Page 222 - Mechanical Behavior of Materials

Section 5.5 Summary 223

Discussion In (a), note that choosing the smaller value, V r = 0.427, gives E Y = 183.6GPa
from Eq. 5.64. Hence, this choice fails the E Y = 200 GPa requirement. But the V r = 0.512
choice gives E X = 273.8 GPa from Eq. 5.55, so that the E X = 250 GPa requirement is
exceeded, and this choice is suitable.

5.5 SUMMARY

Deformations may be classified according to physical mechanisms and analogies with rheological
models as elastic, plastic, or creep deformations. The latter category may be further subdivided
into steady-state creep and transient creep. The simplest rheological models for each are shown in
Fig. 5.1.
Elastic deformation is associated with stretching the chemical bonds in solids so that the
distances between atoms increases. The deformation is not time dependent and is recovered
immediately upon unloading. Stress–strain curves for metals, especially, but also for many other
materials, exhibit a distinct elastic region where the stress–strain behavior is linear.
Plastic deformation is associated with the relative movement of planes of atoms, or of
chainlike molecules, and is not strongly time dependent. Rheological models containing frictional
sliders have behavior analogous to plastic deformation, sharing the following characteristics with
plastically deforming materials: (1) Departure from linear behavior occurs that results in permanent
deformation if the load is removed. (2) Compressive stressing is required to achieve a return to
zero strain after yielding. (3) There is a memory effect on reloading after elastic unloading, in that
yielding occurs at the same stress and strain from which unloading occurred.
Creep is time-dependent deformation that may or may not be recovered after unloading.
The physical mechanisms include vacancy and dislocation motions, grain boundary sliding, and
flow as a viscous fluid. Such mechanisms acting in metals, ceramics, and glasses produce creep
deformations that are mostly not recovered after unloading. However, considerable recovery of
creep deformation may occur in polymers as a result of interactions among the long carbon-chain
molecules. Rheological models built up of springs and dashpots can be used to study creep behavior.
In the simplest form of such models, strain rates are proportional to applied stresses, a situation
termed linear viscoelasticity. If a strain is applied and held constant, creep behavior in the material
causes the stress to decrease, a phenomenon termed relaxation.
Elastic deformation occurs in all materials at all temperatures. Plastic deformation is important
in strengthened metal alloys at room temperature, whereas creep effects are small. Significant
creep occurs at room temperature in low-melting-temperature metals, and in many polymers. At
sufficiently high temperature, creep becomes an important factor for strengthened metal alloys and
even for ceramics.
If a material is both isotropic and homogeneous, the elastic strains for the general three-
dimensional case are related to stresses by the generalized Hooke’s law:

τ xy
1
ε x = σ x − ν σ y + σ z , γ xy = (5.67)
E G