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288                               Chapter 7  Yielding and Fracture under Combined Stresses


                 (b) To achieve a safety factor of X = 2.0 with a modified value of diameter d, we need

                                   ¯ σ S = σ o /X = (260 MPa)/2.0 = 130 MPa

             Substituting this and the given P and T into the equation for ¯σ S developed earlier gives


                                                                            2
                                                                  6
                                     4                   8(1.50 × 10 N·mm)

                                                    2
                          130 MPa =    2  (200,000 N) +
                                    πd                           d
             This cannot be solved for d in a closed-form manner, so trial and error or other iterative procedure
             is required to obtain
                                              d = 54.1mm                              Ans.


              As might be expected, increasing the safety factor to 2.0 from that found in (a) requires a larger
             diameter.



            7.5 OCTAHEDRAL SHEAR STRESS YIELD CRITERION

            Another yield criterion often used for ductile metals is the prediction that yielding occurs when the
            shear stress on the octahedral planes reaches the critical value


                                         τ h = τ ho  (at yielding)                    (7.28)

            where τ ho is the value of octahedral shear stress τ h necessary to cause yielding. The resulting
            octahedral shear stress yield criterion, also often called either the von Mises or the distortion energy
            criterion, represents an alternative to the maximum shear criterion.
               A physical justification for such an approach is as follows: Since hydrostatic stress σ h is
            observed not to affect yielding, it is logical to find the plane where this occurs as the normal
            stress, and then to use the remaining stress τ h as the failure criterion. Another justification is to note
            that, although yielding is caused by shear stresses, τ max occurs on only two planes in the material,
            whereas τ h is never very much smaller and occurs on four planes. (Compare Figs. 6.8 and 6.15.)
            Hence, on a statistical basis, τ h has a greater chance of finding crystal planes that are favorably
            oriented for slip, and this may overcome its disadvantage of being slightly smaller than τ max .


            7.5.1 Development of the Octahedral Shear Stress Criterion
            From the previous chapter, Eq. 6.33, the shear stress on the octahedral planes is

                                     1          2          2          2
                                τ h =   (σ 1 − σ 2 ) + (σ 2 − σ 3 ) + (σ 3 − σ 1 )    (7.29)
                                     3
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