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ISSN Print: 1543-1649
ISSN Online: 1940-4352
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EVALUATION OF GENERALIZED CONTINUUM SUBSTITUTION MODELS FOR HETEROGENEOUS MATERIALS
ABSTRACT
Several extensions of standard homogenization methods for composite materials have been proposed in the literature that rely on the use of polynomial boundary conditions enhancing the classical affine conditions on the unit cell. Depending on the choice of the polynomial, overall Cosserat, second gradient, or micromorphic homogeneous substitution media are obtained. They can be used to compute the response of the composite when the characteristic length associated with the variation of the applied loading conditions becomes of the order of the size of the material inhomogeneities. A significant difference between the available methods is the nature of the fluctuation field added to the polynomial expansion of the displacement field in the unit cell, which results in different definitions of the overall stress and strain measures and higher order elastic moduli. The overall higher order elastic moduli obtained from some of these methods are compared in the present contribution in the case of a specific periodic two-phase composite material. The performance of the obtained overall substitution media is evaluated for a chosen boundary value problem at the macroscopic scale for which a reference finite element solution is available. Several unsatisfactory features of the available theories are pointed out, even though some model predictions turn out to be highly relevant. Improvement of the prediction can be obtained by a precise estimation of the fluctuation at the boundary of the unit cell.
-
Auffray, N., Bouchet, R., and Brechet, Y., Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior. DOI: 10.1016/j.ijsolstr.2008.09.009
-
Auffray, N., Bouchet, R., and Brechet, Y., Strain gradient elastic homogenization of bidimensional cellular media. DOI: 10.1016/j.ijsolstr.2010.03.011
-
Bacigalupo, A. and Gambarotta, L., Second-order computational homogenization of heterogeneous materials with periodic microstructure. DOI: 10.1002/zamm.201000031
-
Bigoni, D. and Drugan, W. J., Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. DOI: 10.1115/1.2711225
-
Boutin, C., Microstructural effects in elastic composites. DOI: 10.1016/0020-7683(95)00089-5
-
Bouyge, F., Jasiuk, I., and Ostoja-Starzewski, M., A micromechanically based couple-stress model of an elastic two-phase composite. DOI: 10.1016/S0020-7683(00)00132-3
-
De Bellis, M. L. and Addessi, D., A Cosserat based multi-scale model for masonry structures. DOI: 10.1615/IntJMultCompEng.2011002758
-
Dell'Isola, F., Rosa, L., and Wozniak, C., A micro-structured continuum modelling compacting fluid-saturated grounds: The effects of pore-size scale parameter. DOI: 10.1007/BF01170371
-
Dillard, T., Forest, S., and Ienny, P., Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. DOI: 10.1016/j.euromechsol.2005.11.006
-
Ebinger, T., Steeb, H., and Diebels, S., Modeling macroscopic extended continua with the aid of numerical homogenization schemes. DOI: 10.1016/j.commatsci.2004.09.034
-
Eringen, A. C. and Suhubi, E. S., Nonlinear theory of simple microelastic solids. DOI: 10.1016/0020-7225(64)90004-7
-
Feyel, F., A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua. DOI: 10.1016/S0045-7825(03)00348-7
-
Fish, J. and Kuznetsov, S., Computational continua. DOI: 10.1002/nme.2918
-
Forest, S., Mechanics of generalized continua: Construction by homogenization. DOI: 10.1051/jp4:1998405
-
Forest, S., Aufbau und identifikation von stoffgleichungen für höhere kontinua mittels homogenisierungsmethoden.
-
Forest, S., Homogenization methods and the mechanics of generalized continua—Part 2. DOI: 10.2298/TAM0229113F
-
Forest, S., The micromorphic approach for gradient elasticity, viscoplasticity and damage. DOI: 10.1061/(ASCE)0733-9399(2009)135:3(117)
-
Forest, S. and Sab, K., Cosserat overall modeling of heterogeneous materials. DOI: 10.1016/S0093-6413(98)00059-7
-
Forest, S. and Trinh, D. K., Generalized continua and non-homogeneous boundary conditions in homogenization methods. DOI: 10.1002/zamm.201000109
-
Geers, M. G. D., Kouznetsova, V. G., and Brekelmans, W. A. M., Gradient-enhanced computational homogenization for the micro-macro scale transition. DOI: 10.1051/jp4:2001518
-
Gologanu, M., Leblond, J. B., and Devaux, J., Continuum micromechanics, Recent extensions of Gurson's model for porous ductile metals.
