Chapter 3

 

Constitutive Models: Relations between Stress and Strain

 

 

 

3.2.  Linear Elastic Constitutive Equations

 

3.2.1.      Using the table of values given in Section 3.2.4, find values of bulk modulus, Lame modulus, and shear modulus for steel, aluminum and rubber.

 

 

3.2.2.      A specimen of an isotropic, linear elastic material with Young’s modulus E and is placed inside a rigid box that prevents the material from stretching in any direction.   This means that the strains in the specimen are zero.  The specimen is then heated to increase its temperature by ΔT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGubaaaa@3515@ .  Find a formula for the stress in the specimen.  Find a formula for the strain energy density.  How much strain energy would be stored in a 1c m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIXaGaam4yaiaad2gadaahaaWcbe qaaiaaiodaaaaaaa@3645@  sample of steel if its temperature were increased by 100C?  Compare the strain energy with the heat required to change the temperature by 100C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the specific heat capacity of steel is about 470 J/(kg-C)

 

 

3.2.3.      A specimen of an isotropic, linear elastic solid is free of stress, and is heated to increase its temperature by ΔT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGubaaaa@3515@ .  Find expressions for the strain and displacement fields in the solid.

 

 

3.2.4.      A thin isotropic, linear elastic thin film with Young’s modulus E, Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqyVd4gaaa@3295@  and thermal expansion coefficient α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqySdegaaa@327C@  is bonded to a stiff substrate.  The film is stress free at some initial temperature, and then heated to increase its temperature by T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8YjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamivaaaa@31B6@ .  The substrate prevents the film from stretching in its own plane, so that ε 11 = ε 22 = ε 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGymaiaaig daaeqaaOGaeyypa0JaeqyTdu2aaSbaaSqaaiaaikdacaaIYaaabeaa kiabg2da9iabew7aLnaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9a qpcaaIWaaaaa@408E@ , while the surface is traction free, so that the film deforms in a state of plane stress. Calculate the stresses in the film in terms of material properties and temperature, and deduce an expression for the strain energy density in the film.

 

 

 

3.2.5.      A cubic material may be characterized either by its moduli as

[ σ 11 σ 22 σ 33 σ 12 σ 13 σ 23 ]=[ 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 12 2 ε 13 2 ε 23 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqbae qabyqaaaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaaaOqa aiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacqaHdpWCda WgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaa igdacaaIYaaabeaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaG4maa qabaaakeaacqaHdpWCdaWgaaWcbaGaaGOmaiaaiodaaeqaaaaaaOGa ay5waiaaw2faaiabg2da9maadmaabaqbaeqabyGbaaaaaeaacaWGJb WaaSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadogadaWgaaWcbaGa aGymaiaaikdaaeqaaaGcbaGaam4yamaaBaaaleaacaaIXaGaaGOmaa qabaaakeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaaaeaacaWGJbWa aSbaaSqaaiaaigdacaaIXaaabeaaaOqaaiaadogadaWgaaWcbaGaaG ymaiaaikdaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacaaIWaaabaaa baaabaGaam4yamaaBaaaleaacaaIXaGaaGymaaqabaaakeaacaaIWa aabaGaaGimaaqaaiaaicdaaeaaaeaacaWGZbGaamyEaiaad2gaaeaa aeaacaWGJbWaaSbaaSqaaiaaisdacaaI0aaabeaaaOqaaiaaicdaae aacaaIWaaabaaabaaabaaabaGaaGimaaqaaiaadogadaWgaaWcbaGa aGinaiaaisdaaeqaaaGcbaGaaGimaaqaaaqaaaqaaaqaaiaaicdaae aacaaIWaaabaGaam4yamaaBaaaleaacaaI0aGaaGinaaqabaaaaaGc caGLBbGaayzxaaWaamWaaeaafaqabeGbbaaaaeaacqaH1oqzdaWgaa WcbaGaaGymaiaaigdaaeqaaaGcbaGaeqyTdu2aaSbaaSqaaiaaikda caaIYaaabeaaaOqaaiabew7aLnaaBaaaleaacaaIZaGaaG4maaqaba aakeaacaaIYaGaeqyTdu2aaSbaaSqaaiaaigdacaaIYaaabeaaaOqa aiaaikdacqaH1oqzdaWgaaWcbaGaaGymaiaaiodaaeqaaaGcbaGaaG Omaiabew7aLnaaBaaaleaacaaIYaGaaG4maaqabaaaaaGccaGLBbGa ayzxaaaaaa@8C35@

