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2 April 2008 A rate-dependent incremental variational formulation of ferroelectricity
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The paper presents continuous and discrete variational formulations for the treatment of the non-linear response of piezoceramics under electrical loading. The point of departure is a general internal variable formulation that determines the hysteretic response of the material as a generalized standard medium in terms of an energy storage and a rate-dependent dissipation function. Consistent with this type of standard dissipative continua, we develop an incremental variational formulation of the coupled electromechanical boundary value problem. We specify the variational formulation for a setting based on a smooth rate-dependent dissipation function which governs the hysteretic response. Such a formulation allows us to reproduce the dielectric and butterfly hysteresis responses characteristic of the ferroelectric materials together with their rate-dependency and to account for macroscopically non-uniform distribution of the polarization in the specimen. An important aspect is the numerical implementation of the coupled problem. The discretization of the two-field problem appears, as a consequence of the proposed incremental variational principle, in a symmetric format. The performance of the proposed methods is demonstrated by means of benchmark problems.
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Daniele Rosato and Christian Miehe "A rate-dependent incremental variational formulation of ferroelectricity", Proc. SPIE 6929, Behavior and Mechanics of Multifunctional and Composite Materials 2008, 69290E (2 April 2008);

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