Smart materials are active and multifunctional materials, which play an important part for sensor and actuator applications.
These materials have the potential to transform passive structures into adaptive systems. However, a prerequisite for the
design and the optimization of these materials is, that reliable models exist, which incorporate the interaction between the
different combinations of thermal, electrical, magnetic, optical and mechanical effects. Polymeric electroelastic materials,
so-called electroactive polymer (EAP), own the characteristic to deform if an electric field is applied. EAP's possesses the
benefit that they share the characteristic of polymers, these are lightweight, inexpensive, fracture tolerant, elastic, and the
chemical and physical structure is well understood. However, the description "electroactive polymer" is a generic term
for many kinds of different microscopic mechanisms and polymeric materials. Based on the laws of electromagnetism
and elasticity, a visco-electroelastic model is developed and implemented into the finite element method (FEM). The
presented three-dimensional solid element has eight nodes and trilinear interpolation functions for the displacement and
the electric potential. The continuum mechanics model contains finite deformations, the time dependency and the nearly
incompressible behavior of the material. To describe the possible, large time dependent deformations, a finite viscoelastic
model with a split of the deformation gradient is used. Thereby the time dependent characteristic of polymeric materials
is incorporated through the free energy function. The electromechanical interactions are considered by the electrostatic
forces and inside the energy function.
Smart materials and structures play an important role for sensor and actuator applications. For the simulation
of such systems it is essential to predict the material and system behavior as precisely as possible. A reliable
simulation may provide an easier, faster and cheaper development of such devices. In a wide range of technical
applications piezoelectric sensors and actuators typically have a shell-like structure. This motivates the present
contribution to deal with the consistent approximation of a piezoelectric shell formulation. A physical description
leads to a system of electromechanical differential equations. Due to the constitutive relations the strains and the
electric field are coupled. In case of bending dominated problems incompatible approximation functions of these
fields cause incorrect results. This effect occurs in standard finite element formulations, where the mechanical
and electrical degrees of freedom are interpolated with lowest order functions. The formulation presented in
this paper is based on the classical Reissner-Mindlin shell theory extended by a piezoelectric part. The shell
element has four nodes and bilinear interpolation functions. The eight degrees of freedom per node are three
displacements, three rotations and the electric potential on top and bottom of the shell. The finite shell element
incorporates a 3D-material law and is able to model arbitrary curved shell geometries of piezoelectric devices. In
order to overcome the described problem of incompatible approximation spaces a mixed multi-field variational
approach is introduced. Six independent fields are employed. These are the displacement, strain, stress, electric
potential, dielectric displacement and the electric field. It allows for approximations of the electric field and the
strains independent of the bilinear interpolation functions. A quadratic approach for the shear strains and the
corresponding electric field is proposed through the shell thickness. This leads to well balanced approximation
functions regarding coupling of electrical and mechanical fields. The numerical results are confirmed by analytical
considerations and an example illustrates the more precise results of the present formulation in contrast to
standard elements.
The paper is concerned with a geometrically nonlinear finite element formulation to analyze piezoelectric shell
structures. The classical shell assumptions are extended to the electromechanical coupled problem. The consideration
of geometrical nonlinearity includes the analysis of stability problems and other nonlinear effects. The
formulation is based on the mixed field functional of Hu-Washizu. The independent fields are displacements,
electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an
interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when
using nonlinear 3D material laws. With respect to the numerical approximation an arbitrary reference surface
of the shell is modeled with a four node element. Each node possesses six mechanical and one electrical degree
of freedom. Some simulations demonstrate the applicability of the present piezoelectric shell element.
This paper is concerned with a macroscopic constitutive law for domain switching
effects, which occur in piezoelectric ceramics. The thermodynamical framework of the law is
based on two scalar valued functions: the electric free Gibbs energy and a switching surface.
In common usage, the remanent polarization and the remanent strain are employed as internal
variables. The novel aspect of the present work is to introduce an irreversible electric field,
which serves besides the irreversible strain as internal variable. The irreversible electric field
has only theoretical meaning, but it makes the formulation very suitable for a finite element implementation,
where displacements and the electric potential are the nodal degrees of freedom.
The constitutive model reproduces the ferroelastic and the ferroelectric hysteresis as well as the
butterfly hysteresis and it accounts for the mechanical depolarization effect.
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