two scale γ convergence of integral functionals and its application to homogenisation of nonlinear high contrast periodic composites an analytical framework is developed for passing to the homogenisation limit in not necessarily convex variational problems for composites whose material properties oscillate with a small period e and that exhibit high contrast of order varepsilon 1 between the constitutive stress strain response on different parts of the period cell the approach of this article is based on the concept of two scale convergence which is a kind of hybrid of the classical convergence de giorgi and franzoni in atti accad naz lincei rend cl sci fis mat natur 8 58 842 850 1975 and the more recent two scale convergence nguetseng in siam j math anal 20 608 623 1989 the present study focuses on a basic high contrast model where soft inclusions are embedded in a stiff matrix it is shown that the standard convergence in the lp space fails to yield the correct limit problem as varepsilon to 0 due to the underlying lack of lp compactness for minimising sequences using an appropriate two scale compactness statement as an alternative starting point the two scale limit of the original family of functionals is determined via a combination of techniques from classical homogenisation the theory of quasiconvex functions and multiscale analysis the related result can be thought of as a non classical two scale extension of the well known theorem by muller arch rational mech anal 99 189 212 1987

quantitative homogenization in nonlinear elasticity for small loads we study quantitative periodic homogenization of integral functionals in the context of nonlinear elasticity under suitable assumptions on the energy densities in particular frame indifference minimality non degeneracy and smoothness at the identity p geqq d growth from below and regularity of the microstructure we show that in a neighborhood of the set of rotations the multi cell homogenization formula of non convex homogenization reduces to a single cell formula the latter can be expressed with the help of correctors we prove that the homogenized integrand admits a quadratic taylor expansion in an open neighborhood of the rotations a result that can be interpreted as the fact that homogenization and linearization commute close to the rotations moreover for small applied loads we provide an estimate on the homogenization error in terms of a quantitative two scale expansion

a multilevel computational strategy for handling microscopic and macroscopic instabilities this paper presents a numerical technique to deal with instability phenomena in the context of heterogeneous materials where buckling may occur at both macroscopic and or microscopic scales we limit ourselves to elastic materials but geometrical nonlinearity is taken into account at both scales the proposed approach combines the multilevel finite element analysis fe2 and the asymptotic method anm in that framework the unknown nonlinear constitutive relationship at the macroscale is found by solving a local finite element problem at the microscale in contrast with fe2 the use of the asymptotic development allows to transform the nonlinear microscopic problems into a sequence of linear problems thus a direct analogy with classical linear homogenization can be made to construct a localisation tensor at each step of the asymptotic development and an explicit macroscopic constitutive relationship can be constructed at each step furthermore the salient features of the anm allow treating instabilities and limit points in a very simple way at both scales the method is tested and illustrated through numerical examples involving local instabilities which have ignificant influence on the macroscopic behavior

multiscale aggregating discontinuities a method for circumventing loss of material stability new methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described these methods called multiscale aggregating discontinuities are based on the concept of perforated unit cells which exclude subdomains that are unstable i e exhibit loss of material stability using this concept it is possible to compute an equivalent discontinuity at the coarser scale including both the direction of the discontinuity and the magnitude of the jump these variables are then passed to the coarse scale model along with the stress in the unit cell the discontinuity is injected at the coarser scale by the extended finite element method analysis of the procedure shows that the method is consistent in power and yields a bulk stress strain response that is stable applications of this procedure to crack growth in heterogeneous materials are given copyright 2007 john wiley sons ltd

an adaptive method for homogenization in orthotropic nonlinear elasticity a database methodology is developed in order to characterize the constitutive behavior of macroscopically orthotropic nonlinearly elastic composites this methodology discretizes the material test space in terms of the eigenvalues of the macroscopic strain tensor and the orientation of its eigenvectors with respect to the axes of orthotropy the macroscopic properties of the composite are extracted at points of this test space by subjecting a statistically representative sample of the composite to appropriate boundary conditions and are subsequently employed in an interpolation procedure which creates the so called material map that characterizes the macroscopic behavior the macroscopic tangent required in the finite element calculations are derived either by using the tangent induced by the map for the macroscopic stress or by explicitly computing a map for the tangent the range of deformations over which a material map is applicable to a problem is enhanced by an adaptive scheme where a multi level finite element method is employed at highly deformed regions of a finite element mesh where the map limits are exceeded

relaxation and optimization for linear growth convex integral functionals under pde constraints abstract we give necessary and sufficient conditions for the minimality of generalized minimizers of linear growth integral functionals of the form f u f x u x d x u r d r n where f r n r is a convex integrand and u is an integrable function satisfying a general pde constraint our analysis is based on two ideas a relaxation argument into a subspace of the space of bounded vector valued radon measures m r n and the introduction of a set valued pairing on m r n l r n by these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle point conditions in particular our results can be applied to relaxation and minimization problems in bv bd and divergence free spaces

