character correspondences and irreducible induction and restriction let s and g be linitc groups such that s acts on g and is ici 1 and let c denote the fixed point subgroup c c s it is well known in this situation that there exists a natural bijection from the set irr g of s invariant irreducible characters of g onto irr c the full set of irreducible characters of c when s is solvable this bijection was constructed by g glauberman 2 and in the remaining case where g is solvable of odd order a totally different construction 3 gives the desired character correspondence it was proved by t r wolf g that when both constructions apply when s is solvable and ig is odd the two constructions yield exactly the same map if 31 e irr g we write x e irr c to denote the uniquely determined corresponding character similarly if he g is s invariant and 0 e irr h we write f3 e irr hn c to denote the image of 8 under the character bijection induced by the action of s on h our main result is the following

interaction integrals for fracture analysis of functionally graded piezoelectric materials this paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials conservation integrals of j type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium the additional terms play an important role of ensuring domain independence of the presented interaction integrals comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples

preconditioning of full waveform inversion in viscoacoustic media intrinsic absorption in the earth affects the amplitude and phase spectra of the seismic wavefields and records and may degrade significantly the results of acoustic full waveform inversion amplitude distortion affects the strength of the scatterers and decreases the resolution phase distortion may result in mislocated interfaces we show that viscoacoustic gradient based inversion algorithms e g steepest descent or conjugate gradients compensate for the effects of phase distortion but not for the effects of amplitude distortion to solve this problem at a reasonable numerical cost we have designed two new forms of preconditioning derived from an analysis of the inverse hessian operator the first type of preconditioning is a frequency dependent compensation for dispersion and attenuation which involves two extra modeling steps with inverse absorption amplification at each iteration the second type only corrects the strength of the recovered scatterers and requires two extra modeling steps at the first iteration only the new preconditioning methods have been incorporated into a finite difference inversion scheme for viscoacoustic media numerical tests on noise free synthetic data illustrate and support the theory

compensating q effects in viscoelastic media by adjoint based least squares reverse time migration low values of p and s wave quality factors qp and qs result in strong intrinsic seismic attenuation in both seismic modeling and imaging we use a linearized waveform inversion approach by generalizing least squares reverse time migration lsrtm for viscoelastic media q lsrtm to compensate the attenuation loss for both p and s images we use the first order particle velocity stress and memory variable equations with explicit q in the formulations based on the generalized standard linear solid gsls as the forward modeling operator the linearized two way viscoelastic modeling operator is obtained with modulus perturbations introduced for the relaxed p and s moduli the viscoelastic adjoint operator and the p and s imaging conditions for modulus perturbations are derived using the adjoint state method and an augmented lagrangian functional q lsrtm solves the viscoelastic linearized modeling operator for generating synthetic data and the adjoint operator is used for back propagating the data re

multiple collinear cracks in a piezoelectric material abstract in this paper an exact solution of multiple collinear cracks in piezoelectric material is obtained the permittivity of air environment is considered two cases have been studied in the first case the permittivity of air is far less than that of piezoelectric material therefore the electric induction in the air is negligible in the second case the permittivity of air is comparable with that of piezoelectric material the problem is deduced into riemann hilbert problem and solved by the way the electric boundary conditions are discussed this result demonstrates that the consideration of air in crack gap reduces the stress intensity factor

consistency and recovery in electroelasticity part i standard finite elements developing finite elements for electroelastic analysis requires a special care as the coupling in the discrete equations depends on the matching between the approximations assumed for mechanical and electrical variables to provide a formal basis to this intuitive remark a notion of consistency is established and a rigorous consistency analysis is presented for the standard finite element model based on assumed displacement and electric potential in this way specific analytical requirements are obtained which serve as a guide to select the interpolation functions for primary variables moreover the analysis shows that violating consistency can be reflected as spurious outcomes upsetting the local distributions of secondary variables indeed this undesirable effect is shown to be typical of the standard approach if stress and electric flux density are computed via the constitutive equations to cure the trouble an alternative recovery procedure is devised based on the consistency analysis the procedure is variationally correct and reconstitutes in a consistent manner the distributions of stress and electric flux density

a level set based iga formulation for topology optimization of flexoelectric materials abstract this paper presents a design methodology based on a combination of isogeometric analysis iga level set and point wise density mapping techniques for topology optimization of piezoelectric flexoelectric materials the fourth order partial differential equations pdes of flexoelectricity which require at least c 1 continuous approximations are discretized using non uniform rational b spline nurbs the point wise density mapping technique with consistent derivatives is directly used in the weak form of the governing equations the boundary of the design domain is implicitly represented by a level set function the accuracy of the iga model is confirmed through numerical examples including a cantilever beam under a point load and a truncated pyramid under compression with different electrical boundary conditions finally we provide numerical examples demonstrating the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization

a further investigation of green s functions for a piezoelectric material with a cavity or a crack abstract in this paper the green s functions of an infinite two dimensional piezoelectric material containing an elliptical cavity are re investigated by introducing exact electric boundary conditions on the hole boundary and the corresponding modified solutions are obtained by setting 0 the dielectric constant in the cavity to be zero the modified green s functions are returned to the conventional ones lu and williams 1998 furthermore the hoop stresses along the hole boundary under the action of a set of generalized concentrated forces are obtained when the hole is reduced to a slit crack the expressions of generalized stress intensity factors are also provided

