analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder in this work we consider a cylindrical structure composed of a nonlinear core inhomogeneity surrounded by a different nonlinear shell matrix we elaborate a technique for determining its linear elastic moduli second order elastic constants and the nonlinear elastic moduli which are called landau coefficients third order elastic constants firstly we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems then we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure the following step consists in the exact solution of such a linear problem by means of the complex elastic potentials as result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli which are valid for any volume fraction of the core embedded in the external shell

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

translation and related bounds for the response of a nonlinear composite conductor lower bounds for the overall energy functions of a class of nonlinear composite conductors are established by two different methods the translation method and the use of a comparison linear composite when the properties of the comparison linear composite are themselves estimated by the translation method the resulting comparison bound is demonstrated never to be superior to the bound obtained by direct application of the translation method to the nonlinear composite examples show however that the bounds obtained by these two methods in fact often coincide when they do the use of the comparison medium permits the development of more refined bounds by exploiting better bounds for the linear composite particular emphasis is placed on a limiting case of material behaviour for which explicit results can be obtained a composite dielectric that displays breakdown when the local electric field exceeds a critical value the formalism delivers an upper bound for the value of the mean field in the composite which induces breakdown in the composite bounds are also established for a composite with a power law relation between current and electric field

improved bounds for the overall properties of a nonlinear composite dielectric the construction of generalized hashin shtrikman bounds for nonlinear composite problems relies on the introduction of a comparison material in recent work a nonlinear comparison medium has been used however this requires detailed knowledge of the properties of the trial fields that are employed the fields used have the property of bounded mean oscillation and this enables the size of the penalty exacted by using a nonlinear comparison material to be controlled some recent results concerning riesz transforms and the beurling operator are used here to reduce the effect of the penalty and hence to generate improved bounds in addition an exact solution is established for a particular class of composites

on periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure abstract due to high computational costs associated with stochastic computational homogenization a highly complex random material microstructure is often replaced by simplified parametric ergodic and sometimes periodic models this replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions in the current contribution we perform a validation of both assumptions through various numerical examples to this end we compare large scale non simplified and non ergodic models with simplified ergodic and periodic solutions in addition we analyze the hill mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions as an example we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning the effect of these novel periodic boundary conditions is also analyzed by comparing with non ergodic simulation results

relaxation of three solenoidal wells and characterization of extremal three phase h measures we fully characterize quasiconvex hulls for three arbitrary solenoidal divergence free wells in dimension three with this aim we establish weak lower semicontinuity of certain functionals with integrands restricted to generic two dimensional planes and convex in up to three rank 2 directions within the planes within the framework of the theory of compensated compactness the latter represents an example when the differential constraints fail the constant rank condition but nevertheless the so called convexity still implies lower semicontinuity and fancyscript a quasiconvexity which essentially means that rank 2 convexity implies s quasiconvexity that is quasiconvexity in the sense of the divergence free differential constraints on the planes the proof employs a version of muller s estimates of haar wavelet projections in terms of the riesz transform the above semicontinuity result is then applied to the three solenoidal wells problem via analogs of sverak s nontrivial quasiconvex functions and connectedness properties of the rank 2 envelopes as another application of the semicontinuity result we obtain a geometric result of a more general nature characterization of certain extremal three point h measures for three phase mixtures of three characteristic functions in dimension three we also discuss the applicability of the results to problems with other differential constrains in particular to three linear elastic wells and further generalizations

homogenization of two phase elasto plastic composite materials and structures study of tangent operators cyclic plasticity and numerical algorithms we develop homogenization schemes and numerical algorithms for two phase elasto plastic composite materials and structures a hill type incremental formulation enables the simulation of unloading and cyclic loadings it also allows to handle any rate independent model for each phase we study the crucial issue of tangent operators elasto plastic or continuum versus algorithmic or consistent and anisotropic versus isotropic we apply two methods of extraction of isotropic tangent moduli we compare mathematically the stiffnesses of various tangent operators all rate equations are discretized in time using implicit integration we implemented two homogenization schemes mori tanaka and a double inclusion model and two plasticity models classical j 2 plasticity and chaboche s model with non linear kinematic and isotropic hardenings we consider composites with different properties and present several discriminating numerical simulations in many cases the results are validated against finite element fe or experimental data we integrated our homogenization code into the fe program abaqus using a user material interface umat a two scale procedure allows to compute realistic structures made of non linear composite materials within reasonable cpu time and memory usage examples are shown c 2002 elsevier science ltd all rights reserved

effective properties of composite materials with periodic microstructure a computational approach abstract this study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents the theoretical background of the method is recalled first two different families of numerical methods are considered to solve the problem the first is based on the finite element method the concept of macroscopic degrees of freedom is presented the implementation of periodicity conditions is discussed a general framework permitting either a strain or stress control is proposed the second numerical method is based on fast fourier transforms it considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain the solution of this problem is based on the explicit form of the periodic green s function of the reference medium the relative merits of the two methods are compared and several examples are discussed both methods give very comparable results on test examples and their domains of applications appear to be complementary

determination of rve size for random composites with local volume fraction variation abstract in this paper a novel computational procedure is proposed for the determination of the representative volume element rve size for random composites which is the basis of homogenization methods the proposed approach takes into account the local volume fraction variation by processing computer simulated images of composites with randomly scattered inclusions a variable number of microstructure models are derived directly from the images using the moving window technique each microstructure model contains different amount of inclusions measured by image analysis tools the proposed computational procedure is based on the solution of multiple boundary value problems under kinematic and static uniform boundary conditions using the extended finite element method xfem coupled with monte carlo simulation mcs in this way the statistical characteristics of the upper and lower bounds of the apparent material properties are computed the rve is attained within a prescribed tolerance by examining the convergence of these two bounds with respect to the mesoscale size the numerical results clearly demonstrate the significant effect of matrix inclusion stiffness ratio and volume fraction on the convergence rate the proposed computational procedure leads to rves which can give more realistic results when used in homogenization problems mesoscale random fields describing the spatial variation of the components of the apparent elasticity tensor are also determined using the proposed approach

