geometric invariant theory and the generalized eigenvalue problem let g be a connected reductive subgroup of a complex connected reductive group hat g fix maximal tori and borel subgroups of g and hat g consider the cone mathcal lr g hat g generated by the pairs nu hat nu of dominant characters such that v nu is a submodule of v hat nu with usual notation here we give a minimal set of inequalities describing mathcal lr g hat g as a part of the dominant chamber in other words we describe the facets of mathcal lr g hat g which intersect the interior of the dominant chamber we also describe smaller faces finally we are interested in some classical redundant inequalities

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

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

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

quiver generalization of a conjecture of king tollu and toumazet abstract stretching the parameters of a littlewood richardson coefficient of value 2 by a factor of n results in a coefficient of value n 1 12 9 19 we give a geometric proof of a generalization for representations of quivers

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

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

a generalization of fulton s conjecture for arbitrary groups we prove a generalization of fulton s conjecture which relates intersection theory on an arbitrary flag variety to invariant theory

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

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

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

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

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 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

the multiplicative eigenvalue problem and deformed quantum cohomology abstract we construct deformations of the small quantum cohomology rings of homogeneous spaces g p and obtain an irredundant set of inequalities determining the multiplicative eigen polytope for the compact form k of g

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

effectively categorical abelian groups abstract we study effective categoricity of computable abelian groups of the form i h where h is a subgroup of q such groups are called homogeneous completely decomposable it is well known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite we study n 0 categoricity in this class of groups for n 1 we introduce a new algebraic concept of s independence which is a generalization of the well known notion of p independence we develop the theory of s independent sets we apply these techniques to show that every homogeneous completely decomposable group is 3 0 categorical we prove that a homogeneous completely decomposable group of infinite rank is 2 0 categorical if and only if it is isomorphic to the free module over the localization of z by a computably enumerable set of primes p with the semi low complement within the set of all primes we apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers including the free abelian group

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

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

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

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

quantum generalization of the horn conjecture recall that the lie algebra of the special unitary group su n is isomorphic to the real vector space of traceless hermitian matrices as representations of su n and hence the terminology additive eigenvalue problem for su n in 1962 horn 17 gave a conjectural solution to a problem equivalent to the above additive eigenvalue problem for su n by a recursively determined system of inequalities in 19 klyachko gave a solution to the additive eigenvalue problem for su n in terms of a certain system of inequalities to write down this system we need to know which structure constants in the cohomology of gr r n written in the schubert basis 0 2 be a positive integer characterize the possible eigenvalues a1 a2 as of matrices a a 2 a e su n which satisfy a a a 1

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

scaling of conformal blocks and generalized theta functions over overline mathcal m _ g n by way of intersection theory on overline mathcal m g n we show that geometric interpretations for conformal blocks as sections of ample line bundles over projective varieties do not have to hold at points on the boundary we show such a translation would imply certain recursion relations for first chern classes of these bundles while recursions can fail geometric interpretations are shown to hold under certain conditions

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

geometric proof of a conjecture of fulton we give a geometric proof of a conjecture of fulton on the multiplicities of irreducible representations in a tensor product of irreducible representations for gl r

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

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

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

eigenvalues of majorized hermitian matrices and littlewood richardson coefficients abstract answering a question raised by s friedland we show that the possible eigenvalues of hermitian matrices or compact operators a b and c with c a b are given by the same inequalities as in klyachko s theorem for the case where c a b except that the equality corresponding to tr c tr a tr b is replaced by the inequality corresponding to tr c tr a tr b the possible types of finitely generated torsion modules a b and c over a discrete valuation ring such that there is an exact sequence b c a are characterized by the same inequalities

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

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

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

an eccentric crack in a piezoelectric strip under anti plane shear impact loading we consider the problem of determining the singular stress and electric fields in a piezoelectric ceramic strip containing a griffith eccentric crack off the center line under anti plane shear impact loading with the theory of linear piezoelectricity laplace transform and fourier transforms are used to reduce the problem to a pair of dual integral equations which are expressed to a fredholm integral equation of the second kind numerical values on the dynamic stress intensity factor and the dynamic energy release rate are obtained