Formulation of Constitutive Equations - Continuum Mechanics of Anisotropic Materials - page 107

5.1 Guidelines for the Formulation of Constitutive Equations
The conservation principles of mass, linear momentum, angular momentum, and
energy do not yield, in general, a sufficient number of equations to determine all the
unknown variables for a physical system. These conservation principles must hold
for all materials and therefore they give no information about the particular material
of which the system is composed, be it fluid or solid, bone, concrete or steel, blood,
oil, honey or water. Additional equations must be developed to describe the
material of the system and to complete the set of equations involving the variables
of the system so that the set of equations consisting of these additional equations
and the conservation equations are solvable for the variables.
Equations that characterize the physical properties of the material of a system are
called constitutive equations. Each material has a different constitutive equation to
describe each of its physical properties. Thus there is one constitutive equation to
describe the mechanical response of steel to applied stress and another to describe
the mechanical response of water to applied stress. Constitutive equations are
contrasted with conservation principles in that conservation principles must hold
for all materials while constitutive equations only hold for a particular property of a
particular material. The purpose of this chapter is to present the guidelines generally
used in the formulation of constitutive equations, and to illustrate their application
by developing four classical continuum constitutive relations, namely Darcy's law
for mass transport in a porous medium, Hooke's law for elastic materials, the
Newtonian law of viscosity, and the constitutive relations for viscoelastic materials.
5.2 Constitutive Ideas
The basis for a constitutive equation is a constitutive idea, that is to say an idea
taken from physical experience or experiment that describes how real materials
behave under a specified set of conditions. For example, the constitutive idea of the
elongation of a bar being proportional to the axial force applied to the ends of the
bar is expressed mathematically by the constitutive equation called Hooke's law.
Another example of a constitutive idea is that, in a saturated porous medium the
fluid flows from regions of higher pressure to regions of lower pressure; this idea is
expressed mathematically by the constitutive equation called Darcy's law for fluid
transport in a porous medium. It is not a simple task to formulate a constitutive
equation from a constitutive idea. The constitutive idea expresses a notion
concerning some aspect of the behavior of real materials, a notion based on the
physics of the situation that might be called physical insight. The art of formulating
constitutive equations is to turn the physical insight into a mathematical equation.
The conversion of insight into equation can never be exact because the equation is
precise and limited in the amount of information it can embody while the constitu-
tive idea is embedded in one's entire understanding of the physical situation.