Chad M. Landis

A constitutive model for ferroelectric polycrystals

Abstract A constitutive model is developed for the non-linear switching of ferroelectric polycrystals under a combination of mechanical stress and electric field. It is envisaged that the polycrystal consists of a set of bonded crystals and that each crystal comprises a set of distinct crystal variants. Within each crystal the switching event, which converts one crystal variant into another, gives rise to a progressive change in remanent strain and polarisation and to a change in the average linear electromechanical properties. It is further assumed that switching is resisted by the dissipative motion of domain walls. The constitutive model for the progressive switching of each crystal draws upon elastic–plastic crystal plasticity theory, and a prescription is given for the tangent moduli of the crystal, for any assumed set of potentially active transformation systems. A self-consistent analysis is used to estimate the macroscopic response of tetragonal crystals (representative of lead titanate) under a variety of loading paths. Also, the evolution of the switching surface in stress-electric field space is calculated. Many of the qualitative features of ferroelectric switching, such as butterfly hysteresis loops, are predicted by the analysis.

Electrostatic forces and stored energy for deformable dielectric materials

An isothermal energy balance is formulated for a system consisting of deformable dielectric bodies, electrodes, and the surrounding space. The formulation in this paper is obtained in the electrostatic limit but with the possibility of arbitrarily large deformations of polarizable material. The energy balance recognizes that charges may be driven onto or off of the electrodes, a process accompanied by external electrical work; mechanical loads may be applied to the bodies, thereby doing work through displacements; energy is stored in the material by such features as elasticity of the lattice, piezoelectricity, and dielectric and electrostatic interactions; and nonlinear reversible material behavior such as electrostriction may occur. Thus the external work is balanced by (I) internal energy consisting of stress doing work on strain increments, (2) the energy associated with permeating free space with an electric field, and (3) by the electric field doing work on increments of electric displacement or, equivalently, polarization. For a conservative system, the internal work is stored reversibly in the body and in the underlying and surrounding space. The resulting work statement for a conservative system is considered in the special cases of isotropic deformable dielectrics and piezoelectric materials. We identify the electrostatic stress, which provides measurable information quantifying the electrostatic effects within the system, and find that it is intimately tied to the constitutive formulation for the material and the associated stored energy and cannot be independent of them. The Maxwell stress, which is related to the force exerted by the electric field on charges in the system, cannot be automatically identified with the electrostatic stress and is difficult to measure. Two well-known and one novel formula for the electrostatic stress are identified and related to specific but differing constitutive assumptions for isotropic materials. The electrostatic stress is then obtained for a specific set of assumptions in regard to a piezoelectric material. An exploration of the behavior of an actuator composed of a deformable, electroactive polymer is presented based on the formulation of the paper.

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.

Micromechanical simulation of the failure of fiber reinforced composites

Abstract The strength of unidirectionally reinforced fiber composites is simulated using the three dimensional shear lag model of Landis, C. M., McGlockton, M. A. and McMeeking, R. M. (1999) (An improved shear lag model for broken fibers in composites. J. Comp. Mat. 33, 667–680) and Weibull fiber statistics. The governing differential equations for the fiber displacements and stresses are solved exactly for any configuration of breaks using an influence superposition technique. The model predicts the tensile strength of well bonded, elastic fiber/matrix systems with fibers arranged in a square array. Length and strength scalings are used which are relevant for elastic, local load sharing composites. Several hundred Monte Carlo simulations were executed to determine the statistical strength distributions of the composite for three values of the fiber Weibull modulus, m =5, 10 and 20. Stress–strain behavior and the evolution of fiber damage are studied. Bundle sizes of 10×10, 15×15, 20×20, 25×25, 30×30 and 35×35 fibers of various lengths are investigated to determine the dependence of strength on the composite size. The validity of weakest link statistics for composite strength is examined as well.

A phenomenological multi-axial constitutive law for switching in polycrystalline ferroelectric ceramics

A phenomenological constitutive law for ferroelectric switching due to multi-axial mechanical and electrical loading of a polycrystalline material is developed. The framework of the law is based on kinematic hardening plasticity theory and has a switching surface in the space of mechanical stress and electric field that determines when non-linear response is possible. The size and shape of the switching surface in a modified electric field space remains fixed during non-linear behavior but its center moves around and thus is controlled by a kinematical hardening process. In general, the remanent polarization and the remanent strain are used as the internal variables that control how the center of the switching surface moves. However, the form presented in this paper has a one-to-one relationship between the remanent strain and the remanent polarization, simplifying the constitutive law and allowing remanent polarization to be used as the only internal variable controlling the kinematic effects. The constitutive law successfully reproduces hysteresis and butterfly loops for ferroelectric ceramics. The hysteresis and butterfly loops respond appropriately to the application of a fixed compressive stress parallel to the electric field. In addition, the law successfully handles remanent polarization rotation due to the application of electric field at an angle to the polarization direction.

