Linear Piezoelectric Solids Research Articles

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.

Extended finite element simulation of stationary dynamic cracks in piezoelectric solids under impact loading

This work presents a transient dynamic analysis of stationary cracks in two-dimensional, homogeneous and linear piezoelectric solids subjected to coupled electromechanical impact loads using the extended finite element method (X-FEM). To serve this purpose, a dynamic X-FEM computer code using quadrilateral elements in conjunction with the level set method to accurately describe the crack geometry is developed. The sixfold basis enrichment functions particularly suitable for cracks in piezoelectric materials are adopted to fully capture the singular fields at the crack-tips in piezoelectricity. The governing equations are transformed into a weak-form and the time-dependent system of discrete equations is then obtained, which is solved by the unconditionally implicit time integration method at each time-step. To accurately assess the relevant dynamic mechanical stress and electric displacement intensity factors precisely and efficiently, domain-form of the contour integration integral taking the inertial effect into account in conjunction with the asymptotic near crack-tip fields of piezoelectric materials is presented. Four numerical examples for stationary cracks in homogeneous piezoelectric solids with impermeable crack-face boundary condition under impact loads are considered, respectively. Validation of the present method is made by comparing the present results with reference solutions available in the literature, and very good agreements are obtained. The effects of different poling directions and combined electromechanical impact loads are analyzed in details.

Analysis of interface cracks in layered piezoelectric composites by a SGBEM

AbstractPiezoelectric materials offer many possibilities in advanced engineering structures due to their inherent coupling effects between mechanical and electrical fields and are widely applied in smart devices and structures like transducers, actuators and sensors [2]. An important application of piezoelectric materials is related to layered or laminated composites because they can be optimized to satisfy the high‐performance requirements according to different in‐service conditions. Beside cracks inside homogeneous domains, one of the most dominant failure mechanisms in layered or laminated composites is the interface failure. Interface cracks and interface debonding may be induced by the mismatch of the mechanical, electrical and thermal properties of the material constituents during the manufacturing process and the in‐service loading conditions. This paper presents a hypersingular symmetric Galerkin boundary element method (SGBEM) for crack analysis in two‐dimensional (2D), layered and linear piezoelectric solids. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic crack analysis in piezoelectric solids with non-linear electrical and mechanical boundary conditions by a time-domain BEM

This paper presents advanced transient dynamic crack analysis in two-dimensional (2D), homogeneous and linear piezoelectric solids using non-linear mechanical and electrical crack-face boundary conditions. Stationary cracks in infinite and finite piezoelectric solids subjected to impact loadings are considered. For this purpose a time-domain boundary element method (TDBEM) is developed. A Galerkin-method is implemented for the spatial discretization, while a collocation method is applied for the temporal discretization. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. An iterative solution algorithm is developed to consider the non-linear semi-permeable electrical crack-face boundary conditions. Furthermore, an additional iteration scheme for crack-face contact analysis is implemented at time-steps when a physically meaningless crack-face intersection occurs. Several numerical examples are presented and discussed to show the effects of the electrical crack-face boundary conditions on the dynamic intensity factors.

2-D transient dynamic analysis of cracked piezoelectric solids by a time-domain BEM

Transient dynamic crack analysis of two-dimensional (2-D), homogeneous and linear piezoelectric solids is presented in this paper. For this purpose, a time-domain boundary element method (BEM) is developed. The method uses a combination of the strongly singular displacement boundary integral equations (BIEs) and the hypersingular traction BIEs. On the external boundary of the cracked solid strongly singular displacement BIEs are used, while on the crack-faces hypersingular traction BIEs are applied. A convolution quadrature formula is applied for the temporal discretization, while a collocation method is adopted for the spatial discretization. The accuracy of the present time-domain BEM is verified by using numerical results obtained by the finite element method (FEM). Numerical examples are presented and discussed to show the effects of the mechanical and the electrical impact loading on the dynamic intensity factors.

