaaBaaaleaacaaI YaGaaGOmaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaG4maiaaiodaae qaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiab eo8aZnaaBaaaleaacaaIXaGaaG4maaqabaaakeaacqaHdpWCdaWgaa WcbaGaaGymaiaaikdaaeqaaaaaaOGaay5waiaaw2faaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaC4qaiabg2da9maadmaabaqb aeqabyGbaaaaaeaacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaaaO qaaiaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaam4yamaa BaaaleaacaaIXaGaaG4maaqabaaakeaacaWGJbWaaSbaaSqaaiaaig dacaaI0aaabeaaaOqaaiaadogadaWgaaWcbaGaaGymaiaaiwdaaeqa aaGcbaGaam4yamaaBaaaleaacaaIXaGaaGOnaaqabaaakeaacaWGJb WaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadogadaWgaaWcbaGa aGOmaiaaikdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIYaGaaG4maa qabaaakeaacaWGJbWaaSbaaSqaaiaaikdacaaI0aaabeaaaOqaaiaa dogadaWgaaWcbaGaaGOmaiaaiwdaaeqaaaGcbaGaam4yamaaBaaale aacaaIYaGaaGOnaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaI ZaaabeaaaOqaaiaadogadaWgaaWcbaGaaGOmaiaaiodaaeqaaaGcba Gaam4yamaaBaaaleaacaaIZaGaaG4maaqabaaakeaacaWGJbWaaSba aSqaaiaaiodacaaI0aaabeaaaOqaaiaadogadaWgaaWcbaGaaG4mai aaiwdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIZaGaaGOnaaqabaaa keaacaWGJbWaaSbaaSqaaiaaigdacaaI0aaabeaaaOqaaiaadogada WgaaWcbaGaaGOmaiaaisdaaeqaaaGcbaGaam4yamaaBaaaleaacaaI ZaGaaGinaaqabaaakeaacaWGJbWaaSbaaSqaaiaaisdacaaI0aaabe aaaOqaaiaadogadaWgaaWcbaGaaGinaiaaiwdaaeqaaaGcbaGaam4y amaaBaaaleaacaaI0aGaaGOnaaqabaaakeaacaWGJbWaaSbaaSqaai aaigdacaaI1aaabeaaaOqaaiaadogadaWgaaWcbaGaaGOmaiaaiwda aeqaaaGcbaGaam4yamaaBaaaleaacaaIZaGaaGynaaqabaaakeaaca WGJbWaaSbaaSqaaiaaisdacaaI1aaabeaaaOqaaiaadogadaWgaaWc baGaaGynaiaaiwdaaeqaaaGcbaGaam4yamaaBaaaleaacaaI1aGaaG OnaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaI2aaabeaaaOqa aiaadogadaWgaaWcbaGaaGOmaiaaiAdaaeqaaaGcbaGaam4yamaaBa aaleaacaaIZaGaaGOnaaqabaaakeaacaWGJbWaaSbaaSqaaiaaisda caaI2aaabeaaaOqaaiaadogadaWgaaWcbaGaaGynaiaaiAdaaeqaaa GcbaGaam4yamaaBaaaleaacaaI2aGaaGOnaaqabaaaaaGccaGLBbGa ayzxaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWH1oGaeyypa0ZaamWaaeaafaqa beGbbaaaaeaacqaH1oqzdaWgaaWcbaGaaGymaiaaigdaaeqaaaGcba GaeqyTdu2aaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabew7aLnaa BaaaleaacaaIZaGaaG4maaqabaaakeaacaaIYaGaeqyTdu2aaSbaaS qaaiaaikdacaaIZaaabeaaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGa aGymaiaaiodaaeqaaaGcbaGaaGOmaiabew7aLnaaBaaaleaacaaIXa GaaGOmaaqabaaaaaGccaGLBbGaayzxaaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWHXoGaeyypa0ZaamWaae aafaqabeGbbaaaaeaacqaHXoqydaWgaaWcbaGaaGymaiaaigdaaeqa aaGcbaGaeqySde2aaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabeg 7aHnaaBaaaleaacaaIZaGaaG4maaqabaaakeaacaaIYaGaeqySde2a aSbaaSqaaiaaikdacaaIZaaabeaaaOqaaiaaikdacqaHXoqydaWgaa WcbaGaaGymaiaaiodaaeqaaaGcbaGaaGOmaiabeg7aHnaaBaaaleaa caaIXaGaaGOmaaqabaaaaaGccaGLBbGaayzxaaaaaaa@20B5@

where c 11 C 1111 c 12 C 1122 = C 2211 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXa aabeaakiabggMi6kaadoeadaWgaaWcbaGaaGymaiaaigdacaaIXaGa aGymaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8Uaam4yamaaBaaale aacaaIXaGaaGOmaaqabaGccqGHHjIUcaWGdbWaaSbaaSqaaiaaigda caaIXaGaaGOmaiaaikdaaeqaaOGaeyypa0Jaam4qamaaBaaaleaaca aIYaGaaGOmaiaaigdacaaIXaaabeaaaaa@4E6B@ , etc are the elastic stiffnesses of the material.  The inverse has the form

ε=Sσ+αΔT S=[ s 11 s 12 s 13 s 14 s 15 s 16 s 12 s 22 s 23 s 24 s 25 s 26 s 13 s 23 s 33 s 34 s 35 s 36 s 14 s 24 s 34 s 44 s 45 s 46 s 15 s 25 s 35 s 45 s 55 s 56 s 16 s 26 s 36 s 46 s 56 s 66 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahw7acqGH9aqpcaWHtbGaaC 4WdiabgUcaRiaahg7acqqHuoarcaWGubaabaGaaC4uaiabg2da9maa dmaabaqbaeqabyGbaaaaaeaacaWGZbWaaSbaaSqaaiaaigdacaaIXa aabeaaaOqaaiaadohadaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGa am4CamaaBaaaleaacaaIXaGaaG4maaqabaaakeaacaWGZbWaaSbaaS qaaiaaigdacaaI0aaabeaaaOqaaiaadohadaWgaaWcbaGaaGymaiaa iwdaaeqaaaGcbaGaam4CamaaBaaaleaacaaIXaGaaGOnaaqabaaake aacaWGZbWaaSbaaSqaaiaaigdacaaIYaaabeaaaOqaaiaadohadaWg aaWcbaGaaGOmaiaaikdaaeqaaaGcbaGaam4CamaaBaaaleaacaaIYa GaaG4maaqabaaakeaacaWGZbWaaSbaaSqaaiaaikdacaaI0aaabeaa aOqaaiaadohadaWgaaWcbaGaaGOmaiaaiwdaaeqaaaGcbaGaam4Cam aaBaaaleaacaaIYaGaaGOnaaqabaaakeaacaWGZbWaaSbaaSqaaiaa igdacaaIZaaabeaaaOqaaiaadohadaWgaaWcbaGaaGOmaiaaiodaae qaaaGcbaGaam4CamaaBaaaleaacaaIZaGaaG4maaqabaaakeaacaWG ZbWaaSbaaSqaaiaaiodacaaI0aaabeaaaOqaaiaadohadaWgaaWcba GaaG4maiaaiwdaaeqaaaGcbaGaam4CamaaBaaaleaacaaIZaGaaGOn aaqabaaakeaacaWGZbWaaSbaaSqaaiaaigdacaaI0aaabeaaaOqaai aadohadaWgaaWcbaGaaGOmaiaaisdaaeqaaaGcbaGaam4CamaaBaaa leaacaaIZaGaaGinaaqabaaakeaacaWGZbWaaSbaaSqaaiaaisdaca aI0aaabeaaaOqaaiaadohadaWgaaWcbaGaaGinaiaaiwdaaeqaaaGc baGaam4CamaaBaaaleaacaaI0aGaaGOnaaqabaaakeaacaWGZbWaaS baaSqaaiaaigdacaaI1aaabeaaaOqaaiaadohadaWgaaWcbaGaaGOm aiaaiwdaaeqaaaGcbaGaam4CamaaBaaaleaacaaIZaGaaGynaaqaba aakeaacaWGZbWaaSbaaSqaaiaaisdacaaI1aaabeaaaOqaaiaadoha daWgaaWcbaGaaGynaiaaiwdaaeqaaaGcbaGaam4CamaaBaaaleaaca aI1aGaaGOnaaqabaaakeaacaWGZbWaaSbaaSqaaiaaigdacaaI2aaa beaaaOqaaiaadohadaWgaaWcbaGaaGOmaiaaiAdaaeqaaaGcbaGaam 4CamaaBaaaleaacaaIZaGaaGOnaaqabaaakeaacaWGZbWaaSbaaSqa aiaaisdacaaI2aaabeaaaOqaaiaadohadaWgaaWcbaGaaGynaiaaiA daaeqaaaGcbaGaam4CamaaBaaaleaacaaI2aGaaGOnaaqabaaaaaGc caGLBbGaayzxaaaaaaa@9F7C@

where s 11 = S 1111 , s 12 = S 1122 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohadaWgaaWcbaGaaGymaiaaigdaae qaaOGaeyypa0Jaam4uamaaBaaaleaacaaIXaGaaGymaiaaigdacaaI XaaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caWGZbWaaSbaaSqaai aaigdacaaIYaaabeaakiabg2da9iaadofadaWgaaWcbaGaaGymaiaa igdacaaIYaGaaGOmaaqabaaaaa@450A@ , etc are the elastic compliances of the material.

 

To satisfy Drucker stability, the eigenvalues of the elastic stiffness and compliance matrices must all be greater than zero.

 

HEALTH WARNING: The shear strain and shear stress components are not always listed in the order given when defining the elastic and compliance matrices.  The conventions used here are common and are particularly convenient in analytical calculations involving anisotropic solids.  But many sources use other conventions.  Be careful to enter material data in the correct order when specifying properties for anisotropic solids.

 

 

 

 

3.2.9 Physical Interpretation of the Anisotropic Elastic Constants.

 

It is easiest to interpret s 11 , s 12 .... s 66 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdacaaIXa aabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8Uaam4CamaaBaaa leaacaaIXaGaaGOmaaqabaGccaGGUaGaaiOlaiaac6cacaGGUaGaam 4CamaaBaaaleaacaaI2aGaaGOnaaqabaaaaa@4456@ , rather than c 11 , c `12 ,... c 66 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXa aabeaakiaacYcacaaMc8UaaGPaVlaadogadaWgaaWcbaGaaiiyaiaa igdacaaIYaaabeaakiaacYcacaGGUaGaaiOlaiaac6cacaWGJbWaaS baaSqaaiaaiAdacaaI2aaabeaaaaa@41F2@ .  Imagine applying a uniaxial stress, say σ 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaig daaeqaaaaa@362A@ , to an anisotropic specimen.  In general, this would induce both extensional and shear deformation in the solid, as shown in the figure.

