on the numerical evaluation of electrostatic fields in dense random dispersions of cylinders we consider the interface boundary value problem which arises in the evaluation of electrostatic fields in composite materials consisting of dense random dispersions of cylinders in a uniform background this is a well studied problem from the viewpoint of homogenization and effective medium theory but one for which accurate numerical simulations have been difficult to obtain size effects in particular have been neglected due to the expense of solving the field equation in the presence of large numbers of close to touching inclusions such features require very fine discretizations even with the use of adaptive gridding and cause the linear systems which arise to be highly ill conditioned in this paper we present a new integral equation method for the solution of the interface problem which uses a recently developed method of images to resolve the close to touching interactions and the fast multipole method to compute far field interactions only minutes of workstation time are needed to solve the field equation with thousands of inclusions allowing us to carry out large scale statistical studies of the effective conductivity of random two phase materials at a variety of volume fractions and contrast ratios

new solutions for effective elastic moduli of microcracked solids abstract a new energy balance equation is proposed to evaluate the effective moduli of solids with randomly dispersed cracks in this study it is assumed that the potential energy released by embedding a circular or spherical rve with microcracks into an infinite matrix is equal to that induced by introducing its effective medium into the identical infinite matrix new non interacting and self consistent solutions for the effective moduli of linear elastic isotropic solids with randomly oriented microcracks were evaluated on the basis of the present energy balance equation the linearization of the two present solutions leads to the dilute solution in the limit of small crack density comparison studies with various existing solutions are also presented in detail

bounds for effective parameters of multicomponent media by analytic continuation recently d bergman introduced a method for obtaining bounds for the effective dielectric constant or conductivity of a two component medium this method does not rely on a variational principle but instead exploits the properties of the effective parameter as an analytic function of the ratio of the component parameters we extend the method to multicomponent media using techniques of several complex variables

addition formulas and the rayleigh identity for arrays of elliptical cylinders we apply the rayleigh method to solve the problem where a uniform electrostatic field is imposed upon a rectangular array of elliptical cylinders embedded in a matrix of unit dielectric constant this new formulation overcomes geometric restrictions inherent in previous methods and is shown in principle and in various examples to converge for all possible geometries of the array and inclusion also presented are forms of both the interior and exterior addition formulas for harmonic functions in elliptical coordinates that possess optimal regions of convergence

a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d we present a new method for the evolution of inextensible vesicles immersed in a stokesian fluid we use a boundary integral formulation for the fluid that results in a set of nonlinear integro differential equations for the vesicle dynamics the motion of the vesicles is determined by balancing the non local hydrodynamic forces with the elastic forces due to bending and tension numerical simulations of such vesicle motions are quite challenging on one hand explicit time stepping schemes suffer from a severe stability constraint due to the stiffness related to high order spatial derivatives and a milder constraint due to a transport like stability condition on the other hand an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step we present two semi implicit schemes that circumvent the severe stability constraints on the time step and whose computational cost per time step is comparable to that of an explicit scheme we discretize the equations by using a spectral method in space and a multistep third order accurate scheme in time we use the fast multipole method fmm to efficiently compute vesicle vesicle interaction forces in a suspension with a large number of vesicles we report results from numerical experiments that demonstrate the convergence and algorithmic complexity properties of our scheme

a numerical simulation for effective elastic moduli of plates with various distributions and sizes of cracks abstract a fast convergent numerical model is developed to calculate the effective moduli of plates with various distributions and sizes of cracks in which the crack line is divided into m parts to obtain the unknown traction on the crack line when m 1 the model reduces to kachanov s approximation method int j solids struct 23 1987 23 six types of crack distributions and three kinds of crack sizes are considered which are four regular equilateral triangle equilateral hexagon rectangle and diamond and two random distributions random location and orientation and parallel orientation and random location and one two and random crack sizes some typical examples are also analyzed using the finite element method fem to validate the present model then the effective moduli associated with the crack distributions and sizes are calculated in detail the present results for the regular distributions show some very interesting phenomena that have not been revealed before and for the two random distributions as the effective moduli depend on samples due to the randomness the effect of the sample size and number are analyzed first then effective moduli for plates with the three sizes of cracks are calculated it is found that the effect of crack sizes on the effective moduli is significant for high crack densities and small for low crack densities and the random crack size leads to the lowest effective moduli the present numerical results are compared with several popular micromechanics models to determine which one can provide the optimum estimation of the effective moduli of cracked plates with general crack densities furthermore some existing numerical results are analyzed and discussed