a model order reduction method for computational homogenization at finite strains on regular grids using hyperelastic laminates to approximate interfaces abstract fft based homogenization methods operate on regular voxel grids and cannot resolve interfaces exactly in general in this article we study hyperelastic laminates and associate their effective properties to voxels containing interfaces extending previous ideas successfully applied in the framework of linear elasticity more precisely we show that for strictly rank one convex energy densities of the constituents there is precisely one phase wise affine minimizer of the laminate problem a result of independent interest we demonstrate that furnishing interface voxels with appropriately rotated effective hyperelastic properties of a two phase laminate significantly enhances both the local solution quality and the accuracy of the computed effective elastic properties with only a small computational overhead compared to using classical fft based homogenization in particular for equal accuracy problems with a lower number of degrees of freedom suffice

the overall constitutive behaviour of nonlinearly elastic composites this work uses the general framework of hill for the definition of overall mechanical properties of composite materials undergoing large deformations together with ball s constitutive assumption of polyconvexity and ideas based on a hashin shtrikman variational structure developed recently by talbot willis for nonlinear problems to obtain rigorous first order depending only on the initial volume fractions of the phases and second order depending also on the overall isotropy of the composite bounds on the overall properties of composite materials with nonlinearly elastic phases although the proposed first order upper bound or voigt bound agrees with a previous result the corresponding lower bound or reuss bound is significantly tighter than the previously available result the proposed second order bounds are completely new and are of course tighter than the first order bounds additionally a self consistent estimate is given for the case when the phases are neohookean

computational homogenization analysis in finite elasticity material and structural instabilities on the micro and macro scales of periodic composites and their interaction the paper investigates instability phenomena in the context of homogenization based micro to macro transitions of heterogeneous materials at finite strains this covers structural instability effects buckling and material instability effects localization which can occur on both the macro as well as the micro scale and may influence each other we develop a general framework for the theoretical and computational treatment of these instability problems for elastic composites with given periodic fine scale micro structures the key methodology is an investigation of the properties and the interaction of two coupled minimization principles which govern a micro heterogeneous material in the context of a classical homogenization procedure the principle of minimum potential energy of the macro structure and the principle of minimum average energy of the micro structure the first variational problem determines the deformation field of the homogenized continuum the latter yields the fine scale fluctuation field on a composite micro structure that is assumed to be attached to each local point of the macro continuum global stability and the existence of solutions are based on weak convexity properties of these variational functionals the convergence of non convex homogenization functionals is based on the limit of periodic heterogeneous micro structures it defines the relevant micro structure as an a priori unknown critical ensemble of periodic cells that catches a possible minimizing buckling mode we summarize these basic results and recast them in a consistent notation suitable for numerical implementation furthermore we in detail discuss the discretization of the coupled minimization problems by means of finite element methods and point out numerical concepts for the detection of material and structural instabilities the treatment provides a comprehensive guide to the classification and computation of instabilities in micro heterogeneous solids focus is put on the interaction of micro and macro instability phenomena such as the loss of macroscopic material stability localization induced by a microscopic structural instability buckling the performance of the proposed computational methods is demonstrated for representative numerical model problems which treat coupled instability phenomena in micro heterogeneous elastic composites

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

a continuum model for nonlinear lattices under large deformations abstract a continuum model is developed for hexagonal lattices composed of a set of masses connected by linear axial and angular springs with nonlinearity arising solely from geometric effects for a set of lattice parameters these lattices exhibit complex deformation patterns under uniform loading conditions due to instabilities a continuum model accounting for these instabilities is developed from explicit expressions of the potential energy functional of a unit cell this functional is non convex it captures the bistable nature of the lattice and is used to derive its effective constitutive behavior finite element simulations of continuum medium illustrate the formation of microstructural patterns with discontinuous displacement gradients similar to the features observed in nonlinear elasticity and finite deformation plasticity a comparison of discrete lattice simulations and finite element analysis under general loading conditions illustrates that the continuum model captures the effective behavior due to instabilities within the lattice

homogenization of first order equations with u varepsilon periodic hamiltonians part i local equations in this paper we present a result of homogenization of first order hamilton jacobi equations with u varepsilon periodic hamiltonians on the one hand under a coercivity assumption on the hamiltonian and some natural regularity assumptions we prove an ergodicity property of this equation and the existence of non periodic approximate correctors on the other hand the proof of the convergence of the solution usually based on the introduction of a perturbed test function in the spirit of evans work uses here a twisted perturbed test function for a higher dimensional problem

on the commutability of homogenization and linearization in finite elasticity we consider a family of non convex integral functionals frac 1 h 2 int omega w x varepsilon rm id h nabla g x rm d x quad g in w 1 p omega mathbb r n where w is a caratheodory function periodic in its first variable and non degenerate in its second we prove under suitable conditions that the limits corresponding to linearization h 0 and homogenization varepsilon rightarrow 0 commute provided w is minimal at the identity and admits a quadratic taylor expansion at the identity moreover we show that the homogenized integrand which is determined by a multi cell homogenization formula has a quadratic taylor expansion with a quadratic term that is given by the homogenization of the second variation of w

