Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 115

q ¼ fr f v =r o ¼ H ð p Þr p ð x ; t Þ; (5.7D)

T

¼ C E

ð x ; t Þ

(5.7H)

T ¼

pU þ N Dðx; t Þ (5.7N)

and

1

T G

ÞD ¼ ð s ð x ; t s Þ d s ; (5.7V)

s¼ 0

Note that, in the above constitutive expressions , not only has the dependence of the
material coefficient tensors been removed by eliminating their dependence upon the
particle X , but also X has been replaced by x everywhere else. For the two constitutive
relations restricted to infinitesimal motions, ( 5.7H )and( 5.7V ), the constitutive
relations based on a rigid continuum, ( 5.7D ) and Eulerian viscous fluid theory
( 5.7N ), there is no difference between X and x (see Sect. 2.4), hence x could have
been used from the beginning of the chapter. For the Newtonian law of viscosity
however, the assumption of homogeneity is much more significant because it permits
the elimination of X from the entire constitutive relation, a constitutive relation that is
not restricted to infinitesimal deformations. Thus, even though ( 5.7N ) applies for large
deformations, it is independent of X . The Newtonian law is different from the other
four constitutive relations in another way, as detailed in the next section.

5.9 Restrictions Due to Material Symmetry

The results of the previous chapter are used in this section to further specify the
form of the constitutive relations. Isotropy or any type of anisotropy is possible for
the three constitutive relations, ( 5.7D ), ( 5.7H ), and ( 5.7V ), that are, or may be,
applied to solid or semi-solid materials. The type of anisotropy is expressed in the
form of the tensors of material coefficients, H , C , and Gð Þ , respectively. Once the

type of anisotropy possessed by the solid or semi-solid material to be modeled has
been determined, the appropriate form of H may be selected from Table 4.3 or the
form of C or G

s

from Tables 4.4 and 4.5. Thus, for these four constitutive relations
any type of material symmetry is possible. In this section and in the first paragraph
of the next section the results summarized in Tables 4.3, 4.4 and 4.5 are cited. The
derivation of these results is presented in Chap. 4.
The concepts of anisotropy and inhomogeneity of materials are sometimes
confused. A constitutive relation is inhomogeneous or homogeneous depending
upon whether the material coefficients (i.e., H ,

ð s Þ

C , and G

ð s Þ ) depend upon X or