-
Janicke, R., Micromorphic media: Interpretation by homogenisation.
-
Janicke, R. and Diebels, S., A numerical homogenisation strategy for micromorphic continua. DOI: 10.1393/ncc/i2009-10348-1
-
Janicke, R., Diebels, S., Sehlhorst, H.-G., and Duster, A., Two-scale modelling of micromorphic continua. DOI: 10.1007/s00161-009-0114-4
-
Kaczmarczyk, L., Pearce, C. J., and Bicanic, N., Scale transition and enforcement of RVE boundary conditions in second-order computational homogenization. DOI: 10.1002/nme.2188
-
Kanit, T., Forest, S., Galliet, I., Mounoury, V., and Jeulin, D., Determination of the size of the representative volume element for random composites: Statistical and numerical approach. DOI: 10.1016/S0020-7683(03)00143-4
-
Kouznetsova, V., Geers, M. G. C., and Brekelmans, W. A. M., Size of a RVE in a second order computational homozenization framework.
-
Kouznetsova, V. G., Geers, M. G. D., and Brekelmans, W. A. M., Multi-scale second-order computational homogenization of multi-phase materials: A nested finite element solution strategy. DOI: 10.1016/j.cma.2003.12.073
-
Larsson, R. and Diebels, S., A second-order homogenization procedure for multi-scale analysis based on micropolar kinematics. DOI: 10.1002/nme.1854
-
Masiani, R. and Trovalusci, P., Cosserat and Cauchy materials as continuum models of brick masonry. DOI: 10.1007/BF00429930
-
Mindlin, R. D., Micro-structure in linear elasticity. DOI: 10.1007/BF00248490
-
Mindlin, R. D. and Eshel, N. N., On first strain gradient theories in linear elasticity. DOI: 10.1016/0020-7683(68)90036-X
-
Muhlich, U., Zybell, L., and Kuna, M., Micromechanical modelling of size effects in failure of porous elastic solids using first oder plane strain gradient eslaticity.
-
Ostoja-Starzewski, M., Boccara, S. D., and Jasiuk, I., Couple-stress moduli and characteristic length of two-phase composite. DOI: 10.1016/S0093-6413(99)00039-7
-
Pham, T. T. T., Un modèle d'endommagement à gradient de déformation à partir de la méthode d'homogénéisation pour les matériaux fragiles.
-
Sansalone, V., Trovalusci, P., and Cleri, F., Multiscale modeling of composite materials by a multifield finite element approach. DOI: 10.1615/IntJMultCompEng.v3.i4.50
-
Tekoglu, C. and Onck, P. R., Size effects in two-dimensional Voronoi foams: A comparison between generalized continua and discrete models. DOI: 10.1016/j.jmps.2008.06.007
-
Triantafyllidis, N. and Bardenhagen, S., The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models. DOI: 10.1016/0022-5096(96)00047-6
-
Trovalusci, P. and Masiani, R., Non-linear micropolar and classical continua for anisotropic discontinuous materials. DOI: 10.1016/S0020-7683(02)00584-X
-
Yuan, X., Tomita, Y., and Andou, T., A micromechanical approach of nonlocal modeling for media with periodic microstructures. DOI: 10.1016/j.mechrescom.2007.07.004
-
Auffray N., Le Quang H., He Q.C., Matrix representations for 3D strain-gradient elasticity, Journal of the Mechanics and Physics of Solids, 61, 5, 2013. Crossref
-
Trovalusci Patrizia, Pau Annamaria, Derivation of microstructured continua from lattice systems via principle of virtual works: the case of masonry-like materials as micropolar, second gradient and classical continua, Acta Mechanica, 225, 1, 2014. Crossref
-
Auffray Nicolas, Analytical expressions for odd-order anisotropic tensor dimension, Comptes Rendus Mécanique, 342, 5, 2014. Crossref
-
Jänicke R., Quintal B., Steeb H., Numerical homogenization of mesoscopic loss in poroelastic media, European Journal of Mechanics - A/Solids, 49, 2015. Crossref
-