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

the invariant factors of the incidence matrices of points and subspaces in operatorname pg n q and operatorname ag n q we determine the smith normal forms of the incidence matrices of points and projective r 1 dimensional subspaces of pg n q and of the incidence matrices of points and r dimensional affine subspaces of ag n q for all n r and arbitrary prime power q

error estimation and adaptive meshing in strongly nonlinear dynamic problems we present work aimed at developing a general framework for mesh adaption in strongly nonlinear possibly dynamic problems we begin by showing that the solutions of the incremental boundary value problem for a wide class of materials including nonlinear elastic materials compressible newtonian fluids and viscoplastic solids obey a minimum principle provided that the constitutive updates are formulated appropriately this minimum principle can be taken as a basis for asymptotic error estimation in particular we chose to monitor the error of a lower order projection of the finite element solution the optimal mesh size distribution then follows from a posteriori error indicators which are purely local i e can be computed element by element we demonstrate the robustness and versatility of the computational framework with the aid of convergence studies and selected examples of application

parallel algorithms for solving systems of nonlinear equations abstract in this paper parallel algorithms are proposed for solving both systems of nonlinear algebraic equations and unconstrained optimization problems theoretical results for their convergence and orders of convergence are also presented the proposed algorithms are analytically compared in terms of their time efficiencies and speed up ratios

incidence modules for symplectic spaces in characteristic two abstract we study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the alternating form a general formula is obtained for the 2 rank of the incidence matrix for the inclusion of one dimensional subspaces in the distinguished subspaces of a fixed dimension

the elementary divisors of incidence matrices between certain subspaces of a finite symplectic space abstract the elementary divisors of the incidence matrices between 1 dimensional subspaces and r dimensional isotropic subspaces of a finite symplectic space are computed

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

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

duality relations for nonlinear incompressible two dimensional elasticity keller s phase interchange relation which computes the overall conductivity of a two dimensional checkerboard with alternating conductivity is revisited in the context of nonlinear incompressible elasticity a general phase interchange relation is obtained in a monotone setting through h convergence and in a convex setting through variational methods several keller like applications are presented

optimal bounds on the in plane shear moduli for orthotropic elastic composites composite and porous materials often appear in nature many composites may be considered orthotropic such as wood or bone the elastic behaviour of these composites under shear stresses is characterized by three independent shear moduli the authors consider the totality of orthotropic composites made from two isotropic linearly elastic components in fixed proportion for a prescribed triple of shear stresses optimal bounds are found on the strongest and weakest orthotropic composites mathematically this problem is one of constrained optimization the set of constraints is related to the convex hull of a surface in three dimensions for given values of the component elasticities the bounds are computed numerically

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

strain based transient gradient damage model for failure analyses a transient gradient enhanced damage model has been developed for the numerical modelling of the damage and fracture process within a continuum damage mechanics framework some deficiencies of existing gradient enhanced damage formulations for the simulation of macroscopic crack propagation are pointed out the transient gradient approach assumes a direct coupling between the material length parameter and the local strain state of the material which leads to a transient behaviour of the nonlocal effect details of the method are presented and fully elaborated in an incremental iterative solution scheme mesh objectivity and physical relevance of the method are analysed by one dimensional and two dimensional numerical examples

a finite strain incremental secant homogenization model for elasto plastic composites abstract this paper presents a finite strain extension of the incremental secant mean field homogenization mfh formulation for two phase elasto plastic composites the formulation of the local finite strain elasto plastic constitutive equations of each phase is based on a multiplicative decomposition of the deformation gradient as suggested by simo in computer methods in applied mechanics and engineering 99 1 61 112 1992 the latter has proposed algorithms which preserve the classical return mapping schemes of the infinitesimal theory by using principal kirchhoff stresses and logarithmic eigenvalues of the left elastic cauchy green strain relying on this property we show that by considering a quadratic logarithmic free energy and j2 flow theory at the local level infinitesimal strain incremental secant mfh is readily extended to finite strains the proposed formulation and corresponding numerical algorithms are then presented finally the predictions are illustrated with several numerical simulations which are verified against full field finite element simulations of composite cells demonstrating that the micro mechanically based approach is able to predict the influence of the micro structure and of its evolution on the macroscopic properties in a very cost effective manner

