Page 319 - Mechanical Behavior of Materials

320 Chapter 7 Yielding and Fracture under Combined Stresses

σ
(a) ductile 3 yield

σ
2

fracture
σ
1
σ
3 yield

(b) brittle
σ
2

σ
1
fracture

Figure 7.24 Relationships of the limiting surfaces for yielding and fracture for materials that
usually behave in a ductile manner, and also for materials that usually behave in a brittle
manner.

yielding (at least for metals) is taken to be a cylinder or other prismatic shape that is symmetrical
about the line σ 1 = σ 2 = σ 3 , such as the surfaces of Figs. 7.6 and 7.9. Limiting surfaces for fracture
are, in general, similar to those for the modified Mohr theory (as discussed previously and illustrated
in Fig. 7.21), although the boundaries may actually be smooth curves.
The situation is illustrated in Fig. 7.24. For certain states of stress, the yield surface is encoun-
tered first, whereas for others the fracture surface is encountered first. The relative dimensions of
the two surfaces change for different materials. For normally ductile materials, fracture prior to
yielding is not expected, except for stress states involving a large hydrostatic tension. The stress
may be increased by varying amounts beyond yielding before fracture occurs, depending on the
amount of hydrostatic compression. However, for normally brittle materials, there is a contrasting
behavior, as fracture occurs prior to yielding, except for stress states involving a large hydrostatic
compression. Thus, if a wide range of stress states are of interest for any material, it is important to
consider the possibility that either yielding or fracture may occur first.

7.9.2 Time-Dependent Effects of Cracks
As already noted, normally brittle materials usually contain, or easily develop, small flaws or other
geometric features that are equivalent to small cracks. Brittle failure generally occurs as a result
of such cracks growing and joining. This process is often time dependent, principally because it