A study of electromechanical switching in ferroelectric single crystals

This article documents both modeling and experimental studies developed to investigate the switching behavior of ferroelectric single crystals. The theoretical model makes a priori ansatz that switching follows the evolution of a particular domain pattern. The choice of this configuration is dictated by the requirement that domains remain compatible during evolution, giving rise to a low-energy path for the overall switching. The construction of this pattern is achieved using multirank laminates. It offers an advantage of specifying different types of domain wall movements, leading to a distinction for the switching types. A loading experiment is performed on a barium titanate (BaTiO₃) single crystal with a constant compressive stress and a cyclic electric field. Both 180 and 90 coercive fields are measured as input parameters required for the theoretical framework. The simulation results show good agreement with the observed strains measured by the present and other available experiments. It is found that depolarization has a non-trivial influence on attainable actuation strains.     

Fully coupled, multi-axial, symmetric constitutive laws for polycrystalline ferroelectric ceramics

In this paper, a general form for multi-axial constitutive laws for ferroelectric ceramics is constructed. The foundation of the theory is an assumed form for the Helmholtz free energy of the material. Switching surfaces and associated flow rules are postulated in a modified stress and electric field space such that a positive dissipation rate during switching is guaranteed. The resulting tangent moduli relating increments of stress and electric field to increments of strain and electric displacement are symmetric since changes in the linear elastic, dielectric and piezoelectric properties of the material are included in the switching surface. Finally, parameters of the model are determined for two uncoupled cases, namely non-remanent straining ferroelectrics and purely ferroelastic switching, and then for the fully coupled ferroelectric case.     

Analysis of the electromechanical behavior of ferroelectric ceramics based on a nonlinear finite element model

A nonlinear finite element (FE) model based on domain switching was proposed to study the electromechanical behavior of ferroelectric ceramics. The incremental FE formulation was improved to avoid any calculation instability. The problems of mesh sensitivity and convergence, and the efficiency of the proposed nonlinear FE technique have been assessed to illustrate the versatility and potential accuracy of the said technique. The nonlinear electromechanical behavior, such as the hysteresis loops and butterfly curves, of ferroelectric ceramics subjected to both a uniform electric field and a point electric potential has been studied numerically. The results obtained are in good agreement with those of the corresponding theoretical and experimental analyses. Furthermore, the electromechanical coupling fields near (a) the boundary of a circular hole, (b) the boundary of an elliptic hole and (c) the tip of a crack, have been analyzed using the proposed nonlinear finite element method (FEM). The proposed nonlinear electromechanically coupled FEM is useful for the analysis of domain switching, deformation and fracture of ferroelectric ceramics.The project supported by the National Natural Science Foundation of China (10025209, 10132010 and 90208002), the Research Grants of the Council of the Hong Kong Special Administrative Region, China (HKU7086/02E) and the Key Grant Project of the Chinese Ministry of Education (0306)     

A rate-dependent two-dimensional free energy model for ferroelectric single crystals

The one-dimensional free energy model for ferroelectric materials developed by Smith et al. [29–31] is generalized to two dimensions. The two-dimensional free energy potential proposed in this paper consists of four energy wells that correspond to four variants of the material. The wells are separated by four saddle points, representing the barriers for 90°-switching processes, and a local maximum, across which 180°-switching processes take place. The free energy potential is combined with evolution equations for the variant fractions based on the theory of thermally activated processes. The model is compared to recent measurements on BaTiO₃ single crystals by Burcsu et al. [8], and predicitions are made concerning the response to the application of in-plane multi-axial electric fields at various frequencies and loading directions. The kinetics of the 90°- and 180°-switching processes are discussed in detail.     

