Plasticity theory of microinhomogeneous materials at large strain gradient

Abstract

A general thermomechanical theory of martensitic phase transformations (PT) in inelastic materials is presented. The results are derived for small and large strains in the reference and actual configurations. PT is treated as a thermomechanical process of growth of transformation strain from the initial to the final value which is accompanied by a change in all material's properties. The theory is developed first of all for a homogeneously deformed material point (neighborhood) undergoing the PT which can belong to a new nucleus or a moving interface. It is shown that a standard thermodynamical approach cannot be directly applied. It can be applied after the averaging of thermodynamical parameters, related to PT, over a PT duration. PT criterion is derived which takes into account the plastic dissipation, temperature variation due to the PT and variation of internal variables. It is shown that the temperature gradient does not contribute to PT criterion. The twinning criterion is derived as a particular case of the PT criterion. Temperature variation in the course of PT is determined with the help of the entropy balance equation under the assumption that the process is adiabatic. After the averaging of the PT criterion over the transforming volume the nucleation and interface propagation criteria, as well as conditions of nondissappearance of nucleus are derived. Using the postulate of realizability [Levitas, V.I. (1992a) Post-bifurcation behavior in finite elastoplasticity. Applications to strain localization and phase transitions, Universität Hannover. Institut für Baumechanik und Numerische Mechanik, IBNM-Bericht 92/5. (1995a) The postulate of realizability: formulation and applications to post-bifurcation behavior and phase transitions in elastoplastic materials. Part I and II. International Journal of Engineering Science, 33, 921–971.], the extremum principle for the determination of all unknown parameters (e.g. position, shape and orientation of nuclei, transformation strain and so on) is derived. It is shown that for the PT in elastic materials the proposed approach gives alternative, but equivalent to the principle of minimum of Gibbs energy formulation. Some aspects of the formulation of boundary-value problem (BVP) are analyzed. Some possible ways of formulation of constitutive relations for inelastic deformations in the course of the PT are discussed. It is obtained that the dissipative threshold in the PT criterion is proportional to yield stress. The thermomechanical theory developed is extended to the case with displacement discontinuities across an interface (noncoherence and fracture).