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Failures Resulting from Static Loading 231
5–7 Failure of Ductile Materials Summary

Having studied some of the various theories of failure, we shall now evaluate them and
show how they are applied in design and analysis. In this section we limit our studies to
materials and parts that are known to fail in a ductile manner. Materials that fail in a brit-
tle manner will be considered separately because these require different failure theories.
6
To help decide on appropriate and workable theories of failure, Marin collected
data from many sources. Some of the data points used to select failure theories for duc-
7
tile materials are shown in Fig. 5–15. Mann also collected many data for copper and
nickel alloys; if shown, the data points for these would be mingled with those already
diagrammed. Figure 5–15 shows that either the maximum-shear-stress theory or the
distortion-energy theory is acceptable for design and analysis of materials that would
fail in a ductile manner.
The selection of one or the other of these two theories is something that you, the
engineer, must decide. For design purposes the maximum-shear-stress theory is easy,
quick to use, and conservative. If the problem is to learn why a part failed, then the
distortion-energy theory may be the best to use; Fig. 5–15 shows that the plot of the
distortion-energy theory passes closer to the central area of the data points, and thus is
generally a better predictor of failure. However, keep in mind that though a failure curve
passing through the center of the experimental data is typical of the data, its reliability
from a statistical standpoint is about 50 percent. For design purposes, a larger factor of
safety may be warranted when using such a failure theory.
Figure 5–15 /S c Oct. shear Yielding (S = S )
2
y
c
Experimental data superposed 1.0 Ni-Cr-Mo steel
on failure theories. (From
AISI 1023 steel
Fig. 7.11, p. 257, Mechanical
2024-T4 Al
Behavior of Materials, 2nd ed.,
3S-H Al
N. E. Dowling, Prentice Hall,
Englewood Cliffs, N.J., 1999.
Modified to show only ductile
failures.)
Max. shear
–1.0
/S c
0 1.0 1

–1.0

6 Joseph Marin was one of the pioneers in the collection, development, and dissemination of material on the
failure of engineering elements. He has published many books and papers on the subject. Here the
reference used is Joseph Marin, Engineering Materials, Prentice-Hall, Englewood Cliffs, N.J., 1952.
(See pp. 156 and 157 for some data points used here.)
7 Note that some data in Fig. 5–15 are displayed along the top horizontal boundary where σ B ≥ σ A . This is often
done with failure data to thin out congested data points by plotting on the mirror image of the line σ B = σ A .