Piezoelectric Solids Research Articles

A novel 3D mixed finite element for flexoelectricity in piezoelectric materials

AbstractFlexoelectricity is the intrinsic length‐scale dependent higher‐order electromechanical response of all centro‐ and non‐centro‐symmetric dielectrics, including piezoelectrics. Direct flexoelectricity is defined as the appearance of an electric field due to induced strain gradients. The numerical modeling of flexoelectricity is largely carried out using mixed FE, which has its historical foundations in strain gradient theories. However, existing finite elements are either limited to 2D or have inherited numerical instabilities due to the known saddle‐point structuring. The current work presents a numerically robust three‐dimensional mixed FE for higher‐order electromechanical applications without the use of stabilization or penalty parameters. After its verification, the new finite element is applied to the new problem of truncated semicone torsion, taking into account flexoelectricity in piezoelectric solids, and the original findings are reported. Current research reveals the complex interaction between first‐order (piezoelectricity) and higher‐order (flexoelectricity) electromechanical coupling.

Nonlocal modelling of piezoelectric solid with cracks based on peridynamic differential operator

The paper establishes a nonlocal model for piezoelectric solids using the Peridynamic differential operator. The method transforms the displacement-electric potential equation, constitutive equation, and boundary conditions of piezoelectric solids from a local differential form to a nonlocal integral form. The solution for the displacement and electric potential distributions in piezoelectric solids is obtained using variational principles and the Lagrange multiplier method. The correctness and effectiveness of the model are validated through a comparison of the deformation analysis of piezoelectric solid square plates with analytical solutions. Analysis of piezoelectric solid square plates with edge cracks proves the applicability of this model for solving discontinuous problems. Furthermore, the paper explores the suitability of the model in analysing problems involving multiple cracks in piezoelectric solid plates, highlighting its capability to address defective piezoelectric solid problems. This model provides a new perspective for solving problems involving defects under electro-mechanical coupling conditions.

A peridynamic model for electromechanical fracture and crack propagation in piezoelectric solids

This paper presents for the first time an approach to the problem of electromechanical fracture and crack propagation in piezoelectric solids with peridynamics. Fracture in piezoelectric ceramics is an important problem in engineering due to its complexity. The majority of the computational models available on the literature rely on sharp crack representations, which have difficulties dealing with complex crack patterns, or rely on diffusive crack models, such as the phase-field fracture model, that can predict complex patterns but does not represent cracks in a discrete way. On the other hand, peridynamics has been designed to solve this problems and is able to simulate discrete cracks as well as complex crack phenomena like fragmentation and branching. We present a novel approach to the implementation of different electric crack face boundary conditions in a peridynamic way, by a field selective bond breaking approach. Moreover, it is also presented the first electromechanical failure criterion for fracture in piezoelectric ceramics. We develop an energy based failure criterion due to its generality and wide range of applications. These approaches are presented in a way such that they are valid for any type of peridynamic formulation. The used numerical model is implemented using correspondence-based peridynamics with the linear piezoelectricity constitutive setting, using a meshfree discretization and solved quasi-statically with an implicit incremental-iterative scheme. Numerical examples are presented to demonstrate the capabilities of these new approaches and assess the performance of the model against experimental results. The model is capable of numerically predicting complicated fracture behaviour and crack propagation in an intrinsic/autonomous way, as demonstrated by the results.

Fractal contact analysis for transversely isotropic piezoelectric materials: Theoretical and numerical predictions

This paper aims to investigate the micro frictionless and non-adhesive rough contact for transversely isotropic piezoelectric materials. The surface topography is formulated by the two-variable Weierstrass–Mandelbrot fractal function. The system of contact is subjected to light generalized loads. By virtue of the Hertzian solution of transversely isotropic piezoelectric materials and the size distribution function, the generalized loads-area relation is developed. The smallest truncated contact area in the theoretical model is determined by comparing the theoretical and simulation results. The present numerical solutions via the finite element method are validated in comparison to the numerical results in literature. The influence of various fractal parameters on the contact behavior of transversely isotropic piezoelectric solids are studied.