multicomponent composites electrical networks and new types of continued fraction i the development of bounds on the complex effective conductivity tensor that relates the average current to the average electric field in a multicomponent composite has been hindered by lack of a suitable continued fraction representation for here a new field equation recursion method is developed which gives an expression for as a continued fraction of a novel form incorporating as coefficients the component conductivities and a set of fundamental geometric parameters reflecting the composite geometry a hierarchy of field equations is set up such that the solutions of the j 1 th order equation generate the solutions of thejth order equation consequently the effective tensor j associated with thejth order field equation is expressible as a fractional linear matrix transformation of j 1 these transformations combine to form the continued fraction expansion for 0 which is exploited in the following paper part ii to obtain bounds crude bounds on j forj 1 give narrow bounds on the continued fraction is a generalization to multivariate functions of the continued fraction expansion of single variable stieltjes functions that proved important in the development of the theory of pade approximants asymptotic analysis and the theory of orthogonal polynomials in the last century the results extend to other transport problems including conduction in polycrystalline media the viscoelasticity of composites and the response of multicomponent multiterminal linear electrical networks

numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage this paper deals with a class of rate independent metal plasticity models which exhibit non linear isotropic hardening non linear kinematic hardening chaboche marquis model and ductile damage lemaitre chaboche model the backward euler scheme is used to integrate the rate constitutive relations the non linear equations obtained are solved by the newton method the consistent tangent operator is obtained by exact linearization of the algorithm despite the complexity of the constitutive equations closed form expressions are derived without any approximations analytical numerical and experimental results are presented acid discussed

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

extended framework of hamiltons principle for thermoelastic continua based upon the extended framework of hamiltons principle a variational formulation for fully coupled thermoelasticity is presented the resulting formulation can properly account for all the governing differential equations as well as initial boundary conditions thus it provides the basis for a class of unified spacetime finite element methods by employing bar elements in one dimensional space along with linear shape functions temporally the simplest spacetime finite element method is presented herein with representative examples for its validity

a switching method for nonlinear systems a new iterative method is proposed for the solution of nonlinear systems the method does not use explicit derivative informations and at each iteration automatically selects one of two distinct iterative schemes a direct search method and a damped approximate newton s method so the method is referred as switching method it is shown that the method is a global method with quadratic convergence numerical results show the very good practical performance of the method

effective properties of composite materials containing voids recent results of theoretical and practical importance prove that the two dimensional in plane effective average young s modulus for an isotropic elastic material containing voids is independent of the poisson s ratio of the matrix material this result is true regardless of the shape and morphology of the voids so long as isotropy is maintained the present work uses this proof to obtain explicit analytical forms for the effective young s modulus property forms which simplify greatly because of this characteristic in some cases the optimal morphology for the voids can be identified giving the shapes of the voids at fixed volume that maximize the effective young s modulus in the two dimensional situation recognizing that two dimensional isotropy is a subset of three dimensional transversely isotropic media it is shown in this more general case that three of the five properties are independent of poisson s ratio leaving only two that depend upon it for three dimensionally isotropic composite media containing voids it is shown that a somewhat comparable situation exists whereby the three dimensional young s modulus is insensitive to variations in poisson s ratio v m over the range 0 v m although the same is not true for negative values of v m this further extends the practical usefulness of the two dimensional result to three dimensional conditions for realistic values of v m

multiphase conductors realizing aleksandrov s mean it is shown that aleksandrov s mean for the effective linear conductivity of multiphase conductors is exact for a class of material systems with two dimensional isotropic microstructures the systems are constructed by sucessive use of iterated dilute homogenization the result provides a more rigorous basis for the use of aleksandrov s mean to estimate effective properties

a new variational principle and its application to nonlinear heterogeneous systems a new variational principle is proposed for estimating the effective properties of nonlinear heterogeneous systems the variational principle expresses the effective energy density function of a given nonlinear heterogeneous dielectric in terms of an infinite dimensional optimization problem involving the effective energy density functions of the class of linear comparison materials with variable dielectric cofficients the variational principle can alternatively be used in an approximate fashion by optimizing over finite dimensional subspaces of the original space of comparison dielectric coefficients to obtain bounds on the effective dielectric properties of nonlinear composite materials with isotropic constituent phases in given proportions the application of the procedure is illustrated by computing an exact estimate for the effective energy density function of a nonlinear laminated dielectric material which has overall anisotropic effective behavior it is also demonstrated that the talbot willis

high performance model reduction techniques in computational multiscale homogenization a novel model order reduction technique for the solution of the fine scale equilibrium problem appearing in computational homogenization is presented the reduced set of empirical shape functions is obtained using a partitioned version that accounts for the elastic inelastic character of the solution of the proper orthogonal decomposition pod on the other hand it is shown that the standard approach of replacing the nonaffine termby an interpolant constructed using only pod modes leads to ill posed formulations we demonstrate that this ill posedness can be avoided by enriching the approximation space with the span of the gradient of the empirical shape functions furthermore interpolation points are chosen guided not only by accuracy requirements but also by stability considerations the approach is assessed in the homogenization of a highly complex porous metal material computed results show that computational complexity is independent of the size and geometrical complexity of the representative volume element the speedup factor is over three orders of magnitude as compared with finite element analysis whereas the maximum error in stresses is less than 10

correctors and field fluctuations for the p x laplacian with rough exponents we provide a corrector theory for the strong approximation of fields inside composites made from two materials with different power law behavior the correctors are used to develop bounds on the local singularity strength for gradient fields inside micro structured media the bounds are multi scale in nature and can be used to measure the amplification of applied macroscopic fields by the micro structure

