Page 232 - Introduction to Continuum Mechanics

5

The Elastic Solid

So far we have studied the kinematics of deformation, the description of the state of stress
and four basic principles of continuum physics: the principle of conservation of mass
[Eq. (3.15.2)], the principle of linear momentum [Eq. (4.7.2)], the principle of moment of
momentum [Eq. (4.4.1)] and the principle of conservation of energy [Eq. (4.14.1)]. All these
relations are valid for every continuum, indeed no mention was made of any material in the
derivations.
These equations are however not sufficient to describe the response of a specific material
due to a given loading. We know froiu experience that under the same loading conditions, the
response of steel is different from that of water. Furthermore, for a given material, it varies
with different loading conditions. For example, for moderate loadings, the deformation in
steel caused by the application of loads disappears with the removal of the loads. This aspect
of the material behavior is known as elasticity. Beyond a certain level of loading, there will
be permanent deformations, or even fracture exhibiting behavior quite different from that of
elasticity. In this chapter, we shall study idealized materials which model the elastic behavior
of real solids. The linear isotropic elastic model will be presented in part A, followed by the
linear anisotropic elastic model in part B and an incompressible isotropic nonlinear elastic
model in part C.

5.1 Mechanical Properties

We want to establish some appreciation of the mechanical behavior of solid materials. To
do this, we perform some thought experiments modeled after real laboratory experiments.
Suppose from a block of material, we cut out a slender cylindrical test specimen of
cross-sectional area^t The bar is now statically tensed by an axially applied load /*, and the
elongation A/, over some axial gage length/, is measured. A typical plot of tensile force against
elongation is shown in Fig. 5.1. Within the linear portion OA (sometimes called the propor-
tional range), if the load is reduced to zero (i.e., unloading), then the line OA is retraced back
to O and the specimen has exhibited an elasticity. Applying a load that is greater than A and
then unloading, we typically traverse OABC and find that there is a "permanent elongation"
OC, Reapplication of the load from C indicates elastic behavior with the same slope as OA,
but with an increased proportional limit. The material is said to have work-hardened.

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