 

The strain induced by  the uniaxial stress would be

ε 11 = s 11 σ 11 , ε 22 = s 12 σ 11 , ε 33 = s 13 σ 11 ε 23 = s 14 σ 11 , ε 13 = s 15 σ 11 , ε 12 = s 16 σ 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabew7aLnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpcaWGZbWaaSbaaSqaaiaaigdacaaIXaaa beaakiabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccaGGSaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaH 1oqzdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0Jaam4CamaaBa aaleaacaaIXaGaaGOmaaqabaGccqaHdpWCdaWgaaWcbaGaaGymaiaa igdaaeqaaOGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabew7aLnaaBaaaleaacaaIZaGaaG4m aaqabaGccqGH9aqpcaWGZbWaaSbaaSqaaiaaigdacaaIZaaabeaaki abeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaaakeaacqaH1oqzdaWg aaWcbaGaaGOmaiaaiodaaeqaaOGaeyypa0Jaam4CamaaBaaaleaaca aIXaGaaGinaaqabaGccqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqa aOGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaeqyTdu2aaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da 9iaadohadaWgaaWcbaGaaGymaiaaiwdaaeqaaOGaeq4Wdm3aaSbaaS qaaiaaigdacaaIXaaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyTdu2aaS baaSqaaiaaigdacaaIYaaabeaakiabg2da9iaadohadaWgaaWcbaGa aGymaiaaiAdaaeqaaOGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabe aaaaaa@AA26@

All the constants have dimensions m 2 /N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGTbWaaWbaaSqabeaacaaIYaaaaO Gaai4laiaad6eaaaa@3630@ .  The constant s 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdacaaIXa aabeaaaaa@355F@  looks like a uniaxial compliance, (like 1/E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIXaGaai4laiaadweaaaa@34FD@  ), while the ratios s 12 / s 11 , s 13 / s 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdacaaIYa aabeaakiaac+cacaWGZbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaa cYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaadohadaWgaaWcbaGaaGymaiaaiodaaeqaaOGaai4laiaadoha daWgaaWcbaGaaGymaiaaigdaaeqaaaaa@4BBC@  are generalized versions of Poisson’s ratio: they quantify the lateral contraction of a uniaxial tensile specimen.   The shear terms are new MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in an isotropic material, no shear strain is induced by uniaxial tension.

 

 

3.2.10 Strain energy density for anisotropic, linear elastic solids

 

The strain energy density of an anisotropic material is

U= 1 2 C ijkl ( ε ij α ij ΔT )( ε kl α kl ΔT ) = 1 2 S ijkl σ ij σ kl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyvaiabg2da9maalaaabaGaaG ymaaqaaiaaikdaaaGaam4qamaaBaaaleaacaWGPbGaamOAaiaadUga caWGSbaabeaakmaabmaabaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQb aabeaakiabgkHiTiabeg7aHnaaBaaaleaacaWGPbGaamOAaaqabaGc cqqHuoarcaWGubaacaGLOaGaayzkaaWaaeWaaeaacqaH1oqzdaWgaa WcbaGaam4AaiaadYgaaeqaaOGaeyOeI0IaeqySde2aaSbaaSqaaiaa dUgacaWGSbaabeaakiabfs5aejaadsfaaiaawIcacaGLPaaaaeaaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH9aqpdaWcaaqa aiaaigdaaeaacaaIYaaaaiaadofadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaGccqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqa aOGaeq4Wdm3aaSbaaSqaaiaadUgacaWGSbaabeaaaaaa@691D@

 

 

 

3.2.11 Basis change formulas for anisotropic elastic constants

 

The material constants c ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35B7@  or s ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35C7@  for a particular material are usually specified in a basis with coordinate axes aligned with particular symmetry planes (if any) in the material.  When solving problems involving anisotropic materials it is frequently necessary to transform these values to a coordinate system that is oriented in some convenient way relative to the boundaries of the solid.  Since C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3778@  is a fourth rank tensor, the basis change formulas are highly tedious, unfortunately. 

 

Suppose that the components of the stiffness tensor are given in a basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@ , and we wish to determine its components in a second basis, { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@ .  We define the usual transformation tensor Ω ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@365D@  with components Ω ij = m i e j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0JaaCyBamaaBaaaleaacaWGPbaabeaakiabgwSi xlaahwgadaWgaaWcbaGaamOAaaqabaaaaa@3DDA@ , or in matrix form

[ Ω ]=[ m 1 e 1 m 1 e 2 m 1 e 3 m 2 e 1 m 2 e 2 m 2 e 3 m 3 e 1 m 3 e 2 m 3 e 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabfM6axbGaay5waiaaw2 faaiabg2da9maadmaaeaqabeaacaWHTbWaaSbaaSqaaiaaigdaaeqa aOGaeyyXICTaaCyzamaaBaaaleaacaaIXaaabeaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaGymaaqa baGccqGHflY1caWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIXaaa beaakiabgwSixlaahwgadaWgaaWcbaGaaG4maaqabaaakeaacaWHTb WaaSbaaSqaaiaaikdaaeqaaOGaeyyXICTaaCyzamaaBaaaleaacaaI XaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaah2 gadaWgaaWcbaGaaGOmaaqabaGccqGHflY1caWHLbWaaSbaaSqaaiaa ikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaC yBamaaBaaaleaacaaIYaaabeaakiabgwSixlaahwgadaWgaaWcbaGa aG4maaqabaaakeaacaWHTbWaaSbaaSqaaiaaiodaaeqaaOGaeyyXIC TaaCyzamaaBaaaleaacaaIXaaabeaakiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaah2gadaWgaaWcbaGaaG4maaqabaGccqGHfl Y1caWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaCyBamaaBaaaleaacaaIZaaabeaakiabgw SixlaahwgadaWgaaWcbaGaaG4maaqabaaaaOGaay5waiaaw2faaaaa @A773@

This is an orthogonal matrix satisfying Ω Ω T = Ω T Ω=I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcqqHPoWvdaahaaWcbeqaai aadsfaaaGccqGH9aqpcqqHPoWvdaahaaWcbeqaaiaadsfaaaGccqqH PoWvcqGH9aqpcaWHjbaaaa@3DFC@ . In practice, the matrix can be computed in terms of the angles between the basis vectors. It is straightforward to show that stress, strain, thermal expansion and elasticity tensors transform as

σ ij (m) = Ω ik σ kl (e) Ω jl ε ij (m) = Ω ik ε kl (e) Ω jl α ij (m) = Ω ik α kl (e) Ω jl C ijkl (m) = Ω ip Ω jq C pqrs (e) Ω kr Ω ls MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaDaaaleaacaWGPb GaamOAaaqaaiaacIcacaWHTbGaaiykaaaakiabg2da9iabfM6axnaa BaaaleaacaWGPbGaam4AaaqabaGccqaHdpWCdaqhaaWcbaGaam4Aai aadYgaaeaacaGGOaGaaCyzaiaacMcaaaGccqqHPoWvdaWgaaWcbaGa amOAaiaadYgaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaaiikaiaah2ga caGGPaaaaOGaeyypa0JaeuyQdC1aaSbaaSqaaiaadMgacaWGRbaabe aakiabew7aLnaaDaaaleaacaWGRbGaamiBaaqaaiaacIcacaWHLbGa aiykaaaakiabfM6axnaaBaaaleaacaWGQbGaamiBaaqabaGccaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqySde2aa0ba aSqaaiaadMgacaWGQbaabaGaaiikaiaah2gacaGGPaaaaOGaeyypa0 JaeuyQdC1aaSbaaSqaaiaadMgacaWGRbaabeaakiabeg7aHnaaDaaa leaacaWGRbGaamiBaaqaaiaacIcacaWHLbGaaiykaaaakiabfM6axn aaBaaaleaacaWGQbGaamiBaaqabaaakeaacaWGdbWaa0baaSqaaiaa dMgacaWGQbGaam4AaiaadYgaaeaacaGGOaGaaCyBaiaacMcaaaGccq GH9aqpcqqHPoWvdaWgaaWcbaGaamyAaiaadchaaeqaaOGaeuyQdC1a aSbaaSqaaiaadQgacaWGXbaabeaakiaadoeadaqhaaWcbaGaamiCai aadghacaWGYbGaam4CaaqaaiaacIcacaWHLbGaaiykaaaakiabfM6a xnaaBaaaleaacaWGRbGaamOCaaqabaGccqqHPoWvdaWgaaWcbaGaam iBaiaadohaaeqaaaaaaa@A1E8@

 

The basis change formula for the elasticity tensor in matrix form can be expressed as