or by the engineering constants

[ ε 11 ε 22 ε 33 2 ε 12 2 ε 13 2 ε 23 ]=[ 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 12 σ 13 σ 23 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaadmaabaqbae qabyqaaaaabaGaeqyTdu2aaSbaaSqaaiaaigdacaaIXaaabeaaaOqa aiabew7aLnaaBaaaleaacaaIYaGaaGOmaaqabaaakeaacqaH1oqzda WgaaWcbaGaaG4maiaaiodaaeqaaaGcbaGaaGOmaiabew7aLnaaBaaa leaacaaIXaGaaGOmaaqabaaakeaacaaIYaGaeqyTdu2aaSbaaSqaai aaigdacaaIZaaabeaaaOqaaiaaikdacqaH1oqzdaWgaaWcbaGaaGOm aiaaiodaaeqaaaaaaOGaay5waiaaw2faaiabg2da9maadmaabaqbae qabyGbaaaaaeaacaaIXaGaai4laiaadweaaeaacqGHsislcqaH9oGB caGGVaGaamyraaqaaiabgkHiTiabe27aUjaac+cacaWGfbaabaGaaG imaaqaaiaaicdaaeaacaaIWaaabaGaeyOeI0IaeqyVd4Maai4laiaa dweaaeaacaaIXaGaai4laiaadweaaeaacqGHsislcqaH9oGBcaGGVa GaamyraaqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiabgkHiTiab e27aUjaac+cacaWGfbaabaGaeyOeI0IaeqyVd4Maai4laiaadweaae aacaaIXaGaai4laiaadweaaeaacaaIWaaabaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIXaGaai4laiabeY 7aTbqaaiaaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaI WaaabaGaaGimaaqaaiaaigdacaGGVaGaeqiVd0gabaGaaGimaaqaai aaicdaaeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGa aGymaiaac+cacqaH8oqBaaaacaGLBbGaayzxaaWaamWaaeaafaqabe GbbaaaaeaacqaHdpWCdaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGa eq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaaaOqaaiabeo8aZnaaBa aaleaacaaIZaGaaG4maaqabaaakeaacqaHdpWCdaWgaaWcbaGaaGym aiaaikdaaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIZaaabe aaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaG4maaqabaaaaaGccaGL BbGaayzxaaaaaa@A16E@

Calculate formulas relating c ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35B7@  to E,ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbGaaiilaiabe27aUbaa@34E1@  and μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBaaa@3365@ , and deduce an expression for the anisotropy factor A=2μ(1+ν)/E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamyqaiabg2da9iaaikdacqaH8oqBca GGOaGaaGymaiabgUcaRiabe27aUjaacMcacaGGVaGaamyraaaa@3B38@  in terms of c ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35B7@ .

 

 

3.2.6.      Suppose that the stress-strain relation for a linear elastic solid is expressed in matrix form as [ σ ]=[ C ][ ε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaaiabeo8aZbGaay5waiaaw2 faaiabg2da9maadmaabaGaam4qaaGaay5waiaaw2faamaadmaabaGa eqyTdugacaGLBbGaayzxaaaaaa@3DD4@ , where  [ σ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq aHdpWCaiaawUfacaGLDbaaaaa@399C@ , [ ε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq aH1oqzaiaawUfacaGLDbaaaaa@3980@  and [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38A1@  represent the stress and strain vectors and the matrix of elastic constants defined in Section 3.2.8.  Show that the material has a positive definite strain energy density ( U>0[ ε ]0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiabg6 da+iaaicdacaaMc8UaaGPaVlaaykW7cqGHaiIidaWadaqaaiabew7a LbGaay5waiaaw2faaiabgcMi5kaaicdaaaa@440E@  ) if and only if the eigenvalues of [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38A1@  are all positive.

 

 

 

3.2.7.      Write down an expression for the increment in stress resulting from an increment in strain applied to a linear elastic material, in terms of the matrix of elastic constants [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38A1@ .  Hence show that, for a linear elastic material to be stable in the sense of Drucker, the eigenvalues of the matrix of elastic constants [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38A1@  must all be positive or zero.

