Keynote Lecturer: Vassilis Panoskaltsis Decompositions of the stress and the rate of deformation ten- sors for materials undergoing phase transformations V International Conference on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2013 S. Idelsohn, M. Papadrakakis and B. Schrefler (Eds) DECOMPOSITIONS OF THE STRESS AND THE RATE OF DEFORMATION TENSORS FOR MATERIALS UNDERGOING PHASE TRANSFORMATIONS VASSILIS P. PANOSKALTSIS * * Department of Civil Engineering Demokritos University of Thrace 12 Vassilissis Sofias Street, Xanthi 67100, Greece e-mail: vpanoska@civil.duth.gr Key words: Phase Transformations, Shape Memory Alloys, Multi Surface Generalized Plasticity, Loading – Unloading Criteria, Lie Derivative, Finite Deformations, Internal Variables, Duhamel – Neumann Hypothesis, Rate of Deformation Tensor. Summary: An extension of the “Duhamel-Neumann hypothesis” for materials undergoing phase transformations and for arbitrary magnitudes of strains and rotations is provided. 1 INTRODUCTION Generalized plasticity theory has been successfully used by the author and his co- workers in order to model phase transformations of shape memory alloy materials. Our work has been initially carried out within small strains (Panoskaltsis et al. [1], Ramanathan et al. [2]) and very recently within finite deformations and rotations and under both isothermal and non-isothermal conditions (Panoskaltsis [3]). In this paper we will develop, for the first time, two important decompositions applied to materials undergoing phase transformations. In the first decomposition (Theorem 1) the rate of the Kirchhoff stress tensor is given in terms of the rate of deformation tensor and the rate of the temperature. In the second decomposition (Theorem 2), which can be thought of as a conjugate to the first one, the rate of deformation tensor is expressed as a sum of the (objective) rate of the stress tensor and the rate of the temperature. In the next section we will review the formulation of 445