infinite sequence of parallel cracks in an anisotropic piezoelectric solid in this paper the problem of an infinite sequence of parallel cracks in an infinite extended piezoelectric solid is analyzed a system of singular integral equations is formulated for general anisotropic piezoelectric materials of which the kernel functions developed are in complex form for commonly used transversely isotropic piezoelectric materials the kernel functions are given in real forms furthermore the obtained real kernel functions may be reduced respectively to those kernel functions for purely elastic and purely electric problems when the coupled mechanical and electric effects disappear the system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors

analysis of stress and electric fields in a rectangular piezoelectric body with a center crack under anti plane shear loading abstract the singular stress and electric fields in a rectangular piezoelectric ceramic body containing a griffith center crack under anti plane shear loading are obtained by the theory of linear piezoelectricity fourier transforms and fourier sine series are used to reduce the problem to a pair of dual integral equations which is expressed to a fredholm integral equation of the second kind numerical values on the stress intensity factor and the energy release rate are obtained to show the influence of the electric field

on the dynamic behavior of a functionally graded piezoelectric strip with periodic cracks vertical to the boundary abstract the dynamic anti plane problem for a functionally graded piezoelectric strip containing a periodic array of parallel cracks which are perpendicular to the boundary is considered integral transforms techniques are employed to reduce the problem to the solution of singular integral equations numerical results are presented to show the influences of geometry electromechanical combination factor and material gradient parameter on the fracture behavior

three dimensional piezoelectric boundary element method the boundary element method is applied to problems of three dimensional linear piezoelectricity the continuum equations for conservation of linear momentum and charge are combined into one governing equation for piezoelectricity a single boundary integral equation is developed from this combined e eld equation and green s solution for a piezoelectric medium green s function and its derivatives are derived using the radon transform and the resulting solution is represented by a line integral that is evaluated numerically using standard gaussian quadrature the boundary integral equation is discretized using eight node isoparametric quadratic elements resulting in a matrixsystem ofequations thesolution of theboundary problem forpiezoelectricmaterialsconsistsof elastic displacements tractions electric potentials and normal charge e ux densities the complete e eld solutions can be obtained once all boundary values have been determined the accuracy of this linear piezoelectric boundary element method is illustrated with two numerical examples the e rst involves a unit cube of material with an applied mechanical load thesecond exampleconsistsof a spherical holein an ine nitepiezoelectricbody loaded by a unit traction on itsboundary comparisonsare made to the analytical solution for the cube and an axisymmetric e nite element solution for the spherical hole the boundary element method is shown to be an accurate solution procedure for general three dimensional linear piezoelectric material problems i introduction t hebrotherspierreandjacquescuriediscoveredpiezoelectricity in 1880 during their investigation of crystals a coupling occurs between the electrical and mechanical e elds in these materials such that application of an external mechanical loading induces achangeinpolarizationproportionaltoit conversely applicationof an electric e eld produces a proportional material deformation this phenomenon was discovered in ceramics in the 1940s and the subsequent manufactured piezoceramic materialshave higher coupling properties than their naturally occurring predecessors this strong piezoelectric effect has made these materials an important part of the emerging technologies of smart materials and structures smart structures are those that incorporate actuators and sensors that are integrated into the structure and have structural functionality the sensors and actuators are used to ine uence the structure s states

decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems abstract this paper presents the formulation of piezoelectric elasticity under generalized plane deformation derived from the three dimensional theory there are four decoupled classes in the generalized plane deformation formulation i e when l 3 l 2 0 l 3 l 3 0 l 3 l 2 0 or l 3 l 3 l 2 0 only the inplane fields of the first class and the antiplane field of the second class include the piezoelectric effect several examples of wedge problem often encountered in smart structures such as sensors or actuators are studied to examine the stress singularity near the apex of the structure the bonded materials to the pzt 4 wedge are pzt 5 graphite epoxy or aluminum conductor the influencing factors on the singular behavior of the electro elastic fields include the wedge angle material type poling direction and the boundary and interface conditions the numerical results of the first case are compared with xu s graphs and some comments are made in detail in addition some new results regarding the antiplane stress singularity of the second class are obtained via the case study the coupled singularity solutions under generalized plane deformation are also investigated to seek the conditions of the weakest or vanishing singular stress fields

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 6mm 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

governing equations and general solutions of plane elasticity of one dimensional quasicrystals plane elasticity theory of one dimensional quasicrystals dealing with all point groups is investigated systematically the governing equations of elastic fields and their general solutions are derived by the complex variable functions method as an example the elastic fields of a straight dislocation along the quasiperiodic axis of an orthorhombic quasicystal are calculated the relevant singularity of the stress field for the dislocation in the quasicrystals is also discussed