interacting elliptic inclusions by the method of complex potentials abstract an accurate series solution has been obtained for a piece homogeneous elastic plane containing a finite array of non overlapping elliptic inclusions of arbitrary size aspect ratio location and elastic properties the method combines standard muskhelishvili s representation of general solution in terms of complex potentials with the superposition principle and newly derived re expansion formulae to obtain a complete solution of the many inclusion problem by exact satisfaction of all the interface conditions a primary boundary value problem stated on a complicated heterogeneous domain has been reduced to an ordinary well posed set of linear algebraic equations a properly chosen form of potentials provides a remarkably simple form of solution and thus an efficient computational algorithm the theory developed is rather general and can be applied to solve a variety of composite mechanics problems the advanced models of composite involving up to several hundred inclusions and providing an accurate account for the microstructure statistics and fiber fiber interactions can be considered in this way the numerical examples are given showing high accuracy and numerical efficiency of the method developed and disclosing the way and extent to which the selected structural parameters influence the stress concentration at the matrix inclusion interface

integral equation methods and numerical solutions of crack and inclusion problems in planar elastostatics we present algorithms for crack and inclusion problems in planar linear elastostatics the algorithms are based on new integral equations for the pure crack problem the integral equations are of fredholm s second kind our algorithms show great stability and allow for solutions to problems more complex than have previously been possible our results are orders of magnitudes more accurate than those of previous investigators which rely on integral equations of fredholm s first kind

on transformation strains and uniform fields in multiphase elastic media the effect of local eigenstrain and eigenstress fields or transformation fields on the local strains and stresses is explored in multiphase elastic solids of arbitrary geometry and material symmetry the residual local fields caused by such transformation fields are sought in terms of certain transformation influence functions and transformation concentration factor tensors general properties of these functions and concentration factors and their relation to the analogous mechanical influence functions and concentration factors are established in part with the help of uniform strain fields in multiphase media specific estimates of the transformation concentration factor tensors are evaluated by the self consistent and mori tanaka methods it is found here that although the two methods use different constraint tensors in solutions of the respective dilute problems their estimates of the mechanical thermal and transformation concentration factor tensors and of the overall stiffness of multiphase media have a similar structure proofs that guarantee that these methods comply with the general properties of the transformation influence functions and provide diagonally symmetric estimates of the overall elastic stiffness are given for two phase and multiphase systems consisting of or reinforced by inclusions of similar shape and alignment one of the possible applications of the results in analysis of overall instantaneous properties and local fields in inelastic composite materials is described in the following paper

solids with periodically distributed cracks abstract a systematic method is presented for estimating the overall properties of solids with periodically distributed cracks in view of the periodicity the displacement strain and stress fields of the cracked solid can be expressed in fourier series elastic solids with periodically distributed flat voids are considered first the results for cracks are then obtained by letting the thickness aspect ratio of the void approach zero this limiting process is performed with care the only approximation involved is the distribution of the homogenization eigenstrains which is assumed to be piecewise constant the estimate of overall elastic moduli crack opening displacements and stress intensity factors eventually reduces to the calculation of several infinite series the formulation is valid for elliptic as well as two dimensional line slit like cracks and cracks with arbitrary shapes it fully includes the interaction effects

a simple technique for finding effective elastic constants of cracked solids for arbitrary crack orientation statistics abstract the effective elastic properties of cracked solids are anisotropic if the cracks have preferred orientations in this paper a simple scheme for evaluating the elastic stiffness tensor for an arbitrary orientation distribution of cracks at finite crack densities is presented the approach is based on a tensorial transformation of the effective elastic constants for isotropic orientation statistics through the use of a second order crack density tensor

elastic equilibrium of a medium containing a finite number of aligned spheroidal inclusions abstract the strict solution in series is obtained of the elasticity theory problem for an unbounded domain containing some aligned spheroidal inhomogeneities under uniform far field loads the essence of the method used is the representation of the displacement field in a multiply connected domain as a sum of general solutions for corresponding single connected domains each term of this sum in turn is expanded into series on vectorial partial solutions of lame s equation in a local spheroidal basis in order to satisfy exactly all interfacial boundary conditions the re expansion formulae addition theorems for external partial solutions are used as a result the primary boundary value problem of elasticity theory is reduced to an infinite set of linear algebraic equations the convergence rate of the proposed solution procedure is evaluated numerically some numerical results demonstrating the influence on stress distribution of material properties spatial position of inclusions and external load are presented

