Page 199 - Mechanical Behavior of Materials

200 Chapter 5 Stress–Strain Relationships and Behavior

Example 5.1
Derive Eq. 5.21, which describes stress relaxation in the elastic, steady-state creep model, as
illustrated by 1–2 in Fig. 5.6(b). Base this on the other equations given just prior to Eq. 5.21.

Solution Differentiate both sides of Eqs. 5.18 and 5.19 with respect to time, noting that

dε /dt = 0as ε is held constant:
dσ
0 =˙ε e +˙ε c , =˙σ = E 1 ˙ε e
dt
Substitute ˙ε e from the second equation, and also ˙ε c from Eq. 5.20, into the first equation to obtain

1 dσ σ
+ = 0
E 1 dt η 1
Separate the variables σ and t and integrate both sides of the equation, resulting in

E 1
ln σ =− t + C
η 1
where C is a constant of integration that can be evaluated by noting that the creep strain is

initially zero, so that σ = E 1 ε at t = 0. This gives C = ln E 1 ε . Substituting for C and solving
for σ then produces the desired result:
σ = E 1 ε e Ans.
−E 1 t/η 1

5.2.4 Discussion
We have discussed models of three major types of deformation, namely, elastic, plastic, and
creep deformation. These are characterized in Table 5.1. Elastic strain is the result of stretching
of chemical bonds. It is not considered to be time dependent and is recovered immediately on
unloading. Plastic strain is also not considered to be dependent on time and is permanent, due to its
being caused by the relative sliding of crystal planes through the incremental process of dislocation
motion. Note that perfectly plastic behavior, as in the models of Fig. 5.3(a) and (b), will result
in unstable rapid deformation if a stress above σ o is maintained. However, some degree of strain
hardening usually occurs in real materials.
Creep strain may be divided into steady-state and transient types, according to whether the rate
is constant or decreases with time. In the ideal model of Fig. 5.5(b), all transient strain is recovered.
However, a portion of this may be permanent in real materials. The recovered portion of the creep
strain may be quite large in polymers due to chain molecules interfering with one another in such a
way that they slowly reestablish their original configuration after removal of the stress, causing the
strain to slowly disappear. For example, creep strains as large as 100% in flexible vinyl (plasticized
PVC) may be mostly recovered after unloading. In metals and ceramics, large creep strains can