– A piece of a ductile metal is confined on two sides by a rigid die, as shown in Fig. P7.25. A uniform compressive stress σz is applied to the surface of the metal. Assume that there is no friction against the die, and also that the material behaves in an elastic, perfectly to plastic manner with uniaxial yield strength σo. Derive an equation for the value of σz necessary to cause yielding in terms of σo and the elastic constants of the material. Is the value of σz that causes yielding affected significantly by Poisson’s ratio? Employ (a) the maximum shear stress criterion, and (b) the octahedral shear stress criterion. (c) For each yield criterion, what stress σz is expected to cause yielding if the material is ductile cast iron A536 (65-45-12)?

Repeat Prob. 7.25(c) for the case where the die material is AISI 4142 steel.

Consider the situation of Fig. E7.5, where a piece of material is stressed in two directions and restrained from deforming in the third direction by rigid, but smooth, walls. Generalize the loading such that σx = λσy , where λ may vary between +1 and −1, and derive the corresponding expression for σy at yielding that is analogous to the result of Ex. 7.5(a). Comment on how the stress at yielding is affected by λ.