The Effect of Adhesion on Indentation Behavior of Various Smart Materials

The nanoindentation technique plays a significant role in characterizing the mechanical properties of materials at nanoscale, where the adhesion effect becomes very prominent due to the high surface-to-volume ratio. For this paper, the classical adhesion theories were generalized to study the contact behaviors of various piezoelectric materials indented by conical punches with different electric properties. With the use of the Hankel integral transform, dual integral equations, and superposing principle, the closed-form solutions of the physical fields for the Johnson-Kendall-Roberts (JKR) and Maugis-Dugdale (M-D) models were obtained, respectively. The contribution of the electrical energy to the energy release rate under the conducting punch was taken into consideration. The relationships between the contact radius, the indentation load, and the indentation depth were set up using the total energy method for the JKR model and the Griffith energy balance for the M-D model, respectively. Numerical results indicate that increasing the half cone angle of the conical punch enhances the adhesion effect, which can significantly affect the accuracy of the results of characterization in nanoindentation tests. It was found that the effect of electric potential on adhesion behaviors is sensitive to different material properties, which are not revealed in the existing studies of axisymmetric adhesive contact of piezoelectric materials and multiferroic composite materials. The load-displacement curves under conical punches with different half cone angles have very different slopes. These results indicate that the half cone angle has a prominent effect on the characterization of mechanical properties of piezoelectric solids in nanoindentation tests.

Multibody contact of two-dimensional anisotropic elastic/piezoelectric/magneto-electro-elastic solids

A unified expression for the generalized two-dimensional linear anisotropic elastic, piezoelectric, and magneto-electro-elastic (MEE) solids is established to simplify the connection between these three different kinds of materials. Here, the two-dimensional assumption includes the most general cases that in-plane and anti-plane deformations are allowed to couple each other. Based upon this unified expression, in this paper a recently developed boundary element method (BEM) for contact of two anisotropic elastic bodies is extended to solve the problems of multibody contact in which the bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids. When a special case of contact between multiple rigid punches and an anisotropic elastic, piezoelectric or MEE half-plane is considered, further reduction can be made by introducing three degrees of freedom (two rigid body translations and one rigid body rotation) for each punch. With this consideration, no meshes are required on the rigid punches. The usual contact modes of separation, stick, and slip all include the situation of conductor and insulator if any one of the multibody is piezoelectric or MEE. In other words, in this paper we successfully develop two BEMs to solve the generalized two-dimensional problems of multibody contact. One is for the general cases whose bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids, and the other is for the special cases of multiple rigid punches indenting on an anisotropic elastic, piezoelectric or MEE half-plane.

A phase field model for fatigue fracture in piezoelectric solids: A residual controlled staggered scheme

Prediction of the fatigue life of piezoelectric devices and structures is of scientific and engineering significance. In this work, a phase field model for fatigue fracture in piezoelectric materials is developed by means of the Hamilton principle. Two classical phase field fracture models are unified in the present framework. A cumulative history variable is introduced to quantify the degradation of the fracture toughness under cyclic loading. The proposed model accounts for three representative electric boundary conditions on crack faces describing the fatigue fracture behaviors in different physical situations. The residual controlled staggered scheme is extended to address the fatigue fracture problems in the context of electro-elasticity. The present algorithm overcomes the convergence problem for the monolithic approach and shows higher accuracy with lower computational effort in comparison to the single-pass staggered scheme. Systematic numerical simulations are carried out in 2D and 3D cases. The effects of the electric boundary condition, external electric field, and the geometry of the crack tip on the fatigue fracture behaviors of piezoelectric solids have been discussed. The effect of the electric boundary conditions on the fatigue life varies depending on the direction of the external electric field. The present work is beneficial to assess the lifetimes of piezoelectric devices in practical applications.

Acoustic wave tunneling across a vacuum gap between two piezoelectric crystals with arbitrary symmetry and orientation

It is not widely appreciated that an acoustic wave can ``jump'' or ``tunnel'' across a vacuum gap between two piezoelectric solids, nor has the general case been formulated or studied in detail. Here, we remedy that situation, by presenting a general formalism and approach to study such an acoustic tunneling effect between two arbitrarily oriented anisotropic piezoelectric semi-infinite crystals. The approach allows one to solve for the reflection and transmission coefficients of all the partial-wave modes, and is amenable to practical numerical or even analytical implementation, as we demonstrate by a few chosen examples. The formalism can be used in the future for quantitative studies of the tunneling effect in connection not only with the manipulation of acoustic waves, but with many other areas of physics of vibrations such as heat transport, for example.

Thermally nonlinear non-Fourier piezoelectric thermoelasticity problems with temperature-dependent elastic constants and thermal conductivity and nonlinear finite element analysis

In non-isothermal conditions, numerous experimental and theoretical investigations reveal that the elastic constants and thermal conductivity in piezoelectric solids depend on temperature distribution. This work aims to investigate thermally nonlinear non-Fourier piezoelectric thermoelasticity problems with temperature-dependent elastic constants and thermal conductivity. The nonlinear time-domain finite element method is developed to directly solve nonlinear finite element governing equations, which maximally avoids the precision losses within the applications of the integrated transformation method. As a numerical example, the developed method is applied to analyze nonlinear transient thermo-electromechanical responses of a two-dimensional orthotropic piezoelectric plate of crystal class mm2. The achieved results reveal that temperature-dependent thermal conductivity and elastic constants remarkably affect the structural dynamic responses, whilst thermal wave or elastic wave will travel faster and the electrical energy harvesting ability is significantly elevated.