the boundary due terms in the green operator of inclusion patterns from distant to contact and to connected situations using radon transforms illustration for spheroid alignments in isotropic media abstract we examine the boundary due components in the mean modified green operator integral green operator for short of an inclusion pattern in distant contact and contact connection transitions the direct rt and inverse irt radon transforms which allow specification of the different contributions to the mean green operator of the pattern in simple geometrical terms are used the already well documented case of axially symmetric alignments of equidistant identical oblate spheroids in an infinite matrix of isotropic elastic like or dielectric like properties is treated up to infinite alignments and for any aspect ratio from unity spheres to infinitesimal platelets simple closed forms for this mean green operator and for its different parts are newly obtained these closed forms allow an easy parametric study of the operator variations in terms of the alignment characteristics from distant to contact situations from contact to connection of the inclusions the changes in the green operator s contributions are pointed what provides relevant operator forms for the connected patterns these results are of interest in problems where phase percolation connectivity inversions or co continuity are implied

green s tensors and lattice sums for electrostatics and elastodynamics we present expressions for the green9s tensors in terms of lattice sums for general two dimensional arrays in both elastostatics and elastodynamics we represent the lattice sums in rapidly convergent forms by fourier transform methods we compare the static lattice sums with recently published highly accurate values for square arrays we also establish rayleigh identities for elastostatic and elastodynamic problems generalizing previous work in electrostatics and electromagnetism the rayleigh identities may be used in studies of phononic band gaps

the numerical calculation of two dimensional effective moduli for microcracked solids abstract there exist several micromechanics models for the determination of the effective moduli of microcracked solids and crack density is the only parameter in these models that characterizes the effect of microcracking a numerical hybrid bem method in conjunction with a unit cell model is proposed in the present paper to evaluate these micromechanics models a unit cell which can be considered as a representative block in the solid contains randomly distributed microcracks the unit cell is then assumed to be periodic in the solid so as to account for interactions between cracks inside and outside the cell there are stochastic variations of the estimated moduli for different microcrack distributions two groups of microcracks with the same crack density one with a low number of large cracks and the other with a large number of small cracks show the same range of stochastic variations and the same mean of effective moduli for random distributions of microcracks the effective moduli based on this numerical method for randomly distributed cracks and parallel cracks are compared with those from various micromechanics models while the differential method provides the closest estimation to the mean of the numerical results at low crack density the generalized self consistent method is much more accurate at relatively high crack density

stress intensity factor and effective stiffness of a solid containing aligned penny shaped cracks abstract the stress state and effective elastic moduli of an isotropic solid containing equally oriented penny shaped cracks are evaluated accurately the geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of arbitrarily placed aligned circular cracks a rigorous analytical solution of the boundary value problem of the elasticity theory has been obtained using the technique of triply periodic solutions of the lame equation by exact satisfaction of the boundary conditions on the cracks surfaces the primary problem is reduced to solving an infinite set of linear algebraic equations an asymptotic analysis of the stress field has been performed and the exact formulae for the stress intensity factor sif and effective elasticity tensor are obtained the numerical results are presented demonstrating the effect of the crack density parameter and arrangement type on sif and overall elastic response of a solid and comparison is made with known approximate theories

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

existence of local sufficiently smooth solutions to the complex monge ampere equation we prove the c local solvability of the n dimensional complex monge ampere equation det u ij f z u u f 0 in a neighborhood of any point z 0 where f z 0 0

transverse conductivity and longitudinal shear of elliptic fiber composite with imperfect interface abstract the paper addresses the problem of calculating the local fields and effective transport properties and longitudinal shear stiffness of elliptic fiber composite with imperfect interface the rayleigh type representative unit cell approach has been used the micro geometry of composite is modeled by a periodic structure with a unit cell containing multiple elliptic inclusions the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions an appropriate choice of the potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the local gradient and flux fields the convergence of solution has been verified and the parametric study of the model has been performed the obtained accurate statistically meaningful results illustrate a substantial effect of imperfect interface on the effective behavior of composite