Micro-potential model for stress-strain hysteresis of micro-cracked materials

While developing models for nonlinear mechanical and acoustical behavior of micro-cracked materials, it is common to start from a purely phenomenological approach. Most approaches essentially assume the material to have certain given “mathematical” properties, that lead to an acceptable equation of state (stress-strain relation) containing nonlinearity and hysteresis. In this paper, we formulate a deeper physical insight on the subject of mechanical hysteresis based on physical and measurable material properties. The theory developed in this paper interprets real images of crack networks in micro-inhomogeneous materials, obtained via electron microscopy, and uses this interpretation to build up a micro-potential model for a medium containing elementary cracks with known properties. It is found that the hysteretic contribution of each crack strongly depends on its average rest opening and its asperity. As a result, a distribution of cracks with different properties yields the physical basis for a slightly more complex version of the commonly used Preisach-Mayergoyz space in rock mechanics. The effect of a uniform distribution of the crack properties on the stress-strain relation is shown as an example.     

Friction in unconforming grain contacts as a mechanism for tensorial stress-strain hysteresis

Materials composed of consolidated grains and/or containing internal contacts are widespread in everyday life (e.g. rocks, geomaterials, concretes, slates, ceramics, composites, etc.). For any simulation of the elastic behavior of this class of solids, be it in seismology, in NDT, or in the modeling of building constructions, the stress-strain constitutive equations are indispensable. Since the most common loading patterns in nature considerably deviate from simple uniaxial compression, the problem of tensorial stress-strain representation arises. In simple loading cases it may be sufficient to use a phenomenological constitutive model. However, in a more general case, phenomenological approaches encounter serious difficulties due to the high number of unknown parameters and the complexity of the model itself. Simplification of the phenomenology can help only partly, since it may require artificial assumptions. For instance, is it enough just to link the volumetric stress to the volumetric strain, or do we have to include shear components as well, and if yes, in what form? We therefore propose a physical tensorial stress-strain model, based on the consideration of plane cracks with friction. To do this, we combine known relations for normal displacements of crack faces given by contact mechanics, the classical Amonton's law of dry friction for lateral displacements, and the equations of elasticity theory for a collection of non-interacting cracks with given orientation. The major advantages of this model consist in the full tensorial representation, the realistic stress-strain curves for uniaxial stress compression and quantitative comparison with experimental data, and a profound account for hysteretic memory effects.     

Microcontact-based theory for acoustics in microdamaged materials

A universal theory describing the wide range of mechanical and acoustic phenomena in solids with internal contacts such as rocks, concrete, ceramics and composites is quite complex to develop. The goal of this paper is to demonstrate the potential to deduce the macroscopic stress-strain constitutive equation for a material as a whole starting from the microscopic hysteretic force-displacement relationship of individual asperities in contact. The material considered in the proposed model contains a large number of isotropic oriented penny-shaped cracks with rough internal surfaces. The stress-strain relationship we obtained for such a material is based on physical principles and laws. Even so, it displays close resemblance to the phenomenological Preisach-Mayergoyz model adopted for mechanical hysteresis and nonlinearity. This constitutive relationship is then used to simulate an experiment with standing acoustic waves in a resonant bar, and to compare model predictions to actual observations. We show that the most important experimentally measurable nonlinear features of these materials, such as the typical classical and nonclassical shifting behavior of the resonant frequency, the dependencies of the amplitudes of the generated harmonics, the softening due to intensive straining, and the subsequent relaxation effect (slow dynamics) can be attributed and explained in terms of the mechanics and the statistics of the internal contacts. The present model bridges the gap between three scales: macroscopic (material as a whole), mesoscopic (structure of intergranular contacts and cracks) and microscopic scale (contacts of individual asperities).     

Energy principle of ferroelectric ceramics and single domain mechanical model

Many physical experiments have shown that the domain switching in a ferroelectric material is a complicated evolution process of the domain wall with the variation of stress and electric field. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of ferroelectric ceramic and used to study the nonlinear constitutive behavior of ferroelectric body in this paper. The principle of stationary total energy is put forward in which the basic unknown quantities are the displacement u _i , electric displacement D _i and volume fraction ρ _I of the domain switching for the variant I. Mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion. On the basis of the domain switching criterion, a set of linear algebraic equations for the volume fraction ρ _I of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. Then a single domain mechanical model is proposed in this paper. The poled ferroelectric specimen is considered as a transversely isotropic single domain. By using the partial experimental results, the hardening relation between the driving force of domain switching and the volume fraction of domain switching can be calibrated. Then the electromechanical response can be calculated on the basis of the calibrated hardening relation. The results involve the electric butterfly shaped curves of axial strain versus axial electric field, the hysteresis loops of electric displacement versus electric filed and the evolution process of the domain switching in the ferroelectric specimens under uniaxial coupled stress and electric field loading. The present theoretic prediction agrees reasonably with the experimental results given by Lynch. The project supported by the National Natural Science Foundation of China (10572138).     

