Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 112

where the tensor of displacement gradients u ( X , t ) has been replaced by

its symmetric part, the infinitesimal strain tensor E ( X , t ) defined by (2.49), and its
skew symmetric part, the infinitesimal rotation tensor Y ( X , t ), also defined by
(2.49). In a similar way the tensor of velocity gradients

∇

v ( X , t ) may be
replaced by its symmetric part, the rate of deformation tensor D ( X , t ) defined by
(2.32), and its skew symmetric part, the spin tensor W ( X , t ), also defined by (2.32).

∇

T

pU þ T v ð ¼ v ð X ; t Þ; D ð X ; t Þ; W ð X ; t Þ X Þ: (5.4N)

s) as in the
case of the Newtonian law of viscosity, the viscoelastic constitutive relation ( 5.3V )
takes the form

Finally, decomposing the tensor of velocity gradients ∇ v ( X , t

1

T T

¼ ð v ð X ; t s Þ; D ð X ; t s Þ; W ð X ; t s Þ; X ; s Þ d s : (5.4V)

s¼ 0

This guideline requires that constitutive equations remain unchanged by super-
posed rigid object motions, thus measures of translational motion, like the displace-
ment u ( X , t ) and the velocity v ( X , t ), and measures of rotational motion, like the
infinitesimal rotation tensor Y ( X , t ) and the spin tensor W ( X , t ), must be excluded
from the above equations . Using this guideline the form of these three constitutive
ideas is then reduced to

T

¼ T ðE ð X ; t Þ; X Þ; (5.5H)

T

pU þ T v ðD

¼ ð X ; t Þ; X Þ; (5.5N)

and

1

T T

ðD

¼ ð X ; t s Þ; X ; s Þ d s ; (5.5V)

s ¼ 0

where the three-dimensional tensors E ( X , t ) and D ( X , t ) have been replaced by their
six-dimensional vector equivalents, E ( X , t ) and D ( X , t ), respectively.

5.5 Determinism

A constitutive equation valid for a material at a time t must depend upon events that
are occurring to the material at the instant t and upon events that have occurred to
the material in the past. The constitutive equation cannot depend upon events that