Page 205 - Materials Science and Engineering An Introduction

6.5 Elastic Properties of Materials • 177

6.4 ANELASTICITY

To this point, it has been assumed that elastic deformation is time independent—that
is, that an applied stress produces an instantaneous elastic strain that remains constant
over the period of time the stress is maintained. It has also been assumed that upon
release of the load, the strain is totally recovered—that is, that the strain immedi-
ately returns to zero. In most engineering materials, however, there will also exist a
time-dependent elastic strain component—that is, elastic deformation will continue
after the stress application, and upon load release, some finite time is required for
anelasticity complete recovery. This time-dependent elastic behavior is known as anelasticity, and
it is due to time-dependent microscopic and atomistic processes that are attendant to
the deformation. For metals, the anelastic component is normally small and is often
neglected. However, for some polymeric materials, its magnitude is significant; in this
case it is termed viscoelastic behavior, which is the discussion topic of Section 15.4.

EXAMPLE PROBLEM 6.1

Elongation (Elastic) Computation

A piece of copper originally 305 mm (12 in.) long is pulled in tension with a stress of 276 MPa
(40,000 psi). If the deformation is entirely elastic, what will be the resultant elongation?

Solution
Because the deformation is elastic, strain is dependent on stress according to Equation 6.5.
Furthermore, the elongation ≤l is related to the original length l 0 through Equation 6.2.
Combining these two expressions and solving for ≤l yields

l
s = PE = a bE
l 0
sl 0
l =
E
The values of s and l 0 are given as 276 MPa and 305 mm, respectively, and the magnitude of E
6
for copper from Table 6.1 is 110 GPa (16 * 10 psi). Elongation is obtained by substitution into
the preceding expression as
(276 MPa)(305 mm)
l = 3 = 0.77 mm (0.03 in.)
110 * 10 MPa

6.5 ELASTIC PROPERTIES OF MATERIALS
When a tensile stress is imposed on a metal specimen, an elastic elongation and ac-
companying strain P z result in the direction of the applied stress (arbitrarily taken to
be the z direction), as indicated in Figure 6.9. As a result of this elongation, there will
be constrictions in the lateral (x and y) directions perpendicular to the applied stress;
from these contractions, the compressive strains P x and P y may be determined. If the
applied stress is uniaxial (only in the z direction) and the material is isotropic, then
Poisson’s ratio P x = P y . A parameter termed Poisson’s ratio n is defined as the ratio of the lateral and
axial strains, or
Definition of
Poisson’s ratio in P x P y (6.8)
terms of lateral n = - P z = - P z
and axial strains