Isotropic Elastic Material

Two kinds of isotropic elastic materials are considered:
  • Linear elastic materials with Hooke’s law,
  • Nonlinear elastic materials with Ogden, Mooney-Rivlin and Arruda-Boyce laws.

Linear Elastic Material (LAW1)

This material law is used to model purely elastic materials, or materials that remain in the elastic range. The Hooke's law requires only two values to be defined; the Young's or elastic modulus E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@ , and Poisson's ratio, υ . The law represents a linear relation between stress and strain.

Ogden Materials (LAW42, LAW69 and LAW82)

Ogden's law is applied to slightly compressible materials as rubber or elastomer foams undergoing large deformation with an elastic behavior. The detailed theory for Odgen material models can be found in 1. The strain energy W MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@ is expressed in a general form as a function of W ( λ 1 , λ 2 , λ 3 ) :(1) W ( λ 1 , λ 2 , λ 3 ) = p = 1 5 μ p α p ( λ ¯ 1 α p + λ ¯ 2 α p + λ ¯ 3 α p 3 ) + K 2 ( J 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaae WaaeaacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2a aSbaaSqaaiaaikdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZa aabeaaaOGaayjkaiaawMcaaiabg2da9maaqahabaWaaSaaaeaacqaH 8oqBdaWgaaWcbaGaamiCaaqabaaakeaacqaHXoqydaWgaaWcbaGaam iCaaqabaaaaOWaaeWaaeaadaqdaaqaaiabeU7aSbaadaWgaaWcbaGa aGymaaqabaGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabe aaaaGccqGHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaGOmaaqa baGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccq GHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaG4maaqabaGcdaah aaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccqGHsislca aIZaaacaGLOaGaayzkaaaaleaacaWGWbGaeyypa0JaaGymaaqaaiaa iwdaa0GaeyyeIuoakiabgUcaRmaalaaabaGaam4saaqaaiaaikdaaa WaaeWaaeaacaWGkbGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaa leqabaGaaGOmaaaaaaa@6C02@

Where, λ 1 , ith principal stretch

λ i = 1 + ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyAaaqabaGccqGH9aqpcaaIXaGaey4kaSIaeqyTdu2a aSbaaSqaaiaadMgaaeqaaaaa@3E9A@ , with ε i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH1oqzda WgaaWcbaGaamyAaaqabaaaaa@391F@ being the ith principal engineering strain

J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbaaaa@372D@ is relative volume with:(2) J = λ 1 λ 2 λ 3 = ρ 0 ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGkbGaey ypa0Jaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOGaeyyXICTaeq4UdW2a aSbaaSqaaiaaikdaaeqaaOGaeyyXICTaeq4UdW2aaSbaaSqaaiaaio daaeqaaOGaeyypa0ZaaSaaaeaacqaHbpGCdaWgaaWcbaGaaGimaaqa baaakeaacqaHbpGCaaaaaa@4A3F@
λ ¯ i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBga qeamaaBaaaleaacaWGPbaabeaaaaa@3944@ is the deviatoric stretch(3) λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbae badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGkbWaaWbaaSqabeaa cqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaaaakiabeU7aSnaaBa aaleaacaWGPbaabeaaaaa@4036@

α p and μ p are the material constants.

p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiCaaaa@36ED@ is order of Ogden model and defines the number of coefficients pairs ( α p , μ p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqydaWgaaWcbaGaamiCaaqabaGccaGGSaGaeqiVd02aaSbaaSqa aiaadchaaeqaaaGccaGLOaGaayzkaaaaaa@3DDB@ .

This law is very general due to the choice of coefficient pair ( α p , μ p ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqydaWgaaWcbaGaamiCaaqabaGccaGGSaGaeqiVd02aaSbaaSqa aiaadchaaeqaaaGccaGLOaGaayzkaaaaaa@3DDB@ .
  • If p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiCaaaa@36ED@ =1, then one pair ( α 1 , μ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqydaWgaaWcbaGaaGymaaqabaGccaGGSaGaeqiVd02aaSbaaSqa aiaaigdaaeqaaaGccaGLOaGaayzkaaaaaa@3D67@ of material constants is needed andin this case if α 1 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ then it becomes a Neo-hookean material model.
  • If p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiCaaaa@36ED@ =2 then two pairs ( α 1 , μ 1 ) , ( α 2 , μ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqydaWgaaWcbaGaaGymaaqabaGccaGGSaGaeqiVd02aaSbaaSqa aiaaigdaaeqaaaGccaGLOaGaayzkaaGaaiilamaabmaabaGaeqySde 2aaSbaaSqaaiaaikdaaeqaaOGaaiilaiabeY7aTnaaBaaaleaacaaI YaaabeaaaOGaayjkaiaawMcaaaaa@4589@ of material constants are needed and in this case if α 1 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ and α 1 = -2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ then it becomes a Mooney-Rivlin material model
For uniform dilitation:(4) λ 1 = λ 2 = λ 3 = λ
The strain energy function can be decomposed into deviatoric part W ¯ ( λ ¯ 1 , λ ¯ 2 , λ ¯ 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rq1rVepeea0xe9Lq=Je9 vqaqVepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGxbGbae bacaGGOaGafq4UdWMbaebadaWgaaWcbaGaaGymaaqabaGccaGGSaGa fq4UdWMbaebadaWgaaWcbaGaaGOmaaqabaGccaGGSaGafq4UdWMbae badaWgaaWcbaGaaG4maaqabaGccaGGPaaaaa@426B@ and spherical part U ( J ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rq1rVepeea0xe9Lq=Je9 vqaqVepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGvbGaai ikaiaadQeacaGGPaaaaa@3986@ :(5) W = W ¯ ( λ ¯ 1 , λ ¯ 2 , λ ¯ 3 ) + U ( J )

With:

