Thermomechanical modeling of polycrystalline SMAs under cyclic loading, Part I: theoretical derivations

A generic form of Gibbs free energy for polycrystalline Shape Memory Alloys (SMAs) is first obtained in this work, by forming the increments of both elastic potential energy and Gibbs chemical energy over a Representative Volume Element (RVE) with respect to an infinitesimal increment of martensite. A set of internal state variables, i.e., martensitic volume fraction, macro-transformation strain, and back and drag stresses due to both martensitic phase transformation and its interaction with plastic strains, are introduced. The evolution of these internal state variables during phase transformation is proposed based on the micromechanical analysis over the RVE. Four primary mechanisms governing the transformation induced hardening effect are discussed. It is concluded that the back and drag stresses related to plastic deformation do not remain constant, but they vary during phase transformation, even though the local plastic residual stresses are assumed to be constant. The initial material heterogeneity of SMAs, which is essential for the initiation of the phase transformation, is modeled by an initial residual stress field, which can be described by a probability distribution function. In this Part I of a four-part paper, the theoretical derivations are presented. Specific cases of the thermomechanical response of SMAs predicted by the model will be presented in Part II, together with experimental results for phase transformation at constant plastic strains. Experimental results and model predictions for cyclic loading of SMAs with evolving plastic strains will be considered in Part III, while the modeling of minor hysteresis loops of SMAs will be presented in Part IV of this series of four papers on SMAs.

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