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The experimental-analytical model of nonlinear homogenization of matrix composites is developed on the basis of the Mori-Tanaka secant model. The calculation results of the analytical model are "fitted" to the experimental results. The fitting parameter is the ratio of constrained equivalent strains of the matrices of the experimental sample and the Mori-Tanaka model. Identification of the fitting parameter is performed using the experimental stress-strain curve for the basic composite composition. A linear dependence of the fitting parameter on the volume fraction of particles is used for a composite of arbitrary composition. The calculation results correspond well to the experimental results. [ABSTRACT FROM AUTHOR]

A simple modeling is proposed to estimate the elastic-plastic behavior of two-phase isotropic composites made of interpenetrated co-continuous phases. This nonlinear modeling, without equivalent so far, is based on the extension of an explicit Laminate System (LS) scheme proposed previously for linear elastic behavior of such co-continuous composites. Considering monotonic radial loadings, the effective nonlinear stiffness of bi-continuous elastic-plastic composites is estimated from stepwise linearizing the composite overall and phase behaviors using a classical secant linearization procedure. The efficiency of this proposed modeling is checked on some comparisons with rare experimental literature data and with other classical analytical homogenization estimate schemes. [ABSTRACT FROM AUTHOR]

Dielectric metasurfaces open new avenues in nonlinear optics through their remarkable capability of boosting frequency conversion efficiency of nonlinear optical interactions. Here, a metasurface consisting of a square array of cruciform-shaped silicon building blocks covered by a monolayer MoS2 is proposed. By designing the metasurface so that it supports optical bound states in the continuum (BICs) at the fundamental frequency and second harmonic, nearly 600× enhancement of the second-harmonic generation (SHG) in the MoS2 monolayer as compared to that of the same MoS2 monolayer suspended in air is achieved. To gain deeper insights into the physics of the metasurface-induced enhancement of nonlinear optical interactions, an eigenmode expansion method is employed to analytically investigate the main characteristics of SHG and the results show a good agreement with the results obtained via full-wave numerical simulations. In addition, a versatile nonlinear homogenization approach is used to highlight and understand the interplay between the BICs of the metasurface and the efficiency of the SHG process. This work suggests a promising method to enhance the nonlinear optical processes in two-dimensional materials, enabling the development of advanced photonic nanodevices.

Predicting failure initiation in nonlinear composite materials, often referred to as metamaterials, is a fundamental challenge in nonlinear solid mechanics. Microstructural failure mechanisms encompass fracture, decohesion, cavitation, compression-induced contact and instabilities, affecting their unconventional static and dynamic performances. To fully take advantage of these materials, especially in extreme applications, it is imperative to predict their nonlinear behaviour using reliable, accurate and computationally efficient numerical methodologies. This study presents an innovative nonlinear homogenization-based theoretical framework for characterizing the failure behaviour of periodic reinforced hyperelastic composites induced by reinforcement/matrix decohesion and interaction between contact mechanisms and microscopic instabilities. Debonding and unilateral contact between different phases are incorporated by employing an enhanced cohesive/contact model, which features a special nonlinear interface constitutive law and an accurate contact formulation within the context of finite strain continuum mechanics. The theoretical formulation is demonstrated using periodically layered composites subjected to macroscopic compressive loading conditions along the lamination direction. Numerical results illustrate the ways in which debonding phenomena, in conjunction with fibre microbuckling, may influence the critical loads of the examined composite solid. The sensitivity of the results obtained through the proposed contact–cohesive model at finite strain with respect to its implementation is also explored. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'. [ABSTRACT FROM AUTHOR]

Dielectric metasurfaces open new avenues in nonlinear optics through their remarkable capability of boosting frequency conversion efficiency of nonlinear optical interactions. Here, a metasurface consisting of a square array of cruciform-shaped silicon building blocks covered by a monolayer MoS2 is proposed. By designing the metasurface so that it supports optical bound states in the continuum (BICs) at the fundamental frequency and second harmonic, nearly 600× enhancement of the second-harmonic generation (SHG) in the MoS2 monolayer as compared to that of the same MoS2 monolayer suspended in air is achieved. To gain deeper insights into the physics of the metasurface-induced enhancement of nonlinear optical interactions, an eigenmode expansion method is employed to analytically investigate the main characteristics of SHG and the results show a good agreement with the results obtained via full-wave numerical simulations. In addition, a versatile nonlinear homogenization approach is used to highlight and understand the interplay between the BICs of the metasurface and the efficiency of the SHG process. This work suggests a promising method to enhance the nonlinear optical processes in two-dimensional materials, enabling the development of advanced photonic nanodevices. [ABSTRACT FROM AUTHOR]