two dimensional piezoelectricity part i eigensolutions of nondegenerate and degenerate materials abstract general solutions of two dimensional piezoelectricity which yield all solutions of 2 d boundary values problems are obtained by combining four complex conjugate pairs of independent eigensolutions each containing an arbitrary analytic function the forms of representation are fundamentally different for 14 different classes of nondegenerate and degenerate piezoelectric materials as determined by the multiplicity and types of eigenvalues degenerate materials possess high order eigensolutions in which the eigenvectors of equal and lower orders are intrinsically coupled such coupling is nonexistent in nondegenerate cases including the well known and analytically simple case with no multiple eigenvalues the present analysis is drastically simplified by using the compliance based formalism instead of the stiffness based extended eshelby stroh formalism explicit expressions are obtained for the eigensolutions the pseudometrics and the intrinsic tensors characterizing piezoelectric materials of every type

full waveform inversion of plane wave seismograms in stratified acoustic media theory and feasibility we have implemented an inversion procedure for obtaining velocity and density profiles from multioffset data in a layered acoustic medium using an iterative modeling technique and the p representation the procedure is derived from the nonlinear least squares formalism of tarantola and valette the feasibility of this method depends upon obtaining frechet derivatives during the modeling process and on vectorizing the thompson haskell reflectivity algorithm test data sets for this study are gathers of two to four plane wave synthetic seismograms which may contain both precritical and postcritical arrivals generated by a seismic wavelet these examples demonstrate convergence to the correct velocity and density profiles with reasonable accuracy

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

the extended finite element method with new crack tip enrichment functions for an interface crack between two dissimilar piezoelectric materials summary this paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method x fem with new crack tip enrichment functions in the x fem crack modeling is facilitated by adding a discontinuous function and crack tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity in this work the coupled effects of an elastic field and an electric field in piezoelectricity are considered corresponding to the two classes of singularities of the aforementioned interface crack problem namely class and class two classes of crack tip enrichment functions are newly derived and the former that exhibits oscillating feature at the crack tip is numerically investigated computation of the fracture parameter i e the j integral using the domain form of the contour integral is presented excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical fem moreover it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency copyright 2015 john wiley sons ltd

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

some explicit expressions of extended stroh formalism for two dimensional piezoelectric anisotropic elasticity abstract since the extended stroh formalism for two dimensional piezoelectric anisotropic elasticity preserves essential features of stroh formalism for pure elastic materials it becomes important to get the corresponding explicit expressions of some important matrices frequently appeared in stroh formalism in this paper explicit expressions are obtained for the fundamental matrix n material eigenvector matrices a and b and barnett lothe tensors l s and h although the explicit expressions are presented under the generalized plane strain and short circuit condition by suitable replacement of the material constants they are still valid for the other two dimensional states to provide a clear picture of these expressions two typical examples are presented which are piezoelectric ceramics with two different poling axes

a coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate part ii numerical solution and applications this two part contribution presents a novel and efficient method to analyze the two dimensional 2 d electromechanical fields of a piezoelectric layer bonded to an elastic substrate which takes into account the fully coupled electric and mechanical behaviors in part i we have obtained a system of governing integro differential equations for the structure via a variational principle this part presents a numerical solution algorithm of the integro differential equations and the numerical results of some applications a numerical algorithm for solving the system of four integro differential equations with strongly singular kernels is developed the convergence of the numerical algorithm is discussed the numerical results suggest that the fully coupled electromechanical analysis is helpful for a better understanding of the performance of the piezoelectric sensor and actuator the interfacial normal stress is much higher than the interfacial shear stress suggesting that the interfacial normal stress causes a delamination initiation

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

penny shaped crack in a piezoelectric cylinder surrounded by an elastic medium subjected to combined in plane mechanical and electrical loads the fracture problem of a penny shaped crack in a piezoelectric ceramic cylinder surrounded by an infinite elastic medium under in plane normal mechanical and electrical loads is considered with the electric continuous boundary conditions on the crack surface by using the potential theory and hankel transform a system of dual integral equations is obtained and expressed to a fredholm integral equation of the second kind the mechanical and electrical field equations and all sorts of field intensity factors of mode i are obtained and the numerical values of various field intensity factors for pzt 6b piezoelectric ceramic surrounded by several different elastic media are graphically shown for a uniform load and a ring shaped load respectively and the effects of the size of the piezoelectric cylinder and the elastic material properties on various field intensity factors are obtained

a density result in two dimensional linearized elasticity and applications we show that in a two dimensional bounded open set whose complement has a finite number of connected components the vector fields u h 1 2 are dense in the space of fields whose symmetrized gradient e u is in l 2 4 this allows us to show the continuity of some linearized elasticity problems with respect to variations of the set with applications to shape optimization or the study of crack evolution

a boundary integral formulation and 2d fundamental solutions for piezoelectric media abstract first based on the basic equations of piezoelectricity the boundary integral formulation is derived by utilizing the reciprocal work theorem second for the plane problem of piezoelectric media one general solution in terms of harmonic functions is derived and the fundamental solutions are therefore obtained numerical calculations by boundary element method bem are performed to give the stress concentration coefficient of an infinite piezoelectric plane with a circular hole as well as the stress and electric intensity factors of a central crack locating on an infinite plate results are found to agree well with the exact solutions

piezoelectric solid with an elliptic inclusion or hole the two dimensional problem of an elliptic hole in a solid of anisotropic piezoelectric material is studied the stroh formalism is adopted here real form solutions are obtained along the hole boundary in the case of an arbitrarily prescribed vector field on the hole surface for an elliptic rigid inclusion of electric conductor subjected to a line force a torque and a line charge a real form solution at the interface is obtained finally general solutions for an elliptic piezoelectric inclusion with uniform loading at infinity are investigated