a circular inclusion with circumferentially inhomogeneous interface in antiplane shear a general method is developed for the rigorous solution of a problem associated with a circular inclusion embedded within an infinite matrix in antiplane shear the bonding at the inclusion matrix interface is assumed to be imperfect most significant is the fact that the imperfection in the interface is assumed to be circumferentially inhomogeneous using analytic continuation the basic boundary value problem for two analytic functions is reduced to a first order differential equation for a single analytic function and the closed form solution is obtained the method is illustrated using several specific examples the results from these examples are compared to the corresponding results when the imperfection in the interface is homogeneous these comparisons illustrate how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the stresses induced within the inclusion

a sensitive interval of imperfect interface parameters based on the analysis of general solution for anisotropic matrix containing an elliptic inhomogeneity abstract a general solution is proposed to solve the plane problem for infinite anisotropic medium containing an elliptic inhomogeneity with imperfect interface the imperfect interface is usually described as a spring model with vanishing thickness based on the assumption that the tractions are continuous but the normal or tangential displacements are discontinuous due to a jump at the interface by means of the series expansion of the complex stress functions and the factor functions for the imperfect elliptic interface a general procedure to determine the coefficients in the series is illustrated and the convergent solutions are obtained by truncating finite number of terms in the series the present solutions are verified with available analytical results for the cases of perfect interface and debonded interface or hole the patterns of the stresses in the anisotropic medium or matrix and inhomogeneity due to the eigenstrains and far filed stresses are presented respectively a sensitive interval of interface parameters is suggested in which the influence of the change of interface parameters on the stress field is very obvious monotonic and non monotonic change of peak stresses on a subset of interface parameters is also discussed the method and the procedure proposed in this work can be used in the analysis of strength and failure of anisotropic materials containing an elliptic inhomogeneity with imperfect interface

the transport properties of layers of elliptical cylinders the rayleigh method has been used in previous work to determine the effective transport properties of a rectangular array of elliptical cylinders for all cases where the ellipses are non intersecting however the calculation of the elliptic lattice sums that naturally arose was numerically inefficient and analytically cumbersome in this present work we use integral transforms to obtain rapidly convergent series for one dimensional elliptical lattice sums and use these lattice sums to determine the transport properties of composites constructed from multiple layers of elliptical cylinders we study the convergence of effective transport properties as a function of the number of layers and show that this convergence is extremely rapid further we use the integral transform representation to exhibit the simplest form of the interior addition formula for harmonic functions in elliptical coordinates

elastic equilibrium of a half plane containing a finite array of elliptic inclusions abstract an accurate analytical method has been proposed to solve for stress in a half plane containing a finite array of elliptic inclusions the last being a model of near surface zone of the fibrous composite part the method combines the muskhelishvili s method of complex potentials with the fourier integral transform technique by accurate satisfaction of all the boundary conditions a primary boundary value elastostatics problem for a piece homogeneous domain has been reduced to an ordinary well posed set of linear algebraic equations a properly chosen form of potentials provides a remarkably simple form of equations and thus an efficient computational algorithm the theory developed is rather general and can be applied to solve a variety of elastostatics problems up to several hundred interacting inclusions can be considered in this way in practical simulations which makes the model of composite half plane realistic and flexible enough to account for the microstructure statistics the stress concentration factors and effective thermoelastic properties of random structure composites with dilute concentration of fibers are estimated in the vicinity of a free edge the numerical examples are given showing accuracy and numerical efficiency of the developed method and disclosing the way and extent to which the nearby free or loaded boundary influences the local and mean stress concentration in the fibrous composite

a renormalization method for the evaluation of lattice sums a number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums sums over all sites of an infinite perfect lattice of some potential energy function one classical example the calculation of lattice sums of circular and spherical harmonics dates back to the last century to lord rayleigh s work on computing the effective conductivity of a simple composite while lord rayleigh presented an efficient asymptotic method for two dimensional problems he resorted to direct evaluation of the lattice sums in the three dimensional case more recent methods based on ewald summation have been developed by nijboer and de wette schmidt and lee and others in this article a fast method for evaluating lattice sums which is based on a new renormalization identity is described

interaction of two elliptic inclusions abstract the paper presents the solution of a boundary value problem involving two interacting elliptical inhomogeneities in an infinite elastic body the loading cases considered include a far field biaxial tension and a thermally induced residual field that is modeled by uniform eigenstrains sustained by the inhomogeneities the mathematical formulation is based upon the papkovich neuber displacement approach certain interesting aspects of the solution and the effects of perfectly bonded and slipping interfaces are discussed in some detail