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

cluster expansion for the dielectric constant of a polarizable suspension we derive a cluster expansion for the electric susceptibility kernel of a dielectric suspension of spherically symmetric inclusions in a uniform background this also leads to a cluster expansion for the effective dielectric constant it is shown that the cluster integrals of any order are absolutely convergent so that the dielectric constant is well defined and independent of the shape of the sample in the limit of a large system we compare with virial expansions derived earlier in statistical mechanics for the dielectric constant of a nonpolar gas in these expansions the virial coefficients are given by integrals which are only conditionally convergent

rigidity for the four gradient problem we prove a general rigidity result for the maximal possible number of gradients by showing that any lipschitz function using four pairwise not rank one connected gradients is necessarily affine this exhibits an interesting difference between rigidity features of approximate and exact solutions of differential inclusions

optimal g closure bounds via stability under lamination optimal bounds for a two dimensional g closure problem are proved they are obtained by combining together bounds derived using fine properties of quasiconformal mappings and a stability property of the g closure which is known as stability under lamination

generalizations of the bruggeman equation and a concept of shape distributed particle composites we consider generalizations of the classical symmetrical bruggeman equation based on the concept of shape distributed particle systems the use of the beta distribution for the particle shape is shown to result in some known as well as unknown equations of the effective medium theory however these equations yield no percolation threshold on the other hand the use of one and two dimensional steplike distributions of spheroidal ellipsoidal shapes yields a percolation threshold depending on the distribution parameters the problem of finding the percolation threshold to fit the systems under consideration as well as the applicability area of the generalized bruggeman equation and its relation to the bergman representation are discussed

on the two state problem for general differential operators abstract in this note we generalize the ball james rigidity theorem for gradient differential inclusions to the setting of a general linear differential constraint in particular we prove the rigidity for approximate solutions to the two state inclusion with incompatible states for merely l 1 bounded sequences in this way our theorem can be seen as a result of compensated compactness in the linear growth setting

upper and lower bounds for the overall properties of a nonlinear composite dielectric ii periodic microgeometry the study of the effective properties of a nonlinear composite dielectric begun in the companion to this paper is continued in the context of a simple cubic array of inclusions embedded in a matrix use of a nonlinear comparison material in the nonlinear hashin shtrikman variational principles permits the generation of both upper and lower bounds for any composite it is straightforward to apply the new approach to the simple composite considered here as the trial electric field which is substituted into the hashin shtrikman variational principle is finite everywhere the results obtained indicate the likely improvement over the simple classical bounds that can be obtained using the hashin shtrikman principles

rigidity and lack of rigidity for solenoidal matrix fields the problem of finding a curl free matrix valued field e with values in an assigned set of matrices has received considerable attention given a bounded connected open set and a compact set of m n matrices in this paper we establish existence or non existence results for the following problem find b l m n such that div b 0 in in the sense of distributions under the constraint that b x almost everywhere in we consider the case of a b rank a b n and we establish non existence both for the case of exact solutions described above and for the case of approximate solutions described in 1 we also prove existence of approximate solutions for a suitably chosen triple a 1 a 2 a 3 of matrices with rank a i a j n i j and i j 1 2 3 we give examples when the differential constraints are of a different type and present some applications to composites

application of volterra wiener series for bounding the overall conductivity of hetergeneous media i general procedure a general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed it utilizes truncated volterra wiener functional series generated by the random conductivity field k x as classes of trial functions for the classical variational principles the method simplifies unifies and or generalizes the earlier proposed variational techniques of beran 3 dederichs and zeller 10 hori 15 kr6ner 17 and prager 25 the general procedure is displayed in detail in the simplest case of interest the construction of the optimal third order bounds that requires knowledge of the two and three point correlation functions for k x the evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions a perturbation solution to this equation is given for miller s cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of beran miller

variational estimates for the creep behaviour of polycrystals a variational procedure is developed for estimating the effective constitutive behaviour of polycrystalline materials undergoing high temperature creep the procedure is based on a new variational principle allowing the determination of the effective potential function of a given nonlinear polycrystal in terms of the corre sponding potential for a linear comparison polycrystal with an identical geometric arrangements of its constituent single crystal grains as such it constitutes an extension to locally anisotropic behaviour of the variational procedure devel oped by ponte castaneda 1991 for nonlinear heterogeneous media with locally isotropic behaviour by way of an example the procedure is applied to the de termination of bounds of the hashin shtrikman type for the effective potentials of statistically isotropic nonlinear polycrystals the bounds are computed for the special class of untextured fcc polycrystals with isotropic pure power law viscous behaviour first considered by hutchinson 1976 in the context of a calculation of the self consistent type the new bounds are found to be more restrictive than the corresponding classical taylor bishop hill bounds and also more re strictive if only slightly so than related bounds of the hashin shtrikman type by dendievel et al 1991 the new procedure has the advantage over the self consistent procedure of hutchinson 1976 that it may be applied without any essential complications to aggregates of crystals with slip systems exhibiting dif ferent creep rules with for example different power exponents and to general loading conditions however the distinctive feature of the new variational proce dure is that it may be used in conjunction with other types of known bounds and estimates for linear polycrystals to generate corresponding bounds and estimates for nonlinear polycrystals

identification of polynomial chaos representations in high dimension from a set of realizations this paper deals with the identification in high dimensions of a polynomial chaos expansion of random vectors from a set of realizations due to numerical and memory constraints the usual polynomial chaos identification methods are based on a series of truncations that induce a numerical bias this bias becomes very detrimental to the convergence analysis of polynomial chaos identification in high dimensions this paper therefore proposes a new formulation of the usual polynomial chaos identification algorithms to avoid this numerical bias after a review of the polynomial chaos identification method the influence of the numerical bias on the identification accuracy is quantified the new formulation is then described in detail and illustrated using two examples

