CONTINUUM MECHANICS for ENGINEERS

FIGURE 9.1
Simple shear element representing a material cube undergoing pure shear loading.
{ P} S = 2{ Q}
η
ij ij
σ = 3 K ε
ii ii
(9.2-7a)
(9.2-7b)
for isotropic media. For anisotropic behavior, the operators {P} and {Q} must
be augmented by additional operators up to a total of as many as twelve as
indicated by {P i} and {Q i} with the index i ranging from 1 to 6, and Eq 9.2-7a
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
12 12 12 12
(9.3-1)
where γ 12 is the engineering shear strain as shown in Figure 9.1. If the deformational
response of the material cube is linearly elastic, the operators {P}
and {Q} in Eq 9.3-1 are constants (P = 1, Q = G) and that equation becomes
the familiar
σ12 = Gγ
12
(9.3-2)