Data driven approach in multiphysics framework: Application to coupled electro-mechanical problems

Solving coupled multiphysics problems using finite elements and conventional modeling approach requires building, from coupled finite element formulation often in the same code, the monophysical operators and the multiphysics coupling operators. In this work the data driven, model-free computational approach is fitted in the multiphysics framework adapted for ‘smart’ materials. Thus, in this work we ought to expand the phase space and propose a new norm for the distance based data driven solver adapted to the problem at hands. Given the good material database which naturally encodes the coupling interactions, we show that proposed application of data driven approach enables to avoid the coupling tangent terms altogether. In other words, the data driven approach permits to decouple the different ‘physics’ and to manage the coupling from the data. We illustrate the performance and robustness of the approach on two examples related to lattice model composed of planar, piezoelectric truss network and of finite element discretized linear piezoelectric solids. The numerical tests show good convergence properties related to the number of data points, as well as the choice of material and weighting parameters.

Phase-field modeling of electromechanical fracture in piezoelectric solids: Analytical results and numerical simulations

The optimal design, reliable manufacture and durable utilization of electromechanical systems demand objective modeling of failure under coupled electric field and mechanical actions. This is still a challenging and largely an open issue despite the large volumes of contributions from the discontinuous approach and the phase-field community for brittle fracture. In this work we propose a length scale insensitive phase-field cohesive zone model (PF-CZM) for electromechanical fracture in linear piezoelectric solids, overcoming all the issues of mesh dependence, crack tracking complexity, length scale sensitivity and crack nucleation inconsistency. More specifically, in the electromechanically coupled phase-field constitutive relations, the evolution law for the crack phase-field is established from the geometric regularization of sharp cracks and the energetic criterion in global form. In order to capture the unsymmetric mechanical and electrical fracture behavior, the mechanical energy release rate contributed from the positive cone of the effective stress in energy norm is employed as the crack driving force, resulting in a hybrid formulation. With a set of phase-field characteristic functions optimal for cohesive fracture, the critical stress upon crack nucleation and the post-peak softening regime are derived in close-form for the first time, and the experimentally observed effects of the electric field on the fracture load are predicted qualitatively. The proposed electromechanical PF-CZM is numerically implemented in the multi-field finite element method and applied to several representative examples. In all cases, as those for purely mechanical problems, the predicted crack patterns and fracture loads are insensitive to the phase-field length scale and the experimental results are well reproduced. In particular, the effect of electric field on the deflection of crack paths can be captured. These merits make the proposed PF-CZM promising in the modeling of electromechanical fracture in piezoelectric solids.

Flexoelectric effect in dielectrics under a dynamic load

Health monitoring of structures can be improved if a more universal two-way electromechanical coupling, the flexoelectric effect, is utilized. It is a coupling between the electric polarization and the strain gradients. In the direct flexoelectricity, the electric polarization is induced by strain gradients, which yield also a finite higher-order stress tensor in the considered phenomenological theory. Because of the size-effect in higher-grade theories of continua, the polarization in piezoelectric solids under a non-uniform strains in nano-sized structures is significantly influenced by flexoelectricity. This effect is substantially enhanced near the crack defects, in regions with large strain gradients. The mixed finite element method (FEM) is developed from the variational formulation of flexoelectric boundary value problems. The C0 continuous approximation is applied independently for both the displacements and displacement gradients. The kinematic constraints between them are satisfied by collocation at some internal points of elements. The developed computational method is applied to general 2D boundary value problems with cracks under a dynamic load. The influence of flexoelectricity on the induced electric potential and the crack opening displacement is investigated.

