Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 108

The fine art in the formulation of constitutive equations is to extract the salient
constitutive idea from the physical situation under consideration and then to express
its essence in an equation.
The four classical constitutive ideas employed here as examples are described in
this paragraph. Darcy's law for mass transport in a porous medium may be
considered as arising from the idea that, in a saturated porous medium, fluid
flows from regions of higher pressure to those of lower pressure. Let

r f denote

the density of the fluid in the pores of the porous medium, r o denote a constant

reference fluid density, and f denote the porosity of the medium. The velocity of
the fluid v passing through the pores is the velocity relative to the solid porous
matrix. This constitutive idea is that the fluid volume flux q ¼ fr f v / r o through the

pores, at a particle X , is a function of the pressure variation in the neighborhood of
X , N ( X ). If p ( X , t ) represents the pressure at the particle X at time t , then this
constitutive idea is expressed as

all X in N q ¼ fr f v =r o ¼ q ð p ð X ; t Þ; X Þ; ð X Þ (5.1D)

Note that q has the dimensions of volume flow per unit area, which means it is
the volume flow rate of fluid across a certain surface area. The volume flow rate q is
the flow rate relative to the solid porous matrix. The constitutive idea for Fourier's
law of heat conduction and Fick's law for diffusion of a solute in a solvent have the
same mathematical structure as Darcy's law for mass transport in a porous medium.
The constitutive idea for Fourier's law of heat conduction is that heat flows from
regions of higher temperature to those of lower temperature. The constitutive idea
for Fick's law for diffusion of a solute in a solvent is that a solute diffuses from
regions of higher solute concentration to those of lower solute concentration. The
developments of the Fourier law and the Fick law are parallel to the development of
Darcy's law. For the Fourier law the heat flux vector replaces the volume flux per
unit area q and the temperature replaces the pressure. For Fick's law, for diffusion
of a solute in a solvent, the diffusion flux vector replaces the volume flux per unit
area q and the pressure is replaced by the concentration of the solute in the solvent.
These substitutions will extend most of what is recorded in this chapter about
Darcy's law to the Fourier law and the Fick law.
The D in the equation number above is to indicate that this equation is associated
with Darcy's law. In this chapter D, H, N, and V will be used in equation numbers to
indicate the constitutive concept that the equation is associated; D is for Darcy, H is
for Hooke, N is for Newton, and V is for viscoelastic.
In the development of the remaining constitutive equations, those that assume
that stress is a function of different kinematic variables, the stress will be denoted as
a vector in six dimensions, T , rather than a tensor in three dimensions, T (see Sect.
A.11). The six-dimensional representation has advantages in the formulation of
constitutive equations. The main advantage in the present chapter is that all the
constitutive ideas to be developed will then have a similar structure except that
some will be in three dimensions and the rest in six dimensions. The constitutive