Internal Stress Generation During Switching of Ferroelectrics
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Internal Stress Generation during Switching of Ferroelectrics Anja Haug, Patrick R. Onck and Erik Van der Giessen Materials Science Center, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands ABSTRACT Progressive switching of grains in ferroelectric materials leads to internal stresses, which may give rise to degradation of the material or even fracture. Here we propose a two-dimensional multigrain model for ferroelectric polycrystal. The numerical computations employ the nonlinear micromechanics model of Huber et al. [1] for each grain. Results for the macroscopic response, butterfly and hysteresis loops, as well as internal stress and electric field distributions are presented. INTRODUCTION Ferroelectrics are crystalline inorganic materials that can be made piezoelectric by poling, i.e. by reorienting domains through application of an electric field. Since the crystallographics axes of the grains are randomly orientated in the material, the remanent strain and polarization develop differently in each grain during poling. Internal mechanical stresses and electric fields develop to compensate for these differences across the grain boundaries. Knowledge of the intergranular stresses and electric fields is a requirement to get more insight in crack initiation and fracture. Therefore movement of domain walls —switching— must be described by constitutive laws. We utilize the micromechanics single crystal model of Huber et al. [1]. The kinematics of the process due to switching systems is developed in analogy to crystal plasticity. A polycrystalline aggregate is built up of hexagonal grains. Each grain has its own crystal orientation and material properties. This multigrain discretization of the material allows to model the interaction of the grains in detail. The problem is solved using finite elements. In this paper we focus on the development of internal stress and electric field in a polycrystal. To aid understanding, we first discuss a finite element calculation of a single crystal. Kamlah et al. [2] already discussed the single crystal response to a poling electric field. However, they did not investigate the effect of the boundary condition on the crystal response. MODEL The model used is described in detail by Huber et al. [1]. It is a continuum theory that divides each material point into different domains, each corresponding to a possible direction of spontaneous polarization. In the case of a tetragonal system (e.g. tetragonal perovskite, BaTiO3 ) six domains (M = 6) variants are possible. The domain size can vary and even vanish. The size of a domain I is characterized by its volume fractions, cI , with values between zero to one. It is further assumed that the strain εi j and the electrical displacement Di are composed of a linear and remanent part, independent of each other. The total response at a material point is calculated by volume averaging over the domains, both for the remanent (superscript R) and the linear part (superscript L), by assuming that the electric field Ek
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