Recent studies have shown that the mechanical properties of bioinspired periodic composite materials can be strongly influenced by finite deformation effects, leading to highly nonlinear static and dynamic behaviors at multiple length scales. For instance, in porous periodic nacre-like microstructures, microscopic and macroscopic instabilities may occur for a given uniaxial loading process and, as a consequence, wave attenuation properties may evolve as a function of the microstructural evolution, designating it as metamaterials. The numerical outcomes provide new opportunities to design bioinspired, soft composite metamaterials characterized by high deformability and enhanced elastic wave attenuation capabilities given by the insertion of voids and lead cores. [ABSTRACT FROM AUTHOR]

We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress local solution manifolds. Our framework is applied to a semilinear elliptic equation with oscillatory media and a nonlinear radiative transfer equation; in both cases, significant improvements in efficacy are observed. This new method does not rely on a detailed analytical understanding of multiscale PDEs, such as their asymptotic limits, and thus is more versatile for general multiscale problems. [ABSTRACT FROM AUTHOR]

Composite materials and structures are inherently in-homogeneous across multiple scales. Multi-scale modelling offers opportunities to apprehend the coupling of material behaviour and characteristics from the micro- to meso- and macro-scales. In this paper, a multi-scale finite element method (FE2) is proposed to compute the modal properties of visco-elastic heterogeneous composite materials in terms of damping frequencies and modal loss factors. In the proposed FE2-based vibration analysis, two finite elements (FE) calculations are carried out in a nested manner, one at the macro-scale and the other at the micro-scale. Unlike conventional analysis, the developed analysis does not require homogenized constitutive properties because these are derived from the micro-scale FE simulations at the representative volume element (RVE) level. The non-linearity at the micro-scale is accounted by using a frequency dependent Young’s modulus. The Asymptotic Numerical Method (ANM) and its automatic differentiation is used to solve the non-linear numerical problem. ANM consists of solving an analytical non-linear problem with a path-following (or continuation) method associated with a high-order perturbation technique. Compared with existing methods, the originality of the proposed approach lies in its ability to account for the frequency dependence of Young’s modulus in visco-elastic microstructure. Using the automatic differentiation makes the proposed approach enough flexible and generic to deal with damped and undamped vibration analyses of composite materials structures.

This article presents a consistent computational framework for multiscale first‐order finite strain homogenization and stability analyses of rate‐independent solids with periodic microstructures. The homogenization formulation is built on a priori discretized microstructure, and algorithms for computing the matrix representations of the homogenized stresses and tangent moduli are consistently derived. The homogenization results lose their validity at the onset of first bifurcation, which can be computed from multiscale stability analysis. The multiscale instabilities include: (a) microscale structural instability calculated by Bloch wave analysis; and (b) macroscale material instability calculated by rank‐1 convexity checks on the homogenized tangent moduli. Implementation details of the Bloch wave analysis are provided, including the selection of wave vector space and the retrieval of real‐valued buckling mode from complex‐valued Bloch wave. Three methods are detailed for solving the resulted constrained eigenvalue problem—two condensation methods and a null‐space projection method. Both the homogenization and stability analyses are verified using numerical examples including hyperelastic and elastoplastic periodic materials. Various microscale buckling phenomena are demonstrated. Aligned with theoretical results, the numerical results show that the microscopic long wavelength buckling can be equivalently detected by the loss of rank‐1 convexity of the homogenized tangent moduli. [ABSTRACT FROM AUTHOR]

• Homogenization estimates for the macroscopic response of nonlinear polycrystals. • Consistent estimates for the first and second moments of the microscopic fields. • Fluctuations of the fields increase with nonlinearity and grain anisotropy. • Intragranular fluctuations of the continuum vorticity do not vanish. • Continuum spin fluctuations have significant impact on elastic spin fluctuations. This paper provides estimates for the stress, strain-rate and spin field statistics in viscoplastic polycrystals. They are obtained by combining the fully optimized second-order (FOSO) homogenization method with the self-consistent estimate for linear polycrystalline aggregates. The FOSO linearization method allows the consistent estimation of the field statistics in nonlinear composites and polycrystals directly from those in suitably optimized linear comparison composites (LCC). For the LCC, a novel generalized approach is used to extract the field statistics of the stress, the strain-rate and also of the continuum-spin fields (including the second moments). Then, the stress-field statistics are used to extract the plastic-spin statistics, which, together with the continuum-spin statistics, provide consistent estimates for the microstructural-spin statistics. Applications to power-law viscoplastic FCC polycrystals and ice-like HCP polycrystals are considered, and the overall and intragranular field fluctuations are computed and found to increase with nonlinearity exponent and grain anisotropy. In particular, the covariance tensors of the continuum-spin fluctuations within the grains of viscoplastic polycrystals, which are characterized here for the first time by means of nonlinear homogenization methods, are found to be quite significant, even when the corresponding averages of the continuum-spin in the grains are negligible, and thus have a strong influence on the corresponding microstructural-spin fluctuations. [ABSTRACT FROM AUTHOR]

