A constitutive law for ferroelectric and ferroelastic piezoceramics is implemented in ABAQUS Standard using the subroutine user element. A linear solid element is defined: it is an eight-node hexahedron having the mechanical displacement components and the electric potential as degrees of freedom for each node. The element is formulated for static analysis and it needs the definition of the contribution of this element to the Jacobian (stiffness) and the definition of an array containing the contributions of this element to the right-hand-side vectors of the overall system of equations The subroutine is called for each element that is of a user-defined element type each time element calculations are required. As an example, the element is used for the simulation of a multilayer actuator made of piezoceramics. In this case, the piezoelectric equations are not valid since the electric loading induces non linear phenomena, which are captured through the constitutive law implemented in the user element.

A thermodynamically consistent phenomenological model for the material behavior of polycrystalline ferroelectric ceramics is presented. The internal state of the material is described by two internal state variables. The first one is a second-order texture tensor, determining a simple orientation distribution function (ODF) for the axes of the crystal unit cells. The second is the vector of relative irreversible polarization. The irreversible strains are derived from the ODF by volume averaging. The polarization saturation states are calculated by summing up the possible contributions of all cells to the overall polarization. An invariant formulation of the piezoelectric law is applied. Analogous to the thermodynamical framework of rate-independent plasticity, driving forces and evolution laws for the internal state variables are established. Saturation and coupling of the switching behavior are governed by energy barrier functions introduced in the electric enthalpy function. Numerical examples illustrate the models capabilities.

This paper addresses the development of an energy-based characterization framework which quantifies stress-induced dipole switching in ferroelectric materials. Helmholtz and Gibbs energy relations that accommodate 90° and 180° dipole orientations as equilibrium states are constructed at the lattice level. For regimes in which thermal relaxation mech- anisms are negligible, minimization of the Gibbs relations provides local polarization and strain relations. Alternatively, behavior such as creep or thermal relaxation can be incorpo- rated by balancing Gibbs and relative thermal energies through Boltzmann principles. In the nal step of the development, stochastic homogenization techniques based on the assump- tion that parameters such as coercive and induced elds are manifestations of underlying distributions are employed to construct macroscopic models suitable for nonhomogeneous polycrystalline compounds. Attributes and limitations of the model are illustrated through comparison with experimental PLZT data.

Mode I steady crack growth is analyzed to determine the toughening due to domain switching in poled ferroelectric ceramics. A multi-axial, electromechanically coupled, incremental constitutive theory is applied to model the material behavior of the ferroelectric ceramic. The constitutive law is then implemented within the finite element method to study steady crack growth. The effects of mechanical and electrical poling on the fracture toughness are investigated. Results for the predicted fracture toughness, remanent strain and remanent polarization distributions, and domain switching zone shapes and sizes are presented. Finally, the model predictions are discussed in comparison to experimental observations.

The determination of the behavior of heterogeneous materials with complex physical and mechanical couplings constitutes a challenge in the design of new materials and the modeling of their effective properties. In real inhomogeneous materials, the simultaneous presence of elastic mechanisms and non linear inelastic ones (viscoplastic, magnetic, ferroelectric, shape memory effect etc.) leads to a complex non linear coupling between the mechanical fields which is tricky to represent in a simple and efficient way. Hence, for many situations the effective global behavior does not follow the same structure than the local constitutive one. Regarding space-time couplings for instance, a heterogeneous material composed of phases described by Maxwell elements can not be considered as a Maxwellian solid at the macro scale.

In this paper, we introduce a new micro-macro approach based on translated fields in its generalized form to be applied to different coupled phenomena. The local total strain (rate) is composed additively of an elastic strain (rate) and an inelastic one which is no more limited to be “stress free” as considered originally by Kröner. An extended (non conventional) self-consistent model is then proposed starting from the integral equation for a translated strain (rate) field and using the projection operators algebra introduced by Kunin. The chosen translated field is the compatible inelastic strain (rate) of the fictitious inelastic heterogeneous medium submitted to a uniform unknown boundary condition. The self-consistency condition amounts to define analytically these boundary conditions so that a relative simple and compact strain (rate) concentration equation is obtained.

In order to illustrate the method, the case of a non linear elastic-viscoplastic coupling is developed and applied to different classes of composites and polycrystals.