the general solution of three dimensional problems in piezoelectric media abstract the governing equations of three dimensional problems in transversely isotropic piezo electric material are composed of four second order partial differential equations in which the displacement components and electric potential functions are the essential unknowns in this paper the so called general solutions of these equations are obtained with the aid of a set of new potential functions which are used to express the components of displacements and the electric potential functions as an application of the general solution the problem of a half space piezoelectrics acted by a concentrated lateral shear force is solved

investigation of the dynamic behavior of two parallel symmetric cracks in piezoelectric materials use of non local theory in this paper the dynamic behavior of two parallel symmetric cracks in piezoelectric materials under harmonic anti plane shear waves is investigated by use of the non local theory for permeable crack surface conditions to overcome the mathematical difficulties a one dimensional non local kernel is used instead of a two dimensional one for the problem to obtain the stress occurs near the crack tips by means of the fourier transform the problem can be solved with the help of two pairs of dual integral equations that the unknown variables are the jumps of the displacement along the crack surfaces these equations are solved using the schmidt method numerical examples are provided contrary to the previous results it is found that no stress and electric displacement singularity is present near the crack tip the non local elastic solutions yield a finite hoop stress near the crack tip thus allowing for a fracture criterion based on the maximum stress hypothesis the finite hoop stress at the crack tip depends on the crack length the frequency of the incident wave the distance between two cracks and the lattice parameter of the materials respectively contrary to the impermeable crack surface condition solution it is found that the dynamic electric displacement for the permeable crack surface conditions is much smaller than the results for the impermeable crack surface conditions the results show that the dynamic field will impede or enhance crack propagation in the piezoelectric materials at different stages of the dynamic load

a meshfree method for inverse wave propagation using collocation and radial basis functions abstract a stable and highly accurate meshfree scheme based on strong form collocation associated with radial basis functions and explicit time integration is proposed to solve ill posed inverse wave propagation problems tikhonov regularization technique and the l curve criterion are employed to deal with the noise measurement data before solving the governing equation to get a stable solution appropriate weights are first derived for imposing boundary conditions and known conditions from the measurement data to yield the optimal convergence for inverse problems compared to the required iterative procedures in common methods no iteration is needed in the proposed method which leads to the time saving computation and avoids the instability resulting from the iterative methods dispersion analysis describes that this method has much less dispersion error than the classical finite element method fem and popular weak form meshfree methods and stability analysis is performed to predict the critical time step for the explicit time integration another outstanding advantage of this method is that it is effortless to extend the solution scheme of one dimensional inverse problem to high dimensional inverse problems furthermore the influences of the high noise into measurement data are evaluated which demonstrate that even handling with very high noise in the input data the proposed algorithm can achieve stable and high accuracy solutions

plane problems in piezoelectric media with defects abstract a two dimensional electroelastic analysis is performed on a transversely isotropic piezoelectric material containing defects a general solution is provided in terms of complex potentials with emphasis being placed on stress concentrations that arise in the vicinity of circular and elliptical holes it is shown that for this genre of problem both mechanical and electrical variables are responsible for the peak stresses

detection of multiple flaws in piezoelectric structures using xfem and level sets abstract an iterative procedure to solve the inverse problem of detecting multiple voids in piezoelectric structure is proposed in each iteration the forward problem is solved for various void configurations and at each iteration the mechanical and electrical responses of a piezoelectric structure is minimized at known specific points along the boundary to match the measured data the extended finite element method xfem is employed for determining the responses as it allows the use of a fixed mesh for varying void geometries the numerical method based on combination of classical shape derivative and of the level set method for front propagation used in structural optimization is utilized to minimize the cost function the results obtained show that this method is effectively able to determine the number of voids in a piezoelectric structure and its corresponding locations and shapes

the behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes in this paper the behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes subjected to an anti plane shear loading is investigated by using schmidt method by using the fourier transform the problem can be solved with the help of two pairs of dual integral equations these equations are solved using the schmidt method this process is quite different from that adopted previously the normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions numerical examples are provided to show the effect of the geometry of the interacting cracks the thickness and the materials constants of the piezoelectric layer upon the stress and the electric displacement intensity factor of the cracks contrary to the impermeable crack surface condition solution it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions

topology optimization with local stress constraint based on level set evolution via reaction diffusion abstract this work focuses the structural topology optimization problem of mass minimization subject to local stress constraints to this aim two related issues are addressed the first one is the successful strategy used to define local stress constraints by means of an augmented lagrangian approach the second and main contribution of the present paper is the use of a reaction diffusion equation to guide via evolution of a level set the design optimization sequence the advantages of this strategy are twofold firstly it allows the creation of new holes during the optimization process a significant feature for a true topological optimization method secondly reinitialization steps usually found in classical hamilton jacobi based evolution are eliminated with a significant improvement in convergence ease a set of benchmark examples in two dimensions are presented numerical results show the efficiency of the algorithm to create new holes identify stress concentrations and to provide stable optimization sequences converging to local minima defined by stress saturated designs

