Page 328 - Mechanical Behavior of Materials

Problems and Questions 329

214 MPa and a compressive strength of 770 MPa. For a safety factor of 3.0 against fracture,
what is the highest torque that can be applied to the shaft? Note that S in Fig. A.12(d) is the
nominal shear stress, and k t S is the shear stress in the bottom of the groove.
7.39 A block of the polycarbonate (PC) plastic of Table 4.3 is loaded in compression and confined
by a rigid die on two sides, as in Fig. P7.25. The compressive yield strength is 20% higher
than the tensile yield strength.
(a) Estimate the value of σ z necessary to cause yielding.
(b) Qualitatively sketch the yield locus for plane stress, in this case σ x = σ 3 = 0, and show
the location of the point corresponding to your answer to (a).
7.40 Specialize the anisotropic yield criterion of Hill, Eq. 7.39, to the case of plane stress. If the
yield strengths σ oX , σ oY , and τ oXY are known, can the needed constants be obtained so that
the criterion can be used? If not, suggest an additional test on the material and explain how
you would use the result to evaluate the needed constants.
7.41 An unusual new material is hypothesized to fail when the absolute value of the hydrostatic
stress exceeds a critical value. That is,

σ x + σ y + σ z
= σ hc

3

However, there is also a possibility that this material obeys either the maximum normal stress
failure criterion or the maximum shear stress failure criterion.
(a) Does the equation given constitute a possible failure theory? Why or why not?
(b) Consider a uniaxial test, and on this basis define a convenient effective stress.
(c) In three-dimensional principal normal stress space, describe the failure surface corre-
sponding to the equation given. Also describe the failure locus for the special case of
plane stress.
(d) Describe a critical experiment, consisting of one or a few mechanical tests, and a min-
imum of experimentation, that provides a definitive choice among the aforementioned
three criteria. Note that some of the mechanical tests that are feasible are uniaxial
tension and compression, torsion of tubes and rods, internal and external pressure of
closed-end tubes, biaxial tension in pressurized diaphragms, and hydrostatic compres-
sion.

Section 7.7
7.42 The results of two tests on diabase rock are given in Table P7.42: (1) a uniaxial compression
test, and (2) a confined compression test with lateral pressure σ 1 = σ 2 .
(a) Assume that the Coulomb–Mohr fracture criterion applies, and use the results of these
tests to determine the slope and intercept constants μ and τ i for Eq. 7.42.
(b) Accurately plot the resulting |τ| versus σ failure envelope line. Also accu-
rately plot the corresponding σ 1 versus σ 2 (biaxial stress) failure locus similar to
Fig. 7.17.