A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 33
or in terms of the Laplace transform
e
ˆσ(s) −t 0s
=κ r ε 0
s(1 + τ ε s) + κ e −t 0s
rτ σ ε 0
1 + τ ε s
e −t 0s ( τσ
) e
−t 0 s
=κ r ε 0 + κ r ε 0 − 1
s τ ε s + 1 .
τ ε
Thus we find
( τσ
)
σ(t) =κ r ε 0 H(t − t 0 ) + κ r ε 0 − 1
τ ε
( τσ
)
=κ r
[1 + − 1 exp
τ ε
[
= κ r + κ 1 exp
[
exp − t − t ]
0
H(t − t 0 )
τ ε
(
− t − t 0
τ ε
)]
ε 0 H(t − t 0 )
(
− t − t 0
τ ε
)]
ε 0 H(t − t 0 ).
This stress relaxation function for the standard linear model (3.18) is illustrated in
Fig. 15.
The creep function is the solution of (3.18) for ε(t) given σ(t) = σ 0 H(t − t 0 ) and
ε(0) = 0. Using the same arguments as above in finding the stress function, we have
ε(t) = 1 ( τε
) (
[1 + − 1 exp − t − t )]
0
σ 0 H(t − t 0 ).
κ r τ σ τ σ
The creep function of the standard linear model (3.18) is illustrated in Fig. 16.
We therefore see that the standard linear model is accurate in predicating both
creep and relaxation responses for many materials of interest.
Figure 15: Stress relaxation function for the standard linear model.
Figure 16: Creep function for the standard linear model.