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• 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]

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]

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]

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]

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]

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]

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]

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]

In this paper, we shall formulate a new micro-mechanical model for describing both local and macroscopic elastic–plastic behavior of porous rocks with a pressure-sensitive and tension–compression asymmetric solid matrix. For this purpose, a new incremental variational mean field homogenization (IVMFH) method is first established. The novelty lies in the fact that the new formulation allows considering solid matrix obeying a nonlinear plastic yield criterion rather than the linear Drucker–Prager one adopted in the previous work Zhao et al. (2019). To this end, a new optimization procedure is proposed to estimate the effective incremental potential involving a stress-dependent relation between volumetric and deviatoric parts of local plastic strain field of solid matrix. The proposed incremental variational model takes into account the effect of non-uniform distribution of local stress and strain fields in solid matrix by incorporating their first and second moments in the homogenization process. The proposed model is able to provide not only macroscopic but also local mechanical responses of porous rocks. Its efficiency is assessed by a series of comparisons with full-field finite element calculations. Application examples are also presented through comparisons with experimental data for two typical porous rocks: Vosges sandstone and Lixhe chalk. [ABSTRACT FROM AUTHOR]

Abstract: The strong-permittivity-fluctuation theory (SPFT) is presented here for an isotropic, nonlinear composite medium under the long-wavelength approximation. The effective permittivity, effective susceptibility and figure of merit (FOM) of the homogenized composite medium are thereby estimated for the composition of the oxide glass and spherical metallic nanoparticles. The nonlinear properties SPFT is found to be insensitive to the choice of the correlation length between particles. [Copyright &y& Elsevier]

Abstract: This paper is devoted to the evaluation of various classical and more recent linearization schemes for nonlinear homogenization in terms of their efficiency to characterize local field fluctuations in nonlinear heterogeneous composites. It relies on an unbiased comparison between field statistics predicted by homogenization theories and those obtained from a reference solution solved using finite element techniques, based on the same microgeometry and boundary conditions and in which local nonlinear constitutive relations are exactly verified at each point. Two categories of linearization methods have been investigated: classical approaches based on a “stress–strain” approach (classical secant, classical and simplified affine) and methods based on “variational principles” (variational and Lahellec–Suquet procedures). For each approach, the maps and the statistical distribution functions of the local fields (strain, stress and incremental work) illustrating the intra- and inter-phase heterogeneities are provided for reinforced and porous power-law composites. This study supplements an earlier study focused on comparisons at the global level () and provides additional information on the accuracy of some available classical and recent linearization procedures. The proposed methodology gives access to a deeper insight on nonlinear homogenization schemes and may eventually lead to improvements of these formulations. [Copyright &y& Elsevier]

Abstract: A multi-scale model for the structural analysis of the in-plane response of masonry panels, characterized by periodic arrangement of bricks and mortar, is presented. The model is based on the use of two scales: at the macroscopic level the Cosserat micropolar continuum is adopted, while at the microscopic scale the classical Cauchy medium is employed. A nonlinear constitutive law is introduced at the microscopic level, which includes damage, friction, crushing and unilateral contact effects for the mortar joints. The nonlinear homogenization is performed employing the Transformation Field Analysis (TFA) technique, properly extended to the macroscopic Cosserat continuum. A numerical procedure is developed and implemented in a Finite Element (FE) code in order to analyze some interesting structural problems. In particular, four numerical applications are presented: the first one analyzes the response of the masonry Representative Volume Element (RVE) subjected to a cyclic loading history; in the other three applications, a comparison between the numerically evaluated response and the micromechanical or experimental one is performed for some masonry panels. [Copyright &y& Elsevier]

Abstract: The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without performing at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated. [Copyright &y& Elsevier]

Abstract: The mechanical response of an elastoplastic polycrystalline aggregate is estimated by means of Taylor and Hill''s incremental self-consistent models in conjunction with two rate-independent crystal-plasticity laws: standard and regularized Schmid laws. With the Taylor model, both constitutive laws lead to the same result whereas for the self-consistent model, the standard Schmid law predicts softer effective response than the regularized Schmid law and higher strain heterogeneity within the polycrystal. It is found that these features are related to the decrease of the shear tangent moduli during deformation when the standard Schmid law is considered. The presented results with the regularized Schmid law are in agreement with previous findings for power-law viscoplastic polycrystals. It is demonstrated that the description of the crystal-plasticity law significantly affects the estimate delivered by the Hill''s incremental self-consistent model for elastoplastic polycrystals. [Copyright &y& Elsevier]