two dimensional nonlinear inversion of seismic waveforms numerical results the nonlinear problem of inversion of seismic waveforms can be set up using least squares methods the inverse problem is then reduced to the problem of minimizing a lp nonquadratic function in a space of many 104to106 variables using gradient methods leads to iterative algorithms each iteration implying a forward propagation generated by the actual sources a backward propagation generated by the data residuals acting as if they were sources and a correlation at each point of the space of the two fields thus obtained which gives the updated model the quality of the results of any inverse method depends heavily on the realism of the forward modeling finite difference schemes are a good choice relative to realism because although they are time consuming they give excellent results numerical tests performed with multioffset synthetic data from a two dimensional model prove the feasibility of the approach if only surface recorded reflections are used the high spatial frequency content of the

improved implementation and robustness study of the x fem for stress analysis around cracks numerical crack propagation schemes were augmented in an elegant manner by the x fem method the use of special tip enrichment functions as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh which is of industrial interest especially when a numerical model for crack propagation is desired this paper improves the implementation of the x fem method for stress analysis around cracks in three ways first the enrichment strategy is revisited the conventional approach uses a topological enrichment only the elements touching the front are enriched we suggest a geometrical enrichment in which a given domain size is enriched the improvements obtained with this enrichment are discussed second the conditioning of the x fem both for topological and geometrical enrichments is studied a preconditioner is introduced so that off the shelf iterative solver packages can be used and perform as well on x fem matrices as on standard fem matrices the preconditioner uses a local nodal cholesky based decomposition third the numerical integration scheme to build the x fem stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme a 2d benchmark problem is designed to show the improvements and the robustness

a first order perturbation analysis of a non ideal crack in a piezoelectric material abstract in this paper a first order perturbation analysis is carried out on a symmetrically perturbed non ideal crack for three kinds of electric boundary conditions namely permeable impermeable and conducting crack boundary condition by using the extended stroh formula the two domain problems are reduced to standard riemann hilbert problems and the singular integral equations of the internal electric field inside the permeable crack are solved the stress and electric intensity factors seifs are determined to the first order of accuracy the results indicate that for a symmetrically perturbed non ideal crack the electro mechanical loading at infinity does not affect the first order solution for the mode i intensity factor for general piezoelectric materials the energy release rate and the seifs are determined by remote mechanical loads only and the perturbation effect on the seifs and energy release rate is small the electric field distribution inside crack is constant for the zeroth order solution and quadratic for the first order solution which is different from the constant electric field distribution for an ideal permeable crack the internal electric concentration near the crack tip caused by the perturbation reveals that the dielectric inside the crack probably breaks down before the matrix does when the matrix is subjected to a not too high electro mechanical load at infinity the seifs and the energy release rate are also given for the non ideal crack under the impermeable and conducting electric boundary condition respectively for all three kinds of electric boundary conditions the lateral stresses 11 13 have no contribution to the seifs to the first order of accuracy

iterative asymptotic inversion in the acoustic approximation we propose an iterative method for the linearized prestack inversion of seismic profiles based on the asymptotic theory of wave propagation for this purpose we designed a very efficient technique for the downward continuation of an acoustic wavefield by ray methods the different ray quantities required for the computation of the asymptotic inverse operator are estimated at each diffracting point where we want to recover the earth image in the linearized inversion we use the background velocity model obtained by velocity analysis we determine the short wavelength components of the impedance distribution by linearized inversion of the seismograms observed at the surface of the model because the inverse operator is not exact and because the source and station distribution is limited the first iteration of our asymptotic inversion technique is not exact we improve the images by an iterative procedure since the background velocity does not change between iterations there is no need to retrace rays

the two dimensional electroelasticity problems for multiconnected bodies situated under electric potential difference action abstract solutions of the two dimensional electroelasticity problems for infinitely long piezoelectric cylinders with cavities and cracks under electrical potential difference action are obtained using the lekhnitskii s generalized complex potentials herewith piezoelectric bodies under consideration are free from mechanical loads some or all openings and maybe the exterior boundary of the cylinders are fully covered by thin electrodes voltage differences are supplied on the electroded surfaces the rest boundary surfaces are charge free in this case the complex potentials are investigated boundary conditions are satisfied by the least squares method there are carried out numerical investigations of behaviour of some basic electroelastic characteristics in a circular hollow cylinder and in an infinite body with two longitudinal cavities and a charge free plane crack new electromechanical regularities of influence of material properties geometrical characteristics of considered regions on values of electroelastic state characteristics and the stress electric displacements and tensions intensity factor k 1 are determined

two dimensional piezoelectricity part ii general solution green s function and interface cracks abstract the complete solution space of a piezoelectric material is the direct sum of several orthogonal eigenspaces one for each distinct eigenvalue each one of the 14 different classes of piezoelectric materials has a distinct form of the general solution expressed in terms of the eigenvectors of the zeroth and higher orders and a kernel matrix containing analytic functions when these functions are chosen to be logarithmic one obtains in a unified way green s function of the infinite space as a single 8 8 matrix function g for the various load cases of concentrated line forces dislocations and a line charge this expression of green s function is valid for all classes of nondegenerate and degenerate materials with an appropriate choice of the parameters it reduces to the solution of a half space with concentrated line forces at a boundary point and with dislocations in the displacements as another application eigenvalues and eigensolutions are obtained for the bimaterial interface crack problem