The generic problem in supervised machine learning is to learn a function f from a collection of samples, with the objective of predicting the value taken by f for any given input. In effect, the learning procedure consists in constructing an explicit function that approximates f in some sense. In this article is introduced a Fourier‐based machine learning method which could be an alternative or a complement to neural networks for applications in engineering. The basic idea is to extend f into a periodic function so as to use partial sums of the Fourier series as approximations. For this approach to be effective in high dimension, it proved necessary to use several ideas and concepts such as regularization, Sobol sequences and hyperbolic crosses. An attractive feature of the proposed method is that the training stage reduces to a quadratic programming problem. The presented method is first applied to some examples of high‐dimensional analytical functions, which allows some comparisons with neural networks to be made. An application to a homogenization problem in nonlinear conduction is discussed in detail. Various examples related to global sensitivity analysis, assessing effective energies of microstructures, and solving boundary value problems are presented. [ABSTRACT FROM AUTHOR]

This study presents a novel computational framework for designing optimal dissipative (damping) metamaterials under time-dependent loading conditions at finite deformations. In this framework, finite strain computational homogenization is integrated with a density-based multimaterial topology optimization. In addition, a thermodynamically consistent finite strain viscoelasticity model is incorporated together with an analytical path-dependent sensitivity analysis. Optimization formulations with and without stiffness and mass constraints are considered, and various new damping metamaterial designs are obtained that combine soft viscoelastic and stiff hyperelastic material phases. Multiscale stability analysis using the Bloch wave analysis and rank-1 convexity checks is also carried out to investigate stability of the optimized designs. Stability analyses demonstrate that the inclusion of voids or soft material phases can make a metamaterial more prone to lose micro and macro-stability. Furthermore, the concept of tunable metamaterials is explored wherein metamaterial's response is steered towards a stable deformation path by tailoring the design with a preselected micro buckling mode. [ABSTRACT FROM AUTHOR]

A novel computational framework for designing metamaterials with negative Poisson's ratio over a large strain range is presented in this work by combining the density-based topology optimization together with a mixed stress/deformation driven nonlinear homogenization method. A measure of Poisson's ratio based on the macro deformations is proposed, which is further validated through direct numerical simulations. With the consistent optimization formulations based on nonlinear homogenization, auxetic metamaterial designs with respect to different loading orientations and with different unit cell domains are systematically explored. In addition, the extension to multimaterial auxetic metamaterial designs is also considered, and stable optimization formulations are presented to obtain discrete metamaterial topologies under finite strains. Various new auxetic designs are obtained based on the proposed framework. To validate the performance of optimized designs, a multiscale stability analysis is carried out using the Bloch analysis and rank-one convexity check. As demonstrated, short and/or long wavelength instabilities can occur during the loading process, leading to a change of periodicity of the microstructure, which can affect the performance of an optimized design. • Novel computational framework for design of nonlinear auxetic metamaterials at finite strains is presented. • Nonlinear homogenization at finite strains is consistently incorporated in the density-based topology optimization. • Optimization formulations with single and multiple hyperelastic phases are considered. • New single and multimaterial auxetic metamaterials designs are discovered. • Multiscale micro and macro stability issues are addressed in the context of metamaterials design. [ABSTRACT FROM AUTHOR]

Homogenized properties of polycrystalline materials are needed in many engineering applications. The present work investigates the effectiveness of computational homogenization approaches based on the Virtual Element Method (VEM). Advantages and/or disadvantages of the VEM formulation with respect to traditional FEM approaches are explored by means of a number of numerical examples. Representative volume elements with different geometrical and material properties are investigated. Both two-and three-dimensional applications, as well as both linear and nonlinear homogenization schemes, are presented. The results show the accuracy of a VEM-based approach. On the contrary, traditional FEM-based homogenization schemes suffer with increasing grains anisotropy, requiring a high number of degree of freedoms for maintaining an acceptable accuracy. In conclusion, VEM is a promising methodology for the homogenization of polycrystalline materials. The advantage of VEM when compared to FEM is of engineering relevance for facing the challenging case of materials with strong and heterogeneous anisotropies. In fact, it is shown that VEM formulations are free from anisotropic locking. • The Virtual Element Method is applied for the homogenization of polycrystalline materials. • Different RVE geometries and anisotropic degrees are investigated. • Linear and nonlinear homogenization schemes are addressed. • In the addressed applications, VEM outperforms FEM. • VEM formulations are free from anisotropic locking. [ABSTRACT FROM AUTHOR]

