A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

30 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
Figure 11: Schematic representation of the Kelvin-Voigt model.
Because the two elements are subject to the same strain, the model is also known as an
iso-strain model. The total stress is the sum of the stress in the spring and the stress in
the dashpot, so that
σ = κε + η dε
dt . (3.17)
We first consider the stress relaxation function and creep function for the Kelvin-Voigt
model (3.17). The stress relaxation function corresponds to the solution of (3.17) when
We find
ε(t) = ε 0 H(t − t 0 ), and σ(0) = 0.
σ(t) = κε 0 H(t − t 0 ) + ηε 0 δ(t − t 0 ).
This stress relaxation function for the Kelvin-Voigt model (3.17) is illustrated in Fig. 12.
The creep function again is the solution ε(t) to (3.17) corresponding to σ(t) =
σ 0 H(t − t 0 ) and ε(0) = 0. We find
κε + η dε
dt = σ 0H(t − t 0 ).
Let ˆε(s) = L {ε(t)}(s), or in terms of the Laplace transform we have
e
ˆε(s) = −t 0s
σ 0
s(κ + ηs) = σ [
0 e
−t 0 s
−
e−t 0s ]
.
κ s s + κ/η
Using the inverse Laplace transform we obtain
ε(t) = 1 κ
[ (
1 − exp − κ )]
η (t − t 0) σ 0 H(t − t 0 ).
The corresponding creep function for the Kelvin-Voigt model (3.17) is illustrated in
Fig. 13 (compare with Fig. 5).
Thus we find that the Kelvin-Voigt model is extremely accurate in modelling creep
in many materials. However, the model has limitations in its ability to describe the
commonly observed relaxation of stress in numerous strained viscoelastic materials.