corner singularities for elliptic problems integral equations graded meshes quadrature and compressed inverse preconditioning we take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points use of non standard integral equations graded meshes interpolatory quadrature and compressed inverse preconditioning are techniques that are explored developed mixed and tested on some familiar problems in materials science the recursive compressed inverse preconditioning the major novelty of the paper turns out to be particularly powerful and when it applies eliminates the need for mesh grading completely in an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 1 2 in another example five times as large but with a conductivity ratio of only a hundred we achieve an estimated relative error of 10 1 4

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

linear viscoelastic analysis of a semi infinite porous medium abstract this paper considers the problem of a semi infinite isotropic linear viscoelastic half plane containing multiple non overlapping circular holes the sizes and the locations of the holes are arbitrary constant or time dependent far field stress acts parallel to the boundary of the half plane and the boundaries of the holes are subjected to uniform pressure three types of loading conditions are assumed at the boundary of the half plane a point force a force uniformly distributed over a segment a force uniformly distributed over the whole boundary of the half plane the solution of the problem is based on the use of the correspondence principle the direct boundary integral method is applied to obtain the governing equation in the laplace domain the unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex fourier series a system of linear equations is obtained by using a taylor series expansion the viscoelastic stresses and displacements at any point of the half plane are found by using the viscoelastic analogs of kolosov muskhelishvili s potentials the solution in the time domain is obtained by the application of the inverse laplace transform all the operations of space integration the laplace transform and its inversion are performed analytically the method described in the paper allows one to adopt a variety of viscoelastic models for the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered the accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ansys

scattering by cavities of arbitrary shape in an infinite plate and associated vibration problems this paper presents a solution for the displacement of a uniform elastic thin plate with an arbitrary cavity modelled using the biharmonic plate equation the problem is formulated as a system of boundary integral equations by factorizing the biharmonic equation with the unknown boundary values expanded in terms of a fourier series at the edge of the cavity we consider free edge simply supported and clamped boundary conditions methods to suppress ill conditioning which occurs at certain frequencies are discussed and the combined boundary integral equation method is implemented to control this problem a connection is made between the problem of an infinite plate with an arbitrary cavity and the vibration problem of an arbitrarily shaped finite plate using the jump discontinuity present in single layer distributions at the boundary the first few frequencies and modes of displacement are computed for circular and elliptic cavities which provide a check on our numerics and results for the displacement of an infinite plate are given for four specific cavity geometries and various boundary conditions

regression to the mean for nonnormal populations summary regression to the mean is the term used to describe the effect by which individuals selected on the basis of an initial measurement which is particularly high or low compared with other members of the population have upon remeasurement values nearer to the population mean for the normal distribution simple expressions for estimates of the magnitude of this effect are available many biological variables however are not normally distributed estimates for regression to the mean for nonnormal data are given based on edgeworth series and saddlepoint approximations

a high order accurate algorithm for electrostatics of overlapping disks a fast and accurate algorithm for the computation of effective electric and mathematically equivalent properties of composites with nonsmooth interfaces is reported the algorithm is based on an integral equation reformulation of the electrostatic partial differential equation and a fast hierarchical technique for potential field evaluation in a numerical example 200 large and strongly inhomogeneous aggregates of randomly overlapping disks are solved with a relative error of 0 0005

methods for assessing whether change depends on initial value statistical analysis of whether the change in a variable depends on its initial value in clinical trials and other studies is complicated by the phenomenon of regression to the mean we review this problem and examine some approaches for handling it macgregor s log log plot fails to correct for the regression effect while oldham s method of plotting the change against the average of initial and final values is shown to give misleading results when the effect of treatment varies between subjects or when subjects are selected for study if their initial observations fall above or below a specified cut off point these results are illustrated by computer simulations we prefer the use of blomqvist s method of correcting the association between change and initial value to allow for the regression effect