stieltjes representation of the 3d bruggeman effective medium and pade approximation the paper deals with bruggeman effective medium approximation ema which is often used to model effective complex permittivity of a two phase composite we derive the stieltjes integral representation of the 3d bruggeman effective medium and use constrained pade approximation method introduced in 39 to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies the problem of reconstruction of the stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite is used to characterize its geometric structure we present two different proofs of the stieltjes analytical representation for the effective complex permittivity in the 3d bruggeman effective medium model one proof is based on direct calculation the other one is the derivation of the representation using stieltjes inversion formula we show that the continuous spectral density in the integral representation for the bruggeman ema model can be efficiently approximated by a rational function a rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition we show results of numerical rational approximation of bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium the volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values

surrogate modeling of multiscale models using kernel methods this work investigates the possibilities of acceleration and approximation of multiscale systems using kernel methods the key element is to learn the interface between the different scales using a fast surrogate for the microscale model which is given by multivariate kernel expansions the expansions are computed using statistically representative samples of in and output of the microscale model as learning methods we apply both support vector machines and a vectorial kernel greedy algorithm we demonstrate the applicability of the resulting surrogate models using two multiscale models from different engineering disciplines first we consider a human spine model coupling a macroscale multibody system with a microscale intervertebral spine disc model and second a model for simulation of saturation overshoots in porous media involving nonclassical shock waves

the permutation representation of sp 2m fp acting on the vectors of its standard module abstract this paper studies the permutation representation of a finite symplectic group over a prime field of odd characteristic on the vectors of its standard module the submodule lattice of this permutation module is determined the results yield additive formulae for the p ranks of various incidence matrices arising from the finite symplectic spaces

extended framework of hamilton s principle for continuum dynamics abstract hamilton s principle is the variational principle for dynamical systems and it has been widely used in mathematical physics and engineering however it has a critical weakness termed end point constraints which means that in the weak form we cannot use the given initial conditions properly by utilizing a mixed formulation and sequentially assigning initial conditions this paper presents a novel extended framework of hamilton s principle for continuum dynamics to resolve such weakness the primary applications lie in an elastic and a j 2 viscoplastic continuum dynamics the framework is simple and initiates the development of a space time finite element method with the proper use of initial conditions non iterative numerical algorithms for both elasticity and j 2 viscoplasticity are presented

the elementary divisors of the incidence matrices of points and linear subspaces in pn fp the elementary divisors of the incidence matrices between points and linear subspaces of fixed dimension in pn fp are computed

the effective conductivity of a periodic array of spheres we consider the problem of determining the effective conductivity k of a composite material consisting of equal sized spheres of conductivity alpha arranged in a cubic array within a homogeneous matrix of unit conductivity we modify zuzovski brenner9s 1977 method and thereby obtain a set of infinite linear equations for the coefficients of the formal solution equivalent to that derived by mckenzie et al 1978 using a method originally devised by rayleigh 1892 on solving these equations we derive expressions for k ast to o c 9 c being the volume fraction of the spheres for simple body centred and face centred cubic arrays and also obtain numerical values for k over the whole range of alpha and c we show that these results for cubic arrays can be used to estimate ast for random arrays of identical spheres for arrays of highly conducting and nearly touching spheres batchelor o9brien 1977 showed that begin equation k ast sim begin cases k 1 ln 1 chi k 2 alpha infty chi c c max frac 1 3 rightarrow 1 2k 1 ln alpha k9 2 quad chi 1 alpha rightarrow infty end cases end equation where c max corresponds to the volume fraction when the spheres are actually touching each other and determined k 1 for the three cubic arrays our numerical results are consistent with the above asymptotic expressions except for the fact that the numerical values for the constants k 2 and k9 2 thereby obtained do not quite satisfy the relation k9 2 k 1 3 9 ln 2 k 2 given by batchelor o9brien we have been unable to find the reason for this slight discrepancy

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

the reduced model multiscale method r3m for the non linear homogenization of hyperelastic media at finite strains this paper presents a new multi scale method for the homogenization analysis of hyperelastic solids undergoing finite strains the key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method in order to alleviate the complexity of multiscale procedures which usually involve a large number of nonlinear nested problems to be solved the problem associated with the representative volume element rve is solved via a model reduction method proper orthogonal decomposition the reduced basis is obtained through pre computations on the rve the technique coined as reduced model multiscale method r3m allows reducing significantly the computation times as no large matrix needs to be inverted and as the convergence of both macro and micro problems is enhanced furthermore the r3m drastically reduces the size of the data base describing the history of the micro problems in order to validate the technique in the context of porous elastomers at finite strains a comparison between a full and a reduced multiscale analysis is performed through numerical examples involving different micro and macro structures as well as different nonlinear models neo hookean mooney rivlin it is shown that the r3m gives good agreement with the full simulations at lower computational and data storage requirements

