The stress-strain relations under a combined state of stresses discussed in Chapters 4 and 5 are typical for polycrystalline metals. It is well known that mild steel exhibits plastic flow under constant stress (Fig. 1.1a). This behavior can be modeled by the theory of perfect plasticity. The more common metals such as aluminum and copper generally fall within the category of hardening materials (Fig. 1.1b). They can best be modeled by the work-hardening theory of plasticity.
One of the remarkable features of metals is that the effects of hydrostatic pressure on yielding and the subsequent plastic deformation are not appreciable. These facts imply that the shear stress alone is critical for yielding and that the plastic volume change is negligible even for large plastic deformations. The maximum shear stress condition of Tresca and the octahedral shear stress condition of von Mises for yielding show a good agreement with experimental data, and the Row rule associated with each of these two yield functions predicts a plastic shear deformation without plastic volume change. The Tresca or von Mises model with or without work hardening is therefore generally adopted in establishing the constitutive laws of metals.
Due to the nonlinear nature of the plastic constitutive relations, analytic solutions of boundary-value problems are difficult to obtain. Up to now, only very few exact solutions of elastic-plastic boundary-value problems are available. Future prospects for exact solutions are far less bright for the case...
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