Page 197 - Mechanical Behavior of Materials

198 Chapter 5 Stress–Strain Relationships and Behavior

In the transient creep model of Fig. 5.5(b), while the stress is applied during 1–2, the elastic
strain in spring E 1 is added to the creep strain in the (η 2 , E 2 ) parallel combination. Hence, Eq. 5.11
again applies. The creep strain can be analyzed by noting that the stress in the (η 2 , E 2 ) stage is the
sum of the separate stresses in the spring and the dashpot:

σ = E 2 ε c + η 2 ˙ε c (5.14)
This gives

dε c σ − E 2 ε c
˙ ε c = = (5.15)
dt η 2

Solving this differential equation for the case of a constant stress σ gives the creep strain versus
time response:

σ
ε c = 1 − e −E 2 t/η 2 (5.16)
E 2
Finally, adding the elastic strain gives the total strain:

σ σ
ε = + 1 − e −E 2 t/η 2 (5.17)
E 1 E 2
Study of this equation shows that the strain rate decreases with time, as shown in Fig. 5.5(b).

Moreover, the creep strain asymptotically approaches the limit σ /E 2 . This occurs as a result of
stress being transferred from the dashpot to the spring as time passes, until the spring must resist all
of the stress at infinite time.
After removal of the stress, the strain in the transient creep model varies as shown by 3–4 in
Fig. 5.5(b). In particular, it decreases toward zero at infinite time due to the spring in the parallel
arrangement pulling on the dashpot. Equations for this recovery response may also be obtained by
solving the differential equations involved.

5.2.3 Relaxation Behavior

So far, we have considered two types of time-dependent behavior. These are creep, which is the
accumulation of strain with time, as under constant stress, and recovery, which is the gradual
disappearance of creep strain that sometimes occurs after removal of the stress. A third type of
behavior is relaxation, which is the decrease in stress when a material is held at constant strain.
Relaxation is illustrated for the steady-state creep plus elastic strain model in Fig. 5.6. Since

the strain ε is suddenly applied, all of this strain is absorbed by the spring as a result of the fact that
the dashpot requires a finite time to respond. With time, motion occurs in the dashpot, and the strain
in the spring decreases, as it must, due to the total strain being held constant. We have

ε = ε e + ε c (5.18)