Bacigalupo Andrea, Gambarotta Luigi, Homogenization of periodic hexa- and tetrachiral cellular solids, Composite Structures, 116, 2014. Crossref
-
Rahali Y., Goda I., Ganghoffer J.F., Numerical identification of classical and nonclassical moduli of 3D woven textiles and analysis of scale effects, Composite Structures, 135, 2016. Crossref
-
Rahali Y., Giorgio I., Ganghoffer J.F., dell'Isola F., Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices, International Journal of Engineering Science, 97, 2015. Crossref
-
Jung A., Diebels S., Modelling of metal foams by a modified elastic law, Mechanics of Materials, 101, 2016. Crossref
-
Forest Samuel, Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472, 2188, 2016. Crossref
-
dell’Isola Francesco, Bucci Sara, Battista Antonio, Against the Fragmentation of Knowledge: The Power of Multidisciplinary Research for the Design of Metamaterials, in Advanced Methods of Continuum Mechanics for Materials and Structures, 60, 2016. Crossref
-
Rosi G., Auffray N., Anisotropic and dispersive wave propagation within strain-gradient framework, Wave Motion, 63, 2016. Crossref
-
Goda Ibrahim, Ganghoffer Jean-François, Construction of first and second order grade anisotropic continuum media for 3D porous and textile composite structures, Composite Structures, 141, 2016. Crossref
-
Eremeyev Victor A., dell’Isola Francesco, Boutin Claude, Steigmann David, Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions, Journal of Elasticity, 132, 2, 2018. Crossref
-
Eugster Simon R., dell'Isola Francesco, Exegesis of Sect. II and III.A from “Fundamentals of the Mechanics of Continua” by E. Hellinger, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 98, 1, 2018. Crossref
-
Delfani M.R., Bagherpour V., Overall properties of particulate composites with periodic microstructure in second strain gradient theory of elasticity, Mechanics of Materials, 113, 2017. Crossref
-
Forest Samuel, Sab Karam, Finite-deformation second-order micromorphic theory and its relations to strain and stress gradient models, Mathematics and Mechanics of Solids, 25, 7, 2020. Crossref
-
Auffray N, Kolev B, Olive M, Handbook of bi-dimensional tensors: Part I: Harmonic decomposition and symmetry classes, Mathematics and Mechanics of Solids, 22, 9, 2017. Crossref
-
Placidi Luca, Giorgio Ivan, Della Corte Alessandro, Scerrato Daria, Euromech 563 Cisterna di Latina 17–21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: A review of presentations and discussions, Mathematics and Mechanics of Solids, 22, 2, 2017. Crossref
-
Berkache K., Deogekar S., Goda I., Picu R.C., Ganghoffer J.-F., Construction of second gradient continuum models for random fibrous networks and analysis of size effects, Composite Structures, 181, 2017. Crossref
-
dell’Isola Francesco, Andreaus Ugo, Cazzani Antonio, Barchiesi Emilio, Introductory remarks about the Volume II of the Complete Works of Gabrio Piola, in The Complete Works of Gabrio Piola: Volume II, 97, 2019. Crossref
-
Ganghoffer Jean-François, Maurice Gérard, Rahali Yosra, Determination of closed form expressions of the second-gradient elastic moduli of multi-layer composites using the periodic unfolding method, Mathematics and Mechanics of Solids, 24, 5, 2019. Crossref
-
Chen Chen, Roberts A J, Bunder J E, Boundary conditions for macroscale waves in an elastic system with microscale heterogeneity, IMA Journal of Applied Mathematics, 83, 3, 2018. Crossref
-
Forest Samuel, Micromorphic Approach to Materials with Internal Length, in Encyclopedia of Continuum Mechanics, 2018. Crossref
-
