Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 117

or

T 11 þ p ¼ l tr D þ 2 m D 11 ; T 22 þ p ¼ l tr D þ 2 m D 22 ; T 33 þ p ¼ l tr D þ 2 m D 33 ;

T 23 ¼ 2 m D 23 ; T 13 ¼ 2 m D 13 ; T 12 ¼ 2 m D 12 ;

l m

where are viscosity coefficients. It is easy to see that the constitutive

relation may be rewritten in three dimensions as

and

T þ p1 ¼ l tr DÞ1 þ 2 mD: (5.11N)

This is the form of the constitutive equation for a viscous fluid, the pressure plus
the Newtonian law of viscosity, which will be used in the remainder of the text.

Problems

5.9.1. Record the explicit matrix form for the constitutive relation for a Darcy
porous medium in an inhomogeneous transversely isotropic material.
5.9.2. Record the explicit matrix form for the constitutive relation for Darcy's law
in a homogeneous isotropic material.
5.9.3. Record the explicit matrix form for the constitutive relation for Hooke's law in
5.9.4. Record the explicit matrix form for the constitutive relation for a trans-
versely isotropic, homogeneous viscoelastic material.
5.9.5. Show that the eigenvalues of ( 5.10N ) are 3

l þ 2 m and 2 m and specify how

many times each is repeated.

5.10 The Symmetry of the Material Coefficient Tensors

In this section the question of the symmetry of the matrices of the tensors of
material coefficients, H , ; N and G (s) is considered. Consider first the tensor of

material coefficients N for a Newtonian viscous fluid. In the previous section it was
assumed that a Newtonian viscous fluid was isotropic, therefore, from Table 4.5, the
tensor of material coefficients N is symmetric. In this case the material symmetry
implied the symmetry of the tensor of material coefficients. A similar symmetry
result emerges for the permeability tensor H if only orthotropic symmetry or greater
symmetry is considered. To see that material symmetry implies the symmetry of the
tensor of material coefficients H , if only orthotropic symmetry or greater symmetry
is considered, one need only consult Table 4.3. The symmetry of H is also true for
symmetries less than orthotropy, namely monoclinic and triclinic, but the proof will
not be given here. Finally, G (s) is never symmetric unless the viscoelastic model is
in the limiting cases of G (0) or G

C

) where the material behavior is elastic.
The symmetry of the tensor of elastic material coefficients C is the only coeffi-
cient tensor symmetry point remaining to be demonstrated in this section. In this

ð1