W ¯ ( λ ¯ 1 , λ ¯ 2 , λ ¯ 3 ) = p μ p α p ( λ ¯ 1 α p + λ ¯ 2 α p + λ ¯ 3 α p 3 ) U ( J ) = K 2 ( J 1 ) 2

The stress σ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgaaeqaaaaa@38D4@ corresponding to this strain energy is given by:(6) σ i = λ i J W λ i
which can be written as:(7) σ i = λ i J W λ i = λ i J ( j = 1 3 W ¯ λ ¯ j λ ¯ j λ i + U J J λ i )

Since λ 1 J λ i = J and λ ¯ j λ i = 2 3 J 1 3 for i=j and λ ¯ j λ i = 1 3 J 1 3 λ j λ i for i≠j

Equation 7 is simplified to:(8) σ i = 1 J ( λ ¯ i W ¯ λ ¯ i ( 1 3 j = 1 3 λ ¯ j W ¯ λ ¯ j J U J ) )
For which the deviator of the Cauchy stress tensor s i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGPbaabeaaaaa@3809@ , and the pressure P MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaaaa@36CC@ would be:(9) s i = 1 J ( λ ¯ i W ¯ λ ¯ i 1 3 j = 1 3 λ ¯ j W ¯ λ ¯ j ) (10) p = 1 3 j = 1 3 σ j = U J
Only the deviatoric stress above is retained, and the pressure is computed independently:(11) P = K F s c a l e b l k f b l k ( J ) ( J 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaey ypa0Jaam4saiabgwSixlaadAeacaWGZbGaam4yaiaadggacaWGSbGa amyzamaaBaaaleaacaWGIbGaamiBaiaadUgaaeqaaOGaeyyXICTaci OzamaaBaaaleaacaWGIbGaamiBaiaadUgaaeqaaOWaaeWaaeaacaWG kbaacaGLOaGaayzkaaGaeyyXIC9aaeWaaeaacaWGkbGaeyOeI0IaaG ymaaGaayjkaiaawMcaaaaa@5293@
Where, f b l k ( J ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGMbWaaS baaSqaaiaadkgacaWGSbGaam4AaaqabaGcdaqadaqaaiaadQeaaiaa wIcacaGLPaaaaaa@3CA0@ a user-defined function related to the bulk modulus K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGlbaaaa@39A7@ in LAW42 and LAW69:(12) K = μ 2 ( 1 + ν ) 3 ( 1 2 ν ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9iabeY7aTjabgwSixpaalaaabaGaaGOmamaabmaabaGaaGymaiab gUcaRiabe27aUbGaayjkaiaawMcaaaqaaiaaiodadaqadaqaaiaaig dacqGHsislcaaIYaGaeqyVd4gacaGLOaGaayzkaaaaaaaa@47D9@
For an imcompressible material ( ν 0.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH9oGBcqGHijYUcaaIWaGaaiOlaiaaiwdaaiaawIcacaGLPaaaaaa@3D13@ , J = 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iaaigdaaaa@3886@ and no pressure in material.(13) μ = p μ p α p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0ZaaSaaaeaadaaeqbqaaiabeY7aTnaaBaaaleaacaWGWbaabeaa kiabgwSixlabeg7aHnaaBaaaleaacaWGWbaabeaaaeaacaWGWbaabe qdcqGHris5aaGcbaGaaGOmaaaaaaa@4481@
With μ being the initial shear modulus, and υ the Poisson's ratio.
Note: For an incompressible material you have υ 0.5 . However, υ 0.495 is a good compromise to avoid too small time steps in explicit codes.
A particular case of the Ogden material model is the Mooney-Rivlin material law which has two basic assumptions:
  • The rubber is incompressible and isotropic in unstrained state
  • The strain energy expression depends on the invariants of Cauchy tensor
The three invariants of the Cauchy-Green tensor are:(14) I 1 = λ 1 2 + λ 2 2 + λ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaBa aaleaacaaIXaaabeaakiabg2da9iabeU7aSnaaBaaaleaacaaIXaaa beaakmaaCaaaleqabaGaaGOmaaaakiabgUcaRiabeU7aSnaaBaaale aacaaIYaaabeaakmaaCaaaleqabaGaaGOmaaaakiabgUcaRiabeU7a SnaaBaaaleaacaaIZaaabeaakmaaCaaaleqabaGaaGOmaaaaaaa@4541@ (15) I 2 = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaS baaSqaaiaaikdaaeqaaOGaeyypa0Jaeq4UdW2aaSbaaSqaaiaaigda aeqaaOWaaWbaaSqabeaacaaIYaaaaOGaeq4UdW2aaSbaaSqaaiaaik daaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeq4UdW2aaSba aSqaaiaaikdaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaeq4UdW2aaS baaSqaaiaaiodaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIa eq4UdW2aaSbaaSqaaiaaiodaaeqaaOWaaWbaaSqabeaacaaIYaaaaO Gaeq4UdW2aaSbaaSqaaiaaigdaaeqaaOWaaWbaaSqabeaacaaIYaaa aaaa@517E@
For incompressible material:(16) I 3 = λ 1 2 λ 2 2 λ 3 2 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaS baaSqaaiaaiodaaeqaaOGaeyypa0Jaeq4UdW2aaSbaaSqaaiaaigda aeqaaOWaaWbaaSqabeaacaaIYaaaaOGaeq4UdW2aaSbaaSqaaiaaik daaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaeq4UdW2aaSbaaSqaaiaa iodaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JaaGymaaaa@46BB@
The Mooney-Rivlin law gives the closed expression of strain energy as:(17) W = C 10 ( I 1 3 ) + C 01 ( I 2 3 )
with:(18) μ 1 = 2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIYaGaeyyXICTaam4qamaa BaaaleaacaaIXaGaaGimaaqabaaaaa@4084@
μ 2 = 2 C 01 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGHsislcaaIYaGaeyyXICTa am4qamaaBaaaleaacaaIWaGaaGymaaqabaaaaa@4172@
α 1 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda WgaaWcbaGaaGymaaqabaGccqGH9aqpcaaIYaaaaa@3BBA@
α 2 = 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqyda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcqGHsislcaaIYaaaaa@3CA8@

The model can be generalized for a compressible material.

