Page 317 - Mechanical Behavior of Materials

318 Chapter 7 Yielding and Fracture under Combined Stresses

where X CM = 1.00 is substituted so that the point of fracture is analyzed. Solving for τ xy yields

τ xy = 134.6MPa Ans.

This value gives σ 1 ,σ 2 =−150 ± τ xy =−15.4, −284.6 MPa. Hence, Eq. 7.65 gives ¯σ NP = 0
and infinite X NP , so the maximum normal stress component does not control, and the preceding
solution is valid.
(b) Proceeding as before, except for substituting X CM = 2.00, gives τ xy = 129.1MPa
(Ans.). This value corresponds to σ 1 ,σ 2 =−20.9, −279.1 MPa, so Eq. 7.65 again gives ¯σ NP =
0, and this solution is also valid.

Comment In the solution for (b), the safety factor of X CM = 2.00 is, in effect, applied
both to the pressure and to τ xy . Due to increased pressure making fracture more difficult, it
turns out that only a small decrease in τ xy is needed to achieve the safety factor. From an
engineering viewpoint, if the pressure is considered not to vary, it would be wise to apply
the desired safety factor of X = 2.00 to only the shear stress that is allowed to vary, so that
the solution for (b) becomes τ xy = 134.6/2.00 = 67.3 MPa. This would be the same as a load
factor design approach, with Y p = 1.00 applied to the pressure and Y τ = 2.00 applied to τ xy .
(See Section 7.6.2.)

7.9 ADDITIONAL COMMENTS ON FAILURE CRITERIA

To gain additional perspective on the subject of this chapter, we will engage in some limited further
discussion on brittle versus ductile behavior and on time-dependent effects.

7.9.1 Brittle Versus Ductile Behavior
Engineering materials that are commonly classed as ductile are those for which the static strength
in engineering applications is generally limited by yielding. Many metals and polymers fit into this
category. In contrast, the usefulness of materials commonly classed as brittle is generally limited by
fracture. In a tension test, brittle materials exhibit no well-defined yielding behavior, and they fail
after only a small elongation, on the order of 5% or less. Examples are gray cast iron and certain
other cast metals, and also stone, concrete, other ceramics, and glasses.
However, normally brittle materials may exhibit considerable ductility when tested under
loading such that the hydrostatic component σ h of the applied stress is highly compressive. Such
an experiment can be conducted by testing the material in a chamber that is already pressurized,
as in Fig. 4.27. The surprising result of large plastic deformations in a normally brittle material is
illustrated by some stress–strain curves for limestone in Fig. 7.22.
Also, materials normally considered ductile fail with increased ductility if the hydrostatic stress
is compressive, or reduced ductility if it is tensile. For example, although the initial yielding of
metals is insensitive to hydrostatic stress, the point of fracture is affected. Data showing this for
a steel are given in Fig. 7.23, where the true fracture stress and strain are seen to increase with