C (m) =K C (e) K T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHdbWaaWbaaSqabeaacaGGOaGaaC yBaiaacMcaaaGccqGH9aqpcaWHlbGaaC4qamaaCaaaleqabaGaaiik aiaahwgacaGGPaaaaOGaaC4samaaCaaaleqabaGaamivaaaaaaa@3D16@

where the basis change matrix K is computed as

K=[ K (1) 2 K (2) K (3) K (4) ] K ij (1) = Ω ij 2 K ij (2) = Ω imod(j+1,3) Ω imod(j+2,3) K ij (3) = Ω mod(i+1,3)j Ω mod(i+2,3)j K ij (4) = Ω mod(i+1,3)mod(j+1,3) Ω mod(i+2,3)mod(j+2,3) + Ω mod(i+1,3)mod(j+2,3) Ω mod(i+2,3)mod(j+1,3) }i,j=1..3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaahUeacqGH9aqpdaWadaqaau aabeqaciaaaeaacaWHlbWaaWbaaSqabeaacaGGOaGaaGymaiaacMca aaaakeaacaaIYaGaaC4samaaCaaaleqabaGaaiikaiaaikdacaGGPa aaaaGcbaGaaC4samaaCaaaleqabaGaaiikaiaaiodacaGGPaaaaaGc baGaaC4samaaCaaaleqabaGaaiikaiaaisdacaGGPaaaaaaaaOGaay 5waiaaw2faaaqaamaaciaaeaqabeaacaWGlbWaa0baaSqaaiaadMga caWGQbaabaGaaiikaiaaigdacaGGPaaaaOGaeyypa0JaeuyQdC1aa0 baaSqaaiaadMgacaWGQbaabaGaaGOmaaaakiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4samaaDaaaleaaca WGPbGaamOAaaqaaiaacIcacaaIYaGaaiykaaaakiabg2da9iabfM6a xnaaBaaaleaacaWGPbGaciyBaiaac+gacaGGKbGaaiikaiaadQgacq GHRaWkcaaIXaGaaiilaiaaiodacaGGPaaabeaakiabfM6axnaaBaaa leaacaWGPbGaciyBaiaac+gacaGGKbGaaiikaiaadQgacqGHRaWkca aIYaGaaiilaiaaiodacaGGPaaabeaaaOqaaiaadUeadaqhaaWcbaGa amyAaiaadQgaaeaacaGGOaGaaG4maiaacMcaaaGccqGH9aqpcqqHPo WvdaWgaaWcbaGaciyBaiaac+gacaGGKbGaaiikaiaadMgacqGHRaWk caaIXaGaaiilaiaaiodacaGGPaGaamOAaaqabaGccqqHPoWvdaWgaa WcbaGaciyBaiaac+gacaGGKbGaaiikaiaadMgacqGHRaWkcaaIYaGa aiilaiaaiodacaGGPaGaamOAaaqabaaakeaacaWGlbWaa0baaSqaai aadMgacaWGQbaabaGaaiikaiaaisdacaGGPaaaaOGaeyypa0JaeuyQ dC1aaSbaaSqaaiGac2gacaGGVbGaaiizaiaacIcacaWGPbGaey4kaS IaaGymaiaacYcacaaIZaGaaiykaiGac2gacaGGVbGaaiizaiaacIca caWGQbGaey4kaSIaaGymaiaacYcacaaIZaGaaiykaaqabaGccqqHPo WvdaWgaaWcbaGaciyBaiaac+gacaGGKbGaaiikaiaadMgacqGHRaWk caaIYaGaaiilaiaaiodacaGGPaGaciyBaiaac+gacaGGKbGaaiikai aadQgacqGHRaWkcaaIYaGaaiilaiaaiodacaGGPaaabeaakiaaykW7 caaMc8UaaGPaVdqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgUcaRiaa ykW7caaMc8UaaGPaVlabfM6axnaaBaaaleaaciGGTbGaai4Baiaacs gacaGGOaGaamyAaiabgUcaRiaaigdacaGGSaGaaG4maiaacMcaciGG TbGaai4BaiaacsgacaGGOaGaamOAaiabgUcaRiaaikdacaGGSaGaaG 4maiaacMcaaeqaaOGaeuyQdC1aaSbaaSqaaiGac2gacaGGVbGaaiiz aiaacIcacaWGPbGaey4kaSIaaGOmaiaacYcacaaIZaGaaiykaiGac2 gacaGGVbGaaiizaiaacIcacaWGQbGaey4kaSIaaGymaiaacYcacaaI ZaGaaiykaaqabaaaaOGaayzFaaGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWGPbGaaiilaiaadQgacqGH9aqp caaIXaGaaiOlaiaac6cacaaIZaaaaaa@108A@

and the modulo function satisfies

mod(i,3)={ ii3 i3i>3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGTbGaai4BaiaacsgacaGGOaGaam yAaiaacYcacaaIZaGaaiykaiabg2da9maaceaabaqbaeqabiqaaaqa aiaadMgacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamyAaiabgsMi JkaaiodaaeaacaWGPbGaeyOeI0IaaG4maiaaykW7caaMc8UaaGPaVl aaykW7caWGPbGaeyOpa4JaaG4maaaaaiaawUhaaaaa@5F49@

Although these expressions look cumbersome they are quite convenient for computer implementation.

 

The basis change for the compliance tensor follows as

S (m) = K T S (e) K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHtbWaaWbaaSqabeaacaGGOaGaaC yBaiaacMcaaaGccqGH9aqpcaWHlbWaaWbaaSqabeaacqGHsislcaWG ubaaaOGaaC4uamaaCaaaleqabaGaaiikaiaahwgacaGGPaaaaOGaaC 4samaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@4002@

where

K T =[ K (1) K (2) 2 K (3) K (4) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHlbWaaWbaaSqabeaacqGHsislca WGubaaaOGaeyypa0ZaamWaaeaafaqabeGacaaabaGaaC4samaaCaaa leqabaGaaiikaiaaigdacaGGPaaaaaGcbaGaaC4samaaCaaaleqaba GaaiikaiaaikdacaGGPaaaaaGcbaGaaGOmaiaahUeadaahaaWcbeqa aiaacIcacaaIZaGaaiykaaaaaOqaaiaahUeadaahaaWcbeqaaiaacI cacaaI0aGaaiykaaaaaaaakiaawUfacaGLDbaaaaa@45DD@

 

The proof of these expressions is merely tiresome algebra and will not be given here.  Ting’s book `Anisotropic Elasticity: Theory and Applications’ OUP (1996) has a nice clear discussion.

 

For the particular case of rotation through an angle θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@347C@  in a counterclockwise sense about the e 1 , e 2 , e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyzamaa BaaaleaacaaIZaaabeaaaaa@39CC@  axes, respectively, the rotation matrix reduces to

[ 1 0 0 0 0 0 0 c 2 s 2 2cs 0 0 0 s 2 c 2 2cs 0 0 0 cs cs c 2 s 2 0 0 0 0 0 0 c s 0 0 0 0 s c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqagyaaaaaabaGaaG ymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaI WaaabaGaaGimaaqaaiaadogadaahaaWcbeqaaiaaikdaaaaakeaaca WGZbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGOmaiaadogacaWGZbaa baGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaam4CamaaCaaaleqaba GaaGOmaaaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaaakeaacqGH sislcaaIYaGaam4yaiaadohaaeaacaaIWaaabaGaaGimaaqaaiaaic daaeaacqGHsislcaWGJbGaam4CaaqaaiaadogacaWGZbaabaGaam4y amaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadohadaahaaWcbeqaai aaikdaaaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaa baGaaGimaaqaaiaaicdaaeaacaWGJbaabaGaeyOeI0Iaam4Caaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGZbaabaGa am4yaaaaaiaawUfacaGLDbaaaaa@6165@    [ c 2 0 s 2 0 2cs 0 0 1 0 0 0 0 s 2 0 c 2 0 2cs 0 0 0 0 c 0 s cs 0 cs 0 c 2 s 2 0 0 0 0 s 0 c ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqagyaaaaaabaGaam 4yamaaCaaaleqabaGaaGOmaaaaaOqaaiaaicdaaeaacaWGZbWaaWba aSqabeaacaaIYaaaaaGcbaGaaGimaaqaaiaaikdacaWGJbGaam4Caa qaaiaaicdaaeaacaaIWaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaa baGaaGimaaqaaiaaicdaaeaacaWGZbWaaWbaaSqabeaacaaIYaaaaa GcbaGaaGimaaqaaiaadogadaahaaWcbeqaaiaaikdaaaaakeaacaaI WaaabaGaeyOeI0IaaGOmaiaadogacaWGZbaabaGaaGimaaqaaiaaic daaeaacaaIWaaabaGaaGimaaqaaiaadogaaeaacaaIWaaabaGaeyOe I0Iaam4CaaqaaiabgkHiTiaadogacaWGZbaabaGaaGimaaqaaiaado gacaWGZbaabaGaaGimaaqaaiaadogadaahaaWcbeqaaiaaikdaaaGc cqGHsislcaWGZbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaadohaaeaacaaIWaaabaGa am4yaaaaaiaawUfacaGLDbaaaaa@6165@       [ c 2 s 2 0 0 0 2cs s 2 c 2 0 0 0 2cs 0 0 1 0 0 0 0 0 0 c s 0 0 0 0 s c 0 cs cs 0 0 0 c 2 s 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqagyaaaaaabaGaam 4yamaaCaaaleqabaGaaGOmaaaaaOqaaiaadohadaahaaWcbeqaaiaa ikdaaaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIYaGaam 4yaiaadohaaeaacaWGZbWaaWbaaSqabeaacaaIYaaaaaGcbaGaam4y amaaCaaaleqabaGaaGOmaaaaaOqaaiaaicdaaeaacaaIWaaabaGaaG imaaqaaiabgkHiTiaaikdacaWGJbGaam4CaaqaaiaaicdaaeaacaaI WaaabaGaaGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaic daaeaacaaIWaaabaGaaGimaaqaaiaadogaaeaacaWGZbaabaGaaGim aaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiabgkHiTiaadohaae aacaWGJbaabaGaaGimaaqaaiabgkHiTiaadogacaWGZbaabaGaam4y aiaadohaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaWGJbWaaW baaSqabeaacaaIYaaaaOGaeyOeI0Iaam4CamaaCaaaleqabaGaaGOm aaaaaaaakiaawUfacaGLDbaaaaa@6165@

where c=cosθs=sinθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbGaeyypa0Jaci4yaiaac+gaca GGZbGaeqiUdeNaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGZbGaeyypa0Jaci4CaiaacMgacaGGUbGaeqiUde haaa@4C31@ . The inverse matrix K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHlbWaaWbaaSqabeaacqGHsislca aIXaaaaaaa@357F@  can be obtained simply by changing the sign of the angle θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@347C@  in each rotation matrix.  Clearly, applying the three rotations successively can produce an arbitrary orientation change.

 

For an isotropic material, the elastic stress-strain relations, the elasticity matrices and thermal expansion coefficient are unaffected by basis changes.

 

 

 

 

3.2.12 The effect of material symmetry on stress-strain relations for anisotropic materials

 

A general anisotropic solid has 21 independent elastic constants. Note that in general, tensile stress may induce shear strain, and shear stress may cause extension.

 

If a material has a symmetry plane, then applying stress normal or parallel to this plane induces only extension in direction normal and parallel to the plane

 

For example, suppose the material contains a single symmetry plane, and let e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaGymaaqabaaaaa@36EA@    be normal to this plane.