 

 

3.2.8.      Let [ σ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq aHdpWCaiaawUfacaGLDbaaaaa@399C@ , [ ε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq aH1oqzaiaawUfacaGLDbaaaaa@3980@  and [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38A1@  represent the stress and strain vectors and the matrix of elastic constants in the isotropic linear elastic constitutive equation

[ 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@

3.2.8.1.            Calculate the eigenvalues of the stiffness matrix [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38B1@  for an isotropic solid in terms of Young’s modulus and Poisson’s ratio.  Hence, show that the eigenvalues are positive (a necessary requirement for the material to be stable MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  see problem ??) if and only if 1<ν<1/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaaIXaGaeyipaWJaeqyVd4 MaeyipaWJaaGymaiaac+cacaaIYaaaaa@3A58@  and E>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbGaeyOpa4JaaGimaaaa@3552@ .

3.2.8.2.            Find the eigenvectors of [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38B1@  and sketch the deformations associated with these eigenvectors.

 

 

3.2.9.      Let [ σ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq aHdpWCaiaawUfacaGLDbaaaaa@399C@ , [ ε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq aH1oqzaiaawUfacaGLDbaaaaa@3980@  and [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38A1@  represent the stress and strain vectors and the matrix of elastic constants in the isotropic linear elastic constitutive equation for a cubic crystal

[ 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@

Calculate the eigenvalues of the stiffness matrix [ C ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGdbaacaGLBbGaayzxaaaaaa@38B1@  and hence find expressions for the admissible ranges of c 11 , c 12 , c 44 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdacaaIXa aabeaakiaacYcacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiaa cYcacaWGJbWaaSbaaSqaaiaaisdacaaI0aaabeaaaaa@3BDF@  for the eigenvalues to be positive.

 

 

 

3.2.10.  Let   C ijkl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeaacaWHLbaaaaaa@3867@  denote the components of the elasticity tensor 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@ . Let   { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  be a second basis, and define Ω ij = m i e j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0JaaCyBamaaBaaaleaacaWGPbaabeaakiabgwSi xlaahwgadaWgaaWcbaGaamOAaaqabaaaaa@3DDA@ .  Recall that the components of the stress and strain tensor in  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaahwgadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyzamaaBaaaleaacaaIYaaabeaakiaacYca caWHLbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3BF7@  and { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  are related by σ ij (m) = Ω ik σ kl (e) Ω jl ε ij (m) = Ω ik ε kl (e) Ω jl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaGGOaGaaCyBaiaacMcaaaGccqGH9aqpcqqHPoWvdaWgaaWc baGaamyAaiaadUgaaeqaaOGaeq4Wdm3aa0baaSqaaiaadUgacaWGSb aabaGaaiikaiaahwgacaGGPaaaaOGaeuyQdC1aaSbaaSqaaiaadQga caWGSbaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaacIcacaWHTbGaaiyk aaaakiabg2da9iabfM6axnaaBaaaleaacaWGPbGaam4AaaqabaGccq aH1oqzdaqhaaWcbaGaam4AaiaadYgaaeaacaGGOaGaaCyzaiaacMca aaGccqqHPoWvdaWgaaWcbaGaamOAaiaadYgaaeqaaaaa@64EE@ . Use this result, together with the elastic constitutive equation, to show that the components of the elasticity tensor in { m 1 , m 2 , m 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGadaqaaiaah2gadaWgaaWcbaGaaG ymaaqabaGccaGGSaGaaCyBamaaBaaaleaacaaIYaaabeaakiaacYca caWHTbWaaSbaaSqaaiaaiodaaeqaaaGccaGL7bGaayzFaaaaaa@3C0F@  can be calculated from

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 caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeaacaGGOaGaaCyBaiaacMcaaaGccqGH9aqpcqqH PoWvdaWgaaWcbaGaamyAaiaadchaaeqaaOGaeuyQdC1aaSbaaSqaai aadQgacaWGXbaabeaakiaadoeadaqhaaWcbaGaamiCaiaadghacaWG YbGaam4CaaqaaiaacIcacaWHLbGaaiykaaaakiabfM6axnaaBaaale aacaWGRbGaamOCaaqabaGccqqHPoWvdaWgaaWcbaGaamiBaiaadoha aeqaaaaa@5096@

 

 

3.2.11.  Consider a cube-shaped specimen of an anisotropic, linear elastic material. The tensor of elastic moduli and the thermal expansion coefficient for the solid (expressed as components in an arbitrary basis) are    C ijkl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaa0baaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeaacaWHLbaaaaaa@3867@ , α ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqydaqhaaWcbaGaamyAaiaadQ gaaeaacaWHLbaaaaaa@375D@ .  The solid is placed inside a rigid box that prevents the material from stretching in any direction.   This means that the strains in the specimen are zero.  The specimen is then heated to increase its temperature by ΔT MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHuoarcaWGubaaaa@3515@ .  Find a formula for the strain energy density, and show that the result is independent of the orientation of the material with respect to the box.

