Page 276 - Mechanical Behavior of Materials

Section 7.2 General Form of Failure Criteria 277

necessary to cause yielding is experimentally observed to be only about half of the value from the
uniaxial test. This result is easily verified by conducting a simple torsion test on a thin-walled tube,
◦
where the desired state of stress exists at an orientation of 45 to the tube axis. (Recall Fig. 4.41.)
Now consider another example, namely, a transverse tension σ x of equal magnitude to σ y ,as
illustrated in (c). Since transverse compression lowered the yield strength, intuition suggests that
transverse tension might increase it. But an experiment will show that the effect of the transverse
tension on yielding is small or absent. The experiment could be done by pressurizing a thin-walled
spherical vessel until it yielded, or by a combination of pressure and tension on a thin-walled tube.
If the material is changed to a brittle one—say, gray cast iron—neither tensile nor compressive
transverse stresses have much effect on its fracture.
An additional experimental fact of interest is that it is difficult, and perhaps impossible, to
yield a ductile material if it is tested under simple hydrostatic stress, where σ x = σ y = σ z , in either
tension or compression. This is illustrated in Fig. 7.1(d). Hydrostatic tension is difficult to achieve
experimentally, but hydrostatic compression consists of simply placing a sample of material in a
pressurized chamber.
Hence, failure criteria are needed that are capable of predicting such effects of combined states
of stress on yielding and fracture. Although both yield and fracture criteria should in general be
employed, materials that typically behave in a ductile manner generally have their usefulness limited
by yielding, and those that typically behave in a brittle manner are usually limited by fracture.

7.1.2 Additional Comments

An alternative to failure criteria based on stress is to specifically analyze cracks in the material by
the use of the special methods of fracture mechanics. Such an approach is not considered in this
chapter, but is instead the sole topic of the next chapter.
In most of the treatment that follows, materials are assumed to be isotropic and homogeneous.
Failure criteria for anisotropic materials is a rather complex topic that is considered only to a limited
extent.
Note that the effect of a complex state of stress on deformation prior to yielding has already
been discussed in Chapter 5. For example, the initial elastic slopes in Fig. 7.1 are readily obtained
from Hooke’s law in the form of Eq. 5.26. The yield criteria considered in this chapter predict the
beginning of plastic deformation, beyond which point Hooke’s law ceases to completely describe the
stress–strain behavior. Detailed treatment of stress–strain behavior beyond yielding is an advanced
topic called plasticity, which is considered to an extent in Chapter 12.
The discussion in this chapter relies rather heavily on the review of complex states of stress
in the previous chapter, specifically, transformation of axes, Mohr’s circle, principal stresses, and
octahedral stresses.

7.2 GENERAL FORM OF FAILURE CRITERIA

In applying a yield criterion, the resistance of a material is given by its yield strength. Yield strengths
are most commonly available as tensile yield strengths σ o , determined from uniaxial tests and based
on a plastic strain offset, as described in Chapter 4. To apply a fracture criterion, the ultimate