Maurice Gérard, Ganghoffer Jean-François, Rahali Yosra, Second gradient homogenization of multilayered composites based on the method of oscillating functions, Mathematics and Mechanics of Solids, 24, 7, 2019. Crossref
-
Khakalo Sergei, Niiranen Jarkko, Form II of Mindlin's second strain gradient theory of elasticity with a simplification: For materials and structures from nano- to macro-scales, European Journal of Mechanics - A/Solids, 71, 2018. Crossref
-
Rocha Felipe Figueredo, Blanco Pablo Javier, Sánchez Pablo Javier, Feijóo Raúl Antonino, Multi-scale modelling of arterial tissue: Linking networks of fibres to continua, Computer Methods in Applied Mechanics and Engineering, 341, 2018. Crossref
-
Dirrenberger Justin, From Architectured Materials to Large-Scale Additive Manufacturing, in Robotic Building, 2018. Crossref
-
Dirrenberger Justin, Forest Samuel, Jeulin Dominique, Computational Homogenization of Architectured Materials, in Architectured Materials in Nature and Engineering, 282, 2019. Crossref
-
Khakalo Sergei, Niiranen Jarkko, Lattice structures as thermoelastic strain gradient metamaterials: Evidence from full-field simulations and applications to functionally step-wise-graded beams, Composites Part B: Engineering, 177, 2019. Crossref
-
Rahali Y, Eremeyev VA, Ganghoffer JF, Surface effects of network materials based on strain gradient homogenized media, Mathematics and Mechanics of Solids, 25, 2, 2020. Crossref
-
dell'Isola Francesco, Steigmann David, Corte Alessandro Della, Synthesis of Fibrous Complex Structures: Designing Microstructure to Deliver Targeted Macroscale Response, Applied Mechanics Reviews, 67, 6, 2015. Crossref
-
Ganghoffer J.F., Rahouadj R., Boisse J., Schiavi J., A phase field approach for bone remodeling based on a second-gradient model, Mechanics Research Communications, 96, 2019. Crossref
-
Berkache Kamel, Deogekar Sai, Goda Ibrahim, Picu R Catalin, Ganghoffer Jean-François, Homogenized elastic response of random fiber networks based on strain gradient continuum models, Mathematics and Mechanics of Solids, 24, 12, 2019. Crossref
-
Neff Patrizio, Eidel Bernhard, d’Agostino Marco Valerio, Madeo Angela, Identification of Scale-Independent Material Parameters in the Relaxed Micromorphic Model Through Model-Adapted First Order Homogenization, Journal of Elasticity, 139, 2, 2020. Crossref
-
Laudato Marco, Di Cosmo Fabio, Drobnicki Rafał, Göransson Peter, Dynamical Vector Fields on Pantographic Sheet: Experimental Observations, in New Achievements in Continuum Mechanics and Thermodynamics, 108, 2019. Crossref
-
Jänicke Ralf, Steeb Holger, Minimal loading conditions for higher-order numerical homogenisation schemes, Archive of Applied Mechanics, 82, 8, 2012. Crossref
-
dell’Isola Francesco, Eremeyev Victor A., Some Introductory and Historical Remarks on Mechanics of Microstructured Materials, in Advances in Mechanics of Microstructured Media and Structures, 87, 2018. Crossref
-
Barchiesi Emilio, Khakalo Sergei, Variational asymptotic homogenization of beam-like square lattice structures, Mathematics and Mechanics of Solids, 24, 10, 2019. Crossref
-
Khakalo Sergei, Niiranen Jarkko, Anisotropic strain gradient thermoelasticity for cellular structures: Plate models, homogenization and isogeometric analysis, Journal of the Mechanics and Physics of Solids, 134, 2020. Crossref
-
Moosavian H., Shodja H. M., Mindlin–Eringen anisotropic micromorphic elasticity and lattice dynamics representation, Philosophical Magazine, 100, 2, 2020. Crossref
-
Forest Samuel, Micromorphic Approach to Materials with Internal Length, in Encyclopedia of Continuum Mechanics, 2020. Crossref
-
