A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

36 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
Figure 17: Schematic representation of the generalized standard linear model.
distinct relaxation property characterized by a relaxation parameter τ j . For a comparison
of models of viscoelastic damping via hysteretic integrals versus internal variable
representations, see [7] and the references therein.
Moreover, we see that model (3.23) with (3.24) can be readily viewed as a special
case of or even an approximation to models with a continuum of relaxation times
(see [12] and the references therein). These models have proved useful in a wide range
of viscoelastic materials. The corresponding stress-strain laws have the form
σ(t; P) = κ r ε(t) + η dε(t)
dt
∫
+ γ
T
ε 1 (t; τ)dP(τ), (3.25)
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
+ 1 τ ε 1(t; τ) = dε(t) , ε 1 (0; τ) = 0. (3.26)
dt
The approach embodied in equations (3.25)-(3.26) offers a computationally tractable
alternative for linear materials with a continuum of relaxation times.
Remark 3.7. The generalized standard linear model (i.e., model (3.23) combined with
(3.24)) with different n has been successfully used to describe the stress relaxation behavior
of a variety of foods. For example, in [57] this model with n = 2 was successfully
used to describe the stress relaxation of lipids such as beeswax, candelilla wax,
carnauba wax and a high melting point milkfat fraction. A comprehensive study on
the ability of the generalized Maxwell model to describe the stress relaxation behavior
of solid food is presented in [18]. In this study, five different food matrices (agar
gel, meat, ripened cheese, ”mozzarella” cheese and white pan bread) were chosen as
representatives of a wide range of foods, and results verify that the proposed model
satisfactorily fits the experimental data.