Review of Drucker’s postulate and the issue of plastic stability in metal forming

Drucker’s postulate defines a class of stable work hardening materials that are classified as non-energetic and is equivalent to the associated flow rule (AFR). The postulate has been shown to be a sufficient condition for plastic stability. However, experiments indicate that plastic deformation of aluminum and steel alloys does not adhere to the constraints of the AFR. Therefore, the requirement for accuracy suggests that the metal forming industry should also consider material models that are based on non-associated flow. But Drucker’s work raises the issue of stability when considering the use of non-associated flow in material models. While this concern is merited and many types of instability arises from certain types of non-associated flow, this has led to a widely accepted view that Drucker’s postulate is a necessary condition for stability. This perception is inhibiting the acceptance or consideration of more accurate material models that are suggested from the experimental observations about violations of the AFR. This paper proposes a specific class of material models based on non-associated flow and derives the constraints on this class of models to ensure stability. The existence of this class of non-AFR models proves that Drucker’s postulate is a sufficient but not necessary condition for stability. Furthermore, the class of models described in this paper is quite general and provides a framework for consideration of potentially more accurate material models while guaranteeing the same level of stability as typically associated with materials that satisfy Drucker’s postulate.