A simulation of the nonlinear electromechanical macroscopic behavior of ferroelectric materials by means of the finite element method is presented. A material point is depicted by a representative volume element, for which homogeneous boundary conditions are valid. The evolution of integral averages over the representative volume element is to homogenize the results. For this homogenization we favor a finite element model in which each Gauss point represents exactly one single crystal. Their number of internal variables is limited to the lattice orientation and the volume fractions of the domains. The former are randomly distributed in space. It is possible to calculate the material behavior for arbitrary coupled and nonlinear electromechanical loading cases, but the model is not effective for the solution of boundary value problems for entire bodies.

The general methodology of calculating linear and nonlinear properties of nanoporous and nanostructured semiconductor materials and composites is presented. A Maxwell-Garnett approach is generalized for the case of porous semiconductor materials composed of a number of differently oriented pore lattices. Specifically, the cases of electrochemically etched mesoporous silicon on (110)-oriented substrate and electrochemically-etched porous InP and GaAs materials on (100) substrates are considered. The observed optical anisotropy of mesoporous Si is explained. A biaxial anisotropy of the porous InP or GaAs material with crystallographic pores is predicted.

Efficient application of piezoelectric sensors and actuators requires extensive investigations of their loading limits, failure-behavior and life-span under service conditions. Here the performance of laminar PZT-patch-modules is studied, applying a combined approach of experimental and numerical tests. Four-point bending tests are used to evaluate the sensor performance. Linear electro-mechanical coupling is implemented in an 8-node brick user element of the research finite element code FEAP to develop a flexible FE-tool for piezoelectric problems. Comparative FE-analyses of the bending test are carried out in order to assess the capability of the implemented FEAP user element versus the commercially available FE-code ABAQUS. FE-results show good agreement with the experimental tests, different element types yield slight deviations which are discussed.

In this work, we present simulations of the thermo-magneto-mechanical performance of a ferromagnetic SMA microactuator in the inhomogeneous field of a CoSm permanent magnet and compare it with experimental results. For the simulations, a combination of different finite element (FEM) programs is used allowing the coupling of electrical, thermal, magnetic and mechanical properties. The field of a permanent magnet is calculated along the trajectory of the microactuator, from which the magnetic forces Fmag and their gradients are derived. Shape memory forces FSME are calculated based on a two-phase macromodel. Both forces, Fmag and FSME, determine the quasi-static behavior of a NiMnGa microscanner, which has been proposed recently.

The one-dimensional free energy model for ferroelectric materials developed in [1-3] is general-ized to two dimensions. The proposed two-dimensional energy potential consists of four energy wells corresponding to four variants of the material, four saddle points representing the barriers for 900 switching processes, and a local energy maximum across which 1800-switching processes take place. The free energy potential is combined with the evolution equations based on the theory of thermally activated processes. The prediction of the model is compared with the recent measurements on a Ba- TiO3 single crystalline ferroelectric in [4]. The responses of the model at various loading frequencies are calculated and the kinetics of 900 and 1800 switching processes are discussed.

Following our previous work on the converse piezoelectric constant- in epitaxial ferroelectric films for MEMS actuator applications, the orientation dependence of the direct piezoelectric constants , and are generally formulated, which can help to predict and optimize the performance of piezoelectric MEMS sensor devices based on ferroelectric thin films. Numerical results are obtained and discussed for Pb(ZrxTi1-x)O3 thin films grow on Si substrates with various compositions and structures.

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.

The rate-dependence of piezoelectric materials resulting from the kinetics of domain switching is an important factor that needs to be included in realistic modeling attempts. This paper provides a systematic study of the rate-dependent hysteresis behavior of a commercially available PZT stack actuator. Experiments covering full as well as minor loops are conducted at different loading rates with polarization and strain recorded. In addition, the creep behavior at different constant levels of the electric field is observed. This provides evidence of kinetics being characterized by strongly varying relaxation times that can be associated with different switching mechanisms.

Initially unpoled soft PZT was subjected to a proportional, coaxial electromechanical loading. The ratio of compressive stress to electric field was changed between the experiments. From this series of nonlinear polarization and strain responses were obtained. Based on an offset method, initial domain switching states in the two-dimensional stress-electric field space were determined. In continuum mechanics, thin walled tubes are used to investigate multi-axial stress states. In this context, thin walled means a ratio of wall thickness to radius of 1:10 or thinner. However, simple linear dielectric analysis indicates an inhomogeneous electric field distribution in such geometries.

