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§2.4 CONTINUUM MECHANICS (SOLIDS)
In this introduction to continuum mechanics we consider the basic equations describing the physical
effects created by external forces acting upon solids and fluids. In addition to the basic equations that
are applicable to all continua, there are equations which are constructed to take into account material
characteristics. These equations are called constitutive equations. For example, in the study of solids the
constitutive equations for a linear elastic material is a set of relations between stress and strain. In the study
of fluids, the constitutive equations consists of a set of relations between stress and rate of strain. Constitutive
equations are usually constructed from some basic axioms. The resulting equations have unknown material
parameters which can be determined from experimental investigations.
One of the basic axioms, used in the study of elastic solids, is that of material invariance. This ax-
iom requires that certain symmetry conditions of solids are to remain invariant under a set of orthogonal
transformations and translations. This axiom is employed in the next section to simplify the constitutive
equations for elasticity. We begin our study of continuum mechanics by investigating the development of
constitutive equations for linear elastic solids.
Generalized Hooke’s Law

If the continuum material is a linear elastic material, we introduce the generalized Hooke’s law in
Cartesian coordinates
σ ij = c ijkl e kl , i,j,k,l =1, 2, 3. (2.4.1)

The Hooke’s law is a statement that the stress is proportional to the gradient of the deformation occurring
in the material. These equations assume a linear relationship exists between the components of the stress
tensor and strain tensor and we say stress is a linear function of strain. Such relations are referred to as a
set of constitutive equations. Constitutive equations serve to describe the material properties of the medium
when it is subjected to external forces.

Constitutive Equations

The equations (2.4.1) are constitutive equations which are applicable for materials exhibiting small
deformations when subjected to external forces. The 81 constants c ijkl are called the elastic stiffness of the
material. The above relations can also be expressed in the form

e ij = s ijkl σ kl , i,j,k,l =1, 2, 3 (2.4.2)

where s ijkl are constants called the elastic compliance of the material. Since the stress σ ij and strain e ij
have been shown to be tensors we can conclude that both the elastic stiffness c ijkl and elastic compliance
s ijkl are fourth order tensors. Due to the symmetry of the stress and strain tensors we find that the elastic
stiffness and elastic compliance tensor must satisfy the relations

c ijkl = c jikl = c ijlk = c jilk
(2.4.3)
s ijkl = s jikl = s ijlk = s jilk
and consequently only 36 of the 81 constants are actually independent. If all 36 of the material (crystal)
constants are independent the material is called triclinic and there are no material symmetries.