We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of R n ( n ≥ 3, p ∈ [ 2 , n) ) periodically perforated by balls of radius O (ε α) where α > 1 and ε is the size of the period. The perforations are distributed along a (n - 1) -dimensional manifold γ , and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter ε - κ , κ ∈ R and ε is a small parameter that we shall make to go to zero. We analyze different relations between the parameters p , n , ε , α and κ , and obtain homogenized problems which are completely new in the literature even for the case p = 2 . [ABSTRACT FROM AUTHOR]

Abstract A one-dimensional beam-like model is proposed to analyze the mechanical behavior of elastic double-layered pipes. Besides the classical kinematic descriptors of the Timoshenko beams, further terms are introduced to describe the local distortion of the cross-section, indeed able to warp, ovalize and perform longitudinal slipping of the two layers. A homogenization procedure is applied, making use of a corresponding three-dimensional model, in order to obtain the nonlinear, coupled, elastic response function of the beam-like model. Case studies for different load conditions are considered, and instability phenomena, which cause loss of the bearing capacity of the structure, are found by means of numerical integration of the relevant boundary value problems. Highlights • Direct one-dimensional model of beam for double-layered pipe. • Three-dimensional refined model constituted by longitudinal fibers and transverse ribs. • Detection of the nonlinear response function through homogenization. • Analysis of the Brazier effect and instability phenomena. • Comparison with a finite element model. [ABSTRACT FROM AUTHOR]

Abstract We investigate the nonlinear behavior of elasto-plastic composites with isotropic and linear kinematic hardening. We first rely on the incremental variational principles introduced by Lahellec and Suquet (2007a). We also take advantage of an alternative formulation, recently proposed by Agoras et al. (2016) for visco-plastic composites without hardening, which consists in a double application of the variational procedure of Ponte-Castañeda. We extend in this paper this approach to elasto-plastic composites with combined linear kinematic and isotropic work-hardening. The first application of the variational procedure linearizes the local behavior, including hardening, and leads to a thermo-elastic Linear Comparison Composite (LCC) with a heterogeneous polarization field inside the phases. The second one deals with the heterogeneity of the polarization and results in a new thermo-elastic LCC with a per-phase homogeneous polarization field, which effective behavior can then be estimated by classical linear homogenization schemes. We develop and implement this new incremental variational procedure for composites comprised of linear elastic spherical particles isotropically distributed in an elasto-plastic matrix. The predictions of the model are compared with results available in the literature for cyclic proportional and non-proportional loadings. New results for elasto-plastic composites with combined isotropic and kinematic hardening are also provided. They are in good agreement with the numerical computations we carried out, at both local and macroscopic scales. [ABSTRACT FROM AUTHOR]

The compressive failure behavior of defected periodic fiber-reinforced composites subjected to macroscopic biaxial loadings is here analyzed from both the theoretical and numerical point of view. The coupled effects of local fiber buckling and matrix or fiber/matrix interface microcracks under unilateral self-contact are included in the analysis by adopting an original full finite deformation approach. The obtained uniqueness and stability conditions associated with a continuum rate formulation of the microscopic equilibrium problem, contain unusual nonlinear contributions arising from crack interface self-contact mechanisms whose significance for an accurate evaluation of macroscopic critical loads at the onset of instability and bifurcation is then investigated in an innovative way. To this end by using a nonlinear FE model and analytical developments, the composite failure domains are determined for radial macroscopic loading paths, ranging from biaxial compression to combined axial compression and lateral expansion, and for different microgeometrical parameters. Comparisons with stability and uniqueness domains obtained when simplified crack contact interface approaches are adopted, show the notable role of the above mentioned crack interface contributions for a realistic prediction of the real failure behavior of the composite solid. [ABSTRACT FROM AUTHOR]