Piezoelectric thin-film superlattices without using piezoelectric materials

In this paper we show that experimentally realizable apparently piezoelectric thin-film superlattices can be created from nonpiezoelectric materials provided an odd-order (e.g., trilayer) stacking sequence is used. The size-dependent mechanism of flexoelectricity, which couples gradients of strain to polarization, allows such a possibility. We present closed-form analytical expressions for the response of various thin-film and superlattice configurations. We also clarify some of the subtleties that arise in considering interface boundary conditions in the theory of flexoelectricity as well as the relationship of flexoelectricity to the frequently used polarization gradient terms used in modeling ferroelectrics. We find that for certain (optimum) material combinations and length scales, thin-film superlattices yielding apparent piezoelectricity close to 75% of ferroelectric barium titanate may be achievable.

Curvature-induced polarization in carbon nanoshells

We investigate the normal polarization induced by bending of graphite shells, which microscopically occurs because of a shift in sp 2 hybridization at each atomic site. Based on a rehybridization model we analytically estimate the dependence on curvature of electronic charge spill into convex region. We further performed DFT calculations and extract the radial atomic dipole by direct electronic charge density integration. A continuum analysis assigns a linear susceptibility tensor to the curvature-induced polarization and shows that this tensor is isotropic for hexagonal lattice, in agreement with microscopic model and computations. The intrinsic polarization effect can be important for nanoscale electronics. 2002 Elsevier Science B.V. All rights reserved.

Crack velocity dependent toughness in rate dependent materials

Mode I, quasi-static, steady-state crack growth is analyzed for rate dependent materials under plane strain conditions in small scale yielding. The solid is characterized by an elastic-viscoplastic constitutive law and the plane ahead of the crack tip is embedded with a rate dependent fracture process zone. The macroscopic work of fracture of the material is computed as a function of the crack velocity and the parameters characterizing the fracture process zone and the solid. With increasing crack velocity a competition exists between the strain rate hardening of the solid, which causes elevated tractions ahead of the crack tip that tend to drive crack propagation, and the rate strengthening of the fracture process zone which tends to resist fracture. Results for material parameters characteristic of polymers show that the toughness of the material can either increase or decrease with increasing crack velocity. To motivate the model, the cohesive zone parameters are discussed in terms of failure mechanisms such as crazing and void growth ahead of the crack tip. The toughness of rubber modified epoxies is explained by employing the fracture model along with micromechanical void cell calculations

A shear-lag model for a broken fiber embedded in a composite with a ductile matrix

A shear-lag model has been developed for the prediction of stress recovery in a broken fiber embedded in a ductile-matrix composite. The model builds on the original shear-lag model of (Cox HL. Br J Appl Phys 1952;3:72‐9) by introducing plasticity constitutive behavior into the matrix. The matrix is assumed to be an elastic/perfectly-plastic material that deforms according to J2 flow theory. The use of a flow rule to govern the matrix deformation in this model diAers from previous attempts to represent plasticity in the matrix. A non-linear partial diAerential equation is obtained from the model. Numerical solutions to the equation are obtained and compared to simpler shear-lag models which assume sliding at the fiber/matrix interface controlled by a uniform shear stress. Axisymmetric finite-element calculations were done to assess the validity of the shear-lag model. It proves to be in good agreement with the finite-element analysis. Predictions of the shear-lag calculations suggest that the global load-sharing (GLS) strength model of (Curtin WA. J Am Ceram Soc 1991;74:2837‐45) is valid for a composite with a yielding matrix that is elastically rigid. # 1999 Elsevier Science Ltd. All rights reserved.

On the fracture toughness of ferroelastic materials

The toughness enhancement due to domain switching near a steadily growing crack in a ferroelastic material is analyzed. The constitutive response of the material is taken to be characteristic of a polycrystalline sample assembled from randomly oriented tetragonal single crystal grains. The constitutive law accounts for the strain saturation, asymmetry in tension versus compression, Bauschinger effects, reverse switching, and strain reorientation that can occur in these materials due to the non-proportional loading that arises near a propagating crack. Crack growth is assumed to proceed at a critical level of the crack tip energy release rate. Detailed finite element calculations are carried out to determine the stress and strain fields near the growing tip, and the ratio of the far field applied energy release rate to the crack tip energy release rate. The results of the finite element calculations are then compared to analytical models that assume the linear isotropic K-field solution holds for either the near tip stress or strain field. Ultimately, the model is able to account for the experimentally observed toughness enhancement in ferroelastic ceramics.