Page 198 - Mechanical Behavior of Materials

Section 5.2 Models for Deformation Behavior 199

(a)
1 2
ε σ
c ε η E ε
ε ' 1 1
ε e
t
0 3 4
(b) 1 (c) 1
σ
σ
2
2
t 0 ε
0 4
4
3 3

Figure 5.6 Relaxation under constant strain for a model with steady-state creep and elastic
behavior. The step in strain (a) causes stress–time behavior as in (b), and stress–strain behavior
as in (c).

where ε is the constant total strain and ε c is the creep strain. Hence, elastic strain is being replaced
by creep strain.
The stress necessary to maintain the constant strain is related to the elastic strain by

σ = E 1 ε e (5.19)
Since ε e is decreasing, this requires that σ must also decrease. The rate of creep strain is related to
the value of σ at any time by

dε c σ
˙ ε c = = (5.20)
dt η 1
Combining these equations and solving the resulting differential equation gives the variation of σ
as it decreases:

σ = E 1 ε e (5.21)
−E 1 t/η 1
This corresponds to a σ-t response that decays—that is, relaxes—with time, as illustrated by curve
1–2inFig.5.6(b).
Relaxation is the same phenomenon as creep, differing only in that it is observed under constant
strain rather than constant stress. Real engineering materials that exhibit creep will also show
relaxation behavior.
In Fig. 5.6, if the strain is returned to zero after a period of relaxation, the stress is forced
into compression. Additional relaxation then occurs, but in the opposite direction, as the relaxation
always proceeds toward zero stress.
Creep and relaxation, and models of the types just discussed, are considered in more detail in
Chapter 15.