Viscous Effects in LAW42

Viscous effects are modeled through the Maxwell model:


Figure 1. Maxwell Model
Where, the shear modulus of the hyper-elastic law μ is exactly the long-term shear modulus G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4ramaaBa aaleaacqGHEisPaeqaaaaa@3860@ .(19) μ = p μ p α p 2 = G MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0ZaaSaaaeaadaaeqbqaaiabeY7aTnaaBaaaleaacaWGWbaabeaa kiabgwSixlabeg7aHnaaBaaaleaacaWGWbaabeaaaeaacaWGWbaabe qdcqGHris5aaGcbaGaaGOmaaaacqGH9aqpcaWGhbWaaSbaaSqaaiab g6HiLcqabaaaaa@47F0@

τ i are relaxation times: τ i = η i G i

Rate effects are modeled through visco-elasticity using convolution integral using Prony series. This corresponds to extension of small deformation theory to finite deformation.

This viscous stress is added to the elastic one.

The visco-Kirchoff stress is given by:(20) τ v = i = 1 M G i 0 t e t s τ i d d s [ dev ( F ¯ F ¯ T ) ] d s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHepWaaW baaSqabeaacaWG2baaaOGaeyypa0ZaaabCaeaacaWGhbWaaSbaaSqa aiaadMgaaeqaaOWaa8qCaeaacaWGLbaaleaacaaIWaaabaGaamiDaa qdcqGHRiI8aaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGnbaaniab ggHiLdGcdaahaaWcbeqaaiabgkHiTmaalaaabaGaamiDaiabgkHiTi aadohaaeaacqaHepaDdaWgaaadbaGaamyAaaqabaaaaaaakmaalaaa baGaamizaaqaaiaadsgacaWGZbaaamaadmGabaGaciizaiaacwgaca GG2bWaaeWaceaaceWHgbGbaebaceWHgbGbaebadaahaaWcbeqaaiaa dsfaaaaakiaawIcacaGLPaaaaiaawUfacaGLDbaacaWGKbGaam4Caa aa@5C91@
Where,
M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaaleaacaWGnbaaaa@3842@
Order of the Maxwell model
F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHgbaaaa@383E@
Deformation gradient matrix
F ¯ = J 1 3 F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWHgbGbae bacqGH9aqpcaWGkbWaaWbaaSqabeaacqGHsisldaWcaaqaaiaaigda aeaacaaIZaaaaaaakiaahAeaaaa@3DA6@
dev ( F ¯ F ¯ T ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGKbGaai yzaiaacAhadaqadiqaaiqahAeagaqeaiqahAeagaqeamaaCaaaleqa baGaamivaaaaaOGaayjkaiaawMcaaaaa@3EA5@
Denotes the deviatoric part of tensor F ¯ F ¯ T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWHgbGbae baceWHgbGbaebadaahaaWcbeqaaiaadsfaaaaaaa@3A43@
The viscous-Cauchy stress is written as:(21) σ v ( t ) = 1 J τ v ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqOqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHdpWaaW baaSqabeaacaWG2baaaOGaaiikaiaadshacaGGPaGaeyypa0ZaaSaa aeaacaaIXaaabaGaamOsaaaacaWHepWaaWbaaSqabeaacaWG2baaaO GaaiikaiaadshacaGGPaaaaa@43B6@

LAW69, Ogden Material Law (Using Test Data as Input)

This law, like /MAT/LAW42 (OGDEN) defines a hyperelastic and incompressible material specified using the Ogden or Mooney-Rivlin material models. Unlike LAW42 where the material parameters are input this law computes the material parameters from an input engineering stress-strain curve from a uniaxial tension and compression tests. This material can be used with shell and solid elements.

The strain energy density formulation used depends on the law_ID.
law_ID =1, Ogden law (Same as LAW42):
W ( λ 1 , λ 2 , λ 3 ) = p = 1 5 μ p α p ( λ ¯ 1 α p + λ ¯ 2 α p + λ ¯ 3 α p 3 ) + K 2 ( J 1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbWaae WaaeaacqaH7oaBdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeq4UdW2a aSbaaSqaaiaaikdaaeqaaOGaaiilaiabeU7aSnaaBaaaleaacaaIZa aabeaaaOGaayjkaiaawMcaaiabg2da9maaqahabaWaaSaaaeaacqaH 8oqBdaWgaaWcbaGaamiCaaqabaaakeaacqaHXoqydaWgaaWcbaGaam iCaaqabaaaaOWaaeWaaeaadaqdaaqaaiabeU7aSbaadaWgaaWcbaGa aGymaaqabaGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabe aaaaGccqGHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaGOmaaqa baGcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccq GHRaWkdaqdaaqaaiabeU7aSbaadaWgaaWcbaGaaG4maaqabaGcdaah aaWcbeqaaiabeg7aHnaaBaaameaacaWGWbaabeaaaaGccqGHsislca aIZaaacaGLOaGaayzkaaaaleaacaWGWbGaeyypa0JaaGymaaqaaiaa iwdaa0GaeyyeIuoakiabgUcaRmaalaaabaGaam4saaqaaiaaikdaaa WaaeWaaeaacaWGkbGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaa leqabaGaaGOmaaaaaaa@6C02@
law_ID =2, Mooney-Rivlin law
W = C 10 ( I 1 3 ) + C 01 ( I 2 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaey ypa0Jaam4qamaaBaaaleaacaaIXaGaaGimaaqabaGcdaqadaqaaiaa dMeadaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIZaaacaGLOaGaay zkaaGaey4kaSIaam4qamaaBaaaleaacaaIWaGaaGymaaqabaGcdaqa daqaaiaadMeadaWgaaWcbaGaaGOmaaqabaGccqGHsislcaaIZaaaca GLOaGaayzkaaaaaa@47ED@
Curve Fitting

After reading the stress-strain curve (fct_ID1), Radioss calculates the corresponding material parameter pairs using a nonlinear least-square fitting algorithm. For classic Ogden law, (law_ID =1), the calculated material parameter pairs are μ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadchaaeqaaaaa@38CE@ and α p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadchaaeqaaaaa@38B7@ where the value of p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiCaaaa@36ED@ is defined via the N_pair input. The maximum value of N_pair = 5 with a default value of 2.