We address homogenization problems for variational inequalities issue from unilateral constraints for the p-Laplacian posed in perforated domains of ℝn, with n ≥ 3 and p ∈ [2, n]. ε is a small parameter which measures the periodicity of the structure while aε ≪ ε measures the size of the perforations. We impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter βε which may be very large, namely, βε → ∞ as ε → 0. We first consider the case where p < n and the domains periodically perforated by tiny balls and we obtain homogenized problems depending on the relations between the different parameters of the problem: p, n, ε, aε and βε. Critical relations for parameters are obtained which mark important changes in the behavior of the solutions. Correctors which provide improved convergence are also computed. Then, we extend the results for p = n and the case of non periodically distributed isoperimetric perforations. We make it clear that in the averaged constants of the problem, the perimeter of the perforations appears for any shape. [ABSTRACT FROM AUTHOR]

Monocalcium aluminate and monocalcium dialuminate are minerals with a monoclinic crystal structure, that result from the fusion of limestone and bauxite. They are involved in many applications including high alumina cement, advanced biomaterials, or cement-polymer composites such as macro-defect-free cement. Despite several theoretical and experimental studies, predicting the nonlinear behavior of cement-polymer composites remains a challenge. Our research goal is to connect the effective strength behavior to the micro- and nano-constituents for cement-polymer composites, using nonlinear micromechanics, computational mechanics and experimental nano-mechanics. To this end, we extend the linear comparison composite method to non-porous granular materials. The nonlinear strength upscaling scheme is integrated into a finite element model so as to represent the mechanical behavior of calcium aluminate-polyvinyl alcohol nanocomposites—or macro-defect-free cements— across multiple length scales, while taking into account the chemistry, internal friction and grain size. At the sub-micron level, grid nano-indentation tests are carried out. An inverse scheme is implemented to predict the behavior of the basic unit, mono-calcium aluminate and monocalcium di-aluminate crystals, at the molecular level. Meanwhile, a forward approach is selected to predict the uniaxial tensile and compressive strength at the macroscopic level. The theoretical predictions agree well with independent experiments reported in the scientific literature. Macro-defect-free cements owe their extraordinary mechanical properties to a high packing density of calcium aluminate crystals, a granular morphology and the lack of porosity. The inter-disciplinary framework built can be applied to yield fundamental insights into the behavior of a broad class of engineered nano-composites. [ABSTRACT FROM AUTHOR]

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general purpose micromechanics approach capable of predicting the effective properties of heterogeneous materials and recovering the local fields. In this study, a novel micromechanics approach has been developed enabling VAMUCH to homogenize heterogeneous microstructure and predict its crack formation through a multi-scale materials genome model. A variational form for homogenization is formulated in combination with a cohesive zone model. The weak form of the problem is derived using an asymptotic method, discretized using finite element formulations, and implemented into VAMUCH. The advantages of the present approach are demonstrated through homogenizing silicon carbide ceramics and predicting its fracture strength. Both the elastic properties and fracture strength can be predicted in a computationally efficient manner using this approach compared with the multi-scale finite element model. [ABSTRACT FROM AUTHOR]

• A comparison is performed between two modelling approaches, the homogenisation approach (macro-modelling) and the micro-modelling approach. • Both modelling approaches provide a reliable prediction of masonry response at the ambient and high temperatures. • These modelling approaches can be used in combination. Refractory masonry with dry joints is widely employed as a protective lining in industrial applications requiring high-temperature treatments. The thermal and mechanical behaviour of alumina spinel refractory masonry is investigated for a wide range of mechanical loading conditions at ambient and high temperature up to 1500 °C within the framework of the ATHOR project. This paper discusses the different numerical analysis approaches for the simulation of the experimental results. Micro and macro modelling approaches show good agreement with the large scale uniaxial and biaxial compression tests for loading and unloading at the ambient temperature. Simulations carried out for large scale uniaxial and biaxial creep tests as well as biaxial relaxation tests at 1500 °C show good agreement. The numerical results indicate the ability of these modelling approaches to represent the complex thermomechanical behaviour of the refractory masonry. Both methods demonstrate an orthotropic and highly nonlinear behaviour of the refractory masonry as observed in the experimental campaign. The numerical outcome, validated with experimental results demonstrate compatibility between micro and macro modelling approach that can be employed to evaluate local and global behaviour of large industrial installations. [ABSTRACT FROM AUTHOR]

