This paper is concerned with the development of a finite element model for fully-coupled magnetomechanical
boundary value problems, which allows the implementation of nonlinear, anisotropic and hysteretic material
models. The formulation is based on a vector-valued magnetic potential and a small strain setting. The derivation
of the numerical algorithm is considered in detail. In order to test its implementation the case of piezomagnetism
is considered, for which a thermodynamically-consistent formulation is presented. The results of the finite element
analysis that have been obtained for two different boundary value problems, which involve the mechanical
and magnetic loading of an elastic matrix material with a transversely isotropic piezomagnetic inclusion, are
discussed.
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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