A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

40 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
We observe that if the relaxation response kernel function K in model (3.27) is defined
as in (3.22), then (3.27) can be written in terms of internal strains similar to (3.23)
with (3.24):
n
σ(t) = g e (ε(t)) + c D ˙ε(t) + ∑ κ j ε j (t), (3.30)
j=1
where the internal strains ε j satisfies
dε j (t)
+ 1 ( )
ε j (t) = g v ε(t), ˙ε(t) , ε j (0) = 0, j = 1, 2, · · · , n. (3.31)
dt τ j
Through comparison with experimental data, in [10] it was discovered that the best
fit to filled elastomer data occurs when g e and g v are cubic. Moreover, model (3.30)
with (3.31) can be readily generalized to models with a continuum of relaxation times
(see [12] and the references therein). These models have been useful in a wide range
of viscoelastic materials. The corresponding stress-strain laws have the form
∫
σ(t; P) = g e (ε(t)) + c D ˙ε(t) + γ ε 1 (t; τ)dP(τ), (3.32)
where P is a probability distribution over the set T of possible relaxation parameters
τ, and ε 1 (t; τ) satisfies, for each τ ∈ T ,
dε 1 (t; τ)
dt
T
+ 1 τ ε 1(t; τ) = g v (ε(t), ˙ε(t)), ε 1 (0; τ) = 0. (3.33)
We also observe that if the reduced relaxation function K(t) in Fung’s QLV model
(3.28) is defined as in (3.22) with
n
κ j = 1,
∑
k=1
then model (3.28) is equivalent to the following internal strain variable formulation
with the internal strains ε j satisfying
dε j (t)
dt
σ(t) =
n
∑ κ j ε j (t), (3.34)
j=1
+ 1 ε j (t) = dσe (λ(t))
, ε j (0) = 0, j = 1, 2, · · · , n. (3.35)
τ j dt
Various internal strain variable models are investigated in [1] and a good agreement
is demonstrated between a two internal strain variable model (i.e., n = 2) and undamped
simulated data based on the Fung kernel (3.29). The corresponding stressstrain
laws for a generalization of these models to have a continuum of relaxation
times have the form
∫
σ(t; P) = ε 1 (t; τ)dP(τ), (3.36)
T