In this study, a deep learning framework for multiscale finite element analysis (FE2) is proposed. To overcome the inefficiency of the concurrent classical FE2 method induced by the repetitive analysis at each macroscopic integration points, the distance-minimizing data-driven computational mechanics is adopted for the FE2 analysis. Macroscopic strain and stress data are directly assigned to the material points of the macroscopic finite element models without constitutive model. Here, the macroscopic problems are solved with the offline macroscopic material genome database, without solving the microscopic problems simultaneously. The novelty of the proposed approach lies in using a deep neural network to enable adaptive sampling points without prior knowledge of the specific mechanical problem. The proposed data augmentation framework updates the sampling points gradually using the distance minimization algorithm with the mechanistic constraints, including the equilibrium and compatibility equations. Particularly, the deep neural network plays a crucial role as a guide in the phase space sampling process, facilitating the efficient use of sparse data. Thus, this method shows high feasibility and significantly improves the computational efficiency of offline computing. [ABSTRACT FROM AUTHOR]

The present paper proposes a hierarchical multiscale approach in order to evaluate the nonlinear constitutive behavior of concrete reinforced with carbon nanotubes (CNTs). To this purpose, representative volume elements, consistent to the microstructural topology of the material, are constructed and analyzed using finite elements. As the dimensions of the CNTs and those of a typical mesoscale concrete RVE differ by orders of magnitude, a hierarchical multiscale analysis strategy is implemented to pass information through scales with different RVEs assigned at each scale of separation. Both elastic and inelastic analyses are performed on all scales for various CNT weight fractions, defining a different nonlinear constitutive stress–strain behavior at each scale of separation. It is shown that the proposed computational approach can be used alternatively to corresponding costly experiments in order to accurately evaluate the nonlinear constitutive behavior of such complex materials. [ABSTRACT FROM AUTHOR]

For linear composite conductors, it is known that the celebrated Hashin–Shtrikman bounds can be recovered by the translation method. We investigate whether the same conclusion extends to nonlinear composites in two dimensions. To that purpose, we consider two-phase composites with perfectly conducting inclusions. In that case, explicit expressions of the various bounds considered can be obtained. The bounds provided by the translation method are compared with the nonlinear Hashin–Shtrikman-type bounds delivered by the Talbot–Willis (1985) [2] and the Ponte Castañeda (1991) [3] procedures. [ABSTRACT FROM AUTHOR]

The analysis and the synthesis of the nonlinear effective response of particulate composite materials are of great importance for developing new systems such as nonlinear elastic and electromagnetic metamaterials, nonlinear waveguides, nonlinear magnetoelectric devices and photonic or phononic crystals. Typically, classical homogenization schemes take into account the shape of the inhomogenieties but neglect the spatial correlation among them, a crucial feature for the above applications. In this paper we develop a nonlinear homogenization technique for dispersions of nonlinear particles in a linear matrix, which is able to take account of spatial correlation by means of the so-called ellipsoidal microstructure. While the linear result corresponds to the well known Ponte Castañeda–Willis estimate, we propose new formulae for the second and third order nonlinear behavior. We finally show applications to the nonlinear elastic Landau coefficients and to the nonlinear hypersusceptibility of transport processes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The computational effort of homogenization schemes based on full-field simulations is prohibitively high for many applications and motivates the use of model order reduction techniques. A reduced order homogenization scheme is presented which allows for a coupled treatment of dissipative mechanisms both in the bulk and on imperfect interfaces within the microstructure. Extending the key idea of the nonuniform transformation field analysis (NTFA), reduced bases are introduced for all fields of internal variables and also for the displacement discontinuities at the interfaces. In the online phase, an initial value problem is solved, which is derived from a mixed incremental variational principle. The variational formulation requires that the underlying constitutive models are given by two potentials, as known from the framework of generalized standard materials. As an example, a unidirectional fiber composite with a viscoplastic matrix and a viscoelastic interface is studied. It is demonstrated that the reduced order model can be used to analyze the rate-dependence of the overall behavior as well as the interface-induced size effect and the impact of varying interface parameters. At a significantly reduced computational effort, good quantitative agreement with finite element reference solutions is found for predictions of global quantities such as the effective stress and the effective mechanical work. For this is a rather simple training strategy sufficient. It is found that the reduced model provides good approximations of the local stress field and of the internal variables within the microstructure, which can benefit from a more comprehensive training. [ABSTRACT FROM AUTHOR]

In this paper, we consider the homogenization of the p -Laplace equation with a periodic coefficient that is perturbed by a local defect. This setting has been introduced in Blanc et al. (2012, 2015) in the linear setting p = 2. We construct the correctors and we derive convergence results to the homogenized solution in the case p > 2 under the assumption that the periodic correctors are non degenerate. [ABSTRACT FROM AUTHOR]