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

stochastic homogenization of nonconvex unbounded integral functionals with convex growth we consider the well trodden ground of the problem of the homogenization of random integral functionals when the integrand has standard growth conditions the qualitative theory is well understood when it comes to unbounded functionals that is when the domain of the integrand is not the whole space and may depend on the space variable there is no satisfactory theory in this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth we first prove that if the integrand is convex and has p growth from below with p d the dimension then it admits homogenization regardless of growth conditions from above this result that crucially relies on the existence and sublinearity at infinity of correctors is also new in the periodic case in the case of nonconvex integrands we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two sided estimate by a convex integrand the domain of which may depend on the space variable that itself admits homogenization this result is of interest to the rigorous derivation of rubber elasticity from polymer physics which involves the stochastic homogenization of such unbounded functionals

numerical multiscale solution strategy for fracturing heterogeneous materials abstract this paper presents a numerical multiscale modelling strategy for simulating fracturing in materials where the fine scale heterogeneities are fully resolved with a particular focus on concrete the fine scale is modelled using a hybrid trefftz stress formulation for modelling propagating cohesive cracks the very large system of algebraic equations that emerges from detailed resolution of the fine scale structure requires an efficient iterative solver with a preconditioner that is appropriate for fracturing heterogeneous materials this paper proposes a two grid strategy for construction of the preconditioner that utilizes scale transition techniques derived for computational homogenization and represents an adaptation of the work of miehe and bayreuther 2 and its extension to fracturing heterogeneous materials for the coarse scale this paper investigates both classical c 0 continuous displacement based finite elements as well as c 1 continuous elements the preconditioned gmres krylov iterative solver with dynamic convergence tolerance is integrated with a constrained newton method with local arc length control and line searches the convergence properties and performance of the parallel implementation of the proposed solution strategy is illustrated on two numerical examples

self contact and fictitious domain using a difference convex approach a numerical approach of contact or self contact of thin structures is performed using a relaxed orientation preserving condition inserted in the difference of convex function framework this method is analyzed on academic examples and cellular media copyright 2007 john wiley sons ltd

local and nonlocal continuum modeling of inelastic periodic networks applied to stretching dominated trusses we present a nonlocal continuum model and its numerical implementation to describe the macroscale response of periodic discrete networks via second order homogenization the scale bridging technique is applied to the specific example of stretching dominated elastic and inelastic periodic truss networks experiments on small scale truss structures have highlighted the importance of nodal connections on the effective stiffness and strength therefore we describe the mechanics of trusses by accounting for the stretching of truss members and the deformation of nodes for the representative 2d examples of lattices having square and triangular architectures and for example bar and nodal constitutive laws we show that a simple continuum model based on affinely deforming a representative unit cell is sufficient to reproduce the nonlinear elastic behavior of discrete trusses by contrast localization that arises e g from inelastic deformation requires a refined model this is where the presented nonlocal continuum model is capable of accurately capturing details of localized deformation we illustrate the performance of the model by comparing the results of example finite element simulations using the continuum constitutive model to discrete lattice calculations with elastic plastic bars optimal performance is achieved when the representative unit cell of the continuum model agrees with the actual size of the discrete truss unit cell which accounts for size effects even in regimes where a separation of scales between finite element size and unit cell size does not strictly apply

two scale kinematics and linearization for simultaneous two scale analysis of periodic heterogeneous solids at finite strain abstract we introduce the notion of two scale kinematics and the procedure of two scale linearization which are indispensable to the simultaneous two scale analysis method for the mechanical behavior of periodic heterogeneous solids at finite strain these are accomplished by formulating the two scale boundary value problem in both material and spatial descriptions with reference to the two scale modeling strategy developed in comput methods appl mech engrg 190 40 41 2001 5427 that utilized the convergence results of mathematical homogenization the formulation brings the intimate relationship between micro and macro scale kinematics in describing the micro macro coupling behavior inherent in heterogeneous media it is also shown that the two scale linearization necessitates the strict consistency with the micro scale equilibrated state and naturally invites the tangential homogenization process for both material and spatial descriptions several numerical examples of simultaneous two scale computations are presented to illustrate the two scale nature of the deformation of a heterogeneous solid at finite strain

homogenization of multiparameter integrals where is a bounded domain in rn u rm 0 and w rn rmn 0 is supposed to be 0 1 periodic with respect to the 7rst variable x rn the distinguishing feature of 1 here is that the integrand is permitted to depend on a vector of parameters 1 k r with k 1 we are interested in the asymptotic behavior of 1 as 0 and 0 integral functionals of this type appear in the mathematical modeling of cellular composite materials when n m 3 1 can be interpreted as the stored strain energy of an elastic and heterogeneous material u being a deformation or displacement 7eld when m 1 u may be a di erence of potential in a condenser in any case the medium under consideration is composed of several materials which are periodically distributed at the microscopic scale given by in the applications we often deal with

