Page 320 - Mechanical Behavior of Materials

Section 7.10 Summary 321

is affected by the presence of moisture (water) or other substances that react chemically with the
material. Time-dependent cracking causes the fracture behavior to be dependent on the loading rate.
Also, if the stress is held constant, failure can occur after some time has elapsed at a stress that
would not cause fracture if maintained for only a short time. Thus, the approaches of this chapter
should be used with some caution to assure that the material properties employed are realistic with
respect to time-related effects.

7.10 SUMMARY

Design to avoid yielding or fracture in nominally uncracked material requires the use of a failure
criterion, which is a procedure for summarizing a complex state of stress as an effective stress ¯σ that
can be compared to the material’s strength. For yielding of ductile materials, the relevant materials
strength property is the yield strength σ o , so that a safety factor can be calculated as

σ o
X = (7.67)
¯ σ
Two yield criteria are available that are reasonably accurate for isotropic materials, namely, the
maximum shear stress criterion and the octahedral shear stress criterion. The effective stresses for
these are, respectively,
¯ σ S = MAX(|σ 1 − σ 2 | , |σ 2 − σ 3 | , |σ 3 − σ 1 |) (7.68)

1 2 2 2
¯ σ H = √ (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 ) (7.69)
2
Effective stresses, and hence safety factors, from these two criteria never differ by more than 15%.
In their basic forms, according to these two equations, both predict that hydrostatic stresses have no
effect. Modifications can be used to predict yielding in anisotropic or pressure-sensitive materials.
The application of safety factors, as just described, is called allowable stress design. An
alternative is load factor design, where the applied loads are increased by factors Y that can vary for
different load inputs, and the failure condition is analyzed. In particular, ¯σ = σ o is employed, where
¯ σ is calculated from stresses that include load factors.
In applying yield criteria to ductile materials, the stresses employed are usually the nominal
ones—that is, the stresses do not include the localized stress raiser effect at notches. This is justified,
as ductile materials can deform beyond yielding in a small region without causing failure of the
component. But this is not the case for brittle materials, for which stress raiser effects should be
considered in fracture criteria.
For brittle materials, no single basic failure criterion suffices to describe the fracture behavior.
The modified Mohr criterion is a reasonable choice. It is a combination of the maximum normal
stress criterion, which is used where the stresses are dominated by tension, and the Coulomb–Mohr
criterion. The latter assumes that fracture occurs when the combination of normal and shear stress
on any plane in the material reaches a critical value given by

|τ| + μσ = τ i (7.70)