A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

26 H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51
The relationship between the relaxation modulus G(t) and dynamic modulus functions
G ′ and G” is given by
∫ ∞
G ′ (ω) = G ∞ + ω
G”(ω) = ω
∫ ∞
The relations may be inverted to obtain
0
0
( G(t) − G∞
)
sin(ωt)dt,
( G(t) − G∞
)
cos(ωt)dt.
or
G(t) = G ∞ + 2 π
G(t) = G ∞ + 2 π
∫ ∞
0
∫ ∞
0
G ′ (ω) − G ∞
ω
G”(ω)
ω
sin(ωt)dω,
cos(ωt)dω.
The interested reader can refer to the recent text [33] for further information.
3.2.3 Linear viscoelastic models: constitutive relationships
The characteristic feature of linear viscoelastic materials is that the stress is linearly
proportional to the strain history, and it is important to note that the property of linearity
of response does not refer to the shape of any material response curve. Linear
viscoelasticity is usually applicable only for small deformations and/or linear materials.
Thus, infinitesimal strain theory should be employed for this case. There are
two standard approaches that have been used to develop constitutive equations for
the linear viscoelastic materials: mechanical analogs and the Boltzmann superposition
principle.
Mechanical analogs
Linear viscoelastic behavior can be conceived as a linear combinations of springs (the
elastic component) and dashpots (the viscous component) as depicted by Fig. 7. The
elastic component is described by
σ = κε,
or
dε
dt = 1 dσ
κ dt ,
where σ is the stress, ε is the strain that occurs under the given stress, and κ is the
elastic modulus of the material with units N/m 2 . The viscous component is modeled
by
σ = η dε
dt ,