For the Mooney-Rivlin law (law_ID =2), the material parameter C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIWaGaaGymaaqabaaaaa@3860@ and C 01 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaBa aaleaacaaIWaGaaGymaaqabaaaaa@3860@ are calculated remembering that μ p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaadchaaeqaaaaa@38CE@ and α p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadchaaeqaaaaa@38B7@ for the LAW42 Ogden law can be calculated using this conversion.

μ 1 = 2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabgwSixlaadoeadaWg aaWcbaGaaGymaiaaicdaaeqaaaaa@3F13@ , μ 2 =2 C 01 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGOmaiabgwSixlaa doeadaWgaaWcbaGaaGimaiaaigdaaeqaaaaa@4001@ , α 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaaaa@3A49@ and α 2 =2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaeyOeI0IaaGOmaaaa@3B37@ .

The minimum test data input should be a uniaxial tension engineering stress strain curve. If uniaxial compression data is available, the engineering strain should increate monotonically from a negative value in compression to a positive value in tension. In compression, the engineering strain should not be less than -1.0 since -100% strain is physically not possible.

This material law is stable when (with p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaWcbaGaamiCaaaa@36ED@ =1,…5) is satisfied for parameter pairs for all loading conditions. By default, Radioss tries to fit the curve by accounting for these conditions (Icheck=2). If a proper fit cannot be found, then Radioss uses a weaker condition (Icheck=1: ) which ensures that the initial shear hyperelastic modulus ( μ ) is positive.

Once the material parameters are calculated by the Radioss Starter in LAW69, all the calculations done by LAW69 in the simulation are the same as LAW42.

LAW82

The Ogden model used in LAW82 is:(22) W = i = 1 N 2 μ i α i 2 ( λ ¯ 1 α i + λ ¯ 2 α i + λ ¯ 3 α i 3 ) + i = 1 N 1 D i ( J 1 ) 2 i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maaqahabaWaaSaaaeaacaaIYaGaeqiVd02aaSbaaSqaaiaadMga aeqaaaGcbaGaeqySde2aaSbaaSqaaiaadMgaaeqaaOWaaWbaaSqabe aacaaIYaaaaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaa0Ga eyyeIuoakmaabmaabaGafq4UdWMbaebadaWgaaWcbaGaaGymaaqaba GcdaahaaWcbeqaaiabeg7aHnaaBaaameaacaWGPbaabeaaaaGccqGH RaWkcuaH7oaBgaqeamaaBaaaleaacaaIYaaabeaakmaaCaaaleqaba GaeqySde2aaSbaaWqaaiaadMgaaeqaaaaakiabgUcaRiqbeU7aSzaa raWaaSbaaSqaaiaaiodaaeqaaOWaaWbaaSqabeaacqaHXoqydaWgaa adbaGaamyAaaqabaaaaOGaeyOeI0IaaG4maaGaayjkaiaawMcaaiab gUcaRmaaqahabaWaaSaaaeaacaaIXaaabaGaamiramaaBaaaleaaca WGPbaabeaaaaGcdaqadaqaaiaadQeacqGHsislcaaIXaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaGaamyAaaaaaeaacaWGPbGaeyypa0 JaaGymaaqaaiaad6eaa0GaeyyeIuoaaaa@6A1C@
The Bulk Modulus is calculated as K = 2 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaa aaaa@3A49@ based on these rules:
  • If ν = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaaaa@396F@ , D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ should be entered.
  • If ν 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey iyIKRaaGimaaaa@3A30@ , D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ input is ignored and will be recalculated and output in the Starter output using:(23) D 1 = 3 ( 1 2 v ) μ ( 1 + v ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaG4maiaacIcacaaI XaGaeyOeI0IaaGOmaiaadAhacaGGPaaabaGaeqiVd0Maaiikaiaaig dacqGHRaWkcaWG2bGaaiykaaaaaaa@43E2@
  • If ν = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaaaa@396F@ and D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ =0, a default value of ν = 0.475 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaiaac6cacaaI0aGaaG4naiaaiwdaaaa@3C5F@ is used and D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ is calculated using Equation 23
LAW88, A simplified hyperelastic material with strain rate effects
This law utilizes tabulated uniaxial tension and compression engineering stress and strain test data at different strain rates to model incompressible materials. It is only compatible with solid elements. The material is based on Ogden’s strain energy density function but does not require curve fitting to extract material constants like most other hyperelastic material models. Strain rate effects can be modeled by including engineering stress strain test data at different strain rates. This can be easier than calculating viscous parameters for traditional hyperelastic material models. The following Ogden strain energy density function is used but instead of extracting material constants via curve fitting this law determines the Ogden function directly from the uniaxial engineering stress strain curve tabulated data. 5(24) W = i = 1 3 j = 1 m μ j α j ( λ ¯ i α j 1 ) d e v i a t o r i c p a r t + K ( J 1 ln J ) s p h e r i c a l p a r t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaey ypa0ZaaGbaaeaadaaeWbqaamaaqahabaWaaSaaaeaacqaH8oqBdaWg aaWcbaGaamOAaaqabaaakeaacqaHXoqydaWgaaWcbaGaamOAaaqaba aaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamyBaaqdcqGHris5aaWc baGaamyAaiabg2da9iaaigdaaeaacaaIZaaaniabggHiLdGcdaqada qaaiqbeU7aSzaaraWaaSbaaSqaaiaadMgaaeqaaOWaaWbaaSqabeaa cqaHXoqydaWgaaadbaGaamOAaaqabaaaaOGaeyOeI0IaaGymaaGaay jkaiaawMcaaaWcbaGaaWizaiaamwgacaaJ2bGaaWyAaiaamggacaaJ 0bGaaW4BaiaamkhacaaJPbGaaW4yaiaamccacaaJWbGaaWyyaiaamk hacaaJ0baakiaawIJ=aiabgUcaRmaayaaabaGaam4samaabmaabaGa amOsaiabgkHiTiaaigdacqGHsislciGGSbGaaiOBaiaadQeaaiaawI cacaGLPaaaaSqaaiaamohacaaJWbGaaWiAaiaamwgacaaJYbGaaWyA aiaamogacaaJHbGaaWiBaiaamccacaaJWbGaaWyyaiaamkhacaaJ0b aakiaawIJ=aaaa@7B85@