a quasistatic evolution model for the interaction between fracture and damage this paper is devoted to the investigation of a quasistatic evolution model for a continuum which undergoes damage and possibly fracture in both cases the model appears to be ill posed so that it is necessary to introduce a relaxed variational evolution preserving the irreversibility of the process the minimality at each time and the energy balance from a mechanical point of view it turns out that the material prefers to form microstructures through the creation of fine mixtures between the damaged and healthy parts of the medium the brutal character of the damage process is then replaced by a progressive one where the original damage internal variable that is the characteristic function of the damaged part is replaced by the local volume fraction the analysis rests on a locality property for mixtures which enables us to use an alternative formula for the lower semicontinuous envelope of the elastic energy in terms of the g closure set

bloch wave homogenization for a spectral problem in fluid solid structures this paper is concerned with the study of the vibrations of a coupled fluid solid periodic structure as the period goes to zero an asymptotic analysis of the spectrum i e the set of eigenfrequencies is performed with the help of a new method the so called bloch wave homogenization method which is a blend of two scale convergence and bloch wave decomposition the limit spectrum is made of three parts the macroscopic or homogenized spectrum the microscopic or bloch spectrum and the boundary layer spectrum the two first parts are completely characterized the homogenized and the bloch spectra are purely essential and have a band structure the boundary layer spectrum is shown to be empty in the special case of periodic boundary condition

computational micro to macro transitions for discretized micro structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy the paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of micro structures undergoing large strain deformations typically these micro structures define representative volumes of nonlinear heterogeneous materials such as inelastic composites polycrystalline aggregates or particle assemblies we consider a priori given discretized micro structures without focussing on details of specific discretization techniques for space and time the key contribution of the paper is the construction of a family of algorithms and matrix representations of overall properties of discretized micro structures which are motivated by a minimization of averaged incremental energy it is shown that the overall stresses and tangent moduli of a typical micro structure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary we focus on deformation driven micro structures where the overall macroscopic deformation is controlled in this context three classical types of boundary conditions are investigated i linear deformation ii uniform tractions and iii periodic deformation and antiperiodic tractions incorporated by lagrangian multiplier methods these conditions generate three classes of constrained minimization problems with associated solution algorithms for the computation of equilibrium states and overall properties of micro structures the proposed algorithms and matrix representations of the overall properties are formally independent of the interior spatial structure and the local constitutive response of the micro structure and are therefore applicable to a broad class of model problems we demonstrate their performance for some representative model problems including finite elastic plastic deformations of composites texture developments in polycrystalline materials and equilibrium states of particle assemblies

a local bifurcation theorem for degenerate elliptic equations with radial symmetry abstract in this work we provide local bifurcation results for equations involving the p laplacian in balls we analyze the continua c n of radial solutions emanating from n p 0 n p being the radial eigenvalues of p first we show that the only nontrivial solutions close to n p 0 lie on a continuous curve thus extending the crandall rabinowitz theorem second it is proved that c n n p 0 splits into two unbounded connected pieces characterized by their nodal properties thus sharpening previous results

functional approximation and projection of stored energy functions in computational homogenization of hyperelastic materials a probabilistic perspective abstract this work is concerned with the construction of a surrogate model for the homogenized stored energy functions defining the effective behavior of nonlinear elastic microstructures here a probabilistic standpoint is adopted and allows for the definition of a nonlinear mapping between the macroscopic deformations and the homogenized potential this functional approximation is specifically obtained by means of a polynomial chaos expansion the coefficients of which are computed through a gauss legendre quadrature rule by invoking well known results related to projections in hilbert spaces closest approximations of arbitrary potentials to standard e g ogden type models are subsequently defined and characterized by appropriate residuals numerical illustrations on various microstructures are finally provided in order to discuss the relevance of the proposed framework in particular it is shown that the surrogate model compares very well with reference results at both the macroscale and the structural scale even for moderate order expansions and that the aforementioned closest approximations constitute very accurate approximation provided that the subset onto which the homogenized response is projected is properly chosen this result readily allows for nonconcurrent coupling using standard constitutive models available in commercial codes and therefore makes the approach very attractive for engineering applications

a class of general algorithms for multi scale analyses of heterogeneous media abstract a class of computational algorithms for multi scale analyses is developed in this paper the two scale modeling scheme for the analysis of heterogeneous media with fine periodic microstructures is generalized by using relevant variational statements instead of the method of two scale asymptotic expansion the mathematical results on the generalized convergence are utilized in the two scale variational descriptions accordingly the global local type computational schemes can be unified in association with the homogenization procedure for general nonlinear problems after formulating the problem in linear elastostatics that with local contact condition and the elastoplastic problem we present representative numerical examples along with the computational algorithm consistent with our two scale modeling strategy as well as some direct approaches

