A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

20 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
where S ij is the (i, j) component of S. By (2.12) and (3.6) we find that the Cauchy stress
tensor σ is given by
σ = 1
|A| A ∂W
∂E AT .
Strain energy function for isotropic elastic materials
For an isotropic material, the configuration gradient A can be expressed uniquely in
terms of the principal stretches (λ i , i = 1, 2, 3) or in terms of the invariants (I 1 , I 2 , I 3 )
of the left Cauchy-Green configuration tensor or right Cauchy-Green configuration
tensor (see Remark 2.3). Hence, we can express the strain energy function in terms of
principal stretches or in terms of invariants. Note that
λ 1 = λ 2 = λ 3 = 1, I 1 = 3, I 2 = 3, and I 3 = 1,
in the initial configuration where we choose W = 0. Thus a general formula for the
strain energy function can be expressed as
or
=
W(λ 1 , λ 2 , λ 3 )
∞
∑
i,j,k=0
a ijk
{[
λ i 1 (λj 2 + λj 3 ) + λi 2(λ j 3 + λj 1 ) + λi 3(λ j 1 + λj 2 ) ](λ 1 λ 2 λ 3 ) k − 6
W(I 1 , I 2 , I 3 ) =
∞
∑ c ijk (I 1 − 3) i (I 2 − 3) j (I 3 − 1) k .
i,j,k=0
}
, (3.7a)
(3.7b)
Due to their ubiquitous approximation properties, polynomial terms are usually chosen
in formulating strain energy functions, but the final forms are typically based on
empirical observations and are material specific for the choice of coefficients and truncations.
For incompressible materials (many rubber or elastomeric materials are often
nearly incompressible), |A| = 1 (which implies that λ 1 λ 2 λ 3 = 1 and I 3 = 1), so (3.7a)
can be reduced to
W(λ 1 , λ 2 , λ 3 ) =
∞ [
∑ a ij λ1 i (λj 2 + λj 3 ) + λi 2(λ j 3 + λj 1 ) + λi 3(λ j 1 + λj 2
], ) − 6 (3.8)
i,j=0
subject to
and (3.7b) can be reduced to
λ 1 λ 2 λ 3 = 1,
W(I 1 , I 2 ) =
Special cases for (3.9) include several materials:
∞
∑ c i,j (I 1 − 3) i (I 2 − 3) j . (3.9)
i,j=0