Ultrahigh Out-of-Plane Piezoelectricity Meets Giant Rashba Effect in 2D Janus Monolayers and Bilayers of Group IV Transition-Metal Trichalcogenides

The simultaneous occurrence of gigantic piezoelectricity and Rashba effect in two-dimensional (2D) materials is unusually scarce. Inversion symmetry occurring in MX3 (M = Ti, Zr, Hf; X = S, Se) monolayers is broken upon constructing their Janus monolayer structures MX2Y (X ≠ Y =S, Se), thereby inducing a large out-of-plane piezoelectric constant d33 (∼68 pm/V) in them. d33 can be further enhanced to a super high value of ∼1000 pm/V upon applying vertical compressive strain in the van der Waals bilayers constituted by interfacing these Janus monolayers. Therefore, d33 in these Janus transition-metal trichalcogenide (TMTC) bilayers reach more than 4-fold times that of bulk ceramic PZT material (∼268 pm/V). The absence of a horizontal mirror symmetry and the presence of strong spin–orbit coupling cause Rashba spin-splitting in electronic bands in these Janus 2D monolayers, which shows up as an ultrahigh Rashba parameter, αR ∼ 1.1 eV Å. It can be raised to 1.41 eV Å via compressive strain. Most of the 2D materials reported to date mainly show in-plane electric polarization, which severely limits their prospects in piezotronic devices. In this present work, the piezoelectricity shown by the Janus monolayers of group IV TMTCs and their bilayers is significantly higher than the ones generally utilized in the form of three-dimensional bulk piezoelectric solids, for example, α-quartz (d11 = 2.3 pm/V), wurtzite-GaN (d33 = 3.1 pm/V), and wurtzite-AlN (d33 = 5.6 pm/V). It is exceedingly higher than that in Janus multilayer/bulk structures of Mo- and W-based transition-metal dichalcogenides, for example, MoSTe (d33 ∼ 10 pm/V). The 2D Janus TMTC monolayers and their bilayers reported herewith straddle giant Rashba spin-splitting and ultrahigh piezoelectricity, thereby making them immensely promising candidates in the next-generation electronics, piezotronics, spintronics, flexible electronics, and piezoelectric devices.

Surface effects of electroelastic tip fields of multiple cracks emanating from a circular hole

The problem of piezoelectric solids containing a nanoscale multi-cracks cracked hole under antiplane mechanical load and inplane electric load is investigated theoretically. The surface electroelastic constants of the defects (crack and hole) are introduced based on the Gurtin-Murdoch surface model. An exact solution of the stress fields and electric displacement fields around the defects are obtained by constructing a conformal mapping function. Closed-form solution of the stress intensity factor and the electric displacement intensity factor at crack tips are presented. Numerical examples are provided to reveal the relationship between the electroelastic fields intensity factors and the defect size, the crack-hole interaction, and the applied mechanical-electrical loads.

Thermo-electro-mechanical fatigue crack growth simulation in piezoelectric solids using XFEM approach

Cracks and multiple voids resulting from manufacturing processes interact with each other and severely affect the fracture phenomenon of piezoelectric components. These geometrical discontinuities degrade the service performance and lower the reliability of the piezoelectric devices. Therefore, the present work predicts fatigue-cracking behavior of piezo-electric structures in existence of multiple geometrical discontinuities under cyclic thermal-electrical-mechanical loads using extended finite element method (XFEM). Thermal interaction integrals approach and Stroh formalism are applied for estimating stress intensity factors (SIFs). Modified hoop SIF criterion has been utilized to predict crack growth direction under quasi-static fatigue loading conditions.

Portable acoustofluidic nebulizer

The field of acoustofluidics studies the effects of acoustic waves transferred from piezoelectric solids to small volumes of fluid. The phenomenon of acoustofluidic atomization has been studied for several decades now, and one of its major proposed applications is in pulmonary drug delivery. It is well known that droplets in the 1 to 5 μm range are optimal for lung delivery. High-frequency, oriface-free, acoustic vibration has been shown to produce droplets in narrow size distributions including this range. Until now, this technology has not been practical for point-of-care. We have created the first portable acoustofluidic nebulizer as a proof of concept for this application. This device has been enabled by improved transducer design that can now be driven by standard COTS (commercial off-the-shelf) integrated circuits for efficient frequency-synthesis and power-delivery in the 5-25 MHz, 0.5–2 W range. We use a thickness mode transducer with reduced dimensions and incorporate an on-board resonance search algorithm in order to obtain a true thickness mode acoustic wave. We supply fluid from a wick in contact with the edge of the transducer and promote wetting of the surface via fine abrasion in order to obtain continuous atomization from a thin film. Our device nebulizes 0.1 ml/min of water in the critical 1 to 5 μm diameter range using a standard 3.7 V, rechargeable, lithium-ion battery.