Unloading can be represented using an unloading function, FscaleunL, or by providing hysteresis, Hys and shape factor, Shape, inputs to a damage model based on energy.

When using the damage model, the loading curves are used for both loading and unloading and the unloading stress tensor is reduced by:(25) σ = ( 1 D ) σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHdpGaey ypa0ZaaeWaaeaacaaIXaGaeyOeI0IaamiraaGaayjkaiaawMcaaiaa ho8aaaa@3DFC@ (26) D = ( 1 H y s ) ( 1 ( W c u r W max ) S h a p e ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey ypa0ZaaeWaaeaacaaIXaGaeyOeI0IaamisaiaadMhacaWGZbaacaGL OaGaayzkaaWaaeWaaeaacaaIXaGaeyOeI0YaaeWaaeaadaWcaaqaai aadEfadaWgaaWcbaGaam4yaiaadwhacaWGYbaabeaaaOqaaiaadEfa daWgaaWcbaGaciyBaiaacggacaGG4baabeaaaaaakiaawIcacaGLPa aadaahaaWcbeqaaiaadofacaWGObGaamyyaiaadchacaWGLbaaaaGc caGLOaGaayzkaaaaaa@4F7D@

If the unloading function, FscaleunL, is entered, unloading is defined based on the unloading flag, Tension and the damage model is not used.

Arruda-Boyce Material (LAW92)

LAW92 describes the Arruda-Boyce material model, which can be used to model hyperelastic behavior. The Arruda-Boyce model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions. It assumes that the chain molecules are located on the average along the diagonals of the cubic in principal stretch space.

The strain energy density function is:(27) W= μ i=1 5 c i ( λ m ) 2i2 ( I ¯ 1 i 3 i ) W( I ¯ 1 ) + 1 D ( J 2 1 2 +ln(J) ) U( J ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbGaey ypa0ZaaGbaaeaacqaH8oqBdaaeWbqaamaalaaabaGaam4yamaaBaaa leaacaWGPbaabeaaaOqaaiaacIcacqaH7oaBdaWgaaWcbaGaamyBaa qabaGccaGGPaWaaWbaaSqabeaacaaIYaGaamyAaiabgkHiTiaaikda aaaaaaqaaiaadMgacqGH9aqpcaaIXaaabaGaaGynaaqdcqGHris5aO WaaeWaaeaaceWGjbGbaebadaqhaaWcbaGaaGymaaqaaiaadMgaaaGc cqGHsislcaaIZaWaaWbaaSqabeaacaWGPbaaaaGccaGLOaGaayzkaa aaleaacaWGxbWaaeWaaeaaceWGjbGbaebadaWgaaadbaGaaGymaaqa baaaliaawIcacaGLPaaaaOGaayjo+dGaey4kaSYaaGbaaeaadaWcaa qaaiaaigdaaeaacaWGebaaamaabmaabaWaaSaaaeaacaWGkbWaaWba aSqabeaacaaIYaaaaOGaeyOeI0IaaGymaaqaaiaaikdaaaGaey4kaS IaciiBaiaac6gaciGGOaGaamOsaiaacMcaaiaawIcacaGLPaaaaSqa aiaadwfadaqadaqaaiaadQeaaiaawIcacaGLPaaaaOGaayjo+daaaa@692A@

Where, Material constant c i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS baaSqaaiaadMgaaeqaaaaa@3861@ are:

c 1 = 1 2 , c 2 = 1 20 , c 3 = 11 1050 , c 4 = 19 7000 , c 5 = 519 673750 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikda aaGaaiilaiaaysW7caWGJbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0 ZaaSaaaeaacaaIXaaabaGaaGOmaiaaicdaaaGaaiilaiaaysW7caWG JbWaaSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaGaaG ymaaqaaiaaigdacaaIWaGaaGynaiaaicdaaaGaaiilaiaaysW7caWG JbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaGaaG yoaaqaaiaaiEdacaaIWaGaaGimaiaaicdaaaGaaGjbVlaacYcacaaM e8Uaam4yamaaBaaaleaacaaI1aaabeaakiabg2da9maalaaabaGaaG ynaiaaigdacaaI5aaabaGaaGOnaiaaiEdacaaIZaGaaG4naiaaiwda caaIWaaaaaaa@625B@
μ
Shear modulus
μ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda WgaaWcbaGaaGimaaqabaaaaa@38FB@
Initial shear modulus
(28) μ 0 =μ( 1+ 3 5 λ m 2 + 99 175 λ m 4 + 513 875 λ m 6 + 42039 67375 λ m 8 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaicdaaeqaaOGaeyypa0JaeqiVd02aaeWaaeaacaaIXaGa ey4kaSYaaSaaaeaacaaIZaaabaGaaGynaiabeU7aSnaaBaaaleaaca WGTbaabeaakmaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqa aiaaiMdacaaI5aaabaGaaGymaiaaiEdacaaI1aGaeq4UdW2aaSbaaS qaaiaad2gaaeqaaOWaaWbaaSqabeaacaaI0aaaaaaakiabgUcaRmaa laaabaGaaGynaiaaigdacaaIZaaabaGaaGioaiaaiEdacaaI1aGaeq 4UdW2aaSbaaSqaaiaad2gaaeqaaOWaaWbaaSqabeaacaaI2aaaaaaa kiabgUcaRmaalaaabaGaaGinaiaaikdacaaIWaGaaG4maiaaiMdaae aacaaI2aGaaG4naiaaiodacaaI3aGaaGynaiabeU7aSnaaBaaaleaa caWGTbaabeaakmaaCaaaleqabaGaaGioaaaaaaaakiaawIcacaGLPa aaaaa@61D5@

λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyBaaqabaaaaa@3931@ is the limit of stretch which describes the beginning of hardening phase in tension (locking strain in tension) and so it is also called the locking stretch.

Arruda-Boyce is always stable if positive values of the shear modulus, μ , and the locking stretch, λ m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaamyBaaqabaaaaa@3931@ are used.

I ¯ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGjbGbae badaWgaaWcbaGaaGymaaqabaaaaa@382C@ is deviatoric part of first strain invarient I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGjbWaaS baaSqaaiaaigdaaeqaaaaa@3814@ (29) I ¯ 1 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 = J 2/3 I 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGjbGbae badaWgaaWcbaGaaGymaaqabaGccqGH9aqpcuaH7oaBgaqeamaaBaaa leaacaaIXaaabeaakmaaCaaaleqabaGaaGOmaaaakiabgUcaRiqbeU 7aSzaaraWaaSbaaSqaaiaaikdaaeqaaOWaaWbaaSqabeaacaaIYaaa aOGaey4kaSIafq4UdWMbaebadaWgaaWcbaGaaG4maaqabaGcdaahaa WcbeqaaiaaikdaaaGccqGH9aqpcaWGkbWaaWbaaSqabeaadaWcgaqa aiabgkHiTiaaikdaaeaacaaIZaaaaaaakiaadMeadaWgaaWcbaGaaG ymaaqabaaaaa@4C50@

with λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbae badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGkbWaaWbaaSqabeaa cqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaaaakiabeU7aSnaaBa aaleaacaWGPbaabeaaaaa@4036@

D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebaaaa@3728@ is a material parameter for the bulk modulus computation given as:(30) D= 2 K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGebGaey ypa0ZaaSaaaeaacaaIYaaabaGaam4saaaaaaa@39CA@
The Cauchy stress corresponding to above strain energy is:(31) σ i = λ i J W λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacqaH7oaBdaWgaaWc baGaamyAaaqabaaakeaacaWGkbaaamaalaaabaGaeyOaIyRaam4vaa qaaiabgkGi2kabeU7aSnaaBaaaleaacaWGPbaabeaaaaaaaa@4421@
For incompressible materials, the Cauchy stress is then given by:
  • Uniaxial test(32) σ = λ W λ = 2 μ ( λ 2 1 λ ) i = 1 5 i c i λ m 2 i 2 ( λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 ) i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jaeq4UdW2aaSaaaeaacqGHciITcaWGxbaabaGaeyOaIyRaeq4U dWgaaiabg2da9iaaikdacqaH8oqBdaqadaqaaiabeU7aSnaaCaaale qabaGaaGOmaaaakiabgkHiTmaalaaabaGaaGymaaqaaiabeU7aSbaa aiaawIcacaGLPaaadaaeWbqaamaalaaabaGaamyAaiabgwSixlaado gadaWgaaWcbaGaamyAaaqabaaakeaacqaH7oaBdaWgaaWcbaGaamyB aaqabaGcdaahaaWcbeqaaiaaikdacaWGPbGaeyOeI0IaaGOmaaaaaa aabaGaamyAaiabg2da9iaaigdaaeaacaaI1aaaniabggHiLdGcdaqa daqaaiqbeU7aSzaaraWaaSbaaSqaaiaaigdaaeqaaOWaaWbaaSqabe aacaaIYaaaaOGaey4kaSIafq4UdWMbaebadaWgaaWcbaGaaGOmaaqa baGcdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcuaH7oaBgaqeamaaBa aaleaacaaIZaaabeaakmaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaa wMcaamaaCaaaleqabaGaamyAaiabgkHiTiaaigdaaaaaaa@6D76@

    with λ 1 = λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBcaaMe8oaaa@3D4B@ and λ 2 = λ 3 = λ 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaH7oaBdaWgaaWcbaGaaG4m aaqabaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiabgkHiTmaalaaaba GaaGymaaqaaiaaikdaaaaaaaaa@420D@ , then I ¯ 1 = λ 2 + 2 λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlqadM eagaqeamaaBaaaleaacaaIXaaabeaakiabg2da9iabeU7aSnaaCaaa leqabaGaaGOmaaaakiabgUcaRmaalaaabaGaaGOmaaqaaiabeU7aSb aaaaa@406A@