2D transient dynamic crack analysis in piezoelectric solids by BEM

This paper presents a two-dimensional (2D) transient dynamic crack analysis in homogeneous and linear piezoelectric solids. A hypersingular traction boundary integral equation formulation is used for this purpose. A finite crack in an infinite piezoelectric solid subjected to an impact crack-face loading is investigated, which could be either an impact mechanical loading, or an impact electrical loading or a combination of both. In the time-domain boundary element method (BEM), the Lubich’s quadrature formula is adopted for approximating the arising convolution integrals. Hypersingular integrals arising in the traction boundary integral equations (BIEs) are computed through a regularization technique, which converts the hypersingular integrals into regular integrals plus strongly singular and another hypersingular integrals. The regular integrals are computed numerically, while the strongly singular and the new hypersingular integrals are integrated analytically. Numerical results for the dynamic intensity factors will be presented and discussed.

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6 mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.

3-D and 2-D Dynamic Green's functions and time-domain BIEs for piezoelectric solids

Dynamic Green's functions for linear piezoelectric solids are derived by using Radon transform. Time-harmonic and Laplace transformed dynamic Green's functions are obtained subsequently by applying the Fourier and the Laplace transform to the time-domain Green's functions. Time-domain boundary integral equation formulations are presented for transient dynamic analysis of linear piezoelectric solids. In particular, hypersingular and non-hypersingular time-domain traction BIEs are derived by two different ways. Their potential application in transient dynamic crack analysis of three-dimensional and two-dimensional piezoelectric solids is discussed.

Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials

This paper is concerned with the investigation of the effect of material inhomogeneity on the decay of Saint–Venant end effects in functionally graded linear piezoelectric materials. Saint–Venant's principle and related results for elasticity theory have received considerable attention in the literature but it is only recently that analogous issues in piezoelectricity have been investigated. The current rapidly developing smart–structures technology provides motivation for the investigation of such problems. We examine the decay of Saint–Venant end effects in the context of anti–plane shear deformations for linear inhomogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations (PDEs) of equilibrium are shown to be a coupled system of second–order PDEs with variable coefficients for the mechanical displacement u and electric potential φ . The traction boundary–value problem with prescribed surface charge is formulated as an oblique derivative boundary–value problem for this elliptic system. The axial decay of solutions on a semi–infinite strip subjected to non–zero boundary conditions only at the near end is investigated. This analysis is carried out for a subclass of special graded materials, namely, laterally inhomogeneous solids. Two specific piezoceramics that have received widespread application are considered. For the first (corresponding to the hexagonal 6 mm crystal–symmetry class), it is shown that the mechanical and electrical problems essentially decouple, and that the decay rates for mechanical and electrical Saint–Venant end effects coincide and are equal to that of linear inhomogeneous isotropic anti–plane elasticity, for which results on the decay rate have been obtained previously. For the second class of materials (corresponding to the cubic 4∼3m symmetry) the situation is quite different. The boundary–value problem involves a full coupling of mechanical and electrical effects. Energy decay estimates using differential–inequality methods are used to obtain an explicit estimated decay rate (a lower bound for the actual decay rate) in terms of a single dimensionless piezoelectric coupling constant d and the decay rate for an associated linear inhomogeneous isotropic anti–plane shear elasticity problem. For fixed material inhomogeneity, the decay rate is shown to be monotone decreasing with increasing values of d In the limit as d→0 we recover the purely mechanical case. Thus, for this class of functionally graded materials, piezoelectric end effects are predicted to penetrate further into the strip than their functionally graded isotropic elastic counterparts , confirming recent results obtained in other contexts in linear piezoelectricity. For both classes of materials, the influence of material inhomogeneity on the decay rate is reflected solely by the shear modulus of an associated inhomogeneous isotropic anti–plane shear elasticity problem. It is shown that, for both material classes, material inhomogeneity has a significant influence on the decay of end effects.

Diffusive surface in stabilities induced by electromechanical loading

It has been recently established that mechanical stresses can induce a diffusive surface instability, which leads to crack formation without any energy barrier. This paper is aimed at investigating this instability mechanism for the more general case of an interface between two linear piezoelectric solids subjected to coupled electromechanical stresses. The energy variation associated with interface roughness is calculated using an extended Stroh formalism in anisotropic elasticity. The solutions obtained are applicable to a wide class of problems involving surfaces or interfaces in material systems with piezoelectric, anisotropic or isotropic properties.