Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials

Abstract

Theoretical and numerical aspects of the formulation of electromechanically coupled, transversely isotropic solids are discussed within the framework of the invariant theory. The main goal is the representation of the governing constitutive equations for reversible material behaviour based on an anisotropic electromechanical enthalpy function, which automatically fulfills the requirements of material symmetry. The introduction of a preferred direction in the argument list of the enthalpy function allows the construction of isotropic tensor functions, which reflect the inherent geometrical and physical symmetries of the polarized medium. After presenting the general framework, we consider two important model problems within this setting: i) the linear piezoelectric solid; and ii) the nonlinear electrostriction. A parameter identification of the invariant-and the common coordinate-dependent formulation is performed for both cases. The tensor generators for the stresses, electric displacements and the moduli are derived in detail, and some representative numerical examples are presented. IntroductionIn certain crystals with unsymmetric structures, mechanical deformation leads to polarization producing electric voltage, which is called the piezoelectric effect. Another important phenomenon occurs, which is the higher-order electromechanical coupling effect, called electrostriction. It is observed in all dielectrics, irrespective of their symmetry, see e.g. [23], which describes the rise of strain proportional to the square of the applied electric field strength. In unsymmteric crystals this effect is usually neglected. Piezoelectric and electrostrictive ceramic devices have a wide range of applications as sensors and actuators in intelligent systems and smart structures, which are e.g. able to realize and to operate in actual environmental conditions within a closed-loop controlled system.For an introduction to the foundations of the linear theory of piezoelectricity, established by W. Voigt at the end of the 19th century, we refer to [13] and the references therein. A general framework of the electrodynamics of continua at finite strains is presented in detail in [4] and [5]; in this context see also the extensive list of references therein. In most formulations, the piezoelectric response is described with respect to an appropriate local orientation of the crystal axis. This is common for the treatment of linear problems. A Taylor series expansion

Computational modeling of rate-dependent domain switching in piezoelectric materials

“…Fundamentals for the problem at hand have been addressed by Allik and Hughes (1970), wherein use of variational principles has been made. Subsequently, (nonlinear) finite element formulations have been investigated in different contexts; see for instance the coupled formulations in Gaudenzi and Bathe (1995) and Schröder and Gross (2004) or, for an overview, the monograph by Silvester and Ferrari (1996). In general, finite element techniques can be used to simulate both micro-and macromechanically motivated models, see for instance McMeeking (1998a, 1998b) or Kamlah and Böhle (2001).…”

Aspects of a direct homogenization procedure for electro‐mechanically coupled problems

“…On the mesoscale we use a transversely isotropic model in a coordinate-invariant formulation, see [4], [5], [6] and [8]. As depicted in Figure 2 the initial total polarization is zero.…”

A hybrid element formulation for electromechanical problems

“…In this contribution the complex electro-mechanical behavior of ferroelectric materials with an oriented internal structure is described with tensor-valued functions governed by the basic and mixed invariants based on the concept of structural tensors. In order to formulate the electric enthalpy function we need a finite set of invariants, which builds the so-called polynomial basis, see [4], [5] and [6].…”

“…Applying a static condensation procedure we obtain a modified finite element formulation governed by the degrees of freedoms associated to the displacements and the electric potential. The anisotropic material behavior is modeled within a coordinate-invariant formulation [6] for an assumed transversely isotropic material [4]. In this context a general return algorithm is applied to compute the remanent quantities at the actual timestep.…”