Page 297 - Mechanical Behavior of Materials

298 Chapter 7 Yielding and Fracture under Combined Stresses

collectively termed notches. (See Appendix A, Section A.6.) Consider components made of ductile
materials, such as most steels, aluminum alloys, titanium alloys, and other structural metals, and
also many polymeric materials. In this case, the material can yield in a small local region without
significantly compromising the strength of the component. This is due to the ability of the material
to deform at the notch and shift some of the stress to adjacent regions, which behavior is called
stress redistribution. Final failure does not occur until yielding spreads over the entire cross section,
as discussed in Section A.7 in the context of fully plastic yielding. (See Figs. A.10 and A.14.)
As a result of a ductile material’s ability to tolerate local yielding, stress raiser effects are
not usually included in applying yield criteria for static design. In other words, net section nominal
stresses, such as S in Figs. A.11 and A.12, are used with the yield criterion, rather than local stresses
σ = k t S that include the notch effect. (However, where cyclic loading may cause fatigue cracking,
stress raiser effects do need to be considered, as treated in detail in Chapters 10, 13, and 14.)
In modern industry, critical components are likely to be analyzed on a digital computer by the
method of finite elements. Linear-elastic behavior is usually assumed, and color-coded plots are
often made of the magnitude of the von Mises stress, which is simply our octahedral effective stress,
¯ σ H . (See the back cover of this book for examples of such plots.) This affords an opportunity to
visualize the size of any regions that exceed the yield strength. Where yielding occurs over regions
of worrisome size, design changes need to be made, and the analysis repeated, to be sure that the
change was successful. It is often not feasible to make a design so conservative as to eliminate all
yielding for severe loading conditions that may occur only rarely.
The preceding argument does not apply to brittle (nonductile) materials, such as glass, stone,
ceramics, PMMA and some other polymeric materials, and gray cast iron and some other cast
metals. Brittle materials are not capable of deforming sufficiently to shift locally high stresses
elsewhere, which is illustrated in Fig. A.10(e). Therefore, the locally elevated stress, σ = k t S,
should be compared with the failure criterion. In tension-dominated situations, brittle materials
fail if the local stress reaches the ultimate tensile strength, according to the maximum normal
stress fracture criterion of Eq. 7.13. As a rough guide, a brittle material can be defined as one
with less than 5% elongation in a tension test. However, there is an interesting exception to the
foregoing recommendation: Where the inherent flaws in a brittle material are relatively large, these
may overwhelm the effect of a small stress raiser so that it has little effect. For example, gray cast
iron is not sensitive to small stress raisers, as its behavior is dominated by relatively large graphite
flakes. (See Fig. 3.7.) In contrast, glass is weakened by a scratch.

7.6.4 Yield Criteria for Anisotropic and Pressure-Sensitive Materials

Several empirical modifications have been suggested so that the octahedral shear stress criterion
can be used for anisotropic or pressure-sensitive materials. Anisotropic materials have different
properties in different directions. Consider anisotropic materials that are orthotropic, possessing
◦
symmetry about three planes oriented 90 to each other. For example, such anisotropy can occur in
rolled plates of metals where the yield strength may differ somewhat between the rolling, transverse,
and thickness directions. The anisotropic yield criterion described in Hill (1998) for this case is

2
2
2
H (σ X − σ Y ) + F (σ Y − σ Z ) + G (σ Z − σ X ) + 2Nτ 2 + 2Lτ 2 + 2Mτ 2 = 1 (7.39)
XY YZ ZX