Summary: We reconsider the basic concepts of piecewise smooth dissipation functions and yield surfaces in classical plasticity. This re-examination is from the perspective of convex analysis, and is applied to an internal variable description of the elastic-plastic solid. First, we recapitulate the arguments in which the yield surface is a derived concept in a theory which assumes the existence of a positive semi-definite quadratic strain energy function and a convex homogeneous dissipation function, in the case where the dissipation function is differentiable everywhere except at the origin. We then consider the case of a multiplicity of dissipation mechanisms, and the determination of the operative mechanism or mechanisms. The global dissipation function is the convex hull of the local dissipation functions. A piecewise smooth yield function follows through a generalization of the transformation for the smooth case.