on the effective viscoelastic moduli of two phase media iii rigorous bounds on the complex shear modulus in two dimensions the translation and the hashin shtrikman methods are used to provide bounds on the effective complex shear modulus of a two phase two dimensional viscoelastic composite they are both given by inequalities that depend on six parameters the best bounds are obtained by optimizing these parameters over the admissible set which is larger for the translation method than for the hashin shtrikman method consequently the translation method generally leads to tighter bounds than would be obtained via the standard hashin shtrikman approach equivalence classes of two dimensional viscoelastic composites directly analogous to similar classes for the pure elastic problem are found combination of the simplified versions of the translation or the hashin shtrikman type bounds and this equivalency results in simple algorithms for computing tight bounds for any choice of phase moduli and volume fractions the bounds constrain the effective shear modulus to lie inside a region of the complex plane bounded by arcs of circles the four points which correspond to the hashin shtrikman walpole bounds on the shear modulus of an elastic composite always lie inside or on the boundary of the bounding region in many cases the bounding region tends to hug the corresponding parallelogram in the complex compliance plane having these four points as vertices

an inverse micro mechanical analysis toward the stochastic homogenization of nonlinear random composites abstract an inverse mean field homogenization mfh process is developed to improve the computational efficiency of non linear stochastic multiscale analyzes by relying on a micro mechanics model first full field simulations of composite stochastic volume element sve realizations are performed to characterize the homogenized stochastic behavior the uncertainties observed in the non linear homogenized response which result from the uncertainties of their micro structures are then translated to an incremental secant mfh formulation by defining the mfh input parameters as random effective properties these effective input parameters which correspond to the micro structure geometrical information and to the material phases model parameters are identified by conducting an inverse analysis from the full field homogenized responses compared to the direct finite element analyzes on sves the resulting stochastic mfh process reduces not only the computational cost but also the order of uncertain parameters in the composite micro structures leading to a stochastic mean field reduced order model mf rom a data driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic mf rom embedded in a stochastic finite element method sfem the two cases of elastic unidirectional ud fibers embedded in an elasto plastic matrix and of elastic ud fibers embedded in a damage enhanced elasto plastic matrix are successively considered in order to illustrate the capabilities of the method the stochastic response of a ply is studied under transverse loading condition

data driven probability concentration and sampling on manifold a new methodology is proposed for generating realizations of a random vector with values in a finite dimensional euclidean space that are statistically consistent with a dataset of observations of this vector the probability distribution of this random vector while a priori not known is presumed to be concentrated on an unknown subset of the euclidean space a random matrix is introduced whose columns are independent copies of the random vector and for which the number of columns is the number of data points in the dataset the approach is based on the use of i the multidimensional kernel density estimation method for estimating the probability distribution of the random matrix ii a mcmc method for generating realizations for the random matrix iii the diffusion maps approach for discovering and characterizing the geometry and the structure of the dataset and iv a reduced order representation of the random matrix which is constructed using the diffusion maps vectors associated with the first eigenvalues of the transition matrix relative to the given dataset the convergence aspects of the proposed methodology are analyzed and a numerical validation is explored through three applications of increasing complexity the proposed method is found to be robust to noise levels and data complexity as well as to the intrinsic dimension of data and the size of experimental datasets both the methodology and the underlying mathematical framework presented in this paper contribute new capabilities and perspectives at the interface of uncertainty quantification statistical data analysis stochastic modeling and associated statistical inverse problems

the permutation action of finite symplectic groups of odd characteristic on their standard modules abstract we study the space of functions on a finite dimensional vector space over a field of odd order as a module for a symplectic group we construct a basis of this module with the following special properties each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule from these properties the dimension and composition factors of the submodule generated by any subset of the basis can be determined these results apply to incidence geometry of the symplectic polar space yielding the symplectic analogue of hamada s additive formula for the p ranks of the incidence matrices between points and flats a special case leads to a closed formula for the p rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order together with earlier results on the 2 ranks this result completes the determination of the p ranks for these quadrangles

a stochastic computational multiscale approach application to mems resonators abstract the aim of this work is to develop a stochastic multiscale model for polycrystalline materials which accounts for the uncertainties in the micro structure at the finest scale we model the micro structure using a random voronoi tessellation each grain being assigned a random orientation then we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso scale a random field of the meso scale elasticity tensor can thus be generated based on the information obtained from the sve simulations finally using a stochastic finite element method these meso scale uncertainties are propagated to the coarser scale as an illustration we study the resonance frequencies of mems micro beams made of poly silicon materials and we show that the stochastic multiscale approach predicts results in agreement with a monte carlo analysis applied directly on the fine finite element model i e with an explicit discretization of the grains

dielectric breakdown optimal bounds we study dielectric breakdown for composites made of two isotropic phases we show that sachs s bound is optimal this simple example is used to illustrate a variational principle which departs from the traditional one we also derive the usual variational principle by elementary means without appealing to the technology of convex duality

some simple explicit bounds for the overall behaviour of nonlinear composites abstract variational expressions developed over the last few years provide bounds for the overall energy functions of a range of nonlinear composite materials the evaluation of a bound requires the solution of a system of nonlinear algebraic equations and this generally involves a computation there are however certain simple composites comprising a nonlinear matrix containing either rigid inclusions or cavities for which very simple explicit formulae can be given these formulae are displayed here at a level of generality greater than in any previous presentation the energy density function of the matrix is arbitrary and the microgeometry of the composite appears through an expression which bounds the energy of a linear composite with the same geometry to the extent that such linear bounds can be developed making allowance for any amount of statistical information on the composite the new nonlinear bounds collect this new results at the level of employing bounds of hashin shtrikman type for the linear problem are given for an incompressible matrix reinforced by aligned rigid platelets or weakened by aligned cracks in the course of the work a recently derived formula more general than any available previously is presented and developed explicitly for any two phase composite