Abstract: 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 and the asymptotic numerical 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 , 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 localization 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 significant influence on the macroscopic behaviour. [Copyright &y& Elsevier]

In this work a formulation for non linear multi-scale analysis of thin shells is presented and a modeling scheme is proposed in a F E 2 method's context. The shells may undergo large deformations and exhibit heterogeneous micro-structures consisting of nonlinear materials and cohesive interfaces. Appropriate use of an attached coordinate system, for the projection of the strain measures, enables the elimination of large rotations from the kinematic constraints that are imposed at the Representative Volume Element (RVE), simplifying this way the boundary value problem (BVP) to be solved at the micro-structural level. This way, the BVP formulation refers to the local coordinate system of the RVE in which the imposition of the plane stress condition becomes totally independent from the current orientation of the midsurface of the shell. Emanating from Hill's averaging theorem, which is satisfied by use of appropriated averaging relations, the principle of virtual work is formulated in the attached auxiliary system basis, where a consistent tangent stiffness matrix is analytically derived. The resulting methodology is assessed against popular benchmarks for thin shells and allows for a straight forward integration of Kirchhoff–Love shells under large deformations in existing F E 2 codes. The simulation possibilities originating from this formulation are countless and are related to the wide applicability of the Kirchhoff Love theory in many fields of engineering. • A formulation for non-linear multiscale modeling of thin shells is presented. • It is appropriate for modeling of thin shells in the context of the F E 2 method. • The shells exhibit heterogeneous micro-structures consisting of nonlinear materials and cohesive interfaces. • Material periodicity holds both spatially as well as through the shell's thickness. [ABSTRACT FROM AUTHOR]

Abstract: The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material. [Copyright &y& Elsevier]

Abstract: The present study is devoted to the development and validation of a nonlinear homogenization approach of the mechanical behavior of Callovo-Oxfordian argillites. The material is modeled as an heterogeneous composite composed of an elastoplastic clay matrix and of linear elastic or elastic damage inclusions. The macroscopic constitutive law is obtained by adapting the incremental method proposed by Hill [Hill, R., 1965. Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13, 89–101]. The approach consists in formulating the macroscopic tangent operator of the material by considering the nonlinear local behavior of each phase. Due to the matrix/inclusion morphology of the microstructure of the argillite, a Mori–Tanaka scheme is considered for the localization step. The developed model is first compared to Finite Element calculations and then validated and applied for the prediction of the macroscopic stress–strain responses of argillites. [Copyright &y& Elsevier]

Abstract: The continuum notions of effective mechanical quantities as well as the conditions that give meaningful deformation processes for homogenization problems with large deformations are reviewed. A continuum homogenization model is presented and recast as a Lagrangian-based approach for heterogeneous media that allows for an extension to discrete systems simulated via molecular dynamics (MD). A novel constitutive relation for the effective stress is derived so that the proposed Lagrangian-based approach can be used for the determination of the “stress–deformation” behavior of particle systems. The paper is concluded with a careful comparison between the proposed method and the Parrinello–Rahman approach to the determination of the “stress–deformation” behavior for MD systems. When compared with the Parrinello–Rahman method, the proposed approach clearly delineates under what conditions the Parrinello–Rahman scheme is valid. [Copyright &y& Elsevier]