Shodja Hossein M., Moosavian Hashem, Weakly nonlocal micromorphic elasticity for diamond structures vis-à-vis lattice dynamics, Mechanics of Materials, 147, 2020. Crossref
-
Solyaev Yury, Lurie Sergey, Ustenko Anastasia, On the Relations between Direct and Energy Based Homogenization Approaches in Second Gradient Elasticity, in Developments and Novel Approaches in Biomechanics and Metamaterials, 132, 2020. Crossref
-
Ganghoffer JF, Do XN, Maurice G, Macrohomogeneity condition for strain gradient homogenization of periodic heterogeneous media with interfacial strong discontinuities, Mathematics and Mechanics of Solids, 26, 3, 2021. Crossref
-
Yang Hua, Timofeev Dmitry, Giorgio Ivan, Müller Wolfgang H., Effective strain gradient continuum model of metamaterials and size effects analysis, Continuum Mechanics and Thermodynamics, 2020. Crossref
-
Rahali Yosra, Reda Hilal, Vieille Benoit, Lakiss Hassan, Ganghoffer Jean-François, Second Gradient Linear and Nonlinear Constitutive Models of Architectured Materials: Static and Dynamic Behaviors, in Mathematical Applications in Continuum and Structural Mechanics, 127, 2021. Crossref
-
Alavi S.E., Ganghoffer J.F., Reda H., Sadighi M., Construction of micromorphic continua by homogenization based on variational principles, Journal of the Mechanics and Physics of Solids, 153, 2021. Crossref
-
Ganghoffer J.F., Reda H., A variational approach of homogenization of heterogeneous materials towards second gradient continua, Mechanics of Materials, 158, 2021. Crossref
-
Rocha Felipe Figueredo, Blanco Pablo Javier, Sánchez Pablo Javier, de Souza Neto Eduardo, Feijóo Raúl Antonino, Damage-driven strain localisation in networks of fibres: A computational homogenisation approach, Computers & Structures, 255, 2021. Crossref
-
Le Quang Hung, He Qi-Chang, Auffray Nicolas, Classification of first strain-gradient elasticity tensors by symmetry planes, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477, 2251, 2021. Crossref
-
Alavi S. E., Nasimsobhan M., Ganghoffer J. F., Sinoimeri A., Sadighi M., Chiral Cosserat model for architected materials constructed by homogenization, Meccanica, 56, 10, 2021. Crossref
-
Reda H., Alavi S.E., Nasimsobhan M., Ganghoffer J.F., Homogenization towards chiral Cosserat continua and applications to enhanced Timoshenko beam theories, Mechanics of Materials, 155, 2021. Crossref
-
Yvonnet J., Auffray N., Monchiet V., Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior, International Journal of Solids and Structures, 191-192, 2020. Crossref
-
Ayad M., Karathanasopoulos N., Reda H., Ganghoffer J.F., Lakiss H., On the role of second gradient constitutive parameters in the static and dynamic analysis of heterogeneous media with micro-inertia effects, International Journal of Solids and Structures, 190, 2020. Crossref
-
Lahbazi Ahmed, Goda Ibrahim, Ganghoffer Jean-François, Size-independent strain gradient effective models based on homogenization methods: Applications to 3D composite materials, pantograph and thin walled lattices, Composite Structures, 284, 2022. Crossref
-
Areias Pedro, Melicio Rui, Carapau Fernando, Carrilho Lopes José, Finite Gradient Models with Enriched RBF-Based Interpolation, Mathematics, 10, 16, 2022. Crossref
-
Mawassy Nagham, Ganghoffer Jean-Francois, Reda Hilal, Alavi S.E., Lakiss Hassan, Analysis of surface effects based on first and second strain gradient mechanics, Mechanics of Materials, 2022. Crossref
-
Alavi S Ehsan, Ganghoffer Jean-François, Sadighi Mojtaba, Chiral Cosserat homogenized constitutive models of architected media based on micromorphic homogenization, Mathematics and Mechanics of Solids, 27, 10, 2022. Crossref