on characterizing the set of possible effective tensors of composites the variational method and the translation method a general algebraic framework is developed for characterizing the set of possible effective tensors of composites a transformation to the polarization problem simplifies the derivation of the hashin shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials a general connection is established between two methods for bounding effective tensors of composites the first method is based on the variational principles of hashin and shtrikman the second method due to tartar murat lurie and cherkaev uses translation operators or equivalently quadratic quasiconvex functions a correspondence is established between these translation operators and bounding operators on the relevant non local projection operator t1 an important class of bounds namely trace bounds on the effective tensors of two component media are given a geometrical interpretation after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors a wide class of translation operators that generate these bounds is found the extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors these extremal translations generate attainable trace bounds even when the tensors of the two components are not well ordered in particular they generate the bounds of walpole on the effective bulk modulus the variational principles of gibiansky and cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by golden and papanicolaou an isomorphism is shown to underlie their variational principles this isomorphism is used to obtain dirichlet type variational principles and bounds for the effective tensors of general non selfadjoint problems it is anticipated that these variational principles which stem from the work of gibiansky and cherkaev will have applications in many fields of science

model order reduction of nonlinear homogenization problems using a hashin shtrikman type finite element method abstract this work presents a computational nonlinear homogenization approach the starting point of which is a model order reduction method based on data clustering to this end the micromechanical data from numerical experiments snapshots is analyzed in order to identify characteristic microstructural deformation patterns these describe how the macroscopic strain typically localizes within the microstructure the outcome of the procedure is a subdivision of the microstructure into a set of clusters of material points within each cluster the strain is then approximated as being constant the mechanical problem is formulated in terms of a three field hashin shtrikman type variational formulation which is based on the introduction of a linear elastic reference medium after discretization most of the global unknowns can be eliminated via static condensation leaving the piecewise constant cluster strains as the primary unknowns the resulting homogenization scheme includes as special cases the finite element method as well as hashin shtrikman and talbot willis type homogenization approaches with phase wise constant trial fields as well as related bounds the limit case finite element method allows to transfer knowledge from finite element technology and thus provides new strategies for the choice of the stiffness of the reference material the method is applied to several nonlinear microstructures with different inclusion volume fractions and varying degree of anisotropy the results are shown to be in good agreement with full field fe simulations furthermore the method is used to compute a refined upper bound of the talbot willis type compared to phase wise constant trial fields which converges to the finite element solution with increasingly refined discretization

reduced basis hybrid computational homogenization based on a mixed incremental formulation a new reduced basis method for the computer assisted homogenization of microheterogeneous materials is proposed key features of the nonuniform transformation field analysis ntfa 1 2 are taken as a point of departure a short coming of the ntfa method is its limitation to simple constitutive models on the microscale a second disadvantage is the possible loss of accuracy when the reduced basis approximating the plastic strains has increasing dimension both issues are related to the simple evolution law for the mode activity coefficients which are the new macroscopic internal variables of the homogenized material in the present contribution a generalization of the ntfa is proposed in which the evolution of the new internal variables is derived from a mixed incremental variational formulation the derivation is based on purely micro mechanical considerations the modification allows for arbitrary generalized standard materials 3 on the microscopic scale including e g crystal visco plasticity additionally the fidelity of the homogenized response is now directly linked to the reduced basis approximating the plastic strain field numerical examples for nonlinear viscous materials and single crystal plasticity outline the accuracy of the proposed multi scale method

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

global existence results for viscoplasticity at finite strain we study a model for rate dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor and investigate the existence of global in time solutions to the related pde system we reveal its underlying structure as a generalized gradient system where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite strain elasticity as well as the multiplicative decomposition of finite strain plasticity moreover the dissipation potential depends on the left invariant plastic rate and thus depends on the plastic state variable the existence theory is developed for a class of abstract nonsmooth and nonconvex gradient systems for which we introduce suitable notions of solutions namely energy dissipation balance and energy dissipation inequality solutions hence we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory

a critical comparison of nonlocal and gradient enhanced softening continua continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients these gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations the effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting a nonlocal damage model and two different gradient damage models are considered in one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation the latter implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal this is not true for the explicit formulation in which the nonlocality is limited to an infinitesimal volume this fundamental difference between the formulations results in quite different behaviour in wave propagation localisation and at crack tips this is shown for the propagation of waves in the models their localisation properties and the behaviour at a crack tip the responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations while the explicit gradient formulation shows an entirely different and sometimes nonphysical response

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media

stochastic finite element analysis of composite structures based on mesoscale random fields of material properties abstract the linking of microstructure uncertainty with the random variation of material properties at higher scales is of paramount importance in the framework of the stochastic finite element method sfem an efficient computational scheme has been recently proposed by the authors for the determination of mesoscale random fields for the apparent properties and of the rve size of particle reinforced composites based on computer simulated images of their microstructure the proposed approach exploits the excellent synergy of the extended finite element method xfem and monte carlo simulation in order to analyze the microstructure models and obtain statistical information probability distribution correlation structure for the apparent properties of the composite in each mesoscale size in this paper sample functions of the statistically homogeneous mesoscale random fields are generated using the spectral representation method in conjunction with translation field theory moreover micro mechanically consistent lower and upper bounds on the macroscopic response of composite structures are computed in the framework of sfem useful conclusions are derived regarding the effect of particle matrix stiffness ratio on the probabilistic characteristics of the mesoscale random fields as well as the influence of mesoscale size on the response variability

partially updated switching method for systems of nonlinear equations abstract a hybrid method for solving systems of n nonlinear equations is given the method does not use derivative information and is especially attractive when good starting points are not available and the given system is expensive to evaluate it is shown that after a few steps each iteration requires 2 k 1 function evaluations where k 1 k n is chosen so as to have an efficient algorithm global convergence results are given and superlinear convergence is established some numerical results show the numerical performance of the proposed method

