Page 299 - Mechanical Behavior of Materials

300 Chapter 7 Yielding and Fracture under Combined Stresses

1.5 σ / σ ot
2

σ oc / σ ot = 1.3 1.0

0.5
σ / σ ot
1
−2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5

−0.5
Yield criterion
PC
−1.0
PVC
HDPE
−1.5

−2.0

Figure 7.12 Biaxial yield data for various polymers compared with a modified octahedral
shear stress theory. (Data from [Raghava 72].)

natural stone have this effect, as do the graphite flakes in gray cast iron. Tensile normal stresses are
expected to open these flaws and therefore to cause them to grow. Thus, failure is expected to occur
on the plane where the maximum tensile normal stress occurs and to be controlled by this stress.
For example, gray cast iron fails normal to the maximum tensile stress in both tension and torsion,
as seen in the photographs of Figs. 4.13 and 4.42.
However, if the dominant stresses are compressive, the planar flaws (cracks, etc.) tend to have
their opposite sides pressed together so that they have less effect on the behavior. This explains the
higher strengths in compression for brittle materials. Also, failure occurs on planes inclined to the
planes of principal normal stress and more nearly aligned with planes of maximum shear. (See the
compressive fractures of gray iron and concrete in Figs. 4.23 and 4.24.)
One possibility for handling the differing behavior of brittle materials in tension and compres-
sion is simply to modify the maximum normal stress criterion so that the compressive and tensile
ultimate strengths differ. This would give the off-center square shown in Fig. 7.13, which still does
not agree with the data. In addition, any successful fracture criterion should predict that even brittle
materials do not fail under hydrostatic compression, which is in agreement with both observation
and intuition.
Therefore, additional failure criteria need to be considered that are capable of predicting the
behavior of brittle materials. A number of such criteria exist, and we will consider two of the simpler
ones in the portions of this chapter that follow.