CONTINUUM MECHANICS for ENGINEERS

˙ε ii ≈ Dii
so that now, from Eq 4.7-20 and Eq 7.1-13b,
˙ηij ≈ βij
(9.2-3b)
(9.2-4)
If the pressure p in Eq 9.2-2b 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
D ii when circumstances require, Eq 9.2-2 may be modified in view of Eq 9.2-3
and Eq 9.2-4 to read
∗
Sij = ij
2 µ η˙
σ = 3 κ ε
* ˙
ii ii
(9.2-5a)
(9.2-5b)
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 κ ∗ . In order to make the generalization complete we add similar differential
operators to the left-hand side of the equations to obtain
{ PS } = 2{ Q}
η
ij ij
{ M} σ = 3{
N}
ε
ii ii
(9.2-6a)
(9.2-6b)
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 p i, q i, m i, and n i representing the
physical properties of the material under consideration. Although these coefficients
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} = 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 engineering
materials behave elastically in dilatation so without serious loss of generality
we may assume the fundamental constitutive equations for linear
viscoelastic behavior in differential operator form to be