Two-scale micromechanics-based probabilistic modeling of domain switching in ferroelectric ceramics

A two-scale micromechanics model is developed in this paper to analyze domain switching in ferroelectric ceramics, using a probabilistic domain switching criterion based on energetic analysis. The microstructure of ferroelectric ceramics at two distinct length scales, domains and grains, has been carefully analyzed. The interaction at domain level is accounted for by energy minimization theory, while the fluctuation at grain level is analyzed using ellipsoidal two-point correlation function. The model has been implemented by Monte Carlo method, and applied to simulate the electric poling and mechanical depoling of Pb(Zr_xTi_1-x)O₃ (PZT) ceramics across morphotropic phase boundary (MPB). The drastically different switching characteristics of PZT ceramics across MPB has been captured, and good agreement with experiments has been observed. The effects of the transformation strains and spontaneous polarizations are highlighted, confirming the proposition of Li et al. [2005. Domain switching in polycrystalline ferroelectric ceramics. Nature Materials 4, 776–781] that the strain compatibility plays a dominant role in domain switching in ferroelectric ceramics.     

Characterization of hysteretic stress-strain behavior using the integrated Preisach density

In this paper, we introduce the concept of Integrated Preisach-Mayergoyz (IPM) density to analyze static uniaxial compression tests at values well below the critical strength, and to characterize the elasticity of materials with hysteresis in their stress-strain relationship. The IPM density can be deduced from a particular force protocol following basic data treatment. The advantage of the IPM density over prior approaches is that no second order differentiation of the data is required which reduces the errors and uncertainties typical for past practice in the specific context of rock elasticity using scanning curves and PM density analysis. The characterization of the elasticity of the material is established in terms of a non-hysteretic strain contribution in the form of a non-linear but reversible equation of state, and a hysteretic contribution represented by the IPM density. The IPM inversion procedure is tested for simulated stress-strain data subjected to additive noise, and the results are compared to the traditional methodology. In addition, we analyze the hysteretic and non-hysteretic characteristics of five natural building stones, and show evidence for a classification based on the inferred properties.     

Effect of electric displacement saturation on the stress intensity factor for a crack in a ferroelectric ceramic

A crack in a ferroelectric ceramic with perfect saturation under electric loading is analyzed. The boundary of the electric displacement saturation zone ahead of the crack tip is assumed to be ellipse in shape. The shape and size of ferroelectric domain switching zone near a crack tip is determined based on the nonlinear electric theory. The stress intensity factor induced by ferroelectric domain switching under small-scale conditions is numerically obtained as a function of the electric saturation zone parameter and the ratio of the coercive electric field to the yield electric field. It is found that the stress intensity factor increases as the ratio of the semi-axes of the saturation ellipse increases.     

Effect of electric boundary conditions on crack propagation in ferroelectric ceramics

In this paper, the effect of electric boundary conditions on Mode I crack propagation in ferroelectric ceramics is studied by using both linear and nonlinear piezoelectric fracture mechanics. In linear analysis, impermeable cracks under open circuit and short circuit are analyzed using the Stroh formalism and a rescaling method. It is shown that the energy release rate in short circuit is larger than that in open circuit. In nonlinear analysis, permeable crack conditions are used and the nonlinear effect of domain switching near a crack tip is considered using an energy-based switching criterion proposed by Hwang et al.（Acta Metal. Mater.,1995）. In open circuit, a large depolarization field induced by domain switching makes switching much more diffcult than that in short circuit. Analysis shows that the energy release rate in short circuit is still larger than that in open circuit, and is also larger than the linear result. Consequently,whether using linear or nonlinear fracture analysis, a crack is found easier to propagate in short circuit than in open circuit, which is consistent with the experimental observations of Kounga Njiwa et al.（Eng. Fract. Mech., 2006）.     