new counterexamples to the cell formula in nonconvex homogenization in this article we show that for the homogenization of multiple integrals the quasiconvexification of the cell formula is different from the asymptotic formula in general to this aim we construct three examples in three different settings the homogenization of a discrete model the homogenization of a composite material and the homogenization of a homogeneous material on a perforated domain

on full two scale expansion of the solutions of nonlinear periodic rapidly oscillating problems and higher order homogenised variational problems we consider a scalar quasilinear equation in the divergence form with periodic rapid oscillations which may be a model of e g nonlinear conducting dielectric or deforming in a restricted way hardening elastic plastic composites with outer periodicity conditions of a fixed large period under some natural growth assumptions on the stored energy function we construct for uniformly elliptic problems a full two scale asymptotic expansion which has a precise double series structure separating the slow and the fast variables in all orders so that its slowly varying part solves asymptotically an infinite order homogenised equation cf bakhvalov n s panasenko g p homogenisation averaging processes in periodic media nauka moscow 1984 in russian english translation kluwer 1989 and whose higher order terms depend on the higher gradients of the slowly varying part we prove the error bound i e that the truncated asymptotic expansion is higher order close to the actual solution in appropriate norms the approach is extended to a non uniformly elliptic case for two dimensional power law potentials we prove the non degeneracy using topological index methods examples and explicit formulae for the higher order terms are given in particular we prove that the first term in the higher order homogenised equations is related to the first order corrector to the mean flux and has in general the form of a fully nonlinear operator which is quadratic with respect to its highest second derivative being a linear combination of the second minors of the hessian with coefficients depending on the first gradient and in dimension two is of monge ampere type we show that this term is present at least for some examples three phase power law laminates

a hybrid variationally consistent homogenization approach based on ritz s method summary multiscale approaches based on homogenization theory provide a suitable framework to incorporate information associated with a small scale microscale problem into the considered large scale macroscopic problem in this connection the present paper proposes a novel computationally efficient hybrid homogenization method its backbone is a variationally consistent fe2 approach in which every aspect is governed by energy minimization in particular scale bridging is realized by the canonical principle of energy equivalence as a direct implementation of the aforementioned variationally consistent fe2 approach is numerically extensive an efficient approximation based on ritz s method is advocated by doing so the material parameters defining an effective macroscopic material model capturing the underlying microstructure can be efficiently computed furthermore the variational scale bridging principle provides some guidance to choose a suitable family of macroscopic material models comparisons between the results predicted by the novel hybrid homogenization method and full field finite element simulations show that the novel method is indeed very promising for multiscale analyses copyright 2013 john wiley sons ltd

quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity in this paper we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star shaped reference configuration rn whose boundary is subjected to an affine deformation i e there exists a constant n n matrix f and a constant n vector b such that x fx b for all x we consider all smooth equilibrium configurations satisfying this boundary condition clearly the homogeneous deformation x fx b for all x in bar omega is one such solution our aim is to prove under suitable hypotheses that it is the only such solution

on multiscale fe analyses of heterogeneous structures from homogenization to multigrid solvers heterogeneous structures like composites often need a fine scale resolution of micro effects which influence the macroscopic overall response this is of particular relevance in the fully non linear range of large strains and inelastic material response of the constituents suitable solution methods introduce a multifield scenario of hierarchically superimposed states on different length scales for big differences of micro and macro scales the argument of scale separation induces the application of homogenization methods such types of physical multiscale approaches can be treated by nested multilevel finite element analyses that discretize both the fine scale micro structure as well as the macroscopic boundary value problem in contrast small scale differences require full resolution of the heterogeneous structure effective solution methods for the resulting large scale problems with strongly oscillating properties are suitably designed geometric multigrid techniques which may be considered as numerical multiscale approaches in both scenarios a key ingredient is the suitable formulation of scale bridging algorithms that govern the transfer between different scales the paper outlines new mesh bridging techniques in a deformation driven context for fully non linear response which exploit in a non trivial manner weak constraints on the average deformation in typical finite element patches the framework is based on an incremental variational structure of finite inelasticity the proposed new formulations provide variational based homogenization algorithms for physical multiscale scenarios and problem dependent optimal finite element grid transfers for numerical multiscale scenarios of heterogeneous materials copyright 2007 john wiley sons ltd

nonlinear structural design using multiscale topology optimization part i static formulation we present a hierarchical multiscale design framework that couples computational homogenization with topology optimization to design a composite structure s microstructure to optimize its nonlinear elastostatic behavior to generate a well posed macroscopic topology optimization problem we use relaxation which requires homogenization to relate the macroscopic homogenized response to its microstructure and because closed form expressions for homogenized properties generally do not exist for materials with nonlinear response we rely on computational homogenization to evaluate them to optimize the homogenized properties of the unit cell we again use topology optimization and to make this unit cell optimization problem well posed we use restriction and thereby obtain a minimum microstructural length scale the coupled nonlinear analyzes and optimization problems are computationally intensive tasks that we resolve with a scalable parallel framework based on a single program multiple data programming paradigm numerical implementation is discussed and examples are provided

