Page 130 - Mechanical Behavior of Materials

Section 4.3 Engineering Stress–Strain Properties 131

The engineering strain at fracture, ε f , is one measure of ductility. This is usually expressed as
a percentage and is then termed the percent elongation at fracture, which we will denote as 100ε f .
This value corresponds to the fracture point on the stress–strain curve, as identified in Figs. 4.9 and
4.10. The ASTM standard for tension testing of polymers includes this property, and it is given as
an option in the standard for tension testing of metals. For this measurement, test standards specify
the gage length as a multiple of the test section diameter or width. For example, for specimens with
round cross sections, a ratio of gage length to diameter of L i /d i = 4 is specified by ASTM for use
in the United States. But international standards generally employ L i /d i = 5.
Another method of determining the elongation that is sometimes used for metals is to employ
marks on the test section. The distance L i between these marks before testing is subtracted from
the distance L f measured after fracture. The resulting length change provides a strain value, which
gives the percent elongation after fracture.

L f − L i
ε pf = , % elongation after fracture = 100ε pf (4.6)
L i
Note that the elastic strain is lost when the stress drops to zero after fracture, so this quantity is a
plastic strain, as identified in Fig. 4.9.
For metals with considerable ductility, the difference between 100ε f and 100ε pf is small, so
that the distinction between these two quantities is not of great importance. However, for metals of
limited ductility, the elastic strain recovered after fracture may constitute a significant fraction of
ε f . Moreover, the elongation may be so small as to be difficult to measure directly from marks on
the sample. It may then be useful to determine ε pf from a value of ε f taken from the fracture point
on the stress–strain record, specifically, by subtracting the elastic strain that is estimated to be lost:

σ f
ε pf = ε f − , % elongation after fracture = 100ε pf (4.7)
E
This gives an estimated after-fracture result consistent with a measurement made on a broken
specimen.
Another measure of ductility is the percent reduction in area, called %RA, which is obtained
by comparing the cross-sectional area after fracture, A f , with the original area:
2
d − d 2
A i − A f i f
%RA = 100 , %RA = 100 2 (a, b) (4.8)
A i d i
Here, form (b) is derived from (a) as a convenience for round cross sections of initial diameter d i
and final diameter d f . As for the elongation, a discrepancy may exist between the area after fracture
and the area that existed at fracture. This presents little problem for ductile metals, but caution is
needed in interpreting area reductions after fracture for polymers.

4.3.4 Discussion of Necking Behavior and Ductility
If the behavior in a tension test is ductile, a phenomenon called necking usually occurs, as illustrated
in Fig. 4.12. The deformation is uniform along the gage length early in the test, as in (a) and (b),
but later begins to concentrate in one region, resulting in the diameter there decreasing more than