A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 25
Note that the in-phase components produce no net work when integrated over a cycle,
while the out-of phase components result in a net dissipation per cycle equal to:
W = πalε 0 σ 0 sin(δ).
Thus, for a purely elastic solid, the stress is in phase with the strain (δ = 0) and no
energy is dissipated. On the other hand, motion in the viscoelastic solid produces
energy loss.
It is a common practice in engineering to use complex variables to describe the
sinusoidal response of viscoelastic materials. Thus, instead of strain history (3.10), we
specify the complex strain as
ε ∗ = ε 0 exp(iωt).
Then we obtain the following complex stress instead of stress described by (3.11)
The above equation can be rewritten as
where G ∗ is defined by
σ ∗ = σ 0 exp ( i(ωt + δ) ) .
σ ∗ = G ∗ ε ∗ ,
G ∗ = σ 0
ε 0
exp(iδ) = σ 0
ε 0
cos(δ) + i σ 0
ε 0
sin(δ). (3.13)
The characteristic parameter G ∗ is referred to as the complex dynamic modulus. We
denote the real part of G ∗ by G ′ and the imaginary part of G ∗ by G”. That is,
where
G ∗ = G ′ + iG”, (3.14)
G ′ = σ 0
ε 0
cos(δ), and G” = σ 0
ε 0
sin(δ).
The coefficient G ′ is called the storage modulus (a measure of energy stored and recovered
per cycle) which corresponds to the in-phase response, and G” is the loss modulus
(a characterization of the energy dissipated in the material by internal damping) corresponding
to the out-of phase response. The in-phase stress and strain results in elastic
energy, which is completely recoverable. The π/2 out-of-phase stress and strain results
in the dissipated energy.
Remark 3.3. The relationship between the two transient functions, relaxation modulus
G(t) and creep compliance J(t), for a viscoelastic material is given by
∫ t
0
J(s)G(t − s)ds = t.