an exact correspondence between plane piezoelectricity and generalized plane strain in elasticity we consider an anisotropic body bounded by a cylindrical surface which is infinitely long in the axial direction suppose the body is loaded in such a way that the field variables do not vary along the generators an exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity in particular we show that by setting a linkage between the two sets of material constants any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa the assertation is true for rectilinearly as well as for cylindrically anisotropic solids the equivalence is found for the most general anisotropic case which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic triclinic elastic solid a few degenerate systems are also identified in addition these correspondences can be extended to inhomogeneous media applied to composite materials or polycrystalline aggregates they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors and vice versa we also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by cherkaev lurie and milton in illustration we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity lekhnitskii9s complex potential approach together with the conformal mapping technique are employed general solutions of the fields inside the inhomogeneity and the matrix are obtained the results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem

an efficient meshless technique for the solution of transversely isotropic two dimensional piezoelectricity in this paper a new meshless technique is presented for the solution of two dimensional piezoelectricity problem the technique derivation is based on the solution of the corresponding analog equation by transforming the original set of differential equations into three poisson equations with unknown right hand side terms boundary discretisation of the resulting integral equations is eliminated by the use of the method of fundamental solutions the right hand side terms are represented in the solution as particular solutions expressed in terms of radial basis functions the problem solution is then rewritten in its new form which involves complementary solution and particular solution the governing partial differential operator for piezoelectricity is applied on the obtained solution form and forced to be satisfied at a set of domain points whereas the prescribed boundary conditions are satisfied at another set of boundary points the proposed technique is implemented into computer code where several numerical examples with different boundary conditions are tested the results demonstrated excellent agreement with those obtained from analytical and fem solutions

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

incorporation of attenuation into time domain computations of seismic wave fields the only numerically tractable way yet found to incorporate attenuation into numerical time domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low order rational function of frequency the coefficients of this function can be determined by the pade approximation our test computations show however that this approximation generally is of poor quality therefore we suggest a new approach which is based on the rheological model of the generalized maxwell body which has a modulus of the desired rational form we choose the relaxation frequencies logarithmically equidistant in the frequency band of interest and determine the weight factors by simple numerical curve fitting to an arbitrary q law this approach is superior to the method above both in accuracy and in computational efficiency for most practical applications approximations of orders 2 or 3 are sufficient the computing time and memory requirements for a finite difference calculation are then approximately twice those of a purely elastic calculation as a first application of the method we compute sh channel waves in discontinuous coal seams with q 50 within the coal the results show that the high frequency airy phase is strongly attenuated this indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data

integral representation and green s functions for 3d time dependent thermo piezoelectricity abstract a boundary integral method is developed for 3d time dependent quasistatic thermo piezoelectricity this time domain formulation involves only surface quantities as a result volume discretization is completely avoided green s functions are derived by resorting to radon transform and presented in integral form reductions to the case of isotropic dielectric solid have been worked out in detail and its corresponding solutions expressed in explicit form are consistent with the existing solutions

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

trefftz solutions for piezoelectricity by lekhnitskii s formalism and boundary collocation method in this paper a solution procedure for plane piezoelectricity is developed by trefftz boundary collocation method starting with the general plane piezoelectricity solution derived by lekhnitskii s formalism the basic sets of trefftz functions which satisfy the homogeneous governing equations are derived moreover special sets of trefftz functions are derived for impermeable elliptical voids impermeable sharp cracks and permeable sharp cracks with arbitrary orientations with respect to the material poling direction the functions in the special sets satisfy not only the homogeneous governing equations but also the boundary conditions at the peripheries of the pertinent defects by adopting trefftz functions as the trial functions multi domain trefftz boundary collocation method is formulated numerical examples are presented to illustrate the efficacy of the formulation copyright 2005 john wiley sons ltd

some problems in plane thermopiezoelectric materials with holes piezoelectric materials have recently attracted considerable attention due to their potential use in intelligent structural systems in this paper we treat the plane problem of thermopiezoelasticity with various holes and subject to coupled mechanical electric and thermal loads an analytical solution is obtained by applying the technique of conformal mapping and some identities in the stroh formalism the solution has a simple unified form for various holes such as ellipse circle triangle and square by way of the solution the expressions for the energy release rate and stress intensity factors of cracks are presented numerical results for concentration coefficients of stress and electric displacement along the hole boundary are given to assess the acceptability of the proposed method

analysis of three dimensional crack initiation and propagation using the extended finite element method an erratum has been published for this article in international journal for numerical methods in engineering 2005 63 8 1228 we present a new formulation and a numerical procedure for the quasi static analysis of three dimensional crack propagation in brittle and quasi brittle solids the extended finite element method xfem is combined with linear tetrahedral elements a viscosity regularized continuum damage constitutive model is used and coupled with the xfem formulation resulting in a regularized crack band version of xfem the evolving discontinuity surface is discretized through a c0 surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two the element s properties allow a closed form integration and a particularly efficient implementation allowing large scale 3d problems to be studied several examples of crack propagation are shown illustrating the good results that can be achieved copyright 2005 john wiley sons ltd

