Numerical Procedure for Polycrystalline Ferroelectric/Ferroelastic Problems Using Landau's Phenomenological Model

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This paper proposes a finite element procedure for the simulations of material nonlinearity so-called domain switching in polycrystalline ferroelectric/ferroelastic bodies. Conventional FEM, which has been used for linear piezoelectric analyses, is not appropriate for solving the nonlinear problem from thermodynamic viewpoint because it corresponds to finding a saddle point of functional. Therefore, we use a recently proposed alternative FEM that corresponds to finding local minimum points of functional. In the alternative FEM, both mechanical displacement and vector potential for electric displacement instead of scalar potential are chosen as unknown variables. As a constitutive law in each crystal, we adopt a class of Landau's potential energy models, which is a popular phenomenological description for phase transitions in solid-state physics. Because the processes of the nonlinear behavior are essentially dynamic phenomena and the total potential of the system may have local minimum solutions, we employ a dynamic formulation for the nonlinear analyses with the consideration of Debye-type dielectric relaxation and mechanical wave propagation. Numerical examples of a two-dimensional polycrystalline ferroelectric model are shown and the results can qualitatively predict ferroelectric behaviors such as hysteresis curves and butterfly curves.
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