    and nominal stress is:(33) N t h = W λ = 2 μ ( λ λ 2 ) i = 1 5 i c i ( λ m ) 2 i 2 ( λ 2 + 2 λ ) i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaCa aaleqabaGaamiDaiaadIgaaaGccqGH9aqpdaWcaaqaaiabgkGi2kaa dEfaaeaacqGHciITcqaH7oaBaaGaeyypa0JaaGOmaiabeY7aTjaacI cacqaH7oaBcqGHsislcqaH7oaBdaahaaWcbeqaaiabgkHiTiaaikda aaGccaGGPaWaaabCaeaadaWcaaqaaiaadMgacqGHflY1caWGJbWaaS baaSqaaiaadMgaaeqaaaGcbaGaaiikaiabeU7aSnaaBaaaleaacaWG TbaabeaakiaacMcadaahaaWcbeqaaiaaikdacaWGPbGaeyOeI0IaaG OmaaaaaaaabaGaamyAaiabg2da9iaaigdaaeaacaaI1aaaniabggHi LdGcdaqadaqaaiabeU7aSnaaCaaaleqabaGaaGOmaaaakiabgUcaRm aalaaabaGaaGOmaaqaaiabeU7aSbaaaiaawIcacaGLPaaadaahaaWc beqaaiaadMgacqGHsislcaaIXaaaaaaa@676C@
  • Equibiaxial test(34) σ = λ W λ = 2 μ ( λ 2 1 λ 4 ) i = 1 5 i c i λ m 2 i 2 ( λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 ) i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jaeq4UdW2aaSaaaeaacqGHciITcaWGxbaabaGaeyOaIyRaeq4U dWgaaiabg2da9iaaikdacqaH8oqBdaqadaqaaiabeU7aSnaaCaaale qabaGaaGOmaaaakiabgkHiTmaalaaabaGaaGymaaqaaiabeU7aSnaa CaaaleqabaGaaGinaaaaaaaakiaawIcacaGLPaaadaaeWbqaamaala aabaGaamyAaiabgwSixlaadogadaWgaaWcbaGaamyAaaqabaaakeaa cqaH7oaBdaWgaaWcbaGaamyBaaqabaGcdaahaaWcbeqaaiaaikdaca WGPbGaeyOeI0IaaGOmaaaaaaaabaGaamyAaiabg2da9iaaigdaaeaa caaI1aaaniabggHiLdGcdaqadaqaaiqbeU7aSzaaraWaaSbaaSqaai aaigdaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIafq4UdWMb aebadaWgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaaikdaaaGccq GHRaWkcuaH7oaBgaqeamaaBaaaleaacaaIZaaabeaakmaaCaaaleqa baGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaamyAaiabgk HiTiaaigdaaaaaaa@6E6B@

    with λ 1 = λ 2 = λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBdaWgaaWcbaGaaGOm aaqabaGccqGH9aqpcqaH7oaBcaaMe8oaaa@40F7@ and λ 3 = λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaG4maaqabaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiab gkHiTiaaikdaaaaaaa@3D96@ , then I ¯ 1 = 2 λ 2 + 1 λ 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmysayaara WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabeU7aSnaaCaaa leqabaGaaGOmaaaakiabgUcaRmaalaaabaGaaGymaaqaaiabeU7aSn aaCaaaleqabaGaaGinaaaaaaaaaa@4083@

    and the nominal stress is:(35) N t h = W λ = 2 μ ( λ λ 5 ) i = 1 5 i c i ( λ m ) 2 i 2 ( 2 λ 2 + 1 λ 4 ) i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaCa aaleqabaGaamiDaiaadIgaaaGccqGH9aqpdaWcaaqaaiabgkGi2kaa dEfaaeaacqGHciITcqaH7oaBaaGaeyypa0JaaGOmaiabeY7aTjaacI cacqaH7oaBcqGHsislcqaH7oaBdaahaaWcbeqaaiabgkHiTiaaiwda aaGccaGGPaWaaabCaeaadaWcaaqaaiaadMgacqGHflY1caWGJbWaaS baaSqaaiaadMgaaeqaaaGcbaGaaiikaiabeU7aSnaaBaaaleaacaWG TbaabeaakiaacMcadaahaaWcbeqaaiaaikdacaWGPbGaeyOeI0IaaG OmaaaaaaaabaGaamyAaiabg2da9iaaigdaaeaacaaI1aaaniabggHi LdGcdaqadaqaaiaaikdacqaH7oaBdaahaaWcbeqaaiaaikdaaaGccq GHRaWkdaWcaaqaaiaaigdaaeaacqaH7oaBdaahaaWcbeqaaiaaisda aaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGPbGaeyOeI0IaaG ymaaaaaaa@691F@
  • Planar test(36) σ = λ W λ = 2 μ ( λ 2 1 λ 2 ) i = 1 5 i c i λ m 2 i 2 ( λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 ) i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0Jaeq4UdW2aaSaaaeaacqGHciITcaWGxbaabaGaeyOaIyRaeq4U dWgaaiabg2da9iaaikdacqaH8oqBdaqadaqaaiabeU7aSnaaCaaale qabaGaaGOmaaaakiabgkHiTmaalaaabaGaaGymaaqaaiabeU7aSnaa CaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaadaaeWbqaamaala aabaGaamyAaiabgwSixlaadogadaWgaaWcbaGaamyAaaqabaaakeaa cqaH7oaBdaWgaaWcbaGaamyBaaqabaGcdaahaaWcbeqaaiaaikdaca WGPbGaeyOeI0IaaGOmaaaaaaaabaGaamyAaiabg2da9iaaigdaaeaa caaI1aaaniabggHiLdGcdaqadaqaaiqbeU7aSzaaraWaaSbaaSqaai aaigdaaeqaaOWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIafq4UdWMb aebadaWgaaWcbaGaaGOmaaqabaGcdaahaaWcbeqaaiaaikdaaaGccq GHRaWkcuaH7oaBgaqeamaaBaaaleaacaaIZaaabeaakmaaCaaaleqa baGaaGOmaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaamyAaiabgk HiTiaaigdaaaaaaa@6E69@