Thermodynamic driving force in ferroelectric crystals with a rank-2 laminated domain pattern, and a study of enhanced electrostriction

The thermodynamic driving force for domain growth in a rank-2 laminated ferroelectric crystal is derived in this article, and we used it, together with a homogenization theory, to study the issue of enhanced electrostrictive actuation recently reported by Burcsu et al. [2004. Large electrostrictive actuation of barium titanate single crystals. J. Mech. Phys. Solids 52, 823-846]. We derived this force from the reduction of Gibbs free energy with respect to the increase of domain concentration. It is shown that both the free energy and the thermodynamic force consist of three parts: the first arises from the difference in M₀ and M₁, the linear electromechanical compliances of the parent and product domains, respectively, at a given level of applied stress and electric field, the second stems from the electromechanical work associated with the change of spontaneous strain and spontaneous polarization during domain switch, and the third from the internal energy due to the distribution of polarizations strain and electric polarization inside the crystal. We prove that the first term is substantially lower than the second one, and the third one is identically zero with compatible domain pattern. The second one is, however, not exactly equal to the commonly written sum of the products of stress with strain, and electric field with polarization during switch, unless both domains have identical moduli in the common global axes. We also show that, with compatible domain patterns and when M₁=M₀, this driving force is identical to Eshelby's driving force acting on a flat interface due to the jump of energy-momentum tensor. Applications of the theory to a BaTiO₃ crystal subjected to a fixed axial compression and decreasing electric field from the [0 0 1] state reveal that the crystal undergoes a three-stage switching process: (i) the 0→90° switch to form a rank-1 laminate, (ii) the 0→180° switch inside the 0° domain to form a laminate I with a concurrent 90°→−90° switch inside the 90° domain to form laminate II, creating a rank-2-laminated domain pattern, and (iii) finally the 90→180° switch. It is the exchange of stability between the 0, 90°, and 180° domains under compression and electric field that is the origin of the enhanced actuation. We illustrate these intrinsic features by showing the evolution of these domains, and demonstrate how the reported large actuation strain can be attained with a rank-2 laminate.     

The influence of mechanical deformation on the electrical properties of single wall carbon nanotubes

Recent experimental studies and atomistic simulations have shown that carbon nanotubes (CNTs) display strong interplay between the mechanical deformation and electrical properties. We have developed a simple and accurate method to determine atom positions in a uniformly deformed CNT via a continuum analysis based on the interatomic potential. A shift vector is introduced to ensure the equilibrium of atoms. Such an approach, involving only three variables for the entire CNT, agrees very well with the molecular mechanics calculations. We then study the effect of mechanical deformation on the band gap change of single wall CNTs under tension, torsion, and combined tension/torsion via the k-space tight-binding method. Prior studies without this shift vector lead to significant overestimation of the band gap change. It is established that the conducting CNTs may easily become semi-conducting ones subject to mechanical deformation, but the semi-conducting CNTs never become conducting ones upon deformation.     

On plastic anisotropy of constitutive model for rate-dependent single crystal

An algorithm for single crystals was developed and implemented to simulate plastic anisotropy using a rate-dependent slip model. The proposed procedure was a slightly modified form of single crystal constitutive model of Sarma and Zacharia. Modified Euler method, together with Newton-Raphson method was used to integrate this equation which was stable and efficient. The model together with the developed algorithm was used to study three problems. First, plastic anisotropy was examined by simulating the crystal deformation in tension and plane strain compression, respectively. Secondly, the orientation effect of some material parameters in the model and applied strain rate on plastic anisotropy for single crystal also is investigated. Thirdly, the influence of loading direction on the active slip system was discussed.     

