Macrohomogeneity Condition Research Articles

Micromechanical approach for the micropolar modeling of heterogeneous periodic media

Computational homogenization is adopted to assess the homogenized two-dimensional response of periodic composite materials where the typical microstructural dimension is not negligible with respect to the structural sizes. A micropolar homogenization is, therefore, considered coupling a Cosserat medium at the macro-level with a Cauchy medium at the micro-level, where a repetitive Unit Cell (UC) is selected. A third order polynomial map is used to apply deformation modes on the repetitive UC consistent with the macro-level strain components. Hence, the perturbation displacement field arising in the heterogeneous medium is characterized. Thus, a newly defined micromechanical approach, based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components, is adopted. Then, to estimate the effective micropolar constitutive response, the well known identification procedure based on the Hill-Mandel macro-homogeneity condition is exploited. Numerical examples for a specific composite with cubic symmetry are shown. The influence of the selection of the UC is analyzed and some critical issues are outlined.

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ɛ; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.

Bounds and scaling laws at finite scales in planar elasticity

Miniaturization and the need for novel materials with unique properties have driven composite materials to the forefront of research in solid mechanics. The response of a composite microstructure is dependent on the properties of individual phases, their distribution, the volume fractions and the scale of observation. The microstructures under investigation are sampled randomly from an infinite two-phase linear elastic planar checkerboard using a binomial distribution. A versatile methodology for investigating the effective response of such microstructures at finite scales is based on the Hill–Mandel macrohomogeneity condition. In this methodology, rigorous bounds are obtained as solutions to stochastic Dirichlet and Neumann boundary value problems from the level of a statistical volume element to that of a representative volume element (RVE). Within the framework of planar elasticity, the concept of a scaling function is introduced which unifies the treatment of several microstructures and quantifies the approach to RVE. It is demonstrated that the scaling function depends on the phase contrast and the mesoscale. Certain exact properties of the scaling function are derived rigorously, and its functional form is established using extensive numerical simulations on 163,728 microstructural realizations at varying contrasts, mesoscale and boundary conditions.

Computational homogenization in dissipative electro-mechanics of functional materials

The homogenization of the complex microscopic response of functional materials with electro-mechanical coupling is a highly demanding research topic and of utmost importance for technological design. Smart materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. In this context, an important challenge is to combine phase field descriptions of microscopic electric domain evolution with homogenization techniques, in order to define the macroscopic overall response. This paper offers a new approach to computational homogenization in dissipative electro-mechanics based on a rigorous exploitation of rate-type and incremental variational principles, governing the scale bridging between micro- and macroscales. It provides a canonical insight in the structure of the coupled electro-mechanical problems, where all field equations appear as the Euler–Lagrange equations of variational principles. A challenge is to link the gradient-type continuum description of the microstructure, governed by a phase field of micro-electric polarization, to a local electro-mechanical macro-continuum. This is achieved by the definition of boundary conditions on the surface of the microstructure, which include the phase field. To this end, Dirichlet-, Neumann-type and periodic boundary constraints are derived from a generalization of the Hill-Mandel macro-homogeneity condition towards the coupled electro-mechanical problem. We outline in detail techniques for the numerical implementation of these boundary conditions by means of Lagrange multiplier and penalty methods. Two- and three-dimensional numerical simulations treat electric domain wall motions in microstructures and their hysteretic homogenized overall response for macroscopic strain- and electric-field- driven volume elements.

Transient computational homogenization for heterogeneous materials under dynamic excitation

This paper presents a novel transient computational homogenization procedure that is suitable for the modelling of the evolution in space and in time of materials with non-steady state microstructure, such as metamaterials. This transient scheme is an extension of the classical (first-order) computational homogenization framework. It is based on an enriched description of the micro–macro kinematics by allowing large spatial fluctuations of the microscopic displacement field in contrast to the macroscopic displacement field, as a result of possible transient phenomena. From the microstructural analysis, the macroscopic stress and the macroscopic linear momentum are obtained from an extended Hill–Mandel macrohomogeneity condition. In particular, the full balance of linear momentum is solved at both scales. Consistent time discretization towards the numerical implementation of the framework as well as a condensed formulation for linear elasticity are provided to ensure a more efficient calculation of the effective response. As an example, the transient computational homogenization approach is applied to the multi-scale analysis of a metamaterial subjected to dynamic uniaxial loading. Consistency of the framework with respect to Direct Numerical Simulations is shown for a large range of loading frequencies and microstructural sizes. Attenuation of the macroscopic waves at specific loading frequencies as well as size effects are highlighted.

