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9
Linear Visco elasticity
9.1
Introduction
The previous chapters have considered constitutive equations that deal pri-
marily with two different types of material behavior: elastic response of
solids and viscous flow of fluids. Examples of materials that behave elasti-
cally under modest loading, and at moderate temperatures, are metals such
as steel, aluminum, and copper, certain polymers, and even cortical bone.
Examples of viscous flow may involve a variety of fluids ranging from water
to polymers under certain conditions of temperature and loading.
Polymers are especially interesting because they may behave (respond) in
either elastic, viscous, or combined manners. At a relative moderate temper-
ature and loading, a polymer such as polymethalmethacrylite (PMMA, plexi-
glass), may be effectively modeled by a linear elastic constitutive equation.
However, at a somewhat elevated temperature, the same material may have
to be modeled as a viscous fluid.
Polymers are by no means the only materials that exhibit different behavior
under altered temperature/frequency conditions. Steel, as well as aluminum,
copper, and other metals, becomes molten at high temperatures and can be
poured into molds to form ingots. Additionally, at a high enough deforma-
tion rate, for example, at the 48 km/hr rate of a vehicle crash, steel will
exhibit considerably altered stiffness properties.
Just as continuum mechanics is the basis for constitutive models as
distinct as elastic solids (stress/strain laws) and viscous fluids
(stress/strain-rate laws), it also serves as the basis for constitutive relations
that describe material behavior over a range of temperature/frequency and
time. One of the simplest models for this combined behavior is that of
linear viscoelasticity.
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9.2
Viscoelastic Constitutive Equations in
Linear Differential
Operator Form
One of the principal features of elastic behavior is the capacity for materials
to store mechanical energy when deformed by loading, and to release this
energy totally upon removal of the loads. Conversely, in viscous flow,
mechanical energy is continuously dissipated with none stored. A number
of important engineering materials simultaneously store and dissipate
mechanical energy when subjected to applied forces. In fact, all actual mate-
rials store and dissipate energy in varying degrees during a loading/unload-
ing cycle. This behavior is referred to as
. In general, viscoelastic
behavior may be imagined as a spectrum with elastic deformation as one
limiting case and viscous flow the other extreme case, with varying combi-
nations of the two spread over the range between. Thus, valid constitutive
equations for viscoelastic behavior embody elastic deformation and viscous
flow as special cases, and at the same time provide for response patterns
that characterize behavior blends of the two. Intrinsically, such equations
will involve not only stress and strain, but time-rates of both stress and strain
as well.
In developing the
viscoelastic
form of constitutive equations
for viscoelastic behavior as presented in Eq 5.12-6 we draw upon the pair of
constitutive equations for elastic behavior, Eq 6.2-12, repeated here,
linear differential operator
SG
ij
=
2
η
(9.2-1
a
)
ij
σ
=
3
K
ε
(9.2-1
b
)
ii
ii
together with those for viscous flow, Eq 7.1-15,
∗
S ij
=
2
µβ
(9.2-2
a
)
ij
(
)
∗
σ
=−
3
pD
−
κ
(9.2-2
b
)
ii
ii
each expressed in terms of their deviatoric and dilatational responses. These
equations are valid for isotropic media only. For linear viscoelastic theory
we assume that displacement gradients,
u
, are small, and as shown by
i,A
Eq 4.10-18, this results in
˙
ε ij
≈
D
(9.2-3
a
)
ij
from which we immediately conclude that
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˙
ε ii
≈
D
(9.2-3
b
)
ii
so that now, from Eq 4.7-20 and Eq 7.1-13
b
,
˙
ηβ
ij
≈
(9.2-4)
ij
If the pressure
is relatively small and may be neglected, or if
we consider the pressure as a uniform dilatational body force that may be
added as required to the dilatational effect of the rate of deformation term
p
in Eq 9.2-2
b
D
when circumstances require, Eq 9.2-2 may be modified in view of Eq 9.2-3
and Eq 9.2-4 to read
ii
∗
˙
S ij
=
2
µη
(9.2-5
a
)
ij
* ˙
σκε
=
3
(9.2-5
b
)
ii
ii
A comparison of Eqs 9.2-5 and Eq 9.2-1 indicates that they differ primarily
in the physical constants listed and in the fact that in Eq 9.2-5 the stress
tensors are expressed in terms of strain rates. Therefore, a generalization of
both sets of equations is provided by introducing linear differential operators
of the form given by Eq 5.12-7 in place of the physical constants
G, K,
µ
,
∗
and
. In order to make the generalization complete we add similar differ-
ential operators to the left-hand side of the equations to obtain
κ
∗
{} = {}
PS
2
Q
η
(9.2-6
a
)
ij
ij
{} =
{}
MN
ii
σ
3
ε
(9.2-6
b
)
ii
where the numerical factors have been retained for convenience in relating
to traditional elasticity and viscous flow equations. As noted, the linear
differential time operators, {
P
}
,
{
Q
}
,
{
M
}
,
and {
N
}, are of the same form as in
Eq 5.12-7 with the associated coefficients
representing the
physical properties of the material under consideration. Although these coef-
ficients may in general be functions of temperature or other parameters, in
the simple linear theory described here they are taken as constants. As stated
at the outset, we verify that for the specific choices of operators {
p
, q
, m
,
and
n
i
i
i
i
P
} = 1,
{
Q
} =
G,
{
M
} = 1, and {
N
} =
K
, Eqs 9.2-6 define elastic behavior, whereas for
{
P
} = 1, {
Q
} =
µ
∂/∂
t
, {
M
} = 1, and {
N
} =
κ
∂/∂
t,
linear viscous behavior is
∗
∗
indicated.
