σ D B D
À Á ¼ 2I
À1=2
3
∂ψ I
∂I 1
þ I 1
∂ψ I
∂I 2
B D À 2I
À1=2
3
∂ψ I
∂I 2
B
2
D
ð46Þ
σ V I 3 I
ð Þ ¼ 2I
1=2
3
∂ψ V
∂I 3
I
ð47Þ
in which the subscripts D and V represent the deviatoric and the volumetric part of
the tensor. Equations (46) and (47) state that:
1. A superimposed deviatoric stress doesn’t produce any volume changes, but only
shape changes (tr σ D ¼ 0).
2. A superimposed pressure uniquely produces a volume change.
Points 1 and 2 are the extension to the nonlinear case of the deviatoric/volumetric stress decomposition introduced in the framework of linear elasticity (and are
consistent with that).
Equation (47) implies that the reference configuration is stress-free if the following restriction on the function ψ V is valid:
∂ψ V
∂I 3
I 3 ¼1
¼ 0
ð48Þ
Therefore, assuming a sufficient regularity for the function ψ V , expression (43)
can be expanded by Taylor series around the undeformed configuration (I 3 ¼ 1) as:
ψ V I 3
ð Þ ¼
X 1
i¼2
1
D i
I 3 À 1
ð
Þ
i
ð49Þ
where
D i
ð Þ
À1 ¼
1
i!
∂
i ψ V
∂I
i
S
I 3 ¼1
ð50Þ
With similar assumptions, one obtain the following expansion around the reference configuration (I 1 ¼ 3, I 2 ¼ 3) for the function ψ I :
ψ I I 1 ; I 2
À
Á ¼
X 1
i, j
c ij I 1 À 3
À
Á i I 2 À 3
À
Á j
ð51Þ
where
If (i, j) ¼ (0, 0)
Modeling of Non-Linear Viscoelastic Behavior of Filled Rubbers
233