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196 Chapter 5 Stress–Strain Relationships and Behavior
Figure 5.4 Loading and unloading behavior of (a) an elastic, perfectly plastic model, (b) an
elastic, linear-hardening model, and (c) a material with nonlinear hardening.
Now consider the response of each model to the situation of the last column in Fig. 5.3, where
the model is reloaded after elastic unloading to σ = 0. In all cases, yielding occurs a second time
when the strain again reaches the value ε 1 from which unloading occurred. It is obvious that the two
perfectly plastic models will again yield at σ = σ o . But the linear-hardening model now yields at a
value σ = σ 1 , which is higher than the initial yield stress. Furthermore, σ 1 is the same value of stress
that was present at ε = ε 1 , when the unloading first began. For all three models, the interpretation
may be made that the model possesses a memory of the point of previous unloading. In particular,
yielding again occurs at the same σ-ε point from which unloading occurred, and the subsequent
response is the same as if there had never been any unloading. Real materials that deform plastically
exhibit a similar memory effect.
We will return to spring and slider models of plastic deformation in Chapter 12, where they will
be considered in more detail and extended to nonlinear hardening cases.
5.2.2 Creep Deformation Models
Significant time-dependent deformation occurs in engineering metals and ceramics at elevated
temperature. This also occurs at room temperature in low-melting-temperature metals, such as
lead, and in many other materials, such as glass, polymers, and concrete. A variety of physical
mechanisms are involved, as discussed to an extent in Chapter 2 and considered again in
Chapter 15.
The creep models of Fig. 5.1(c) and (d) are shown in Fig. 5.5, with springs E 1 added to simulate
elastic strain, as in real materials. Note that in (b) the vertical bar is assumed not to rotate, so that
the parallel spring and dashpot are subjected to the same strain. Also, these models are expressed
in terms of stress and strain, so that springs deform according to ε = σ/E and dashpots according
to ˙ε = σ/η. If a constant stress σ is applied to either model, an elastic strain ε e = σ /E 1 appears
instantly (0–1) and then later disappears (2–3) when the stress is removed.
The use of constant viscosities η in these models results in all strain rates and strains being
proportional to the applied stress, a situation described by the term linear viscoelasticity. Such
idealized linear behavior is sometimes a reasonable approximation for real materials, as for some
polymers, and also for metals and ceramics at high temperature, but low stress. However, for metals
and ceramics at high stress, strain rate is proportional not to the first power of stress, but to a
higher power on the order of five. In such cases, models or equations involving more complex