 

Then the components of the elastic stiffnes matrix c 15 = c 16 = c 25 = c 26 = c 35 = c 36 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadogadaWgaa WcbaGaaGymaiaaiwdaaeqaaOGaeyypa0Jaam4yamaaBaaaleaacaaI XaGaaGOnaaqabaGccqGH9aqpcaWGJbWaaSbaaSqaaiaaikdacaaI1a aabeaakiabg2da9iaadogadaWgaaWcbaGaaGOmaiaaiAdaaeqaaOGa eyypa0Jaam4yamaaBaaaleaacaaIZaGaaGynaaqabaGccqGH9aqpca WGJbWaaSbaaSqaaiaaiodacaaI2aaabeaakiabg2da9iaaicdaaaa@4B8C@  ( C 1112 = C 1113 = C 2212 = C 2213 = C 3312 = C 3313 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadoeadaWgaa WcbaGaaGymaiaaigdacaaIXaGaaGOmaaqabaGccqGH9aqpcaWGdbWa aSbaaSqaaiaaigdacaaIXaGaaGymaiaaiodaaeqaaOGaeyypa0Jaam 4qamaaBaaaleaacaaIYaGaaGOmaiaaigdacaaIYaaabeaakiabg2da 9iaadoeadaWgaaWcbaGaaGOmaiaaikdacaaIXaGaaG4maaqabaGccq GH9aqpcaWGdbWaaSbaaSqaaiaaiodacaaIZaGaaGymaiaaikdaaeqa aOGaeyypa0Jaam4qamaaBaaaleaacaaIZaGaaG4maiaaigdacaaIZa aabeaakiabg2da9iaaicdaaaa@5384@  ). (symmetrical terms also vanish, of course). This leaves 13 independent constants. 

 

Similar restrictions on the thermal expansion coefficient can be determined using symmetry conditions.  Details are left as an exercise.

 

In the following sections, we list the stress-strain relations for anisotropic materials with various numbers of symmetry planes.

 

 

 

 

3.2.13 Stress-strain relations for linear elastic orthotropic materials

 

An orthotropic material has three mutually perpendicular symmetry planes. This type of material has 9 independent material constants.  With basis vectors perpendicular to the symmetry plane, the elastic stiffness matrix has the form

C=[ c 11 c 12 c 13 0 0 0 c 22 c 23 0 0 0 c 33 0 0 0 sym c 44 0 0 0 c 55 0 0 0 c 66 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahoeacqGH9a qpdaWadaqaauaabeqagyaaaaaabaGaam4yamaaBaaaleaacaaIXaGa aGymaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaaaO qaaiaadogadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaaGimaaqa aiaaicdaaeaacaaIWaaabaaabaGaam4yamaaBaaaleaacaaIYaGaaG OmaaqabaaakeaacaWGJbWaaSbaaSqaaiaaikdacaaIZaaabeaaaOqa aiaaicdaaeaacaaIWaaabaGaaGimaaqaaaqaaaqaaiaadogadaWgaa WcbaGaaG4maiaaiodaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaI WaaabaaabaGaam4CaiaadMhacaWGTbaabaaabaGaam4yamaaBaaale aacaaI0aGaaGinaaqabaaakeaacaaIWaaabaGaaGimaaqaaaqaaaqa aaqaaiaaicdaaeaacaWGJbWaaSbaaSqaaiaaiwdacaaI1aaabeaaaO qaaiaaicdaaeaaaeaaaeaaaeaacaaIWaaabaGaaGimaaqaaiaadoga daWgaaWcbaGaaGOnaiaaiAdaaeqaaaaaaOGaay5waiaaw2faaaaa@5E2A@

This relationship is sometimes expressed in inverse form, in terms of generalized Young’s moduli and Poisson’s ratios (which have the same significance as Young’s modulus and Poisson’s ratio for uniaxial loading along the three basis vectors) as follows                                                                    

S=[ 1/ E 1 ν 21 / E 2 ν 31 / E 3 0 0 0 ν 12 / E 1 1/ E 2 ν 32 / E 3 0 0 0 ν 13 / E 1 ν 23 / E 2 1/E 3 0 0 0 0 0 0 1/ μ 23 0 0 0 0 0 0 1/ μ 13 0 0 0 0 0 0 1/ μ 12 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahofacqGH9a qpdaWadaqaauaabeqagyaaaaaabaGaaGymaiaac+cacaWGfbWaaSba aSqaaiaaigdaaeqaaaGcbaGaeyOeI0IaeqyVd42aaSbaaSqaaiaaik dacaaIXaaabeaakiaac+cacaWGfbWaaSbaaSqaaiaaikdaaeqaaaGc baGaeyOeI0IaeqyVd42aaSbaaSqaaiaaiodacaaIXaaabeaakiaac+ cacaWGfbWaaSbaaSqaaiaaiodaaeqaaaGcbaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaeyOeI0IaeqyVd42aaSbaaSqaaiaaigdacaaIYa aabeaakiaac+cacaWGfbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaGym aiaac+cacaWGfbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeyOeI0Iaeq yVd42aaSbaaSqaaiaaiodacaaIYaaabeaakiaac+cacaWGfbWaaSba aSqaaiaaiodaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaIWaaaba GaeyOeI0IaeqyVd42aaSbaaSqaaiaaigdacaaIZaaabeaakiaac+ca caWGfbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeyOeI0IaeqyVd42aaS baaSqaaiaaikdacaaIZaaabeaakiaac+cacaWGfbWaaSbaaSqaaiaa ikdaaeqaaaGcbaGaaGymaiaac+cacaWGfbGaaCjaVpaaBaaaleaaca aIZaaabeaaaOqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaigdacaGGVaGaeqiVd02aaSbaaS qaaiaaikdacaaIZaaabeaaaOqaaiaaicdaaeaacaaIWaaabaGaaGim aaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaigdacaGGVaGaeq iVd02aaSbaaSqaaiaaigdacaaIZaaabeaaaOqaaiaaicdaaeaacaaI WaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaig dacaGGVaGaeqiVd02aaSbaaSqaaiaaigdacaaIYaaabeaaaaaakiaa wUfacaGLDbaaaaa@8C8C@

Here the generalized Poisson’s ratios are not symmetric but instead satisfy ν ij / E i = ν ji / E j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUnaaBa aaleaacaWGPbGaamOAaaqabaGccaGGVaGaamyramaaBaaaleaacaWG Pbaabeaakiabg2da9iabe27aUnaaBaaaleaacaWGQbGaamyAaaqaba GccaGGVaGaamyramaaBaaaleaacaWGQbaabeaaaaa@42ED@  (no sums). This ensures that the stiffness matrix is symmetric.

 

The engineering constants are related to the components of the compliance tensor by

c 11 = E 1 ( 1 ν 23 ν 32 )ϒ c 22 = E 2 ( 1 ν 13 ν 31 )ϒ c 33 = E 3 ( 1 ν 12 ν 21 )ϒ c 12 = E 1 ( ν 21 + ν 31 ν 23 )ϒ= E 2 ( ν 12 + ν 32 ν 13 )ϒ c 13 = E 1 ( ν 31 + ν 21 ν 32 )ϒ= E 3 ( ν 13 + ν 12 ν 23 )ϒ c 23 = E 2 ( ν 32 + ν 12 ν 31 )ϒ= E 3 ( ν 23 + ν 21 ν 13 )ϒ c 44 = μ 23 c 55 = μ 13 c 66 = μ 12 ϒ= 1 1 ν 12 ν 21 ν 23 ν 32 ν 31 ν 13 2 ν 21 ν 32 ν 13 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaam4yam aaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqpcaWGfbWaaSbaaSqa aiaaigdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaSbaaS qaaiaaikdacaaIZaaabeaakiabe27aUnaaBaaaleaacaaIZaGaaGOm aaqabaaakiaawIcacaGLPaaacqqHspqOcaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4yamaa BaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpcaWGfbWaaSbaaSqaai aaikdaaeqaaOWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaSbaaSqa aiaaigdacaaIZaaabeaakiabe27aUnaaBaaaleaacaaIZaGaaGymaa qabaaakiaawIcacaGLPaaacqqHspqOcaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8Uaam4yamaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaaiodaaeqaaOWaaeWaaeaa caaIXaGaeyOeI0IaeqyVd42aaSbaaSqaaiaaigdacaaIYaaabeaaki abe27aUnaaBaaaleaacaaIYaGaaGymaaqabaaakiaawIcacaGLPaaa cqqHspqOaeaacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2 da9iaadweadaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiabe27aUnaa BaaaleaacaaIYaGaaGymaaqabaGccqGHRaWkcqaH9oGBdaWgaaWcba GaaG4maiaaigdaaeqaaOGaeqyVd42aaSbaaSqaaiaaikdacaaIZaaa beaaaOGaayjkaiaawMcaaiabfk9aHkabg2da9iaadweadaWgaaWcba GaaGOmaaqabaGcdaqadaqaaiabe27aUnaaBaaaleaacaaIXaGaaGOm aaqabaGccqGHRaWkcqaH9oGBdaWgaaWcbaGaaG4maiaaikdaaeqaaO GaeqyVd42aaSbaaSqaaiaaigdacaaIZaaabeaaaOGaayjkaiaawMca aiabfk9aHcqaaiaadogadaWgaaWcbaGaaGymaiaaiodaaeqaaOGaey ypa0JaamyramaaBaaaleaacaaIXaaabeaakmaabmaabaGaeqyVd42a aSbaaSqaaiaaiodacaaIXaaabeaakiabgUcaRiabe27aUnaaBaaale aacaaIYaGaaGymaaqabaGccqaH9oGBdaWgaaWcbaGaaG4maiaaikda aeqaaaGccaGLOaGaayzkaaGaeuO0deQaeyypa0JaamyramaaBaaale aacaaIZaaabeaakmaabmaabaGaeqyVd42aaSbaaSqaaiaaigdacaaI ZaaabeaakiabgUcaRiabe27aUnaaBaaaleaacaaIXaGaaGOmaaqaba GccqaH9oGBdaWgaaWcbaGaaGOmaiaaiodaaeqaaaGccaGLOaGaayzk aaGaeuO0dekabaGaam4yamaaBaaaleaacaaIYaGaaG4maaqabaGccq GH9aqpcaWGfbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacqaH9oGB daWgaaWcbaGaaG4maiaaikdaaeqaaOGaey4kaSIaeqyVd42aaSbaaS qaaiaaigdacaaIYaaabeaakiabe27aUnaaBaaaleaacaaIZaGaaGym aaqabaaakiaawIcacaGLPaaacqqHspqOcqGH9aqpcaWGfbWaaSbaaS qaaiaaiodaaeqaaOWaaeWaaeaacqaH9oGBdaWgaaWcbaGaaGOmaiaa iodaaeqaaOGaey4kaSIaeqyVd42aaSbaaSqaaiaaikdacaaIXaaabe aakiabe27aUnaaBaaaleaacaaIXaGaaG4maaqabaaakiaawIcacaGL PaaacqqHspqOaeaacaWGJbWaaSbaaSqaaiaaisdacaaI0aaabeaaki abg2da9iabeY7aTnaaBaaaleaacaaIYaGaaG4maaqabaGccaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadogada WgaaWcbaGaaGynaiaaiwdaaeqaaOGaeyypa0JaeqiVd02aaSbaaSqa aiaaigdacaaIZaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaadogadaWgaaWcbaGaaGOnaiaa iAdaaeqaaOGaeyypa0JaeqiVd02aaSbaaSqaaiaaigdacaaIYaaabe aaaOqaaiabfk9aHkabg2da9maalaaabaGaaGymaaqaaiaaigdacqGH sislcqaH9oGBdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeqyVd42aaS baaSqaaiaaikdacaaIXaaabeaakiabgkHiTiabe27aUnaaBaaaleaa caaIYaGaaG4maaqabaGccqaH9oGBdaWgaaWcbaGaaG4maiaaikdaae qaaOGaeyOeI0IaeqyVd42aaSbaaSqaaiaaiodacaaIXaaabeaakiab e27aUnaaBaaaleaacaaIXaGaaG4maaqabaGccqGHsislcaaIYaGaeq yVd42aaSbaaSqaaiaaikdacaaIXaaabeaakiabe27aUnaaBaaaleaa caaIZaGaaGOmaaqabaGccqaH9oGBdaWgaaWcbaGaaGymaiaaiodaae qaaaaaaaaa@4A28@

