Hyper Visco-elastic Law for Foams (LAW62)

Experimental tests on foam specimens working in compression illustrate that the material behavior is highly nonlinear. The general behavior can be subdivided into three parts related to particular deformation modes of material cells. When the strain is small, the cells working in compression deform in membrane without causing buckling in its lateral thin-walls. In the second step, the lateral thin-walls of the cells buckle while the material undergoes large deformation. Finally, in the last step the cells are completely collapsed and the contact between the lateral thin-walled cells increases the global stiffness of the material.

As the viscous behavior of foams is demonstrated by various tests, it is worthwhile to elaborate a material law including the viscous and hyper elasticity effects. This is developed in 1 where a decoupling between viscous and elastic parts is introduced by using finite transformations. Only the deviatoric part of the stress tensor is concerned by viscous effects.

Material LAW62 corresponds to a hyper-elastic solid material using the Ogden formulation for rubber material. The strain energy functional 2 is given by:(1)
W ( C ) = i = 1 N 2 μ i α i 2 ( λ 1 α i + λ 2 α i + λ 3 α i 3 + 1 β ( J α i β 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4vamaabm aabaGaaC4qaaGaayjkaiaawMcaaiabg2da9maaqahabaWaaSaaaeaa caaIYaGaeqiVd02aaSbaaSqaaiaadMgaaeqaaaGcbaGaeqySde2aaS baaSqaaiaadMgaaeqaaOWaaWbaaSqabeaacaaIYaaaaaaaaeaacaWG PbGaeyypa0JaaGymaaqaaiaad6eaa0GaeyyeIuoakmaabmaabaGaeq 4UdW2aaSbaaSqaaiaaigdaaeqaaOWaaWbaaSqabeaacqaHXoqydaWg aaadbaGaamyAaaqabaaaaOGaey4kaSIaeq4UdW2aaSbaaSqaaiaaik daaeqaaOWaaWbaaSqabeaacqaHXoqydaWgaaadbaGaamyAaaqabaaa aOGaey4kaSIaeq4UdW2aaSbaaSqaaiaaiodaaeqaaOWaaWbaaSqabe aacqaHXoqydaWgaaadbaGaamyAaaqabaaaaOGaeyOeI0IaaG4maiab gUcaRmaalaaabaGaaGymaaqaaiabek7aIbaadaqadaqaaiaadQeada ahaaWcbeqaaiabgkHiTiabeg7aHnaaBaaameaacaWGPbaabeaaliab ek7aIbaakiabgkHiTiaaigdaaiaawIcacaGLPaaaaiaawIcacaGLPa aaaaa@69D8@
Where, C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4qaaaa@36C3@ is the right Cauchy Green Tensor, C = F t F with F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ the deformation gradient matrix, λ i are the eigenvalues of F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36DC@ , J = det F MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOsaiabg2 da9iGacsgacaGGLbGaaiiDaiaadAeaaaa@3B61@ , β = ν ( 1 2 ν ) , ν 0 and ν 1 2
注: For rubber materials which are almost incompressible, the bulk modulus is very large compared to the shear modulus.
The ground shear modulus is given by:(2)
μ = i = 1 N μ i
W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4vaaaa@36D3@ can be written as:(3)
W ( C ) = W ¯ ( C ¯ ) + U ( J )
Where,
C ¯ = F ¯ t F ¯
F ¯ = J 1 / 3 F
C ¯
Deviatoric part of the right Cauchy Green Tensor
U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGvbaaaa@373A@ and W ¯
Volumetric and deviatoric parts of the stored energy functions and S 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIWaaabeaaaaa@37B5@ the second Piola-Kirchhoff stress tensor given by:
(4)
S 0 = W E = 2 W C = 2 W ¯ C + 2 U C = S 0 d e v + S 0 v o l

With E = 1 2 ( C I )

The Green-Lagrange strain tensor:

S 0 d e v = 2 W ¯ C and S 0 v o l = 2 U C are the deviatoric and volumetric parts of the second Piola-Kirchhoff stress tensor S 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIWaaabeaaaaa@37B4@ .

Rate effects are modeled through visco-elasticity using a convolution integral using Prony series. This corresponds to an extension of small strain theory or finite deformation to large strain. The rate effect is applied only to the deviatoric stress. The deviatoric stress is computed as:(5)
S d e v ( t ) = γ S 0 d e v ( t ) J 2 / 3 D E V [ i = 1 M i Q i ( t ) ]
Where, Q i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGPbaabeaaaaa@37E7@ is the internal variable given by the following rate equations:(6)
Q ˙ i ( t ) + 1 τ i Q i ( t ) = γ i τ i D E V [ 2 W ¯ C ¯ ( t ) ]

lim Q i ( t ) = 0 , t

γ i [ 0 , 1 ] , τ i > 0

Q i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaacaWGPbaabeaaaaa@37E7@ is given by the following convolution integral:(7)
Q i ( t ) = γ i τ i t exp [ ( t s ) / τ i ] d [ D E V { 2 C ¯ W ¯ 0 [ C ¯ ( s ) ] } ] d s d s
Where,
γ = G / G 0
1 = γ + i = 1 M i γ i
γ i = G i / G 0
G 0 = G + i = 1 M i G i
d e v ( ) = 1 3 ( : C ) C 1

Where, G 0 is the initial shear modulus; G 0 should be exactly the same as the ground shear modulus μ . G is the long-term shear modulus that can be obtained from long-term material testing. τ i are the relaxation times.

The relation between the second Piola-Kirchhoff stress tensor S = S d e v + S 0 v o l and Cauchy stress tensor σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4Wdaaa@3745@ is:(8)
σ = 1 det F F S F t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHdpGaey ypa0ZaaSaaaeaacaaIXaaabaGaciizaiaacwgacaGG0bGaaCOraaaa caWHgbGaaC4uaiaahAeadaahaaWcbeqaaiaadshaaaaaaa@40B9@
1 Simo J.C., 「On a fully three-dimensional finite strain viscoelastic damage model: Formulation and Computational Aspects」, Computer Methods in Applied Mechanics and Engineering, Vol. 60, pp. 153-173, 1987.
2 Ogden R.W., 「Nonlinear Elastic Deformations」, Ellis Horwood, 1984.