scale effects in the optimal design of a microstructured medium against buckling abstract certain classes of problems in optimal structural design lead naturally to the introduction of periodic microstructured media as the basis material for the construction of a mechanical element the unit cell size of these microstructures cannot be arbitrarily small as suggested by the pertaining optimization analyses up to date and has to be related to the structure s overall dimensions one physically important mechanism that provides such microstructure size limitations is elastic buckling an analytically tractable model of an infinite periodic rectangular planar frame with axially compressed beams is used to study the optimal buckling loads for any given design one can find a critical stress above which buckling instability occurs in addition one can also find the region in the design space for which the optimal critical mode is a global one i e its characteristic length is much larger than the unit cell size in this region of the design space one can safely use the homogenized material properties to describe the medium for they provide all the information needed to predict a global buckling instability in addition to the detailed parametric study for the model problem investigated here implications for the optimal design against buckling of more general structures are also briefly discussed

the spectrum of the periodic p laplacian we consider one dimensional p laplacian eigenvalue problems of the form pu q u p 1sgnu on 0 b together with periodic or separated boundary conditions where p 1 p is the p laplacian q c1 0 b and b 0 r it will be shown that when p 2 the structure of the spectrum in the general periodic case that is with q 0 and periodic boundary conditions can be completely different from those of the following known cases i the general periodic case with p 2 ii the periodic case with p 2 and q 0 and iii the general separated case with any p 1

formulation and implementation of stress driven and or strain driven computational homogenization for finite strain in this paper we present a homogenization approach that can be used in the geometrically nonlinear regime for stress driven and strain driven homogenization and even a combination of both special

heterogeneous thin films of martensitic materials we study the effective behavior of heterogeneous thin films with three competing length scales the film thickness and the length scales of heterogeneity and material microstructure we start with three dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der waals type and derive the effective energy density as all length scales tend to zero with given limiting ratios we do not require any a priori selection of asymptotic expansion or ansatz in deriving our results depending on the dominating length scale the effective energy density can be identified by three procedures averaging homogenization and thin film limit we apply our theory to martensitic materials with multi well energy density and use a model example to show that the shape memory behavior can crucially depend on the ratios of these length scales we comment on the effective conductivity of linear composites and also on multilayers made of shape memory and elastic materials

an investigation on microscopic and macroscopic stability phenomena of composite solids with periodic microstructure an analysis of the effects of microscopic instabilities on the homogenized response of heterogeneous solids with periodic microstructure and incrementally linear constitutive law is here carried out in order to investigate the possibility to obtain a conservative prediction of microscopic primary instability in terms of homogenized properties novel macroscopic constitutive stability measures are introduced corresponding to the positive definiteness of the homogenized moduli tensors relative to a class of conjugate stress strain pairs numerical simulations addressed to hyperelastic microstructural models representing cellular solids and reinforced composites are worked out through the implementation of an innovative one way coupled finite element formulation able to determine sequentially the principal equilibrium solution the incremental equilibrium solutions providing homogenized moduli and the stability eigenvalue problem solution for a given monotonic macrostrain path both uniaxial and equibiaxial loading conditions are considered the exact microscopic stability region in the macrostrain space obtained by taking into account microstructural details is compared with the macroscopic stability regions determined by means of the introduced macroscopic constitutive measures these results highlight how the conservativeness of the adopted macroscopic constitutive stability measure with respect to microscopic primary instability strictly depends on the type of loading condition tensile or compressive and the kind of microstructure 2010 elsevier ltd all rights reserved

exact multiplicity and stability of solutions of a 1 dimensional p laplacian problem with positive convex nonlinearity abstract we consider the p laplacian problem 1 p u f u on 1 1 2 u 1 0 where p 1 p 2 p z z p 1 sgn z z r 0 and f r 0 is c 2 and there exists c 0 q p such that f c 1 q 1 f 0 f 0 0 we show that the set of solutions u of 1 2 consists of a c 2 curve in 0 c 1 1 1 with end point 0 0 r c 1 1 1 and which tends to 0 and has a single turning point thus there exists 0 such that 1 2 has exactly two solutions u u when 0 and no solutions when when 0 the solutions u are equilibria of a related time dependent parabolic problem and in this context their stability is of interest we show that the lower solution u is stable and the upper solution u is unstable and solutions of the parabolic problem with initial values below u converge to u while those with initial values above u are unbounded