xfem based crack detection scheme using a genetic algorithm a new computational tool is developed for the accurate detection and identification of cracks in structures to be used in conjunction with non destructive testing of specimens it is based on the solution of an inverse problem based on some measurements typically along part of the boundary of the structure that describe the response of the structure to vibration in a chosen frequency or a combination of frequencies the goal is to estimate whether the structure contains a crack and if so to find the parameters location size orientation and shape of the crack that produces a response closest to the given measurement data in some chosen norm the inverse problem is solved using a genetic algorithm ga the ga optimization process requires the solution of a very large amount of forward problems the latter are solved via the extended finite element method xfem this enables one to employ the same regular mesh for all the forward problems performance of the method is demonstrated via a number of numerical examples involving a cracked flat membrane various computational aspects of the method are discussed including the a priori estimation of the ill posedness of the crack identification problem copyright 2007 john wiley sons ltd

implementation of regularized isogeometric boundary element methods for gradient based shape optimization in two dimensional linear elasticity the present work addresses shape sensitivity analysis and optimization in two dimensional elasticity with a regularized isogeometric boundary element method igabem non uniform rational b splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations with the advantage of tight integration of design and analysis the application of igabem in shape optimization reduces the mesh generation regeneration burden greatly the work is distinct from the previous literatures in igabem shape optimization mainly in two aspects 1 the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives which were the main implementation difficulty in igabem and 2 although based on the same computer aided design cad model the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh thus achieving a balance between less design variables for efficiency and refined mesh for accuracy this technique was initially used in isogeometric finite element method and was incorporated into the present igabem implementation

sensitivity and uncertainty analysis for flexoelectric nanostructures abstract in this paper sensitivity analysis has been applied to identify the key input parameters influencing the energy conversion factor ecf of flexoelectric materials the governing equations of flexoelectricity are modeled by a nurbs based iga formulation exploiting their higher order continuity and hence avoiding a complex mixed formulation the examined input parameters include model and material properties and the sampling has been obtained using the latin hypercube sampling lhs method in the probability space the sensitivity of the model output to each of the input parameters at different aspect ratios of the beam is quantified by three various common methods i e morris one at a time moat pce sobol and extended fourier amplitude sensitivity test efast the numerical results indicate that the flexoelectric constants are the most dominant factors influencing the uncertainties in the energy conversion factor in particular the transversal flexoelectric coefficient h 12 moreover the model parameters also show considerable interaction effects of the material properties

three dimensional eigenfunction analysis of a crack in a piezoelectric material abstract a three dimensional analysis of a semi infinite crack embedded in a transversely isotropic piezoelectric material was performed by means of the eigenfunction expansions method the results show the characteristic 1 r singular behavior of the stress tensor in the vicinity of the crack a similar behavior is also revealed for the electric field in addition the study demonstrates that coupling or decoupling of the mechanical and electrical variables is a function of the distance to the crack tip as well as of the crack orientation

fracture in magnetoelectroelastic materials using the extended finite element method static fracture analyses in two dimensional linear magnetoelectroelastic mee solids is studied by means of the extended finite element method x fem in the x fem crack modeling is facilitated by adding a discontinuous function and the crack tip asymptotic functions to the standard finite element approximation using the framework of partition of unity in this study media possessing fully coupled piezoelectric piezomagnetic and magnetoelectric effects are considered new enrichment functions for cracks in transversely isotropic mee materials are derived and the computation of fracture parameters using the domain form of the contour interaction integral is presented the convergence rates in energy for topological and geometric enrichments are studied excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method copyright 2011 john wiley sons ltd

frequency wavenumber inversion of acoustic data we describe an inversion procedure for obtaining velocities densities and layer thicknesses from multioffset reflection seismograms generated by a point source excitation in a stratified acoustic medium the reference seismograms are decomposed into cylindrical waves in the frequency wavenumber domain where the inversion is performed by minimizing the difference between the reference data and modeled data using a least squares technique the modeling algorithm is based on the classical reflectivity method we include the effects of limited temporal and spatial apertures by applying a frequency and wavenumber dependent filter in the modeling process this inversion is suited for jacobian matrix methods where the first derivatives of the model generated data with respect to the model parameters are calculated from analytical expressions together with the modeling the numerical examples show that the inversion algorithm recovers with reasonable accuracy the parameters used to generate the reference data the low frequency trends of both the velocity and the density are restored the convergence of the iterative minimization algorithm is fast

direct nonlinear q compensation of seismic primaries reflecting from a stratified two parameter absorptive medium q compensation of seismic primaries that have reflected from a stratified absorptive dispersive medium may be posed as a direct nonlinear inverse scattering problem if the reference medium is chosen to be nonattenuating and homogeneous an inverse scattering q compensation procedure may be derived that is highly nonlinear in the data but which operates in the absence of prior knowledge of the properties of the subsurface including its q structure it is arrived at by 1 performing an order by order inversion of a subset of the born series 2 isolating and extracting a component of the resulting nonlinear inversion equations argued to enact q compensation and 3 mapping the result back to data space once derived the procedure can be understood in terms of nonlinear interaction of the input primary reflection data the attenuation of deeper primaries is corrected by an operator built automatically using the angle and frequency variations of all shallower primaries a simple synthetic exam

