A unified thermodynamic theory of elasticity and plasticity

Abstract

A thermodynamics is developed for a unified theory of elasticity and plasticity in infinitesmal strain. The constitutive equations which relate stress and strain deviators are rate type differential equations. When they satisfy a Lipschitz condition, uniqueness for the initial value problem dictates that the stress and strain will be related through elastic relations. Failure of the Lipschitz condition occurs when a von Mises yield condition is achieved: Plastic yield then occurs and the deviator relations turn into the Prandtl-Reuss equations. The plastic yield solution is stable during loading and unstable during unloading. The requirement that the solution followed during unloading be stable dictates entry into an elastic regime. Appropriate thermodynamic functions are constructed. It then appears that stress deviator (not strain deviator) is a viable state variable, and the thermodynamic relations are constructed in terms of a Gibbs function. The energy balance leads to satisfaction of the Clausius-Duhem inequality (and thus the second law of thermodynamics) in an elastic regime because it is shown that in an elastic regime entropy production is caused only by heat flux. During yield, the proper method of differentiating yields entropy production terms in addition to those arising from heat flux. These terms are positive during loading, whence it is concluded that the requirement that a stable solution be followed leads to satisfaction of the Clausius-Duhem inequality during plastic as well as elastic behavior.