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

multi parameter homogenization by localization and blow up we give an alternative self contained proof of the homogenization theorem for periodic multi parameter integrals that was established by the authors the proof in that paper relies on the so called compactness method for gamma convergence while the one presented here is by direct verification the candidate to be the limit homogenized functional is first exhibited and the definition of gamma convergence is then verified this is done by an extension of bounded gradient sequences using the acerbi et al extension theorem from connected sets and by the adaptation of some localization and blow up techniques developed by fonseca and muller together with de giorgi s slicing method

homogenization for a class of integral functionals in spaces of probability measures abstract we study the homogenization of a class of actions with an underlying lagrangian l defined on the set of absolutely continuous paths in the wasserstein space p p r d we introduce an appropriate topology on this set and obtain the existence of a limit of the rescaled lagrangians our main goal is to provide a representation formula for these limits in terms of the effective lagrangians this allows us to study not only the convexity properties of the effective lagrangian but also the differentiability properties of its legendre transform restricted to constant functions for the case d 1 we obtain partial results in terms of an effective lagrangian defined on l p 0 1 d r d our study provides a way of computing the limit of a family of metrics on the wasserstein space the results of this paper can also be applied to study the homogenization of variational solutions of the one dimensional vlasov poisson system as well as the asymptotic behavior of calibrated curves fathi 2003 6 gangbo and tudorascu 2010 12 whereas our study for the one dimensional case covers a large class of lagrangians that for the higher dimensional case is concerned with special lagrangians such as the ones obtained by regularizing the potential energy of the d dimensional vlasov poisson system

dynamics of localized solutions to the raman extended derivative nonlinear schrodinger equation using virial estimates we study the temporal evolution of localized solutions to the raman extended derivative nonlinear schrodinger r ednls equation with a particular emphasis on collapse and non collapse by means of a lagrangian approach we investigate the possibility of realizing a finite time self similar collapse as it appears in solutions to the usual critical nonlinear schrodinger cnls equation

asymptotic rigidity of layered structures and its application in homogenization theory in the context of elasticity theory rigidity theorems allow one to derive global properties of a deformation from local ones this paper presents a new asymptotic version of rigidity applicable to elastic bodies with sufficiently stiff components arranged into fine parallel layers we show that strict global constraints of anisotropic nature occur in the limit of vanishing layer thickness and give a characterization of the class of effective deformations the optimality of the scaling relation between layer thickness and stiffness is confirmed by suitable bending constructions beyond its theoretical interest this result constitutes a key ingredient for the homogenization of variational problems modeling high contrast bilayered composite materials where the common assumption of strict inclusion of one phase in the other is clearly not satisfied we study a model inspired by hyperelasticity via gamma convergence for which we are able to give an explicit representation of the homogenized limit problem it turns out to be of integral form with its density corresponding to a cell formula

homogenization of a nonlinear elastic fibre reinforced composite a second gradient nonlinear elastic material abstract we study the homogenization of a nonlinear elastic material in contact with periodic parallel nonlinear elastic fibres of higher rigidity the energy density of both materials is of saint venant kirchhoff type the interactions between the matrix and the fibres are described by a local adhesion contact law with interfacial adhesive stiffness parameter depending on the period assuming that the lame constants in the fibres and the stiffness parameter have appropriate orders of magnitude we derive a class of nonlinear elastic energies involving displacement gradient of second order

failure oriented multi scale variational formulation micro structures with nucleation and evolution of softening bands abstract this contribution presents the theoretical foundations of a failure oriented multi scale variational formulation fomf for modeling heterogeneous softening based materials undergoing strain localization phenomena the multi scale model considers two coupled mechanical problems at different physical length scales denoted as macro and micro scales respectively every point at the macro scale is linked to a representative volume element rve and its constitutive response emerges from a consistent homogenization of the micro mechanical problem at the macroscopic level the initially continuum medium admits the nucleation and evolution of cohesive cracks due to progressive strain localization phenomena taking place at the microscopic level and caused by shear bands damage or any other possible failure mechanism a cohesive crack is introduced in the macro model once a specific macroscopic failure criterion is fulfilled the novelty of the present failure oriented multi scale formulation is based on a proper kinematical information transference from the macro to micro scales during the complete loading history even in those points where macro cracks evolve in fact the proposed fomf includes two multi scale sub models consistently coupled i a classical multi scale model clamm valid for the stable macro scale constitutive response ii a novel cohesive multi scale model cohmm valid once a macro discontinuity surface is nucleated for modeling the macro crack evolution when a macro crack is activated two important kinematical assumptions are introduced i a change in the rule that defines how the increments of generalized macro strains are inserted into the micro scale and ii the kinematical admissibility concept from where proper strain homogenization procedures are obtained then as a consequence of the hill mandel variational principle and the proposed kinematical assumptions the fomf provides an adequate homogenization formula for the stresses in the continuum part of the body as well as for the traction acting on the macro discontinuity surface the assumed macro to micro mechanism of kinematical coupling defines a specific admissible rve displacement space which is obtained by incorporating additional boundary conditions non standard boundary conditions nsbc in the new model a consequence of introducing these non standard boundary conditions is that they guarantee the existence of a physically admissible rve size a concept that we call through the paper objectivity of the homogenized constitutive response several numerical examples are presented showing the objectivity of the formulation as well as the capabilities of the new multi scale approach to model material failure problems