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

the influence of strain on confined electronic states in semiconductor quantum structures a continuum finite element technique is adopted to study electronic properties of submicron electronic devices where function hinges on quantum mechanical eaects of particular interest is the influence of mechanical strain on confined electronic states the steady state schrequation which governs the electronic behavior of such devices is modified to include the potential induced by a strain field which is present as a consequence of the fabrication the governing equation is cast in a variational form and it is discretized on a standard finite element mesh which is more refined in regions where large quantum mechanical wave function gradients are expected multiple energy bands and three dimensional structures can be considered and eaects including strain enhanced charge confinement and strain induced energy band mixing are studied as examples a ge 5 0 1 faceted island or quantum dot on a si substrate and a ge v groove quantum wire on a si substrate are considered the technique is used to determine size ranges in which these devices are expected to be most useful the nonuniform mismatch strain field in the structures is found to aaect the energies of experimentally accessible confined states and in some cases to enhance quantum mechanical confine ment 2001 elsevier science ltd all rights reserved

duality relations for non ohmic composites with applications to behavior near percolation keller dykhne and others have exploited duality to derive exact results for the effective behavior of two dimensional ohmic composites this paper addresses similar issues in the non ohmic context we focus primarily on three different types of nonlinearity a the weakly nonlinear regime b power law behavior and c dielectric breakdown we first make the consequences of duality explicit in each setting then we draw conclusions concerning the critical exponents and scaling functions of dual pairs of random non ohmic composites near a percolation threshold these results generalize unify and simplify relations previously derived for nonlinear resistor networks we also discuss some self dual nonlinear composites our treatment is elementary and self contained however we also link it with the more abstract mathematical discussions of duality by jikov and kozlov

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

simulation of the multi scale convergence in computational homogenization approaches although the asymptotic homogenization is known to explicitly predict the thermo mechanical behaviors of an overall structure as well as the microstructures the current developments in engineering fields introduce some kinds of approximation about the microstructural geometry in order for the homogenization method for periodic media to apply for general heterogeneous ones the problems arising from mathematical modeling are examined in the framework of representative volume element rve analyses here the notion of homogenization convergence allows us to eliminate the geometrical periodicity requirement when the size of rve is sufficiently large then the numerical studies in this paper realize the multi scale nature of the convergence of overall material properties as the unit cell size is increased in addition to such dependency of the macroscopic field variables on the selected size of unit cells the convergence nature of microscopic stress values is also studied quantitatively via the computational homogenization method similar discussions are made for the elastoplastic mechanical responses in both macro and microscopic levels in these multi scale numerical analyses the specific effects of the microstructural morphology are reflected by using the digital image based dib finite element fe modeling technique which enables the construction of accurate microstructural models

stochastic finite elements as a bridge between random material microstructure and global response this paper presents a finite element method aimed at the introduction of microstructural material randomness below the level of a single finite element a consideration of dependence of effective moduli on scale and on the defining boundary conditions leads to an identification of a finite element as a mesoscale window or a mesoscale finite element in a stochastic finite element method sfe an estimation of the global response can be obtained through bounds stemming from minimum potential energy and complementary energy principles which involve dirichlet and neumann boundary conditions on all the mesoscale finite elements respectively while in the classical case of a homogeneous material these two bounds converge to each other as the finite element mesh becomes sufficiently fine an optimal mesoscale with respect to the difference between both bounds may exist in the case of a heterogeneous material the proposed sfe method is illustrated with a numerical example of a sample two phase voronoi composite with some 26 000 grains where a reference solution taking into account the entire microstructure without any smearing out is shown to fall between both energy bounds a generalization to an ensemble response is straightforward

computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis this paper is devoted to the computational nonlinear stochastic homogenization of a hyperelastic heterogeneous microstructure using a non concurrent multiscale approach the geometry of the microstructure is random since the non concurrent multiscale approach is based on the use of a tensorial decomposition which is then submitted to the curse of dimensionality we perform an analysis with respect to the stochastic dimension the technique uses a database describing the strain energy density function potential in both the macroscopic cauchy green strain space and the geometrical random parameters domain each value of the potential is numerically computed by means of the finit element method on an elementary cell an interpolation scheme is finall introduced to obtain a continuous explicit form of the potential which by derivation allows to evaluate the macroscopic stress and elastic tangent tensors during the macroscopic structural computations two numerical examples are presented

effective medium approximation and exact formulae for electrokinetic phenomena in porous media electrokinetic phenomena in porous media are studied by application of the effective medium theory and the theory of duality transformation we deduce new exact relations and analytical formulae for the effective constants of the macroscopic tensor we also prove that the effective tensors obtained by these approaches coincide for 2d problems the obtained results for the electrokinetic processes are closely related to similar results derived for piezoelectric composites because of a common mathematical background