There are two basic models for linear viscoelastic modeling: Maxwell and Voigt.
46 The Maxwell model uses a viscous damper and an elastic spring connected in series, so that it can predict relaxation describing how a material deformed by external perturbation returns to equilibrium. However, it cannot predict creep, which is the tendency of a material to deform permanently under constant force. Conversely, the Voigt model consists of a Newtonian damper and Hookean elastic spring connected in parallel that can describe creep accurately, but is poor for predicting relaxation. For the purpose of compensating these limitations, several models have been constructed as linear combinations of springs and dashpots. We elected to employ the Wiechert model, which provides for superposition of linear combinations arbitrarily many spring-dashpot elements (
Fig. 2), and is convenient for finite element analysis. For stress relaxation, the total stress σ(t) transmitted by the Wiechert constitutive model is given by
47 where
t is the time, E
i (i = 0, 1, …) the relaxation modulus of the i-th spring, τ
i (i = 1, 2, …) the relaxation time of the i-th dashpot, and ε
0 is the constant strain applied to the material during the stress relaxation testing. Both sides of
Equation 1 can be divided by
ε0, giving
where E
rel is called the time-dependent relaxation modulus.
Through experiments, a Wiechert model for a viscoelastic material can be defined by determination of constants E
i (i = 0, 1, …, n) and
τi (i = 1, 2, …, n) in the Prony series of
Equation 2. In this study, EOM specimens were characterized by Wiechert model of stress relaxation data.