semi analytic solution for multiple interacting three dimensional inhomogeneous inclusions of arbitrary shape in an infinite space this paper develops a semi analytic solution for multiple arbitrarily shaped three dimensional inhomogeneous inclusions embedded in an infinite isotropic matrix under external load all interactions between the inhomogeneous inclusions are taken into account in this solution the inhomogeneous inclusions are discretized into small cuboidal elements each of which is treated as a cuboidal inclusion with initial eigenstrain plus unknown equivalent eigenstrain according to the equivalent inclusion method all the unknown equivalent eigenstrains are determined by solving a set of simultaneous constitutive equations established for each equivalent cuboidal inclusion the final solution is obtained by summing up the closed form solutions for each individual equivalent cuboidal inclusion in an infinite space the solution evaluation is performed by application of the fast fourier transform algorithm which greatly increases the computational efficiency finally the solution is validated by taking eshelby s analytic solution of an ellipsoidal inhomogeneous inclusion as a benchmark and by the finite element analysis a few sample results are also given to demonstrate the generality of the solution the solution may have potentially significant applications in solving a wide range of inhomogeneity related problems copyright 2011 john wiley sons ltd

general saddlepoint approximations with applications to l statistics abstract saddlepoint approximations are extended to general statistics the technique is applied to derive approximations to the density of linear combinations of order statistics including trimmed means a comparison with exact results shows the accuracy of these approximations even in very small sample sizes

effect of crack orientation statistics on effective stiffness of mircocracked solid this paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics to do so the representative unit cell approach has been used the microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks a sufficient number is taken to account for the microstructure statistics the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions a proper choice of potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields the convergence study has been performed the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks comparison has been made with the selected simple micromechanical models namely non interaction approximation differential scheme and modified differential scheme it is found that among these models the differential scheme provides the best fit of the numerical data

nonparametric regression to the mean available data may reflect a true but unknown random variable of interest plus an additive error which is a nuisance the problem in predicting the unknown random variable arises in many applied situations where measurements are contaminated with errors it is known as the regression to the mean problem there exists a well known solution when both the distributions of the true underlying random variable and the contaminating errors are normal this solution is given by the classical regression to the mean formula which has a data shrinkage interpretation we discuss the extension of this solution to cases where one or both of these distributions are unknown and demonstrate that the fully nonparametric case can be solved for the case of small contaminating errors the resulting nonparametric regression to the mean paradigm can be implemented by a straightforward data sharpening algorithm that is based on local sample means asymptotic justifications and practical illustrations are provided

on the method of images for systems of closely spaced conducting spheres in this paper a new formalism is developed to generate the image of a multipole expansion of arbitrary order in a conducting sphere a method of images based on this formalism is subsequently combined with the method of moments to solve boundary value problems for systems of closely spaced conducting spheres the effectiveness of the new hybrid method is demonstrated through several numerical examples

analysis of a mode i crack perpendicular to an imperfect interface the elastostatic problem of a mode i crack embedded in a bimaterial with an imperfect interface is investigated the crack is in proximity to and perpendicular to the imperfect interface which is governed by linear spring like relations the fourier transform is applied to reduce the associated mixed boundary value problem to a singular integral equation with cauchy kernel by numerically solving the resulting equation stress intensity factors near both crack tips are evaluated obtained results reveal that the stress intensity factors in the presence of the imperfect interface vary between that with a perfect interface and that with a completely debonding interface moreover an increase in the interface parameters decreases the stress intensity factors in particular when crack approaches to the weakened interface closer the stress intensity factors become larger for a sliding interface and become larger or smaller for a winkler interface depending on the crack lying in a stiffer or softer material the influences of the imperfection of the interface on the stress intensity factors for a bimaterial composed of aluminum and steel are presented graphically

asymptotic studies of closely spaced highly conducting cylinders we consider the solution of the scalar transport problem for a pair of nearly touching cylinders of high conductivity we obtain an expression for the set of multipole moments of the potential distribution for this problem in terms of the hypergeometric function we apply this expression in the estimation of truncation errors occurring in the matrix solution of the corresponding transport problem for the square array of cylinders consequently we are able to calculate the array transport coefficient for arbitrarily high cylinder conductivities and arbitrarily small cylinder separations we derive and verify an expression for this coefficient which is uniformly valid throughout the whole asymptotic region when highly conducting cylinders approach touching

