Page 190 - Mechanical Behavior of Materials

Section 5.2 Models for Deformation Behavior 191

elementary mechanics of materials, elastic behavior with a linear stress–strain relationship is
assumed and used in calculating stresses and deflections in simple components such as beams and
shafts. More complex situations of geometry and loading can be analyzed by employing the same
basic assumptions in the form of theory of elasticity. This is now often accomplished by using the
numerical technique called finite element analysis with a digital computer.
Stress–strain relationships need to consider behavior in three dimensions. In addition to elastic
strains, the equations may also need to include plastic strains and creep strains. Treatment of creep
strain requires the introduction of time as an additional variable. Regardless of the method used,
analysis to determine stresses and deflections always requires appropriate stress–strain relationships
for the particular material involved.
For calculations involving stress and strain, we express strain as a dimensionless quantity, as
derived from length change, ε = L/L. Hence, strains given as percentages need to be converted
6
to the dimensionless form, ε = ε % /100, as do strains given as microstrain, ε = ε μ /10 .
In this chapter, we will first consider one-dimensional stress–strain behavior and some
corresponding simple physical models for elastic, plastic, and creep deformation. The discussion
of elastic deformation will then be extended to three dimensions, starting with isotropic behavior,
where the elastic properties are the same in all directions. We will also consider simple cases of
anisotropy, where the elastic properties vary with direction, as in composite materials. However,
discussion of three-dimensional plastic and creep deformation behavior will be postponed to
Chapters 12 and 15, respectively.

5.2 MODELS FOR DEFORMATION BEHAVIOR

Simple mechanical devices, such as linear springs, frictional sliders, and viscous dashpots, can
be used as an aid to understanding the various types of deformation. Four such models and their
responses to an applied force are illustrated in Fig. 5.1. Such devices and combinations of them are
called rheological models.
Elastic deformation, Fig. 5.1(a), is similar to the behavior of a simple linear spring characterized
by its constant k. The deformation is always proportional to the force, x = P/k, and it is recovered
instantly upon unloading. Plastic deformation, Fig. 5.1(b), is similar to the movement of a block of
mass m on a horizontal plane. The static and kinetic coefficients of friction μ areassumedtobe
equal, so that there is a critical force for motion P o = μmg, where g is the acceleration of gravity.

If a constant applied force P is less than the critical value, P < P o , no motion occurs. However, if

it is greater, P > P o , the block moves with an acceleration

P − P o
a = (5.1)
m
2
When the force is removed at time t, the block has moved a distance x = at /2, and it remains at
this new location. Hence, the model behavior produces a permanent deformation, x p .
Creep deformation can be subdivided into two types. Steady-state creep, Fig. 5.1(c), proceeds
at a constant rate under constant force. Such behavior occurs in a linear dashpot, which is an element
where the velocity, ˙x = dx/dt, is proportional to the force. The constant of proportionality is the