Plastic spin associated with a non-normality theory of plasticity

A plasticity model using a vertex-type plastic flow rule on a smooth yield surface for an anisotropic solid has been proposed recently. This model is here completed by incorporating the effect of plastic spin. Simple shear with a large shear strain is one of the hardest tests on finite strain anisotropic plasticity models, and it is here shown which plastic spin expression is needed to avoid unrealistic oscillatory behavior of the shear stress under large shear strains. The idea of using non-normality with a smooth yield surface originates from a recent proposal of using an abrupt strain path change to determine the subsequent yield surface shape. For this method both polycrystal plasticity calculations and experiments have shown a vertex-type response on the apparently smooth yield surface.     

Domain switching effect on fracture of piezoelectric solids

Rational design of smart sensors and actuators that consist of piezoelectric solids requires a thorough understanding of the constitutive behavior of this material under mechanical and electrical loading. Domain switching is the cause of significant nonlinearity in the constitutive behavior of piezoelectric solids, which may be enhanced in the presence of cracks. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The corresponding finite element equations are derived and applied in the solution of the piezoelectric center crack problems. The effects of domain switching are evaluated on the near tip stress intensity factors.     

Analyses on micro-fracture of crack tip under large deformation conditions

In the present paper, Gurson's constitutive equation, which takes into account the development of voids, is used to study the behaviour of the material in the region near crack tip. Furthermore, the effect of void development on Young's modulus, which was not considered by Gurson, is taken into consideration. The analyses on void development, on stress distribution near crack tip, and on the variance of COD for the plane strain mode I problem are carried out with the large elastic-plastic deformation finite element method. The results are compared with those estimated from the Prandtl-Reuss constitutive equation.     

Adiabatic shear band localization of inelastic single crystals in symmetric double-slip process

Summary The main objective of the present paper is the development of a viscoplastic regularization procedure valid for an adiabatic dynamic process for multi-slips of single crystals. The next objective is to focus attention on the investigation of instability criteria, and particularly on shear band localization conditions.To achieve this aim, an analysis of acceleration waves is given, and advantage is taken of the notion of the instantaneous adiabatic acoustic tensor. If zero is an eigenvalue of the acoustic tensor, then the associated discontinuity does not propagate, and one speaks of a stationary discontinuity. This situation is referred to as the strain localization condition, and corresponds to a loss of hyperbolicity of the dynamical equations. It has been proved that for an, adiabatic process of rate-dependent (elastic-viscoplastic) crystal, the wave speed of discontinuity surface always remains real and different from zero. It means that for this case the initial-value problem is well-posed. However, for an adiabatic process of rate-independent(elastic-plastic) crystal, the wave speed of discontinuity surface can be equal zero. Then the necessary condition for a localized plastic deformation along the shear band to be formed is as follows: the determinant of the instantaneous adiabatic acoustic tensor is equal to zero. This condition for localization is equivalent to that obtained by using the standard bifurcation method. Based on this idea, the conditions for adiabatic shear band localization of plastic deformation have been investigated for single crystals. Particular attention has been focused on the discussion of the influence of thermal expansion, thermal plastic, softening and spatial covariance effects on shear band localization criteria for a planar model of an f.c.c. crystal undergoing symmetric primary-conjugate double slip. The results obtained have been compared with available experimental observations.Finally, it is noteworthy that the viscoplasticity regularization procedure can be used in the developing of an unconditionally stable numerical integration algorithm for simulation of adiabatic inelastic flow processes in ductile single crystals, cf. [21].The paper has been prepared within research programme sponsored by the Committee of Scientific Research under Grant 3 P404 031 07.     

Evaluation of a simple nonlinear elastic constitutive form

A nonlinear, two constant stress-deformation form is deduced for elastic materials. At very large stretch ratios of greater than about 3 or 4, the model exhibits the strain stiffening behavior common to many elastomers. The constitutive form is very simple since the two material constants enter it as multiplying constants times certain nonlinear deformation terms. The model is evaluated with respect to data upon natural rubber under both uniaxial and bi-axial stress conditions. The model is also used to evaluate data obtained from a nonlinear membrane inflation experiment. The latter experimental capability and corresponding data are new.Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.