Because of their actuating, sensing and power generating capabilities, the magnetic shape memory alloys (MSMA) have drawn an increasing interest in the past few years. Such materials are capable of attaining large actuation strains with actuation bandwidth which is greater than that of conventional shape memory alloys (SMA). In magnetoelastic magnetostrictive materials deformation can be triggered by magnetic field and the corresponding magnetic field induced strains (MFIS) are nonlinear, stress dependent and exhibit a hysteresis. Linear theory which have been most frequently used in practice and have been incorporated in a number of finite element codes, provides constitutive relations which characterize linear elastic, magnetic and magnetoelastic interactions, but neglects inherent hysteresis and nonliner behaviour of MFIS. On the other hand, three-dimensional theories for magnetostrictive materials which have the capability of acurate modeling domain structure in strain and magnetization together with the corresponding nonlinear response of the homogenized system, are not developed to an extent comparable to similar theories for shape memory materials. In this work we develop a theoretical concept with higher-order energy relations which tends to incorporate some of the above mentioned phenomena. First we construct an appropriate Helmholtz energy relation in terms of temperature and magentization. Then by invoking a double Legendre transform we introduce the Gibbs energy relation as a function of stress, strain, magnetic field, magnetization, and temperature, and by minimization seek the condition for local minimum in strains and magnetization. Derivation is conceptualized on truncated polynomial energy relations and results obtained from applying the described thermodynamic formalism are scrutinized for available experimental data and compared with some other existing models based on the theoretical concepts developed by Jiles-Atherton and Stoner-Wohlfarth.

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