A recently introduced nonlinear homogenization method [J. Mech. Phys. Solids 50 ( 2002) 737–757] is used to estimate the effective behavior and the associated strain and stress fluctuations in two-phase, power-law composites with aligned-fiber microstructures, subjected to anti-plane strain, or in-plane strain loading. Using the Hashin–Shtrikman estimates for the relevant “linear comparison composite,” results are generated for two-phase systems, including fiber-reinforced and fiber-weakened composites. These results, which are known to be exact to second-order in the heterogeneity contrast, are found to satisfy all known bounds. Explicit analytical expressions are obtained for the special case of rigid-ideally plastic composites, including results for arbitrary contrast and fiber concentration. The effective properties, as well as the phase averages and fluctuations predicted for these strongly nonlinear composites appear to be consistent with deformation mechanisms involving shear bands. More specifically, for the case where the fibers are stronger than the matrix, the predictions appear to be consistent with the shear bands tending to avoid the fibers, while the opposite would be true for the case where the fibers are weaker. [Copyright &y& Elsevier]

This work proposes a Bayesian framework for determining the mechanical properties of carbon based nanocomposites. In particular, Bayesian parameter inference is applied to learn the parameters that characterize the CNT/polymer interface in the microscale. These parameters are associated with great uncertainties and their characterization is a difficult task, since microscale measurements are costly and hard to obtain. To overcome this, the present study introduces a computational framework for updating the prior beliefs on the values of these parameters, by using measurements on large-scale structures comprised of the composite. In terms of modeling, the CNT/polymer interface is formulated with the cohesive zone model and a bilinear bond-slip constitutive law. Typically, predicting the response of such systems requires multiscale analysis approaches, such as the F E 2 method, employed in this work. Despite its accuracy, F E 2 is associated with immense computational demands for large-scale problems and, therefore, its application to the Bayesian setting, which requires multiple model evaluations, is prohibitive. To alleviate this cost, a surrogate modeling technique is developed which utilizes artificial neural networks, trained to predict the nonlinear stress–strain relationship of representative volume elements of the microstructure. The data set over which the neural network is trained, is obtained by analyzing a limited number of different RVE configurations using a detailed finite element analysis. The elaborated methodology is first validated through a numerical example from 2D elasticity, which demonstrated its high accuracy and its significant cost reduction capabilities. It is then applied to a more challenging large-scale problem from 3D elasticity. Even though this paper focuses on the characterization of the mechanical properties of composite materials, the proposed numerical procedure is generic and can be straightforwardly applied to other physically analogous phenomena related to nano-composite modeling, such as parameter identification in heat transfer or electrical conduction. • A Bayesian framework is proposed for the mechanical characterization of composites • The proposed framework is specifically tailored to FE2 multiscale analysis • The method uses macrostructure responses to update microstructural parameters • A neural network is used to replace the nonlinear homogenization scheme • This framework is generic and can be extended to other physically analogous phenomena [ABSTRACT FROM AUTHOR]

A topology optimization method is presented for the design of periodic microstructured materials with prescribed homogenized nonlinear constitutive properties over finite strain ranges. Building upon an existing computational homogenization method for periodic materials undergoing finite strain, generalized sensitivity equations are derived for the gradient-based topology optimization of periodic unit cells that account for the nonlinear homogenized response of the unit cell. The mechanical model assumes linear elastic isotropic materials, geometric nonlinearity at finite strain, and a quasi-static response. The optimization problem is solved by a nonlinear programming method and the sensitivities computed via the adjoint method. Several topology optimization examples are presented to evaluate the performance of the developed method. Furthermore, two-dimensional structures identified using this optimization method are additively manufactured and their uniaxial tensile strain response compared with the numerically predicted behavior. The optimization approach herein enables the design and development of lattice-like materials with prescribed nonlinear effective properties, for use in myriad potential applications, ranging from stress wave and vibration mitigation to soft robotics. [ABSTRACT FROM AUTHOR]

