Normality Relations and Convexity of Yield Surfaces for Unstable Materials or Structural Elements

The stress-strain relations for materials and the load-deflection relations for structural elements play corresponding roles in the analysis of three-dimensional continua and of structures, respectively. Mathematically equivalent and phenomenologically quite similar, they are treated simultaneously here. As in previous treatments of stable (rising) plastic stress-strain curves, unstable (falling) curves in simple shear or tension are generalized to all states of stress through the exploration of the work done in a cycle of stress (Drucker) and in a cycle of strain (Ilyushin). The plastic increment of strain is found to be normal to the current yield surface for a wide class of unstable materials in which a continuous variation of strain produces a unique continuous variation of stress and of the shape and position of the yield surface. In the absence of any significant alteration in the (stable) elastic response, each yield surface then is shown to be convex. The degree of concavity possible when the elastic response is stable but is nonlinear and does alter appreciably due to plastic deformation is illustrated by a nonlinear elastic spring and a plastic block in parallel. Such concavity would not be observable in the yield surfaces of common structural metals but might be found for soils, rocks, or concrete and can be quite pronounced for structural elements.