Crack-face frictional contact modelling in cracked piezoelectric materials

Actuators, sensors, micro- and nano-electromechanical systems and other piezoelectirc components are generally constructed in block form or as a thin laminated composites. The study of the integrity of such materials in their various forms and small sizes is still a challenge nowadays. To gain a better understanding of these systems, this work presents a crack surface contact formulation that includes friction and thus makes it possible to study the integrity of these advanced materials under more realistic crack surface multifield operational conditions. The dual boundary element method (BEM) is used for modeling frictional crack surface contact on piezoelectric solids in the presence of electric fields, further taking into account the electrical semipermeable boundary conditions on the crack. The formulation uses contact operators over the augmented Lagrangian to enforce contact constraints on the crack surfaces. The BEM reveals to be a very suitable methodology for these interface interaction problems because it considers only the boundary degrees of freedom, what makes it possible to reduce the number of unknowns and to obtain accurate results with a much lower number of elements than formulations based on the standard finite element method or the eXtended finite element method. The capabilities of this methodology are illustrated by solving some benchmark problems.

Функции податливости электромагнитупругой пьезоэлектрической пьезомагнитной полуплоскости и полупространства с функционально-градиентным или слоистым покрытием

Статья посвящена построению функций податливости осесимметричных и плоских контактных задач электромагнитоупругости для полубесконечных пьезоэлектрических пьезомагнитных тел с функционально градиентными или кусочно-однородными покрытиями. Материалы покрытия и подложки считаются трансверсально-изотропными. Вычисление функций податливости сведено к решению краевых двухточечных задач для систем обыкновенных дифференциальных уравнений с переменными коэффициентами с использованием техники интегральных преобразований. Граничные условия этих краевых задач соответствуют распределенной в некоторой области единичной механической нормальной или касательной, электрической или магнитной нагрузке. Получены парные интегральные уравнения и их системы, описывающие контактные задачи о вдавливании изолированного или проводящего штампов в полупространство (или полуплоскость) с покрытием и задачу об электроде, расположенном на поверхности покрытия. Трансформанты ядер этих интегральных уравнений совпадают с функциями податливости. Предложены специальные аппроксимации трансформант ядер, с использованием которых можно построить замкнутые аналитические решения приближенных парных интегральных уравнений и их систем. Представлены численные результаты, иллюстрирующие свойства всех десяти независимых функций податливости для различных сочетаний свойств покрытия и подложки и законов изменения свойств по глубине. Показано, что при отсутствии касательной нагрузки все функции податливости являются строго положительными. Исследованы условия, при которых функции податливости, соответствующие касательной нагрузке, являются знакопеременными. Проиллюстрированы различия между свойствами функций податливости, соответствующих однородным и функционально градиентным покрытиям.

Three-dimensional boundary element analysis for anisotropic elastic solids and its extension to piezoelectric and magnetoelectroelastic solids

This article presents three-dimensional (3D) analysis of generally anisotropic elastic, piezoelectric, piezomagnetic, and magneto-electro-elastic solids by boundary element method (BEM). The associated Green's functions of displacements and tractions in both complex and real forms together with their first derivatives are derived completely in the first time by using Radon–Stroh formalism. Like the Stroh formalism for two-dimensional problems, the solutions for the solids with these different material types all bear exactly the same mathematical forms distinguished by the contents and dimensions of the related matrices and vectors. This feature provides a big advantage for computer programming. To emphasize the development of 3D-BEM, the system of algebraic equations to calculate the nodal displacements and nodal tractions along the boundary of 3D solids is formulated. The relations for calculating strains and stresses at the boundary nodes as well as at the interior points are also derived. After the employment of Green's functions to the newly formulated 3D-BEM, we found that the 3D-BEM with complex form solution is computationally more efficient than that with real form solution, although the opposite performance was observed for a single point calculation of Green's function. To verify the correctness of our solutions, numerical examples of a cube with or without through hole are presented as illustrations of our successful implementation in the BEM.

Unified and explicit expressions of three-dimensional Green’s functions and their first derivatives for piezoelectric solids with general anisotropy

Unified and explicit expressions of static three-dimensional generally anisotropic Green’s functions and their first derivatives are presented. The application of residue calculus to the line integral form of the Green’s functions and their derivatives yields the conventional explicit expressions. The numerical evaluations of these expressions become unstable in the nearly degenerate case. In contrast to the conventional explicit expressions, the novel unified explicit expressions have no factors like (pi−pj) in any denominator. The novel expressions remain valid for all cases. Their numerical evaluations, therefore, possess very high accuracy and efficiency. Following our previous works in generally anisotropic elasticity, the unified explicit expressions of the Green’s functions and their derivatives for anisotropic piezoelectric materials are presented for the first time. The accuracy and stability of novel unified and explicit expressions are verified numerically. Furthermore, the newly derived explicit expressions are used to investigate the influences of the material anisotropy on the displacements and electrical potential in a piezoelectric solid, when a point force or an electrical charge is imposed. The numerical results suggest that the field quantities can be tuned by the proper choose of the material anisotropy or the orientation of the principal material axes.