A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

32 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
that obey the Biot theory (the two-phase formulation of Biot in [15]). This viscoelastic
model is simpler to use than poroelastic models but yields similar results for a
wide range of soils and dynamic loadings. In addition, the author in [40] developed a
model, the Kelvin-Voigt-Maxwell-Biot model, that splits the soil into two components
(pore fluid and solid frame) where these two masses are connected by a dashpot which
can then be related to permeability. In addition, a mapping between the Kelvin-Voigt
model and the Kelvin-Voigt-Maxwell-Biot model is developed in [40] so that one may
continue to use the Kelvin-Voigt model for saturated soil.
The standard linear solid (SLS) model
The standard linear solid model, also known as the Kelvin model or three-element
model, combines the Maxwell Model and a Hookean spring in parallel as depicted
in Fig. 14.
The stress-strain relationship is given by
dσ
(
σ + τ ε
dt = κ r
ε + τ σ
dε
dt
)
, (3.18)
where
τ ε = η 1
κ 1
, and τ σ = η 1
κ r + κ 1
κ r κ 1
,
from which can be observed that τ σ > τ ε .
The stress relaxation function and the creep function for the standard linear model
(3.18) are obtained in the usual manner. As usual, the stress relaxation function σ(t) is
obtained by solving (3.18) with ε(t) = ε 0 H(t − t 0 ) and σ(0) = 0. We find
σ(t) + τ ε
dσ
dt = κ rε 0 H(t − t 0 ) + κ r τ σ ε 0 δ(t − t 0 ),
Figure 14: Schematic representation of the standard linear model.