Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 113

will occur to the material in the future. For example, the constitutive assumption for
elastic materials is that the stress depends upon the strain between a previous
unstressed reference configuration and the instantaneous configuration of the
object. All four of the constitutive equations satisfy this guideline. The first four
satisfy it because all the variables entering the relationships are at a time t .
The viscoelastic constitutive relation satisfies the guideline by only depending
upon past events.

5.6 Linearization

Each of the constitutive ideas considered has been reduced to the form of a
vector-valued ( q or T ) function or functional of another vector (

p , E or D ), X ,
and some scalar parameters. It is assumed that each of these vector-valued functions
is linear in the vector argument, thus each may be represented by a linear transfor-
mation. For Darcy's law the second order tensor in three dimensions represents the
coefficients of the linear transformation and, due to the dependence of the volume
flow rate upon pressure, this second order tensor admits the functional dependency
indicated:

∇

q ¼ fr f v =r o ¼ H ð p ; X Þr p ð X ; t Þ: (5.6D)

The minus sign was placed in ( 5.6D ) to indicate that the volume fluid flux q
would be directed down the pressure gradient, from domains of higher pore fluid
pressure to domains of lower pressure.
For the three constitutive ideas involving the stress vector T , second order
tensors in six dimensions represent the coefficients of the linear transformation:

T

¼ C ÞE

ð X ð x ; t Þ; (5.6H)

T

pU þ N ÞD

¼ ð X ð X ; t Þ; (5.6N)

and

1

T G

ÞD

¼ ð X ; s ð X ; t s Þ d s : (5.6V)

s ¼ 0

The six-dimensional second order tensors C

and N are for Hooke's law

and the Newtonian law of viscosity, respectively. The six-dimensional second order
tensor function G

ð X Þ ð X Þ

ð X ; s Þ represents the viscoelastic coefficients.