A two-scale finite element formulation of Stokes flow in porous media

Seepage through saturated porous material with an open pore system is modeled as a non-linear Stokes flow through a rigid matrix. Based on variationally consistent homogenization, the resulting macroscale problem becomes a Darcy-type flow. The prolongation of the Darcy flow fulfills a macrohomogeneity condition, which in a Galerkin context implies a symmetric macroscale problem. The homogenization is of 1st order and periodic boundary conditions are adopted on a Representative Volume Element. A nonlinear nested multiscale technique, in which the subscale problem is used as a constitutive model, is devised. In the presented numerical investigation, the effects of varying physical parameters as well as of the discretization are considered. In particular, it is shown that the two-scale results agree well with those of the fully resolved fine-scale problem.

Variational Homogenization of Micro‐Electro‐Elasticity

AbstractUnderstanding of micromechanical mechanisms in functional materials with electro‐mechanical coupling is a highly demanding area of simulation technology and increasing interest has been shown in the last decades. Smart materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. In this context, a key challenge is to combine models for microscopic electric domain evolution with variational principles of homogenization. We outline a variational‐based micro‐electro‐elastic model for the micro‐structural evolution of electric domains in ferroelectric ceramics. The micro‐to‐macro transition is performed on the basis of variational principles, extending purely mechanical formulations to coupled electro‐mechanics. We focus on an electro‐mechanical Boltzmann continuum on the macro‐scale with mechanical displacement and electric potential as primary variables. The material model on the micro‐scale is described by a gradient‐extended continuum formulation taking into account the polarization vector field and its gradient, see Landis [1] and Schrade et al. [2] for conceptually similar approaches. A crucial aspect of the proposed homogenization analysis is the derivation of appropriate boundary conditions on the surface of the representative volume element. In this work we derive stiff Dirichlet‐type, soft Neumann‐type, and periodic boundary constraints starting from a generalized Hill‐Mandel macrohomogeneity condition. Furthermore, we propose techniques to incorporate these boundary conditions in the variational principles of homogenization by means of Lagrange multiplier methods. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Homogenized elastic–viscoplastic behavior of anisotropic open-porous bodies with pore pressure

Constitutive modeling is studied for the homogenized elastic–viscoplastic behavior of pore-pressurized anisotropic open-porous bodies made of metallic base solids at small strains and rotations. For this purpose, by describing micro–macro relations relevant to periodic unit cells of anisotropic open-porous bodies subjected to pore pressure, constitutive features are discussed for the viscoplastic macrostrain rate in steady states. On the basis of the constitutive features found, the viscoplastic macrostrain rate is represented as an anisotropic function of Terzaghi’s effective stress, which is shown using Hill’s macrohomogeneity condition. The resulting viscoplastic equation is used to simulate the homogenized elastic–viscoplastic behavior of an ultrafine plate-fin structure subjected to uniaxial/biaxial loading in addition to pore pressure. The corresponding finite element homogenization analysis is also performed for comparison. It is demonstrated that the developed viscoplastic equation simulates well the anisotropic effect of pore pressure in the viscoplastic range in spite of there being no anisotropic factor and no fitting parameter in Terzaghi’s effective stress itself.

Reduced order computational continua

We present a new multiscale approach, hereafter referred to as reduced order computational continua (RC2), that possesses computational efficiency of phenomenological models for heterogeneous media with accuracy inherent to generalized and nonlocal continua models. The RC2 approach introduces no scale separation, makes no assumption about infinitesimality of the fine-scale structure, does not require higher order continuity, introduces no new degrees-of-freedom, is free of higher order boundary conditions and exploits a pre-computed material database to effectively solve a unit cell (representative volume) problem. It features three building blocks: (i) the nonlocal quadrature scheme, (ii) the coarse-scale stress function and (iii) the residual-free fields. The nonlocal quadrature scheme permits nonlocal interactions to extend over finite neighborhoods and thus introduces nonlocality into the two-scale integrals employed in the multiple-scale asymptotic expansion methods, or alternatively, into the Hill–Mandel macrohomogeneity condition. The coarse-scale stress function, which replaces the classical notion of coarse-scale stress being the average of fine-scale stresses, is constructed to express the governing equations in terms of coarse-scale fields only. Finally, the residual-free fields are constructed to avoid costly discrete equilibrium solution of the unit cell problems, which is known to be the bottleneck of multiscale computations.

