Page 129 - Mechanical Behavior of Materials

130 Chapter 4 Mechanical Testing: Tension Test and Other Basic Tests

engineering member changes its dimensions and/or shape, which is almost always undesirable.
Thus, the first step in engineering design is usually to assure that stresses are sufficiently small
that yielding does not occur, except perhaps in very small regions of a component.
The yielding event can be characterized by several methods. The simplest is to identify the
stress where the first departure from linearity occurs. This is called the proportional limit, σ p , and is
illustrated in Fig. 4.11. Some materials, as in (c), may exhibit a stress–strain curve with a gradually
decreasing slope and no proportional limit. Even where there is a definite linear region, it is difficult
to precisely locate where this ends. Hence, the value of the proportional limit depends on judgment,
so that this is a poorly defined quantity. Another quantity sometimes defined is the elastic limit,
which is the highest stress that does not cause permanent (i.e., plastic) deformation. Determination
of this quantity is difficult, as periodic unloading to check for permanent deformation is necessary.
A third approach is the offset method, which is illustrated by dashed lines in Fig. 4.11. A straight
line is drawn parallel to the elastic slope, E or E t , but offset by an arbitrary amount. The intersection
of this line with the engineering stress–strain curve is a well-defined point that is not affected by
judgment, except in cases where E t is difficult to establish. This is called the offset yield strength,
σ o . The most widely used and standardized offset for engineering metals is a strain of 0.002, that
is, 0.2%, although other values are also used. Note that the offset strain is a plastic strain, such as
ε po = 0.002, as unloading from σ o would follow a dashed line in Fig. 4.11, and this ε po would be
the unrecovered strain.
In some engineering metals, notably in low-carbon steels, there is very little nonlinearity prior
to a dramatic drop in load, as illustrated in Fig. 4.11(b). In such cases, one can identify an upper
yield point, σ ou , and a lower yield point, σ ol . The former is the highest stress reached prior to the
decrease, and the latter is the lowest stress prior to a subsequent increase. Values of the upper yield
point in metals are sensitive to testing rate and to inadvertent small amounts of bending, so that
reported values for a given material vary considerably. The lower yield point is generally similar
to the 0.2% offset yield strength, with the latter having the advantage of being applicable to other
types of stress–strain curve as well. Thus, the offset yield strength is generally the most satisfactory
means of defining the yielding event for engineering metals.
For polymers, offset yield strengths are also used. However, it is more common for polymers to
define a yield point only if there is an early relative maximum (upper yield point) or flat region in the
curve, in which case σ o is the stress where dσ/dε = 0 first occurs. In polymers with an upper yield
point, σ ou , this stress may exceed that at fracture, σ f , but in other cases, it does not. (See Fig. 4.10.)
Hence, the ultimate tensile strength σ u is the higher of either σ ou or σ f . The two situations are distin-
guished by describing the value as either the tensile strength at yield or the tensile strength at break.
In most materials, the proportional limit, elastic limit, and offset yield strength can be
considered to be alternative measures of the beginning of permanent deformation. However, for
a nonlinear elastic material such as rubber, the first two of these measure distinctly different events,
and the offset yield strength loses its significance. (See Fig. 3.17.)

4.3.3 Engineering Measures of Ductility
Ductility is the ability of a material to accommodate inelastic deformation without breaking. In the
case of tension loading, this means the ability to stretch by plastic strain, but with creep strain also
sometimes contributing.