or in inverse form

E 1 =( c 11 c 22 c 33 +2 c 23 c 12 c 13 c 11 c 23 2 c 22 c 13 2 c 33 c 12 2 )/( c 22 c 33 c 23 2 ) E 2 =( c 11 c 22 c 33 +2 c 23 c 12 c 13 c 11 c 23 2 c 22 c 13 2 c 33 c 12 2 )/( c 11 c 33 c 13 2 ) E 3 =( c 11 c 22 c 33 +2 c 23 c 12 c 13 c 11 c 23 2 c 22 c 13 2 c 33 c 12 2 )/( c 11 c 22 c 12 2 ) ν 21 =( c 12 c 33 c 13 c 23 )/( c 11 c 33 c 13 2 ) ν 12 =( c 12 c 33 c 13 c 23 )/( c 22 c 33 c 23 2 ) ν 31 =( c 13 c 22 c 12 c 23 )/( c 11 c 22 c 12 2 ) ν 13 =( c 22 c 13 c 12 c 23 )/( c 22 c 33 c 23 2 ) ν 23 =( c 11 c 23 c 12 c 13 )/( c 11 c 33 c 13 2 ) ν 32 =( c 11 c 23 c 12 c 13 )/( c 11 c 22 c 12 2 ) μ 23 = c 44 , μ 13 = c 55 μ 12 = c 66 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadweadaWgaaWcbaGaaGymaa qabaGccqGH9aqpdaqadaqaaiaadogadaWgaaWcbaGaaGymaiaaigda aeqaaOGaam4yamaaBaaaleaacaaIYaGaaGOmaaqabaGccaWGJbWaaS baaSqaaiaaiodacaaIZaaabeaakiabgUcaRiaaikdacaWGJbWaaSba aSqaaiaaikdacaaIZaaabeaakiaadogadaWgaaWcbaGaaGymaiaaik daaeqaaOGaam4yamaaBaaaleaacaaIXaGaaG4maaqabaGccqGHsisl caWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaadogadaqhaaWcba GaaGOmaiaaiodaaeaacaaIYaaaaOGaeyOeI0Iaam4yamaaBaaaleaa caaIYaGaaGOmaaqabaGccaWGJbWaa0baaSqaaiaaigdacaaIZaaaba GaaGOmaaaakiabgkHiTiaadogadaWgaaWcbaGaaG4maiaaiodaaeqa aOGaam4yamaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaaakiaawI cacaGLPaaacaGGVaWaaeWaaeaacaWGJbWaaSbaaSqaaiaaikdacaaI YaaabeaakiaadogadaWgaaWcbaGaaG4maiaaiodaaeqaaOGaeyOeI0 Iaam4yamaaDaaaleaacaaIYaGaaG4maaqaaiaaikdaaaaakiaawIca caGLPaaaaeaacaWGfbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Zaae WaaeaacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaadogadaWg aaWcbaGaaGOmaiaaikdaaeqaaOGaam4yamaaBaaaleaacaaIZaGaaG 4maaqabaGccqGHRaWkcaaIYaGaam4yamaaBaaaleaacaaIYaGaaG4m aaqabaGccaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaadogada WgaaWcbaGaaGymaiaaiodaaeqaaOGaeyOeI0Iaam4yamaaBaaaleaa caaIXaGaaGymaaqabaGccaWGJbWaa0baaSqaaiaaikdacaaIZaaaba GaaGOmaaaakiabgkHiTiaadogadaWgaaWcbaGaaGOmaiaaikdaaeqa aOGaam4yamaaDaaaleaacaaIXaGaaG4maaqaaiaaikdaaaGccqGHsi slcaWGJbWaaSbaaSqaaiaaiodacaaIZaaabeaakiaadogadaqhaaWc baGaaGymaiaaikdaaeaacaaIYaaaaaGccaGLOaGaayzkaaGaai4lam aabmaabaGaam4yamaaBaaaleaacaaIXaGaaGymaaqabaGccaWGJbWa aSbaaSqaaiaaiodacaaIZaaabeaakiabgkHiTiaadogadaqhaaWcba GaaGymaiaaiodaaeaacaaIYaaaaaGccaGLOaGaayzkaaaabaGaamyr amaaBaaaleaacaaIZaaabeaakiabg2da9maabmaabaGaam4yamaaBa aaleaacaaIXaGaaGymaaqabaGccaWGJbWaaSbaaSqaaiaaikdacaaI YaaabeaakiaadogadaWgaaWcbaGaaG4maiaaiodaaeqaaOGaey4kaS IaaGOmaiaadogadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaam4yamaa BaaaleaacaaIXaGaaGOmaaqabaGccaWGJbWaaSbaaSqaaiaaigdaca aIZaaabeaakiabgkHiTiaadogadaWgaaWcbaGaaGymaiaaigdaaeqa aOGaam4yamaaDaaaleaacaaIYaGaaG4maaqaaiaaikdaaaGccqGHsi slcaWGJbWaaSbaaSqaaiaaikdacaaIYaaabeaakiaadogadaqhaaWc baGaaGymaiaaiodaaeaacaaIYaaaaOGaeyOeI0Iaam4yamaaBaaale aacaaIZaGaaG4maaqabaGccaWGJbWaa0baaSqaaiaaigdacaaIYaaa baGaaGOmaaaaaOGaayjkaiaawMcaaiaac+cadaqadaqaaiaadogada WgaaWcbaGaaGymaiaaigdaaeqaaOGaam4yamaaBaaaleaacaaIYaGa aGOmaaqabaGccqGHsislcaWGJbWaa0baaSqaaiaaigdacaaIYaaaba GaaGOmaaaaaOGaayjkaiaawMcaaaqaaiabe27aUnaaBaaaleaacaaI YaGaaGymaaqabaGccqGH9aqpcaGGOaGaam4yamaaBaaaleaacaaIXa GaaGOmaaqabaGccaWGJbWaaSbaaSqaaiaaiodacaaIZaaabeaakiab gkHiTiaadogadaWgaaWcbaGaaGymaiaaiodaaeqaaOGaam4yamaaBa aaleaacaaIYaGaaG4maaqabaGccaGGPaGaai4laiaacIcacaWGJbWa aSbaaSqaaiaaigdacaaIXaaabeaakiaadogadaWgaaWcbaGaaG4mai aaiodaaeqaaOGaeyOeI0Iaam4yamaaDaaaleaacaaIXaGaaG4maaqa aiaaikdaaaGccaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaeqyVd42aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2 da9iaacIcacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaadoga daWgaaWcbaGaaG4maiaaiodaaeqaaOGaeyOeI0Iaam4yamaaBaaale aacaaIXaGaaG4maaqabaGccaWGJbWaaSbaaSqaaiaaikdacaaIZaaa beaakiaacMcacaGGVaGaaiikaiaadogadaWgaaWcbaGaaGOmaiaaik daaeqaaOGaam4yamaaBaaaleaacaaIZaGaaG4maaqabaGccqGHsisl 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XaaabeaakiaadogadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaeyOeI0 Iaam4yamaaBaaaleaacaaIXaGaaGOmaaqabaGccaWGJbWaaSbaaSqa aiaaigdacaaIZaaabeaakiaacMcacaGGVaGaaiikaiaadogadaWgaa WcbaGaaGymaiaaigdaaeqaaOGaam4yamaaBaaaleaacaaIZaGaaG4m aaqabaGccqGHsislcaWGJbWaa0baaSqaaiaaigdacaaIZaaabaGaaG OmaaaakiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaeqyVd42aaSbaaSqaaiaaiodacaaIYaaabeaakiab g2da9iaacIcacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaado gadaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaeyOeI0Iaam4yamaaBaaa leaacaaIXaGaaGOmaaqabaGccaWGJbWaaSbaaSqaaiaaigdacaaIZa aabeaakiaacMcacaGGVaGaaiikaiaadogadaWgaaWcbaGaaGymaiaa igdaaeqaaOGaam4yamaaBaaaleaacaaIYaGaaGOmaaqabaGccqGHsi slcaWGJbWaa0baaSqaaiaaigdacaaIYaaabaGaaGOmaaaakiaacMca aeaacqaH8oqBdaWgaaWcbaGaaGOmaiaaiodaaeqaaOGaeyypa0Jaam 4yamaaBaaaleaacaaI0aGaaGinaaqabaGccaGGSaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abeY7aTnaaBaaaleaacaaIXaGaaG4maaqabaGccqGH9aqpcaWGJbWa aSbaaSqaaiaaiwdacaaI1aaabeaakiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaeqiVd02aaSbaaSqaaiaaigdacaaIYaaabeaaki abg2da9iaadogadaWgaaWcbaGaaGOnaiaaiAdaaeqaaaaaaa@005D@