OS0101 内圧を受ける異方オープンポーラス構造体の弾粘塑性マクロ構成式

Constitutive modeling is studied for the homogenized elastic-viscoplastic behavior of pore-pressurized anisotropic open-porous bodies made of metallic solids. For this, by describing micro-macro relations relevant to periodic unit cells of anisotropic open-porous bodies subjected to pore pressure, constitutive features are discussed for the viscoplastic macro-strain rate in steady states. On the basis of the constitutive features found, the viscoplastic macro-strain rate is represented as an anisotropic function of Terzaghi's effective stress, which is shown using Hill's macro-homogeneity condition. The resulting homogenized equation is used to simulate the homogenized elastic-visco plastic behavior of an ultrafine plate-fin structure under uniaxial and biaxial loading conditions combined with pore pressure. The corresponding finite element homogenization analysis is also performed for comparison. It is demonstrated that the developed homogenized constitutive equation simulates well the anisotropic effect of pore pressure.

NON-LOCAL COMPUTATIONAL HOMOGENIZATION OF PERIODIC MASONRY

Micro-polar and second-order homogenization procedures for periodic elastic masonry have been implemented to include geometric and material length scales in the constitutive equation. From the evaluation of the numerical response of the unit cell representative of the masonry to properly prescribed displacement boundary conditions related to homogeneous macro-strain fields, the elastic moduli of the higher-order continua are obtained on the basis of an extended Hill-Mandel macro-homogeneity condition. Elastic moduli and internal lengths for the running bond masonry are obtained in the case of Cosserat and second-order homogenization. To evaluate these results, a shear layer problem representative of a masonry wall subjected to a uniform horizontal displacement at points on the top is analyzed as a micro-polar and a second-order continuum and the results are compared to those corresponding with the reference heterogeneous model. From this analysis the second-order homogenization appears to provide better results in comparison with the micro-polar homogenization.

Macrohomogeneity condition in dynamics of micropolar media

We determine the macrohomogeneity (Hill-Mandel type) condition in the dynamic response of inhomogeneous micropolar (Cosserat) materials. The setting calls for small deformation gradients and curvatures, but without restrictions on the constitutive behavior and without any requirements of spatial periodicity. The condition gives admissible boundary loadings, along with extra terms representing kinetic energy contributions of both classical type and micropolar type. The said loadings involve various combinations of average stresses and strains, along with couple-stresses and curvature-torsion tensors. If applied to a specific microstructure in a computational mechanics approach, these boundary loadings will allow one to determine scale-dependent homogenization toward a representative volume element (RVE) of an equivalent homogeneous micropolar medium in either elastic or inelastic settings. By restricting the continuum model to an inhomogeneous Cauchy continuum and/or a quasi-static setting, the macrohomogeneity condition simplifies to conventional versions.

Variationally consistent computational homogenization of transient heat flow

AbstractA framework for variationally consistent homogenization, combined with a generalized macro‐homogeneity condition, is exploited for the analysis of non‐linear transient heat conduction. Within this framework the classical approach of (model‐based) first‐order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are (in practice) introduced in quadrature points in standard fashion. Along with the classical averages, a higher order conservation quantity is obtained. An iterative FE2‐algorithm is devised for the case of non‐linear diffusion and storage coefficients, and it is applied to transient heat conduction in a strongly heterogeneous particle composite. Parametric studies are carried out, in particular with respect to the influence of the ‘internal length’ associated with the second‐order conservation quantity. Copyright © 2009 John Wiley & Sons, Ltd.

Scaling function, anisotropy and the size of RVE in elastic random polycrystals

This article is focused on the identification of the size of the representative volume element (RVE) in linear elastic randomly structured polycrystals made up of cubic single crystals. The RVE is approached by setting up stochastic Dirichlet and Neumann boundary value problems consistent with the Hill(-Mandel) macrohomogeneity condition. Within this framework we introduce a scaling function that relates the single crystal anisotropy to the scale of observation. We derive certain exact characteristics of the scaling function and postulate others based on detailed calculations on copper, lithium, tantalum, magnesium oxide and antimony-yttrium. In deriving the above, we make use of the fact that cubic crystals and polycrystals have a uniquely determined scale-independent bulk modulus. It turns out that the scaling function is exact in the single crystal anisotropy. A methodology to develop a material selection diagram that clearly separates the microscale from the macroscale is proposed. The proposed scaling function not only bridges the length scales but also unifies the treatment of a wide spectrum of cubic crystals. Although the scope of this article is restricted to aggregates made up of cubic-shaped and cubic-symmetry single crystals, the concept of the scaling function can be generalized to other crystal shapes and classes as well as to scaling of other elastic/inelastic properties.