 

 

3.2.12.  The figure shows a cubic crystal.   Basis vectors  { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BD6@  are aligned perpendicular to the faces of the cubic unit cell. A tensile specimen is cut from the cube MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the axis of the specimen lies in the { e 1 , e 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiyFaaaa @3945@  plane and is oriented at an angle θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@348C@  to the e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaigdaaeqaaa aa@34AB@  direction. The specimen is then loaded in uniaxial tension σ nn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOBaiaad6 gaaeqaaaaa@36AB@  parallel to its axis.  This means that the stress components in the basis { n 1 , n 2 , n 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCOBamaaBaaaleaacaaIXa aabeaakiaacYcacaWHUbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa h6gadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BF1@  shown in the picture are

[ σ nn 0 0 0 0 0 0 0 0 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWadaqaauaabeqadmaaaeaacqaHdp WCdaWgaaWcbaGaamOBaiaad6gaaeqaaaGcbaGaaGimaaqaaiaaicda aeaacaaIWaaabaGaaGimaaqaaiaaicdaaeaacaaIWaaabaGaaGimaa qaaiaaicdaaaaacaGLBbGaayzxaaaaaa@3E8E@

3.2.12.1.        Use the basis change formulas for tensors to calculate the components of stress in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BD6@  basis in terms of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@348C@ .

Use the stress-strain equations in Section 3.1.16 to find the strain components in the { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCyzamaaBaaaleaacaaIXa aabeaakiaacYcacaWHLbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hwgadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BD6@  basis, in terms of the engineering constants E,ν,μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbGaaiilaiabe27aUjaacYcacq aH8oqBaaa@386E@  for the cubic crystal. You need only calculate ε 11 , ε 22 , ε 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGymaiaaig daaeqaaOGaaiilaiabew7aLnaaBaaaleaacaaIYaGaaGOmaaqabaGc caGGSaGaeqyTdu2aaSbaaSqaaiaaigdacaaIYaaabeaaaaa@3E28@ .

3.2.12.2.        Use the basis change formulas again to calculate the strain components in the { n 1 , n 2 , n 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGG7bGaaCOBamaaBaaaleaacaaIXa aabeaakiaacYcacaWHUbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa h6gadaWgaaWcbaGaaG4maaqabaGccaGG9baaaa@3BF1@  basis oriented with the specimen.  Again, you need only calculate ε 11 , ε 22 , ε 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaaGymaiaaig daaeqaaOGaaiilaiabew7aLnaaBaaaleaacaaIYaGaaGOmaaqabaGc caGGSaGaeqyTdu2aaSbaaSqaaiaaigdacaaIYaaabeaaaaa@3E28@ .  Check your answer by setting μ=E/2(1+ν) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBcqGH9aqpcaWGfbGaai4lai aaikdacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaaa@3C79@  - this makes the crystal isotropic, and you should recover the isotropic solution.

3.2.12.3.        Define the effective axial Young’s modulus of the tensile specimen as E(θ)= σ nn / ε nn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbGaaiikaiabeI7aXjaacMcacq GH9aqpcqaHdpWCdaWgaaWcbaGaamOBaiaad6gaaeqaaOGaai4laiab ew7aLnaaBaaaleaacaWGUbGaamOBaaqabaaaaa@4000@ , where ε nn =nεn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH1oqzdaWgaaWcbaGaamOBaiaad6 gaaeqaaOGaeyypa0JaaCOBaiabgwSixlaahw7acqGHflY1caWHUbaa aa@3F62@  is the strain component parallel to the n 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbWaaSbaaSqaaiaaigdaaeqaaa aa@34B4@  direction.  Find a formula for E(θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbGaaiikaiabeI7aXjaacMcaaa a@36AF@  in terms of E,ν,μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbGaaiilaiabe27aUjaacYcacq aH8oqBaaa@386E@ .

3.2.12.4.        Using data for copper, plot a graph of E(θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGfbGaaiikaiabeI7aXjaacMcaaa a@36AF@  against θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@348C@ .  For copper, what is the orientation that maximizes the longitudinal stiffness of the specimen?   Which orientation minimizes the stiffness?