 

For an orthotropic material thermal expansion cannot induce shear (in this basis) but the expansion in the three directions need not be equal.  Consequently the thermal expansion coefficient tensor has the form

[ α 1 0 0 0 α 2 0 0 0 α 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaamWaaeaafaqabeWadaaabaGaeqySde 2aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaI WaaabaGaeqySde2aaSbaaSqaaiaaikdaaeqaaaGcbaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaeqySde2aaSbaaSqaaiaaiodaaeqaaaaa aOGaay5waiaaw2faaaaa@3EE7@

 

 

 

 

3.2.14 Stress-strain relations for linear elastic Transversely Isotropic Material

 

A special case of an orthotropic solid is one that contains a plane of isotropy (this implies that the solid can be rotated with respect to the loading direction about one axis without measurable effect on the solid’s response).  Choose e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahwgadaWgaa WcbaGaaG4maaqabaaaaa@36ED@  perpendicular to this symmetry plane.  Then, transverse isotropy requires that c 22 = c 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaikdacaaIYa aabeaakiabg2da9iaadogadaWgaaWcbaGaaGymaiaaigdaaeqaaaaa @38EC@ , c 23 = c 13 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaikdacaaIZa aabeaakiabg2da9iaadogadaWgaaWcbaGaaGymaiaaiodaaeqaaaaa @38EF@ , c 55 = c 44 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaiwdacaaI1a aabeaakiabg2da9iaadogadaWgaaWcbaGaaGinaiaaisdaaeqaaaaa @38F8@ , c 66 =( c 11 c 12 )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaiAdacaaI2a aabeaakiabg2da9iaacIcacaWGJbWaaSbaaSqaaiaaigdacaaIXaaa beaakiabgkHiTiaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaai ykaiaac+cacaaIYaaaaa@3F48@ , so that the stiffness matrix has the form                          

C=[ c 11 c 12 c 13 0 0 0 c 11 c 13 0 0 0 c 33 0 0 0 sym c 44 0 0 0 c 44 0 0 0 ( c 11 c 12 )/2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahoeacqGH9a qpdaWadaqaauaabeqagyaaaaaabaGaam4yamaaBaaaleaacaaIXaGa aGymaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaaaO qaaiaadogadaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaaGimaaqa aiaaicdaaeaacaaIWaaabaaabaGaam4yamaaBaaaleaacaaIXaGaaG ymaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIZaaabeaaaOqa aiaaicdaaeaacaaIWaaabaGaaGimaaqaaaqaaaqaaiaadogadaWgaa WcbaGaaG4maiaaiodaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaI WaaabaaabaGaam4CaiaadMhacaWGTbaabaaabaGaam4yamaaBaaale aacaaI0aGaaGinaaqabaaakeaacaaIWaaabaGaaGimaaqaaaqaaaqa aaqaaiaaicdaaeaacaWGJbWaaSbaaSqaaiaaisdacaaI0aaabeaaaO qaaiaaicdaaeaaaeaaaeaaaeaacaaIWaaabaGaaGimaaqaaiaacIca caWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgkHiTiaadogada WgaaWcbaGaaGymaiaaikdaaeqaaOGaaiykaiaac+cacaaIYaaaaaGa ay5waiaaw2faaaaa@6465@

The engineering constants must satisfy

E 1 = E 2 = E p E 3 = E t ν 12 = ν 21 = ν p ν 31 = ν 32 = ν tp ν 13 = ν 23 = ν pt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyram aaBaaaleaacaaIXaaabeaakiabg2da9iaadweadaWgaaWcbaGaaGOm aaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaadchaaeqaaOGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadweadaWgaaWc baGaaG4maaqabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaadshaaeqaaa GcbaGaeqyVd42aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iab e27aUnaaBaaaleaacaaIYaGaaGymaaqabaGccqGH9aqpcqaH9oGBda WgaaWcbaGaamiCaaqabaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyVd42aaSbaaSqaai aaiodacaaIXaaabeaakiabg2da9iabe27aUnaaBaaaleaacaaIZaGa aGOmaaqabaGccqGH9aqpcqaH9oGBdaWgaaWcbaGaamiDaiaadchaae qaaaGcbaGaeqyVd42aaSbaaSqaaiaaigdacaaIZaaabeaakiabg2da 9iabe27aUnaaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpcqaH9o GBdaWgaaWcbaGaamiCaiaadshaaeqaaOGaaGPaVdaaaa@822D@

and the compliance matrix has the form

S=[ 1/ E p ν p / E p ν tp / E t 0 0 0 ν p / E p 1/ E p ν tp / E t 0 0 0 ν pt / E p ν pt / E p 1/ E t 0 0 0 0 0 0 1/ μ t 0 0 0 0 0 0 1/ μ t 0 0 0 0 0 0 1/ μ p ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahofacqGH9a qpdaWadaqaauaabeqagyaaaaaabaGaaGymaiaac+cacaWGfbWaaSba aSqaaiaadchaaeqaaaGcbaGaeyOeI0IaeqyVd42aaSbaaSqaaiaadc haaeqaaOGaai4laiaadweadaWgaaWcbaGaamiCaaqabaaakeaacqGH sislcqaH9oGBdaWgaaWcbaGaamiDaiaadchaaeqaaOGaai4laiaadw eadaWgaaWcbaGaamiDaaqabaaakeaacaaIWaaabaGaaGimaaqaaiaa icdaaeaacqGHsislcqaH9oGBdaWgaaWcbaGaamiCaaqabaGccaGGVa GaamyramaaBaaaleaacaWGWbaabeaaaOqaaiaaigdacaGGVaGaamyr amaaBaaaleaacaWGWbaabeaaaOqaaiabgkHiTiabe27aUnaaBaaale aacaWG0bGaamiCaaqabaGccaGGVaGaamyramaaBaaaleaacaWG0baa beaaaOqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiabgkHiTiabe2 7aUnaaBaaaleaacaWGWbGaamiDaaqabaGccaGGVaGaamyramaaBaaa leaacaWGWbaabeaaaOqaaiabgkHiTiabe27aUnaaBaaaleaacaWGWb GaamiDaaqabaGccaGGVaGaamyramaaBaaaleaacaWGWbaabeaaaOqa aiaaigdacaGGVaGaamyramaaBaaaleaacaWG0baabeaaaOqaaiaaic daaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGim aaqaaiaaigdacaGGVaGaeqiVd02aaSbaaSqaaiaadshaaeqaaaGcba GaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaa caaIWaaabaGaaGymaiaac+cacqaH8oqBdaWgaaWcbaGaamiDaaqaba aakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGim aaqaaiaaicdaaeaacaaIXaGaai4laiabeY7aTnaaBaaaleaacaWGWb aabeaaaaaakiaawUfacaGLDbaaaaa@8C62@

where μ p = E p /2(1+ ν p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeY7aTnaaBa aaleaacaWGWbaabeaakiabg2da9iaadweadaWgaaWcbaGaamiCaaqa baGccaGGVaGaaGOmaiaacIcacaaIXaGaey4kaSIaeqyVd42aaSbaaS qaaiaadchaaeqaaOGaaiykaaaa@423C@ .  As before the Poisson’s ratios are not symmetric, but satisfy ν tp / E t = ν pt / E p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabe27aUnaaBa aaleaacaWG0bGaamiCaaqabaGccaGGVaGaamyramaaBaaaleaacaWG 0baabeaakiabg2da9iabe27aUnaaBaaaleaacaWGWbGaamiDaaqaba GccaGGVaGaamyramaaBaaaleaacaWGWbaabeaaaaa@4320@

 

The engineering constants and stiffnesses are related by

c 11 = c 22 = E p ( 1 ν pt ν tp )ϒ c 33 = E t ( 1 ν p 2 )ϒ c 12 = E p ( ν p + ν pt ν tp )ϒ c 13 = c 23 = E p ( ν tp + ν p ν tp )ϒ= E t ( ν pt + ν p ν pt )ϒ c 44 = μ t c 66 = μ p ϒ= 1 1 ν p 2 2 ν pt ν tp 2 ν p ν pt ν tp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaam4yam aaBaaaleaacaaIXaGaaGymaaqabaGccqGH9aqpcaWGJbWaaSbaaSqa aiaaikdacaaIYaaabeaakiabg2da9iaadweadaWgaaWcbaGaamiCaa qabaGcdaqadaqaaiaaigdacqGHsislcqaH9oGBdaWgaaWcbaGaamiC aiaadshaaeqaaOGaeqyVd42aaSbaaSqaaiaadshacaWGWbaabeaaaO GaayjkaiaawMcaaiabfk9aHkaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8Uaam4yamaaBaaaleaacaaIZaGaaG4maa qabaGccqGH9aqpcaWGfbWaaSbaaSqaaiaadshaaeqaaOWaaeWaaeaa caaIXaGaeyOeI0IaeqyVd42aa0baaSqaaiaadchaaeaacaaIYaaaaa GccaGLOaGaayzkaaGaeuO0deQaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaO Gaeyypa0JaamyramaaBaaaleaacaWGWbaabeaakmaabmaabaGaeqyV d42aaSbaaSqaaiaadchaaeqaaOGaey4kaSIaeqyVd42aaSbaaSqaai aadchacaWG0baabeaakiabe27aUnaaBaaaleaacaWG0bGaamiCaaqa baaakiaawIcacaGLPaaacqqHspqOaeaacaWGJbWaaSbaaSqaaiaaig dacaaIZaaabeaakiabg2da9iaadogadaWgaaWcbaGaaGOmaiaaioda aeqaaOGaeyypa0JaamyramaaBaaaleaacaWGWbaabeaakmaabmaaba GaeqyVd42aaSbaaSqaaiaadshacaWGWbaabeaakiabgUcaRiabe27a UnaaBaaaleaacaWGWbaabeaakiabe27aUnaaBaaaleaacaWG0bGaam iCaaqabaaakiaawIcacaGLPaaacqqHspqOcqGH9aqpcaWGfbWaaSba aSqaaiaadshaaeqaaOWaaeWaaeaacqaH9oGBdaWgaaWcbaGaamiCai aadshaaeqaaOGaey4kaSIaeqyVd42aaSbaaSqaaiaadchaaeqaaOGa eqyVd42aaSbaaSqaaiaadchacaWG0baabeaaaOGaayjkaiaawMcaai abfk9aHkaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caWGJbWaaSbaaSqaaiaaisdacaaI0aaabe aakiabg2da9iabeY7aTnaaBaaaleaacaWG0baabeaakiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGJbWaaSba aSqaaiaaiAdacaaI2aaabeaakiabg2da9iabeY7aTnaaBaaaleaaca WGWbaabeaaaOqaaiabfk9aHkabg2da9maalaaabaGaaGymaaqaaiaa igdacqGHsislcqaH9oGBdaqhaaWcbaGaamiCaaqaaiaaikdaaaGccq GHsislcaaIYaGaeqyVd42aaSbaaSqaaiaadchacaWG0baabeaakiab e27aUnaaBaaaleaacaWG0bGaamiCaaqabaGccqGHsislcaaIYaGaeq yVd42aaSbaaSqaaiaadchaaeqaaOGaeqyVd42aaSbaaSqaaiaadcha caWG0baabeaakiabe27aUnaaBaaaleaacaWG0bGaamiCaaqabaaaaa aaaa@0362@