    with λ 1 = λ , λ 3 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH7oaBcaaMe8Uaaiilaiaa ysW7cqaH7oaBdaWgaaWcbaGaaG4maaqabaGccqGH9aqpcaaIXaGaaG zbVdaa@457E@ and λ 2 = λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBda WgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaH7oaBdaahaaWcbeqaaiab gkHiTiaaigdaaaGccaaMb8oaaa@3F28@ , then I ¯ 1 = λ 2 + 1 + λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjbVlqadM eagaqeamaaBaaaleaacaaIXaaabeaakiabg2da9iabeU7aSnaaCaaa leqabaGaaGOmaaaakiabgUcaRiaaigdacqGHRaWkcqaH7oaBdaahaa WcbeqaaiabgkHiTiaaikdaaaaaaa@4311@

    and nominal stress is:(37) N t h = W λ = 2 μ ( λ λ 3 ) i = 1 5 i c i ( λ m ) 2 i 2 ( λ 2 + 1 + λ 2 ) i 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaCa aaleqabaGaamiDaiaadIgaaaGccqGH9aqpdaWcaaqaaiabgkGi2kaa dEfaaeaacqGHciITcqaH7oaBaaGaeyypa0JaaGOmaiabeY7aTjaacI cacqaH7oaBcqGHsislcqaH7oaBdaahaaWcbeqaaiabgkHiTiaaioda aaGccaGGPaWaaabCaeaadaWcaaqaaiaadMgacaWGJbWaaSbaaSqaai aadMgaaeqaaaGcbaGaaiikaiabeU7aSnaaBaaaleaacaWGTbaabeaa kiaacMcadaahaaWcbeqaaiaaikdacaWGPbGaeyOeI0IaaGOmaaaaaa aabaGaamyAaiabg2da9iaaigdaaeaacaaI1aaaniabggHiLdGcdaqa daqaaiabeU7aSnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdacq GHRaWkcqaH7oaBdaahaaWcbeqaaiabgkHiTiaaikdaaaaakiaawIca caGLPaaadaahaaWcbeqaaiaadMgacqGHsislcaaIXaaaaaaa@67D4@

Additional information about Arruda-Boyce model. 2 3

Yeoh Material (LAW94)

The Yeoh model (LAW94) 4is a hyperelastic material model that can be used to describe incompressible materials. The strain energy density function of LAW94 only depends on the first strain invariant and is computed as:(38) W = i = 1 3 [ C i 0 ( I ¯ 1 3 ) i W ( I ¯ 1 ) + 1 D i ( J 1 ) 2 i U ( J ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaiabg2 da9maaqahabaWaamWaaeaadaagaaqaaiaadoeadaWgaaWcbaGaamyA aiaaicdaaeqaaOWaaeWaaeaaceWGjbGbaebadaWgaaWcbaGaaGymaa qabaGccqGHsislcaaIZaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWG PbaaaaqaaiaadEfacaGGOaGabmysayaaraWaaSbaaWqaaiaaigdaae qaaSGaaiykaaGccaGL44pacqGHRaWkdaagaaqaamaalaaabaGaaGym aaqaaiaadseadaWgaaWcbaGaamyAaaqabaaaaOWaaeWaaeaacaWGkb GaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaiaa dMgaaaaabaGaamyvaiaacIcacaWGkbGaaiykaaGccaGL44paaiaawU facaGLDbaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaaG4maaqdcqGH ris5aaaa@5CAE@
Where,
I ¯ 1 = λ ¯ 1 2 + λ ¯ 2 2 + λ ¯ 3 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmysayaara WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jafq4UdWMbaebadaqhaaWc baGaaGymaaqaaiaaikdaaaGccqGHRaWkcuaH7oaBgaqeamaaDaaale aacaaIYaaabaGaaGOmaaaakiabgUcaRiqbeU7aSzaaraWaa0baaSqa aiaaiodaaeaacaaIYaaaaaaa@44FF@
First strain invariant
λ ¯ i = J 1 3 λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4UdWMbae badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGkbWaaWbaaSqabeaa cqGHsisldaWcaaqaaiaaigdaaeaacaaIZaaaaaaakiabeU7aSnaaBa aaleaacaWGPbaabeaaaaa@4036@
Deviatoric stretch
The Cauchy stress is computed as:(39) σ i = λ i J W λ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacqaH7oaBdaWgaaWc baGaamyAaaqabaaakeaacaWGkbaaamaalaaabaGaeyOaIyRaam4vaa qaaiabgkGi2kabeU7aSnaaBaaaleaacaWGPbaabeaaaaaaaa@4421@

For incompressible materials with i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E4@ =1 only and D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ are input and the Yeoh model is reduced to a Neo-Hookean model.

The material constant specify the deviatoric part (shape change) of the material and parameters D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ , D 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ , D 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ specify the volumetric change of the material. These six material constants need to be calculated by curve fitting material test data. RD-E: 5600 Hyperelastic Material with Curve Input includes a Yeoh fitting Compose script for uniaxial test data. The Yeoh material model has been shown to model all deformation models even if the curve fit was obtained using only uniaxial test data.

The initial shear modulus and the bulk modulus are computed as:

μ=2 C 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey ypa0JaaGOmaiabgwSixlaadoeadaWgaaWcbaGaaGymaiaaicdaaeqa aaaa@3E22@ and K= 2 D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maalaaabaGaaGOmaaqaaiaadseadaWgaaWcbaGaaGymaaqabaaa aaaa@3A49@

LAW94 is available only as an incompressible material model.

If D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ =0, an incompressible material is considered, where ν = 0.495 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyVd4Maey ypa0JaaGimaiaac6cacaaI0aGaaGyoaiaaiwdaaaa@3C61@ and D 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaaaaa@37A7@ is calculated as:(40) D 1 = 3(12v) μ(1+v) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiramaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaaG4maiaacIcacaaI XaGaeyOeI0IaaGOmaiaadAhacaGGPaaabaGaeqiVd0Maaiikaiaaig dacqGHRaWkcaWG2bGaaiykaaaaaaa@43E2@
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