the coherent potential approximation is a realizable effective medium scheme the effective electrical conductivity of an aggregate composed of grains of various conductivities is frequently estimated by the coherent potential approximation which embodies a local effective medium concept it is proved rigorously that this approximation is exact for a wide class of hierarchical model composites made of spherical grains the starting material 0 in the hierarchy is chosen arbitrarily otherwise materialj 1 2 consists of equisized spheres sayj spheres of arbitrary conductivities embedded in materialj 1 the spatial distribution of thej spheres must satisfy a mild homogeneity condition and their radiusr j must asymptotically increase faster than exponentially withj furthermore the minimum spacing 2s j between thej spheres is such that the ratios j r j diverges on the basis of these and some further ancillary conditions it is established that the coherent potential approximation becomes asymptotically exact for the effective conductivity of materialj the results extend to other effective parameters of the composites including the thermal conductivity dielectric constant and magnetic permeability in addition the model composites and the proof of realizability may be generalized to allow non spherical grains

smoothness up to the boundary for solutions of the nonlinear and nonelliptic dirichlet problem pour le probleme de dirichlet associe a une equation aux derivees partielles d ordre 2 reelle generale f x u u 2 u 0 dans un ensemble ouvert lisse de r n on demontre la regularite jusqu a la frontiere de la solution u pour laquelle la forme caracteristique linearisee est non negative et satisfait une condition de crochets de hormander

the dirichlet problems for a class of semilinear sub elliptic equations where x x1 xm is a system of real smooth vector elds de ned in an open domain m rn n 2 is a bounded open subdomain of m with smooth c x c0 0 x j denote the adjoint of xj we assume that the system of vector elds x x1 xm satis es the following h ormander s condition x1 xm together with their commutators x x 1 x s 1 x s up to some xed length r span the tangent space at each point of m we have the following theorem

an elliptic inclusion with imperfect interface in anti plane shear abstract a semi analytic solution is developed for the problem associated with an elliptic inclusion embedded within an infinite matrix in anti plane shear the bonding at the inclusion matrix interface is assumed to be homogeneously imperfect the interface is modeled as a spring interphase layer with vanishing thickness the behaviour of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface complex variable techniques are used to obtain infinite series representations of the stresses induced within the inclusion the results obtained demonstrate how the non uniform stress field and the average stresses inside the inclusion vary with the aspect ratio of the inclusion and the parameter describing the imperfect interface in addition it is shown that in some cases depending on the aspect ratio of the ellipse it is possible to identify specific values of the interface parameter which correspond to maximum peak stress along the interface

eshelby s problem for two dimensional piezoelectric inclusions of arbitrary shape eshelby s problem for piezoelectric inclusions of arbitrarily shaped crosssection remains a challenging topic in this paper a simple method is presented to obtain an analytic solution for eshelby

transport properties of cylinder pairs and of the square array of cylinders we evaluate the multipole coefficients b 2n 1 in the potential expansion for a perfectly conducting cylinder pair separated by unit potential difference we obtain a simple expression for the b 2n 1 when the cylinders approach each other and show that this expression also holds for a square array of cylinders we derive a formula linking ratios of multipole coefficients from green9s theorem and use this to obtain an estimate for the effect of changing the cylinder conductivity from infinity to a large finite value on the array conductivity

semi analytical computation of laplacian green functions in three dimensional domains with disconnected spherical boundaries abstract the generalized method of separation of variables gmsv is applied to solve boundary value problems for the laplace operator in three dimensional domains with disconnected spherical boundaries e g an arbitrary configuration of non overlapping partially reactive spherical sinks or obstacles we consider both exterior and interior problems and all most common boundary conditions dirichlet neumann robin and conjugate one using the translational addition theorems for solid harmonics to switch between the local spherical coordinates we obtain a semi analytical expression of the green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions although the numerical computation of the coefficients involves series truncation and solution of a system of linear algebraic equations the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the gmsv particularly efficient especially for exterior problems the obtained green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate escape probability harmonic measure residence time and mean first passage time to name but a few the relevant aspects of the numerical implementation and potential applications in chemical physics heat transfer electrostatics and hydrodynamics are discussed

duality relation for the maxwell system this paper is intended to establish a link between the vector maxwell system for three dimensional 3d and 2d finite photonic crystals in the low frequency limit for this we generalize the classical results of keller and dykhne chessboard problem to periodic media described by piecewise continuous permittivity profiles our theorem enlights the result of mendelson polycrystalline and multiphase media in the framework of homogenization theory of elliptic operators in fine we give illustrative examples by using both integral equation and variational approaches via the so called method of fictitious charges and finite element method