Extensive experimental evidence has shown that practically all engineer-
ing materials behave elastically in dilatation so without serious loss of gen-
erality we may assume the fundamental constitutive equations for linear
viscoelastic behavior in differential operator form to be
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FIGURE 9.1
Simple shear element representing a material cube undergoing pure shear loading.
{} =
{}
PS
2
Q
η
(9.2-7
a
)
ij
ij
σ
=
3
K
ε
(9.2-7
b
)
ii
ii
for isotropic media. For anisotropic behavior, the operators {
} must
be augmented by additional operators up to a total of as many as twelve as
indicated by {
P
} and {
Q
P
} and {
Q
} with the index
i
ranging from 1 to 6, and Eq 9.2-7
a
i
i
expanded to six separate equations.
9.3
One-Dimensional Theory, Mechanical Models
Many of the basic ideas of viscoelasticity can be introduced within the
context of a one-dimensional state of stress. For this reason, and because the
viscoelastic response of a material is associated directly with the deviatoric
response as was pointed out in arriving at Eq 9.2-7, we choose the simple
shear state of stress as the logical one for explaining fundamental concepts.
Thus, taking a material cube subjected to simple shear, as shown by
Figure 9.1 , we note that for this case Eq 9.2-7 reduces to the single equation
{}
{}
{}
= {}
P
σ
=
2
Q
η
=
2
Q
ε
Q
γ
(9.3-1)
12
12
12
12
is the engineering shear strain as shown in Figure 9.1 . If the defor-
mational response of the material cube is linearly elastic, the operators {
where
γ
12
P
}
and {
Q
} in Eq 9.3-1 are constants (
P
= 1,
Q
=
G
) and that equation becomes
the familiar
σ
=
G
γ
(9.3-2)
12
12
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FIGURE 9.2
Mechanical analogy for simple shear.
is the elastic shear modulus. This equation has the same form as
the one which relates the elongation (or shortening)
where
G
δ
of a linear mechanical
spring to the applied force
F
as given by the equation
Fk
= δ
(9.3-3)
where
is the spring constant. Because Eqs 9.3-2 and 9.3-3 are identical in
form, the linear spring is adopted as the mechanical analog of simple elastic
shearing with
k
k
assuming the role of
G
. The analogy is depicted graphically
by the plots shown in Figure 9.2 .
In similar fashion, if the response of the material cube is viscous flow,
Eq 9.3-1 is written
∗ ˙
σµγ
=
(9.3-4
a
)
12
12
where
is the coefficient of viscosity. In viscoelastic theory it is a longstand-
ing practice that the coefficient of viscosity be represented by the symbol
µ
∗
η
,
and it is in keeping with this practice that we hereafter use the scalar
η
for
the coefficient of viscosity. Thus, Eq 9.3-4
a
becomes
˙
σ γ
=
(9.3-4
b
)
12
12
in all subsequent sections of this chapter. The mechanical analog for this
situation is the
(a loose fitting piston sliding in a cylinder filled with
a viscous fluid) subjected to an axial force
dashpot
F
. Here
˙
F
= ηδ
(9.3-5)
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