• The wave propagation properties of periodic lightened nacre-like bioinspired composite materials are investigated. • The Bloch-Floquet theory is employed to study the appearance of band gaps for several geometrical and material parameters. • Excellent wave absorption capabilities in nacre-like metamaterials lightened through hollow platelets are obtained. • New opportunities in 3D printed metamaterials design were highlighted exploiting the onset of microscopic instabilities. • The mechanical properties evolution in terms of homogenized tangent moduli tensor were investigated. In this work, the wave propagation properties of lightened bioinspired composite materials are investigated. A new lightened nacre-like microstructure is proposed demonstrating that the incorporation of hollow platelets inside the soft matrix and the application of specific prestress states represent a viable way for tuning the complete band gaps range without varying the geometry length scale and also improving the mechanical properties in terms of homogenized tangent moduli. The wave dispersion relations are obtained by adopting the Bloch-Floquet theory to investigate the appearance of complete band gaps for several geometrical and material parameters (platelets aspect ratio, matrix joints thickness, void volume fraction, and shear modulus contrast). The obtained results reveal new combinations of geometrical and material parameters for which excellent wave absorption capabilities in nacre-like periodic composite microstructure are obtained thanks to the application of prestress states which may provide new opportunities in the microstructural design of 3D printed composite metamaterials. [ABSTRACT FROM AUTHOR]

In this paper a computational two-scale model for sandwich composites with a comb-like core structure is proposed. On the global scale the sandwich panel is modeled with homogenized finite shell elements, while on the local scale a representative volume element (RVE) describes the microstructure of the sandwich through the full thickness. The local model is implemented as a constitutive law for the shell elements on the global scale. As the material response is updated in each iteration step of a nonlinear simulation, local effects of nonlinearity such as face sheet buckling or plastic flow can be described by the model. [ABSTRACT FROM AUTHOR]

This paper extends current concepts of topology optimization to the design of structures made of nonlinear microheterogeneous materials. The objective is to maximize the macroscopic structural stiffness for a prescribed material volume usage while accounting for the nonlinearity and the microstructure of the material. The resulting design problem considers two scales: the macroscopic scale at which the optimization is performed and the microscopic scale at which the material heterogeneities and the nonlinearities are observed. The topology optimization at the macroscopic scale is performed by means of the bi-directional evolutionary structural optimization method. The solution of the macroscopic boundary value problem requires as inputs the effective constitutive response with full consideration of the microstructure. While computational homogenization methods such as the FE2 method could be used to solve the nonlinear multiscale problem, the associated numerical expense (CPU time and memory) is highly unacceptable. In order to regain the computational feasibility of the computational scale transition, a recent model reduction technique of the authors is employed: the potential-based reduced basis model order reduction with graphics processing unit acceleration. Numerical examples show the efficiency of the resulting nonlinear two-scale designs. The impact of different load amplitudes on the design is examined. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study the asymptotic convergence to a periodic steady state of the solution of a nonlinear system of equations with periodic boundary data modeling electrical conduction in biological tissues, both in the microscopic and in the homogenized version. Such model keeps into account the resistive behavior of the intracellular and extracellular domains and also the capacitive/resistive behavior of the lipidic cellular membrane. The rate of convergence is analyzed and the systems of equations satisfied by the asymptotic limits are exhibited, when the resistive behavior of the membrane is described by a strictly monotone and coercive nonlinear function. The special case of homogeneous boundary conditions is also investigated, where the coercivity assumption can be relaxed. [ABSTRACT FROM AUTHOR]

In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three-dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two-scale example of heterogeneous structure with graded nonlinear properties. [ABSTRACT FROM AUTHOR]

Computationally inexpensive nonlinear homogenization methods are much sought after in academia and industry. However, the accuracy and the accessibility of the methods play an important role. Two nonlinear homogenization methods for microstructured solid materials are investigated in this work: the pRBMOR (Fritzen and Leuschner, 2013; Fritzen et al., 2014) and the NUTFA (Sepe et al., 2013). Both methods are based on ideas of the nonuniform transformation field analysis (NTFA; Michel and Suquet, 2003, 2004). A detailed comparison with respect to accuracy, storage requirements and the evolution of the reduced degrees of freedom is carried out. Numerical examples for two- and three-dimensional problems undergoing nonproportional load paths are presented. [ABSTRACT FROM AUTHOR]

This paper revisits the modeling of compressive failure of long fiber composite materials by considering a multiscale finite element approach. It is well known that this failure follows from a fiber microbuckling phenomenon. Fiber microbuckling is governed by both material and geometrical quantities: the elastoplastic shear behavior of the matrix and the fiber misalignment. Although all these parameters are easily accounted by a finite element analysis at the local level, the failure is also influenced by macrostructural quantities. That is why a multilevel finite element model (FE 2 ) is relevant to describe the compressive failure of composite. Furthermore, fiber local buckling leads to a loss of ellipticity of the macroscopic model, which can be a criterion of failure. [ABSTRACT FROM AUTHOR]