[Display omitted] In this work, we shall propose a new micro-mechanical constitutive model for the estimation of effective elastic-plastic behaviors of heterogeneous rocks. A bi-potential based incremental variational (BIV) approach is developed in order to take into account non-uniform local strain fields of constituents. The studied materials are composed of a non-associated and pressure sensitive plastic matrix, elastic inclusions and/or voids. For clarity, the local behavior of matrix is first described by an elastic perfectly-plastic model. Based on the bi-potential theory to dealing with non-associated plastic flow, the solid matrix is considered as pertaining to implicit standard materials (ISMs). The effective incremental bi-potential and macroscopic stress tensor are then estimated through an extension of the incremental variational method initially established for generalized standard materials(GSMs). The accuracy of the BIV model is verified by comparing the model's predictions with the reference results obtained from direct finite element simulations. Furthermore, by assuming that the general formulation obtained for the perfectly plastic matrix remains valid for each loading increment, the BIV model is extended to considering that the solid matrix exhibits an isotropic hardening by using an explicit algorithm. The accuracy of the extended BIV model is also validated by a series of comparisons with the reference solutions obtained by direct finite element simulations for both inclusion-reinforced composites and porous materials. Both local and macroscopic responses are compared. As an example of application, the extended BIV model is finally applied to estimating the mechanical responses of typical claystone and sandstone under different loading paths. [ABSTRACT FROM AUTHOR]

• Computational nonlinear homogenization of refractory masonry is performed. • Effects of joints closure/reopening on the mechanical behavior are considered. • Computationally efficient homogenized elastic-viscoplastic model is developed. • Dry joints close and reopen due to cyclic loading and unloading. • The homogenized mechanical response of the masonry is orthotropic and nonlinear. [Display omitted] Refractory masonry with dry joints is widely used in the steel-making industry for the linings of several high-temperature components (>1500 ∘ C) including steel ladles and furnaces. To properly optimize the design and performance of these linings, thorough numerical models that consider the presence of joints, joints closure and reopening and the nonlinear elastic-viscoplastic behaviour (creep and stress relaxation) of refractories at high temperature are required. The present study reports on the formulation, numerical implementation, and application of a homogenized multi-scale elastic-viscoplastic model for the simulation of refractory masonry linings with dry joints. Refractory bricks are considered to exhibit linear elasticity as well as rate-dependent plasticity. Four joint patterns are predefined based on the state of bed and head joints. The homogenized elastic-viscoplastic behaviour of each joint pattern is determined using finite element based nonlinear homogenization approach. The transition criteria between the four patterns are defined in terms of macroscopic stresses and strains. Verification of the developed homogenized constitutive laws is carried out by comparing the numerical results of the detailed micro models (brick and joints are considered) with the homogeneous equivalent material models. Furthermore, comparisons with experimental results of refractory masonry walls subjected to biaxial compression load at room and high temperature are carried out. Good agreements between the experimental and numerical results are obtained. Then, the validated models are employed to predict the mechanical behavior of refractory masonry structures subjected to different loading conditions. The present numerical model is able to simulate the orthotropic, compressible, rate-dependent homogenized behaviour of mortarless refractory masonry structures, and accounts for joints closure and reopening due to loading and unloading. [ABSTRACT FROM AUTHOR]

In this paper, the mechanical behavior of bio-inspired nacre-like staggered composites is studied. The bio-inspired materials, combining stiff and soft constituents, exhibit superior mechanical properties. Here, the attention is focused on the competing properties: penetration resistance and flexibility of the composites. To this end, a novel hybrid multiscale method is developed, combining a hierarchical multiscale approach with a concurrent approach. The method allows to perform accurate parametric nonlinear analyses at a low computational cost. The influence of the microstructural parameters (i.e., platelet aspect ratio and volume fraction) on the macroscopic mechanical behavior is thus analyzed. Finally, the potential of achieving tailored protective properties and flexibility through microstructural design of the bio-inspired composites is illustrated. [ABSTRACT FROM AUTHOR]

Abstract: A Hill type incremental homogenization method was recently proposed by Abou-Chakra Guéry et al. [Abou-Chakra Guéry, et al., 2008. A micromechanical model of elasto-plastic and damage behavior of a cohesive geomaterial. Int. J. Solid. Struct., 45(5), 1406–1429] for the elasto-plastic damage behavior of a cohesive geomaterial, the Callovo–Oxfordian argillite. The aim of this paper is then to extend the homogenization method to its time-dependent behavior. The argillite is now seen as a three phase composite with an elasto-viscoplasticity matrix and elastic damaged calcite grains and elastic quartz grains. Considering that the incremental formulation of Hill cannot rigorously be used, we proposed a new modified incremental method. A validation of its predictions against experimental data is then conducted. [Copyright &y& Elsevier]