Constructive and destructive interplay between piezoelectricity and flexoelectricity in flexural sensors and actuators

Flexoelectricity is an electromechanical effect coupling polarization to strain gradients. It fundamentally differs from piezoelectricity because of its size-dependence and symmetry. Flexoelectricity is generally perceived as a small effect noticeable only at the nanoscale. Since ferroelectric ceramics have a particularly high flexoelectric coefficient, however, it may play a significant role as piezoelectric transducers shrink to the submicrometer scale. We examine this issue with a continuum model self-consistently treating piezo- and flexoelectricity. We show that in piezoelectric device configurations that induce strain gradients and at small but technologically relevant scales, the electromechanical coupling may be dominated by flexoelectricity. More importantly, depending on the device design flexoelectricity may enhance or reduce the effective piezoelectric effect. Focusing on bimorph configurations, we show that configurations that are equivalent at large scales exhibit dramatically different behavior for thicknesses below 100¿nm for typical piezoelectric materials. Our results suggest flexoelectric-aware designs for small-scale piezoelectric bimorph transducers.Peer ReviewedPostprint (author's final draft

Rayleigh wave correction for the BEM analysis of two-dimensional elastodynamic problems in a half-space

This is the pre-peer reviewed version of the following article: Arias, I.; Achenbach, J. Rayleigh wave correction for the BEM analysis of two-dimensional elastodynamic problems in a half-space. "International journal for numerical methods in engineering", Agost 2004, vol. 60, núm. 13, p. 2131-2146, which has been published in final form at http://www3.interscience.wiley.com/journal/109060861/abstractA simple, elegant approach is proposed to correct the error introduced by the truncation of the infinite boundary in the BEM modelling of two-dimensional wave propagation problems in elastic half-spaces. The proposed method exploits the knowledge of the far-field asymptotic behaviour of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution involving the same boundary integrals that form the original BEM system. The method is formulated for a two-dimensional homogeneous, isotropic, linearly elastic half-space and its implementation in a frequency domain boundary element scheme is discussed in some detail. The formulation is then validated for a free Rayleigh pulse travelling on a half-space and successfully tested for a benchmark problem with a known approximation to the analytical solution.Peer ReviewedPostprint (author’s final draft

Rippling and a phase-transforming mesoscopic model for multiwalled carbon nanotubes

We propose to model thick multiwalled carbon nanotubes as beams with non-convex curvature energy. Such models develop stressed phase mixtures composed of smoothly bent sections and rippled sections. This model is motivated by experimental observations and large-scale atomistic-based simulations. The model is analyzed, validated against large-scale simulations, and exercised in examples of interest. It is shown that modelling MWCNTs as linear elastic beams can result in poor approximations that overestimate the elastic restoring force considerably, particularly for thick tubes. In contrast, the proposed model produces very accurate predictions both of the restoring force and of the phase pattern. The size effect in the bending response of MWCNTs is also discussed.Peer ReviewedPostprint (author’s final draft

Modeling and simulation of conducting crack propagation in ferroelectric single crystals under purely electrical loading

We present a phase-field model of fracture in ferroelectric single crystals for the simulation of conducting crack propagation under purely electrical loading. This is done by introducing the electrical enthalpy of a diffuse conducting layer into the phase-field formulation. Simulation results show an oblique crack propagation and crack branching from a conducting notch in a ferroelectric sample under applied electric fields. Microstructure evolution indicates the formation of 90 domains which results in a charge accumulation around the crack. The charge accumulation, in turn, induces a high electric field and hence a high electrostatic energy for driving the conducting crack.Peer ReviewedPostprint (published version

Numerical simulations of vickers indentation crack growth in ferroelectric single crystals: effect of microstructure on the fracture process

Ferroelectric materials exhibit strong electro-mechanical coupling which make them ideal materials for use in electro-mechanical devices such as sensors, actuators and transducers. To assure optimum reliability of these devices, understanding of the fracture behavior in these materials is essential. The complex nonlinear interactions of the mechanical and electrical fields in the vicinity of the crack, with localized switching phenomena, govern the fracture behavior of ferroelectric materials. Experimental techniques have been used to study fracture in ferroelectrics, including Vickers indentation to investigate the fracture toughness anisotropy [1–5]. Experiments show that cracking along the poling direction of the material has a shorter length and consequently a higher effective fracture toughness than that normal to the poling direction. In this paper we introduce a model able to capture the anisotropic crack growth under Vickers indentation loading. This anisotropy is obtained by linking the crack propagation with the microstructural phenomena. The model treats in a coupled phase-field energetic fashion both the brittle crack propagation and the microstructure evolution. We have recently presented a model, showing that the interaction of the microstructure and the crack leads to a slow-fast crack propagation behavior observed in experiment [6]. In Ref. [7], we have introduced a modification in the formulation to endow the phase-field model with the ability to simulate the aforementioned anisotropic crack growth. We present here the highlights of that work. The theory of the coupled phase-field model is summarized in Section 2. Simulation results are presented and discussed in Section 3. The last Section is the conclusion of this paper

Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy

Crack propagation in brittle materials with anisotropic surface energy is important in applications involving single crystals, extruded polymers, or geological and organic materials. Furthermore, when this anisotropy is strong, the phenomenology of crack propagation becomes very rich, with forbidden crack propagation directions or complex sawtooth crack patterns. This problem interrogates fundamental issues in fracture mechanics, including the principles behind the selection of crack direction. Here, we propose a variational phase-field model for strongly anisotropic fracture, which resorts to the extended Cahn-Hilliard framework proposed in the context of crystal growth. Previous phase-field models for anisotropic fracture were formulated in a framework only allowing for weak anisotropy. We implement numerically our higher-order phase-field model with smooth local maximum entropy approximants in a direct Galerkin method. The numerical results exhibit all the features of strongly anisotropic fracture and reproduce strikingly well recent experimental observations.Peer ReviewedPostprint (author’s final draft

Phase-field modeling of the coupled microstructure and fracture evolution in ferroelectric single crystals

We propose a phase-field model for the coupled simulation of microstructure formation and evolution, and the nucleation and propagation of cracks in single-crystal ferroelectric materials. The model naturally couples two existing energetic phase-field approaches for brittle fracture and ferroelectric domain formation and evolution. The finite-element implementation of the theory in two dimensions (plane-polarization and plane-strain) is described. We perform, to the best of our knowledge, the first crack propagation calculations of ferroelectric single crystals, simultaneously allowing general microstructures to develop. Previously, the microstructure calculations were performed at fixed crack configurations or under the assumption of small-scale switching. Our simulations show that this assumption breaks down as soon as the crack-tip field interacts with the boundaries of the test sample (or, in general, obstacles such as defects or grain boundaries). Then, the microstructure induced by the presence of the crack propagates beyond its vicinity, leading to the formation of twins. Interactions between the twins and the crack are investigated under mechanical and electromechanical loadings, both for permeable and impermeable cracks, with an emphasis on fracture toughening due to domain switching, and compared with experiments.Peer ReviewedPostprint (published version

Crack initiation patterns at electrode edges in multilayer ferroelectric actuators

Peer ReviewedPostprint (author's final draft

Mathematical and computational modeling of flexoelectricity

We first revisit the mathematical modeling of the flexoelectric effect in the context of continuum mechanics at infinitesimal deformations. We establish and clarify the relation between the different formulations, point out theoretical and numerical issues related to the resulting boundary value problems, and present the natural extension to finite deformations. We then present a simple B-spline based computational technique to numerically solve the associated boundary value problems, which can be extended to handle unfitted meshes, hence allowing for arbitrarily-shaped geometries. Several numerical examples illustrate the flexoelectric effect in simple benchmark setups, as well as in new flexoelectric devices and metamaterials engineered for sensing or actuation.Peer ReviewedPostprint (author's final draft

Weak enforcement of interface continuity and generalized periodicity in high-order electromechanical problems

This is the peer reviewed version of the following article: Barcelo, J. [et al.]. Weak enforcement of interface continuity and generalized periodicity in high-order electromechanical problems. "International journal for numerical methods in engineering", 28 Febrer 2022, vol. 123, núm. 4, p. 901-923, which has been published in final form at DOI10.1002/nme.6882. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.We present a formulation for the weak enforcement of continuity conditions atmaterialinterfacesinhigh-orderproblemsbymeansofNitsche’smethod,whichis particularly suited for unfitted discretizations. This formulation is extendedto impose generalized periodicity conditions at the unit cell boundaries of peri-odic structures. The formulation is derived for flexoelectricity, a high-orderelectromechanical coupling between strain gradient and electric field, mathe-matically modeled as a coupled system of fourth-order PDEs. The design of flex-oelectricdevicesrequiresthesolutionofhigh-orderboundaryvalueproblemsoncomplex material architectures, including general multimaterial arrangements.ThiscanbeefficientlyachievedwithanimmersedboundaryB-splinesapproach.Furthermore, the design of flexoelectric metamaterials also involves the anal-ysis of periodic unit cells with generalized periodicity conditions. Optimalhigh-order convergence rates are obtained with an unfitted B-spline approxi-mation, confirming the reliability of the method. The numerical simulationsillustrate the usefulness of the proposed approach toward the design of func-tional electromechanical multimaterial devices and metamaterials harnessingthe flexoelectric effect.Departament d'Universitats, Recerca i Societat de la Informació, 2017-SGR-1278; ICREA Academia; H2020 European Research Council, StG-679451; Secretaría de Estado de Investigación, Desarrollo e Innovación, CEX2018-000797-S; RTI2018-101662-B-I00.Peer ReviewedPostprint (author's final draft