The mechanical response of composite materials is strongly influenced by the nonlinear behavior of the interface between the constituents. In order to make reliable yet computationally efficient predictions for such materials, a reduced order model is developed. Conceptual ideas of the NTFA (Michel and Suquet, Int J Solids Struct 40:6937-6955, , Comput Methods Appl Mech Eng 193:5477-5502, ) and of the pRBMOR (Fritzen, Hodapp and Leuschner Comput Methods Appl Mech Eng 260:143-154, , Fritzen et al., Comput Methods Appl Mech Eng 278:186-217, ) are adopted. The key idea is to parameterize the displacement jumps on the cohesive interfaces by a reduced basis of global ansatz functions. Micromechanical considerations and the potential structure of the constitutive models lead to a variational formulation and reduced equilibrium conditions. The effect of the preanalysis phase on the accuracy is investigated using geometrically optimal training directions. The reduced model is tested for three-dimensional microstructures. Besides the effective stress response, the tension-compression asymmetry and the distribution of the separation of the interface are investigated. Memory savings on the order of $$10^5$$ are realized. The computing time is reduced considerably. [ABSTRACT FROM AUTHOR]

On the basis of non-biased comparative evaluations of various linearization procedures used in nonlinear homogenization, performed both at the global and local scales for power-law composites (Rekik et al., 2005, 2007, 2012), we propose in this paper six ad hoc enhancements of some of the linearization procedures considered in Rekik et al. (2007). Both “stress–strain” approaches (the secant and affine formulations) or “variational formulations” (the tangent second-order method (Ponte Castañeda, 1996)) are considered. The main idea consists in proposing alternatives for the usual reference strains used by the secant, affine and tangent second-order procedures. The new linear comparison composites generated by the linearization step around the chosen alternative descriptors of the strain field statistics explicitly account for either intraphase strain fluctuations or both inter- and intraphase strain fluctuations. As a first illustration, the relevance and limitations of the enhanced linearization procedures are tested for rigidly-reinforced and porous power-law composites. For isochoric loadings, it is shown that two variants of the enhanced tangent second-order formulation lead to accurate estimates of the exact effective response which are in good agreement with the efficient second-order scheme of Ponte Castañeda (2002a). Further, the modified secant formulation provides good results for strongly nonlinear rigidly-reinforced composites away from low particulate volume fraction and the percolation threshold; however some new inherent limitations of secant formulations are also established. At last, a very discriminant situation is tested: it consists of a porous medium submitted to a pure hydrostatic loading at low pore concentrations. It is shown that one variant of the proposed enhanced second-order formulations leads to accurate estimates alike the efficient and more sophisticated formulations proposed in Bilger et al. (2002); Danas et al. (2008). [ABSTRACT FROM AUTHOR]

The variational asymptotic method for unit cell homogenization (VAMUCH) has emerged as a general-purpose micromechanics code capable of predicting the effective properties of heterogeneous materials and recovering the local fields. The objective of this paper is to propose a micromechanics approach enabling VAMUCH to homogenize elasto-viscoplastic heterogeneous materials. An affine formulation of the constitutive relations for an elasto-viscoplastic constituent, which exhibits viscoplastic anisotropy and combined isotropic–kinematic hardening, is derived. The weak form of the problem is derived using an asymptotic method, discretized using finite elements, and implemented into VAMUCH. The new features of VAMUCH are validated with examples such as homogenizing binary, fiber-reinforced, and particle-reinforced composites. VAMUCH is found to be capable of handling various microstructure, complex material models, complex loading conditions, and complex loading paths. More sophisticated material models can be implemented into it. [ABSTRACT FROM AUTHOR]

Highlights: [•] 3rd-Order bound derived for nonlinear composites under hydrostatic deformation. [•] Hashin’s sphere assemblage not an optimal microstate in nonlinear cases. [•] Distinctive behaviors of a dry porous medium from a saturated one. [Copyright &y& Elsevier]

Highlights: [•] We developed an homogenization scheme for nonlinear laminated materials. [•] It is able to work with anisotropic magneto-electro-elastic components. [•] It is based on an extension of the Backus–Tartar method. [•] The procedure takes into account an arbitrary lamination direction. [•] The explicit results can be easily implemented in a software code with standard operations of tensor calculus. [ABSTRACT FROM AUTHOR]