E p =( c 11 2 c 33 +2 c 13 2 c 12 2 c 11 c 13 2 c 33 c 12 2 )/( c 11 c 33 c 13 2 ) E t =( c 11 2 c 33 +2 c 13 2 c 12 2 c 11 c 13 2 c 33 c 12 2 )/( c 11 2 c 12 2 ) ν p =( c 12 c 33 c 13 2 )/( c 11 c 33 c 13 2 ), ν tp =( c 13 c 11 c 12 c 13 )/( c 11 2 c 12 2 ) ν pt =( c 11 c 13 c 12 c 13 )/( c 11 c 33 c 13 2 ), μ 23 = c 44 , μ 13 = c 55 μ 12 = c 66 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadweadaWgaaWcbaGaamiCaa qabaGccqGH9aqpdaqadaqaaiaadogadaqhaaWcbaGaaGymaiaaigda aeaacaaIYaaaaOGaam4yamaaBaaaleaacaaIZaGaaG4maaqabaGccq GHRaWkcaaIYaGaam4yamaaDaaaleaacaaIXaGaaG4maaqaaiaaikda aaGccaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgkHiTiaaik dacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaadogadaqhaaWc baGaaGymaiaaiodaaeaacaaIYaaaaOGaeyOeI0Iaam4yamaaBaaale aacaaIZaGaaG4maaqabaGccaWGJbWaa0baaSqaaiaaigdacaaIYaaa baGaaGOmaaaaaOGaayjkaiaawMcaaiaac+cadaqadaqaaiaadogada WgaaWcbaGaaGymaiaaigdaaeqaaOGaam4yamaaBaaaleaacaaIZaGa aG4maaqabaGccqGHsislcaWGJbWaa0baaSqaaiaaigdacaaIZaaaba GaaGOmaaaaaOGaayjkaiaawMcaaaqaaiaadweadaWgaaWcbaGaamiD aaqabaGccqGH9aqpdaqadaqaaiaadogadaqhaaWcbaGaaGymaiaaig daaeaacaaIYaaaaOGaam4yamaaBaaaleaacaaIZaGaaG4maaqabaGc cqGHRaWkcaaIYaGaam4yamaaDaaaleaacaaIXaGaaG4maaqaaiaaik daaaGccaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgkHiTiaa ikdacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaadogadaqhaa WcbaGaaGymaiaaiodaaeaacaaIYaaaaOGaeyOeI0Iaam4yamaaBaaa leaacaaIZaGaaG4maaqabaGccaWGJbWaa0baaSqaaiaaigdacaaIYa aabaGaaGOmaaaaaOGaayjkaiaawMcaaiaac+cadaqadaqaaiaadoga daqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaOGaeyOeI0Iaam4yam aaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaaaakiaawIcacaGLPaaa aeaacqaH9oGBdaWgaaWcbaGaamiCaaqabaGccqGH9aqpcaGGOaGaam 4yamaaBaaaleaacaaIXaGaaGOmaaqabaGccaWGJbWaaSbaaSqaaiaa iodacaaIZaaabeaakiabgkHiTiaadogadaqhaaWcbaGaaGymaiaaio daaeaacaaIYaaaaOGaaiykaiaac+cacaGGOaGaam4yamaaBaaaleaa caaIXaGaaGymaaqabaGccaWGJbWaaSbaaSqaaiaaiodacaaIZaaabe aakiabgkHiTiaadogadaqhaaWcbaGaaGymaiaaiodaaeaacaaIYaaa aOGaaiykaiaacYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlabe27aUnaaBaaaleaacaWG0bGaamiCaaqabaGc cqGH9aqpcaGGOaGaam4yamaaBaaaleaacaaIXaGaaG4maaqabaGcca WGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgkHiTiaadogadaWg aaWcbaGaaGymaiaaikdaaeqaaOGaam4yamaaBaaaleaacaaIXaGaaG 4maaqabaGccaGGPaGaai4laiaacIcacaWGJbWaa0baaSqaaiaaigda caaIXaaabaGaaGOmaaaakiabgkHiTiaadogadaqhaaWcbaGaaGymai aaikdaaeaacaaIYaaaaOGaaiykaaqaaiabe27aUnaaBaaaleaacaWG WbGaamiDaaqabaGccqGH9aqpcaGGOaGaam4yamaaBaaaleaacaaIXa GaaGymaaqabaGccaWGJbWaaSbaaSqaaiaaigdacaaIZaaabeaakiab gkHiTiaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaam4yamaaBa aaleaacaaIXaGaaG4maaqabaGccaGGPaGaai4laiaacIcacaWGJbWa aSbaaSqaaiaaigdacaaIXaaabeaakiaadogadaWgaaWcbaGaaG4mai aaiodaaeqaaOGaeyOeI0Iaam4yamaaDaaaleaacaaIXaGaaG4maaqa aiaaikdaaaGccaGGPaGaaiilaiaaykW7caaMc8UaaGPaVlabeY7aTn aaBaaaleaacaaIYaGaaG4maaqabaGccqGH9aqpcaWGJbWaaSbaaSqa aiaaisdacaaI0aaabeaakiaacYcacaaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlabeY7aTnaaBaaaleaacaaIXaGaaG4maaqabaGccqGH9aqp caWGJbWaaSbaaSqaaiaaiwdacaaI1aaabeaakiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabeY7aTnaaBaaaleaacaaIXaGaaGOm aaqabaGccqGH9aqpcaWGJbWaaSbaaSqaaiaaiAdacaaI2aaabeaaaa aa@143A@

 

For this material the two thermal expansion coefficients in the symmetry plane must be equal, so the thermal expansion coefficient tensor has the form

[ α 1 0 0 0 α 1 0 0 0 α 3 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaamWaaeaafaqabeWadaaabaGaeqySde 2aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaI WaaabaGaeqySde2aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaeqySde2aaSbaaSqaaiaaiodaaeqaaaaa aOGaay5waiaaw2faaaaa@3EF6@

 

 

 

 

3.2.15 Representative values for elastic constants of transversely isotropic hexagonal close packed crystals

 

Hexagonal close-packed crystals are an example of transversely isotropic materials.  The e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  axis must be taken to be perpendicular to the basal (0001) plane of the crystal, as shown in the picture.  Since the plane perpendicular to e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  is isotropic the orientation of e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa aa@349B@  and e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaikdaaeqaaa aa@349C@  is arbitrary.

 

A table of values of stiffnesses (taken from Freund and Suresh, Thin Film Materials, CUP 2003) is listed below.  F&S list the original sources for their data on page 163.

 

 

 

 

c 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXa aabeaaaaa@3550@  (GPa)

c 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaiodacaaIZa aabeaaaaa@3554@  (GPa)

c 44 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaisdacaaI0a aabeaaaaa@3556@  (GPa)

c 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIYa aabeaaaaa@3551@  (GPa)

c 13 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIZa aabeaaaaa@3552@  (GPa)

Be

292.3

336.4

162.5

26.7

14

C

1160

46.6

2.3

290

109

Cd

115.8

51.4

20.4

39.8

40.6

Co

307

358.1

78.3

165

103

Hf

181.1

196.9

55.7

77.2

66.1

Mg

59.7

61.7

16.4

26.2

21.7

Ti

162.4

180.7

46.7

92

69

Zn

161

61

38.3

34.2

50.1

Zr

143.4

164.8

32

72.8

65.3

ZnO

209.7

210.9

42.5

121.1

105.1

 

The engineering constants can be calculated to be

 

 

E p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaSbaaSqaaiaadchaaeqaaa aa@34B1@  (GPa)

E t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbWaaSbaaSqaaiaadshaaeqaaa aa@34B5@  (GPa)

ν p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBdaWgaaWcbaGaamiCaaqaba aaaa@359F@

 

ν tp MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBdaWgaaWcbaGaamiDaiaadc haaeqaaaaa@3698@

 

ν pt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBdaWgaaWcbaGaamiCaiaads haaeqaaaaa@3698@

 

μ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBdaWgaaWcbaGaamiDaaqaba aaaa@35A1@
(GPa)

μ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBdaWgaaWcbaGaamiCaaqaba aaaa@359D@

(GPa)

Be

289.38

335.17

0.09

0.04

0.04

162.50

132.80

C

903.69

30.21

0.04

0.08

2.25

2.30

435.00

Cd

83.02

30.21

0.09

0.26

0.72

20.40

38.00

Co

211.30

313.15

0.49

0.22

0.15

78.30

71.00

Hf

139.87

163.07

0.35

0.26

0.22

55.70

51.95

Mg

45.45

50.74

0.36

0.25

0.23

16.40

16.75

Ti

104.37

143.27

0.48

0.27

0.20

46.70

35.20

Zn

119.45

35.28

-0.06

0.26

0.87

38.30

63.40

Zr

98.79

125.35

0.40

0.30

0.24

32.00

35.30

ZnO

127.30

144.12

0.44

0.32

0.28

42.50

44.30

 

 

 

 

3.2.16 Linear elastic stress-strain relations for cubic materials

 

A huge number of materials have cubic symmetry MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  all the FCC and BCC metals, for example.  The constitutive law for such a material is particularly simple, and can be parameterized by only 3 material constants.  Pick basis vectors perpendicular to the symmetry planes, as shown.