electroelastic analysis of an anti plane shear crack in a piezoelectric ceramic strip the electroelastic analysis of a piezoelectric strip with an anti plane shear crack whose surfaces are parallel to the strip boundaries is made four cases of combined electromechanical loadings applied at the strip boundaries are considered for the case of a crack lying at the center of a strip the electroelastic field field intensity factors and energy release rate can be determined in explicit form via solving a resulting singular integral equation for other cases they can be given in terms of an auxiliary function which is obtained as a power series of via solving a resulting fredholm integral equation using the iterative method 2 being the ratio of the crack length over the strip width some numerical results are presented graphically to show the influence of crack length and crack position on the normalized energy release rate

an efficient multiscale method for time domain waveform tomography this efficient multiscale method for time domain waveform tomography incorporates filters that are more efficient than hamming window filters a strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain a staggered grid explicit finite difference method with fourth order accuracy in space and second order accuracy in time is used for forward modeling and the adjoint calculation the adjoint method is utilized in inverting for an efficient computation of the gradient directions in the multiscale approach multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography the method is applied successfully to 1d and 2d heterogeneous models it can accurately recover low and high wavenumber components of the velocity models the inversion result for the 2d model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional single scale method

exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack based on the complex potential approach the two dimensional problems in a piezoelectric material containing an elliptic hole subjected to uniform remote loads are studied the explicit closed form solutions satisfying the exact electric boundary condition on the hole surface are given both inside and outside the hole when the elliptic hole degenerates into a crack the field intensity factors are obtained it is shown that the stress intensity factors are the same as that of isotropic material while the electric displacement intensity factor depends on both the material properties and the mechanical loads but not on the electric loads in other words the uniform electric loads have no influence on the field singularities it is also shown that the impermeable crack assumption used previously to simply the electric condition is not valid to crack problems in piezoelectric materials

waveform inversion for attenuation estimation in anisotropic media abstractrobust estimation of attenuation coefficients remains a challenging problem especially for heterogeneous anisotropic media here we apply waveform inversion wi to perform attenuation analysis in heterogeneous vti transversely isotropic with a vertical symmetry axis media a time domain finite difference algorithm based on the standard linear solid model simulates nearly constant quality factor values in a specified frequency band we employ the adjoint state method to derive the gradients of the objective function based on the born approximation four parameters describing the attenuation coefficients of p and sv waves are updated simultaneously with a quasi newton optimization algorithm to remove the time shifts between the modeled and observed data caused by velocity errors we apply a local similarity technique the inversion still requires a sufficiently accurate velocity model to minimize the trade off between the contributions of velocity and attenuation to amplitudes the inversion

the boundary contour method for piezoelectric media with linear boundary elements this paper presents a development of the boundary contour method bcm for piezoelectric media first the divergence free property of the integrand of the piezoelectric boundary element is proved secondly the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and green s functions computer methods in applied mechanics and engineering 1998 158 65 for piezoelectric media the bcm is applied to the problem of piezoelectric media finally numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method bem the numerical results of the bcm coincide very well with the exact solution and the feasibility and efficiency of the method are verified copyright 2003 john wiley sons ltd

new developments concerning piezoelectric materials with defects abstract the problem of a piezoelectric body with an elliptic cavity is revisited within the frame work of two dimensional electro elasticity in contrast to our previous formulation the present analysis is based on the use of exact electric boundary conditions at the rim of the hole thus avoiding the common assumption of electric impermeability expressions for the elastic and electric variables induced inside and outside the cavity are derived in closed form in terms of complex potentials comparisons between present exact and previous approximate models are effected to establish the limitations associated with the assumption of the impermeable hole in particular when it becomes a slit crack

structural boundary design via level set and immersed interface methods we develop and test an algorithmic approach to the boundary design of elastic structures the goal of our approach is two fold first to develop a method which allows one to rapidly solve the two dimensional lam equations in arbitrary domains and compute for example the stresses and second to develop a systematic way of modifying the design to optimize chosen properties at the core our approach relies on two distinct steps given a design we first apply an explicit jump immersed interface method to compute the stresses for a given design shape we then use a narrow band level set method to perturb this shape and progress towards an improved design the equations of 2d linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers this effectively reduces the dimensionality of the problem by one once the stresses are found the level set method which represents the design structure through an embedded implicit function is used in the second step to alter the shape with velocities depending on the stresses in the current design criteria are provided for advancing the shape in an appropriate direction and for correcting the evolving shape when given constraints are violated

analysis of a central crack normal to a piezoelectric orthotropic interface this study is concerned with the general solution of the field intensity factors and energy release rate for a griffith crack normal to the interface between a rectangular piezoelectric ceramic and two same rectangular orthotropic materials of finite length under combined in plane electrical and anti plane mechanical loading both electrically continuous and impermeable crack surface conditions are considered employing fourier transforms and fourier series the problem is reduced to dual integral forms the solution to the resulting expressions is expressed in terms of fredholm integral equation of the second kind analytical solutions are provided for a number of cases to study the influence of material combinations geometry crack surface boundary and loading conditions the work also examines the influence of the crack boundary conditions on the results of the stress intensity factors and the energy release rate