Therefore, the suitability of hollow cylinders (in the range from thick to thin walled tubes) for multi-axial electromechanical experiments has to be investigated. Simulations with a finite element tool based on a phenomenological constitutive model for ferroelectric and ferroelastic hysteresis behavior were performed. The results confirm inhomogeneous distributions of electric fields and stresses after poling. A geometry variation is discussed to minimize these effects.

The paper presents a free energy model for the pseudoelastic behavior of shape memory alloys. It is based on a stochastic homogenization process, which uses distributions in energy barriers and internal stresses to represent effects typically encountered in polycrystalline materials. This concept leads to a realistic desription of the rate-dependent inner loop behavior, but is characterized by rather long computation times. This is prohibitive in regard to a potential implementation into other numerical codes, such as finite element or optimal control programs or a Matlab/Simulink environment. For this purpose a parameterization method is introduced, which is derived from the concept of a representative single crystal. The approach preserves the desirable properties of the original formulation, at the same time reducing the numerical effort significantly. Finally, we show that the method can reproduce the experimentally observed behavior accurately over a large range of strain rates including the minor loop behavior.

Properties and phase transition behaviors of ferroelectric thin films that are different from that of their bulk form is usually referred to as size effect. A first-principles-based scheme is used to investigate the effects of four important factors contributing to the size effects in epitaxial (001) BaTiO3 ultrathin films: misfit strain, existence of surface, film thickness, and electrical boundary conditions.

In this paper, the magnetoelectric (ME) coupling and working modes are defined and discussed. Several types of ME laminate composites based on Terfenol-D, Fe-Ga magnetostrictive layers and piezoelectric Pb(Mg1/3Nb2/3)O3-PbTiO3 crystal, Pb(Zr,Ti)O3 ceramic layers have been developed. Our experiments confirm that these ME composites have giant magnetoelectric voltage coefficients of ∼0.43×103 mV/Oe in the frequency range of 10-2<f<104 Hz under low magnetic bias. When operated under resonant conditions, the ME voltage coefficient is dramatically enhanced to maximum values as high as ∼20×103 mV/Oe. At resonance, our ME laminates have (i) a strong voltage gain effect; and (ii) an ultrahigh magnetic field sensitivity of near 10-12 Tesla.

A new electromechanical finite element is implemented in the Abaqus F.E. code. It takes into account the ferroelectric and ferroelastic behavior. It is represented by a developed phenomenological constitutive law using two internal variables (remnant ferroelastic strain and remnant polarization) and two loading surfaces (electrical and mechanical ones). Saturation phenomena and depolarization under compression mechanical loading are taken into account.

In order to lead to a more accurate description of the domain switching mechanisms, a micromechanical single-crystal ferroelectric and ferroelastic constitutive law was also developed.

The self consistent scale transition technique will be adopted to derive the polycristalline behavior. The results that were obtained by the micromechanical and phenomenological approaches will be compared in order to identify material parameters that will be introduced in the F.E. modeling.

Absract:The microscopic domain structures in 2D ferroelectrics under applied biaxial strains are investigated using a phase field model based on the time-dependent Ginzburg-Landau equation that takes both long-range electric and -elastic interactions into account. The stable polarization patterns are simulated at different temperatures and applied inequiaxial strains. The results show that the ferroelectrics transfer from multi-domain state to single-domain state when temperature surpasses a critical value. On the other hand, the macroscopic equilibrium polarization states are also studied through a nonlinear thermodynamic theory. The corresponding transition from a1, a2 state (p1 ≠ 0, P2 ≠ 0) to a1 state (p1 ≠0, P2 = 0) or a2 state (p2 ≠ 0, P2 = 0) is also found from the “strain-straintemperature” phase diagram, which is constructed by minimizing Helmholtz free energy.

Ferroelectric films are growing in significance as non-volatile memory devices, sensors and microactuators. The stress state of the film, induced by processing or constraints such as the substrate, strongly affects device behaviour. Thus it is important to be able to model the coupled and constrained behaviour of film material. This work presents a preliminary study of the application of micromechanical modelling to ferroelectric films. A self-consistent micromechanics model developed for bulk ferroelectrics is adapted for thin film behaviour by incorporating features such as grain structure, mechanical clamping by the substrate, residual stresses, and crystallographic orientation of the film.