on the computation of the macroscopic tangent for multiscale volumetric homogenization problems in this contribution methods of computing the macroscopic tangent that is required in multiscale volumetric homogenization problems are investigated a condensation procedure that employs the tangent information from the microscale finite element analysis is derived within a special framework where deformation controlled boundary conditions in micromechanical testing are enforced via the penalty method the developed methodology is demonstrated by sample macroscale problems in the context of the multilevel finite element strategy due to the high memory allocation costs that the condensation procedure induces a perturbation procedure is developed based on a minimal number of micromechanical tests results from the condensation and perturbation procedures are compared for sample infinitesimal and finite deformation inelasticity problems and the algorithmic consistency of the macroscopic tangent with the evolution of the macroscopic stress is discussed it is shown that the ability to compute the macroscopic tangent can also be employed to construct a stress controlled micromechanical testing procedure

nonlinear wavefields in optical fibers with finite time response and amplification effects a higher order nonlinear integro differential equation of the schrodinger type modeling the evolution of finite amplitude wavepackets in the presence of optical amplification and finite material and polarization time responses is analyzed with respect to the dynamical evolution of localized solutions nonexistence of localized travelling and stationary wave solutions is pointed out for certain parameter regimes finally the linear stage of the modulational instability of plane waves is studied and it is shown that the finite time responses produce finite bandwidth instabilities

appropriate number of unit cells in a representative volume element for micro structural bifurcation encountered in a multi scale modeling the paper proposes a method to determine the number of unit cells basic structural elements to be employed for a representative volume element rve of the multi scale modeling for a solid with periodic micro structures undergoing bifurcation main difficulties for the multi scale modeling implementing instability are twofold loss of convexity of the total potential energy that should be homogenized and determination of a pertinent rve that contains multiple unit cells in order to resolve these difficulties variational formulation is achieved with the help of convergence theory within the framework of non convex homogenization method while the number of unit cells in an rve is determined by the block diagonalization method of group theoretic bifurcation theory the latter method enables us to identify the most critical bifurcation mode among possible bifurcation patterns for an assembly of arbitrary number of periodic micro structures thus the appropriate number of unit cells to be employed in the rve can be determined in a systematic manner representative numerical examples for a cellular solid show the feasibility of the proposed method and illustrate material instability at a macroscopic point due to geometrical instability in a micro scale

stability of solutions of a 1 dimensional p laplacian problem and the shape of the bifurcation curve abstract we consider the p laplacian boundary value problem 1 p u f u on 1 1 2 u 1 0 where p 1 p 2 p z z p 1 sgn z z r 0 f r r is c 2 and f 0 on r under these conditions the set of solutions u of 1 2 consists of the trivial solution u 0 0 together with a single connected c 2 curve s r c 0 1 1 1 r 0 under additional conditions on f the shape of s can be determined solutions of 1 2 are equilibrium solutions of a related time dependent parabolic problem and in this time dependent setting the stability of these equilibria is of interest it will be shown that the stability of solutions on s is determined by the shape of s this will first be discussed in a general setting and the results will then be applied to the specific case where s is s shaped finally similar results will be obtained for generic without any additional conditions on f

convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations abstract we study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t div a x u f x u 0 on a bounded domain subject to dirichlet boundary and to initial conditions the data are supposed to satisfy suitable regularity and growth conditions our approach to the convergence result and decay estimate is based on the ojasiewicz simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates the abstract results and their applications are discussed also in the framework of orlicz sobolev spaces

regularity in oscillatory nonlinear elliptic systems we give uniform bounds in morrey sobolev spaces h1 for weak solutions ue to oscillatory nonlinear elliptic systems the main novelty is in the precise qualitative description of the e uniform morrey exponent in terms of the ellipticity ratio of the elliptic systems this is achieved by showing that weak solutions of general nonlinear elliptic systems with lipschitz coefficients are of class w2 p locally or globally depending on assumptions for an exponent p 2 that is independent of dimensions and for which p 2 is inversely proportional to the ellipticity ratio

optical solitons in a monomode fiber we discuss the propagation of optical solitons in a monomode fiber as a model of long distance high bit rate transmission system we give several new results which did not appear in our previous papers on this subject such as 1 a derivation of the perturbed nonlinear schrodinger equation from the maxwell equation 2 on the integrability of the perturbed nonlinear schrodinger equation 3 a discussion of the soliton as a stable fixed point of certain infinite dimensional map generated by a transmission system with periodic excitations

numerical studies of localized wavefields governed by the raman extended derivative nonlinear schrodinger equation using a stable pseudospectral multi domain method we investigate the dynamics of localized wavefields in the extended derivative nonlinear schrodinger equation with particular emphasis on the critical mass and structure of the initial conditions that promote wave collapse the results are found to correspond well with theoretical observations based on a lagrangian approach and through comparison with solutions of the critical nonlinear schrodinger equation inclusion of high order nonlinear dissipation due to the self induced raman effect leading to the raman extended derivative nonlinear schrodinger equation is found to inhibit finite time collapse in certain cases