 

Then

[ σ 11 σ 22 σ 33 σ 23 σ 12 σ 13 ]=[ c 11 c 12 c 12 0 0 0 c 11 c 12 0 0 0 c 11 0 0 0 sym c 44 0 0 0 c 44 0 0 0 c 44 ][ ε 11 ε 22 ε 33 2 ε 23 2 ε 12 2 ε 13 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqbae qabyqaaaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaaaOqa aiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacqaHdpWCda WgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaa ikdacaaIZaaabeaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGOmaa qabaaakeaacqaHdpWCdaWgaaWcbaGaaGymaiaaiodaaeqaaaaaaOGa ay5waiaaw2faaiabg2da9maadmaabaqbaeqabyGbaaaaaeaacaWGJb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadogadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIXaGaaGOmaa qabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaaeaacaWGJbWa aSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadogadaWgaaWcbaGaaG ymaiaaikdaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaIWaaabaaa baaabaGaam4yamaaBaaaleaacaaIXaGaaGymaaqabaaakeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaaaeaacaWGZbGaamyEaiaad2gaaeaa aeaacaWGJbWaaSbaaSqaaiaaisdacaaI0aaabeaaaOqaaiaaicdaae aacaaIWaaabaaabaaabaaabaGaaGimaaqaaiaadogadaWgaaWcbaGa aGinaiaaisdaaeqaaaGcbaGaaGimaaqaaaqaaaqaaaqaaiaaicdaae aacaaIWaaabaGaam4yamaaBaaaleaacaaI0aGaaGinaaqabaaaaaGc caGLBbGaayzxaaWaamWaaeaafaqabeGbbaaaaeaacqaH1oqzdaWgaa WcbaGaaGymaiaaigdaaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaikda caaIYaaabeaaaOqaaiabew7aLnaaBaaaleaacaaIZaGaaG4maaqaba aakeaacaaIYaGaeqyTdu2aaSbaaSqaaiaaikdacaaIZaaabeaaaOqa aiaaikdacqaH1oqzdaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaaG Omaiabew7aLnaaBaaaleaacaaIXaGaaG4maaqabaaaaaGccaGLBbGa ayzxaaaaaa@8C35@

or in terms of engineering constants

[ ε 11 ε 22 ε 33 2 ε 23 2 ε 12 2 ε 13 ]=[ 1/E ν/E ν/E 0 0 0 ν/E 1/E ν/E 0 0 0 ν/E ν/E 1/E 0 0 0 0 0 0 1/μ 0 0 0 0 0 0 1/μ 0 0 0 0 0 0 1/μ ][ σ 11 σ 22 σ 33 σ 23 σ 12 σ 13 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqbae qabyqaaaaabaGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaaaOqa aiabew7aLnaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacqaH1oqzda WgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaaGOmaiabew7aLnaaBaaa leaacaaIYaGaaG4maaqabaaakeaacaaIYaGaeqyTdu2aaSbaaSqaai aaigdacaaIYaaabeaaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGaaGym aiaaiodaaeqaaaaaaOGaay5waiaaw2faaiabg2da9maadmaabaqbae qabyGbaaaaaeaacaaIXaGaai4laiaadweaaeaacqGHsislcqaH9oGB caGGVaGaamyraaqaaiabgkHiTiabe27aUjaac+cacaWGfbaabaGaaG imaaqaaiaaicdaaeaacaaIWaaabaGaeyOeI0IaeqyVd4Maai4laiaa dweaaeaacaaIXaGaai4laiaadweaaeaacqGHsislcqaH9oGBcaGGVa GaamyraaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiabgkHiTiab e27aUjaac+cacaWGfbaabaGaeyOeI0IaeqyVd4Maai4laiaadweaae aacaaIXaGaai4laiaadweaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXaGaai4laiabeY 7aTbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaI WaaabaGaaGimaaqaaiaaigdacaGGVaGaeqiVd0gabaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGa aGymaiaac+cacqaH8oqBaaaacaGLBbGaayzxaaWaamWaaeaafaqabe GbbaaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGa eq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabeo8aZnaaBa aaleaacaaIZaGaaG4maaqabaaakeaacqaHdpWCdaWgaaWcbaGaaGOm aiaaiodaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabe aaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaG4maaqabaaaaaGccaGL BbGaayzxaaaaaa@A16E@

This is virtually identical to the constitutive law for an isotropic solid, except that the shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeY7aTbaa@322E@  is not related to the Poisson’s ratio and Young’s modulus through the usual relation given in Section 3.1.6.   In fact, the ratio

A= 2μ(1+ν) E = 2 c 44 c 11 c 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyqaiabg2da9maalaaabaGaaGOmai abeY7aTjaacIcacaaIXaGaey4kaSIaeqyVd4Maaiykaaqaaiaadwea aaGaeyypa0ZaaSaaaeaacaaIYaGaam4yamaaBaaaleaacaaI0aGaaG inaaqabaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiab gkHiTiaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaaaaaaa@450D@

provides a convenient measure of anisotropy.  For A=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyqaiabg2da9iaaigdaaaa@3364@  the material is isotropic. 

 

For this material the thermal expansion coefficient matrix must be isotropic.

 

The relationships between the elastic constants are

E=( c 11 2 + c 12 c 11 2 c 12 2 )/( c 11 + c 12 )ν= c 12 /( c 11 + c 12 )μ= c 44 c 11 =E(1ν)/(1ν2 ν 2 ) c 12 =Eν/(1ν2 ν 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadweacqGH9aqpdaqadaqaai aadogadaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaOGaey4kaSIa am4yamaaBaaaleaacaaIXaGaaGOmaaqabaGccaWGJbWaaSbaaSqaai aaigdacaaIXaaabeaakiabgkHiTiaaikdacaWGJbWaa0baaSqaaiaa igdacaaIYaaabaGaaGOmaaaaaOGaayjkaiaawMcaaiaac+cadaqada qaaiaadogadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaey4kaSIaam4y amaaBaaaleaacaaIXaGaaGOmaaqabaaakiaawIcacaGLPaaacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlabe27aUjabg2da9iaadogadaWgaaWcbaGaaGymai aaikdaaeqaaOGaai4laiaacIcacaWGJbWaaSbaaSqaaiaaigdacaaI XaaabeaakiabgUcaRiaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaO GaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7cqaH8oqBcqGH9aqpcaWGJbWaaSbaaSqaaiaaisdacaaI0aaa beaaaOqaaiaadogadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyypa0 JaamyraiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaiaac+cacaGG OaGaaGymaiabgkHiTiabe27aUjabgkHiTiaaikdacqaH9oGBdaahaa WcbeqaaiaaikdaaaGccaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0Jaamyr aiabe27aUjaac+cacaGGOaGaaGymaiabgkHiTiabe27aUjabgkHiTi aaikdacqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPaaaaaa@BFF6@

 

 

 

 

3.2.17 Representative values for elastic properties of cubic crystals and compounds

 

A table of elastic constants for various cubic crystals and compounds (modified from Simmons and Wang ‘Single Crystal Elastic Constants and Calculated Aggregate Properties’ MIT Press (1970)) is given below

 

Material

 

 

c 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXa aabeaaaaa@3550@  (GPa)

c 44 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaisdacaaI0a aabeaaaaa@3556@  (GPa)

c 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIYa aabeaaaaa@3551@  (GPa)

E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlaadweacaaMc8UaaG PaVdaa@39BC@  (GPa)

ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlabe27aUjaaykW7ca aMc8oaaa@3AAA@

 

μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaMc8UaaGPaVlabeY7aTjaaykW7ca aMc8oaaa@3AA8@  (GPa)

A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGbbaaaa@338C@

 

Ag

(fcc)

124.00

46.10

93.40

43.75

0.43

46.10

3.01

Al

(fcc)

107.30

28.30

60.90

63.20

0.36

28.30

1.22

Au

(fcc)

192.90

41.50

163.80

42.46

0.46

41.50

2.85

Cu

(fcc)

168.40

75.40

121.40

66.69

0.42

75.40

3.21

Ir

(fcc)

580.00

256.00

242.00

437.51

0.29

256.00

1.51

Ni

(fcc)

246.50

127.40

147.30

136.31

0.37

127.40

2.57

Pb

(fcc)

49.50

14.90

42.30

10.52

0.46

14.90

4.14

Pd

(fcc)

227.10

71.70

176.00

73.41

0.44

71.70

2.81

Pt

(fcc)

346.70

76.50

250.70

136.29

0.42

76.50

1.59

Cr

(bcc)

339.80

99.00

58.60

322.56

0.15

99.00

0.70

Fe

(bcc)

231.40

116.40

134.70

132.28

0.37

116.40

2.41

K

(bcc)

4.14

2.63

2.21

2.60

0.35

2.63

2.73

Li

(bcc)

13.50

8.78

11.44

3.00

0.46

8.78

8.52

Mo

(bcc)

440.80

121.70

172.40

343.86

0.28

121.70

0.91

Na

(bcc)

6.15

5.92

4.96

1.72

0.45

5.92

9.95

Nb

(bcc)

240.20

28.20

125.60

153.95

0.34

28.20

0.49

Ta

(bcc)

260.20

82.60

154.50

145.08

0.37

82.60

1.56

V

(bcc)

228.00

42.60

118.70

146.72

0.34

42.60

0.78

W

(bcc)

522.40

160.80

204.40

407.43

0.28

160.80

1.01

C

(dc)

949.00

521.00

151.00

907.54

0.14

521.00

1.31

Ge

(dc)

128.40

66.70

48.20

102.09

0.27

66.70

1.66

Si

(dc)

166.20

79.80

64.40

130.23

0.28

79.80

1.57

GaAs

 

118.80

59.40

53.70

85.37

0.31

59.40

1.82

GaP

 

141.20

70.50

62.50

102.85

0.31

70.50

1.79

InP

 

102.20

46.00

57.60

60.68

0.36

46.00

2.06

KCl

 

39.50

6.30

4.90

38.42

0.11

6.30

0.36

LiF

 

114.00

63.60

47.70

85.86

0.29

63.60

1.92

MgO

 

287.60

151.40

87.40

246.86

0.23

151.40

1.51

NaCl

 

49.60

12.90

12.40

44.64

0.20

12.90

0.69

TiC

 

500.00

175.00

113.00

458.34

0.18

175.00

0.90