MATERIALS ENGINEERING CONCEPTS - Materials for Civil and Construction Engineers - page 10

Figure 1.6(c) shows an elastoplastic response in which the first portion
is an elastic response followed by a combined elastic and plastic response. If
the load is removed after the plastic deformation, the stress-strain relation
will follow a straight line parallel to the elastic portion; consequently, some of
the strain in the material will be removed, and the remainder of the strain
will be permanent. Upon reloading, the material again behaves in a linear
elastic manner up to the stress level that was attained in the previous stress
cycle. After that point the material will follow the original stress-strain
curve. Thus, the stress required to cause plastic deformation actually in-
creases. This process is called strain hardening or work hardening . Strain
hardening is beneficial in some cases, since it allows more stress to be ap-
plied without permanent deformation. Mild steel is an example of material
that experiences strain hardening during plastic deformation.
Some materials exhibit strain softening , in which plastic deformation
causes weakening of the material. Portland cement concrete is a good exam-
ple of such a material. In this case, plastic deformation causes microcracks
at the interface between aggregate and cement paste.

Sample Problem 1.2

An elastoplastic material with strain hardening has the stress-strain relation shown in
Figure 1.6(c). The modulus of elasticity is yield strength is 70 ksi, and
the slope of the strain-hardening portion of the stress-strain diagram is

25 * 10 6 psi,

10 6 psi.

3 *

a. Calculate the strain that corresponds to a stress of 80 ksi.
b. If the 80-ksi stress is removed, calculate the permanent strain.

Solution

10 6 10 6 (a) e = 1 70,000/25 * 2 + [ 1 80,000 - 70,000 2 / 1 3 * 2 ] = 0.0028 + 0.0033

= 0.0061 in./in.

0.0061 - [80,000/(25 x 10 6 )]

(b) e permanent = = 0.0061 - 0.0032

= 0.0029 in./in.

Materials that do not undergo plastic deformation prior to failure, such
as concrete, are said to be brittle , whereas materials that display apprecia-
ble plastic deformation, such as mild steel, are ductile . Generally, ductile
materials are preferred for construction. When a brittle material fails, the
structure can collapse in a catastrophic manner. On the other hand, over-
loading a ductile material will result in distortions of the structure, but the
structure will not necessarily collapse. Thus, the ductile material provides
the designer with a margin of safety.