Abstract
The characterization of the mechanical response of complex materials, like tissues, woven composites, fibre networks etc. often requires multilevel analyses, since the micro structure strongly influences the observable behaviour. Existing approaches can be subdivided essentially into equivalent continuum or multiscale models. Often non-local continuum models are needed, see for instance [1]. The choice of the appropriate mechanical model and of its physical parameters must rely on the experimental or numerical consideration of the real micro-structure of the material. Asymptotic homogenization techniques can be effectively used for obtaining simplified continuum models, since they can be expanded up to the required order able to include the desired physical effects. In the case of lattice microstructures (networks, pantographic structures, tissues,...), discrete homogenization appears particularly useful [2, 3]. The purpose of the work is to extend the method to general, unbalanced, lattice systems, periodic or quasi-periodic, discussing the forms obtained for the constitutive relations and for the micro-rotations. The presentation will be restricted to a 2D case. Although not specifically presented, the procedure will be set within a framework ready for analysing the geometrical non linear case.
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References
dell’Isola, F., Cuomo, M., Greco, L., Della Corte, A.: Bias extension test for pantographic sheets: numerical simulations based on second gradient shear energies. J. Eng. Math. 103, 127–157 (2016)
Mourad, A.: Description topologique de l’architecture fibreuse et modelisation mecanique du myocarde. Ph.D. thesis, I.N.P.L. Grenoble (2003)
Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets Part I. Asymptotic micro-macro models identification. Math. Mech. Complex 5, 127–162 (2017)
Calí, M., Oliveri, S.M., Cella, U., Martorelli, M., Gloria, A., Speranza, D.: Mechanical characterization and modeling of downwind sailcloth in fluid-structure interaction analysis. Ocean Eng. 165, 488–504 (2018)
Boisse, P., Hamila, N., Madeo, A.: The difficulties in modeling the mechanical behavior of textile composite reinforcements with standard continuum mechanics of Cauchy. Some possible remedies. Int. J. Solids Struct. 154, 895–898 (2018)
Steigmann, D., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta Mech. Sin. 31, 373–382 (2015)
Forest, S., Sab, K.: Cosserat overall modeling of heteregeneous materials. Mech. Res. Commun. 25, 449–454 (1998)
Forest, S.: Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135, 117–131 (2009)
Forest, S., Cordero, N.M., Busso, E.P.: First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Comput. Mater. Sci. 50, 1299–1304 (2011)
Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland Publishing Company, Amsterdam (1978)
Sanchez-Palencia, E.: Homogenization method for the study of composite media. In: Asymptotic Analysis II. Lecture Notes in Mathematics, vol. 985, pp. 192–214. Springer, Heidelberg (1983)
Contrafatto, L., Cuomo, M., Greco, L.: Meso-scale simulation of concrete multiaxial behaviour. Eur. J. Environ. Civ. Eng. 21, 896–911 (2017)
Raoult, A., Caillerie, D., Mourad, A.: Elastic lattices: equilibrium, invariant laws and homogenization. Ann. Univ. Ferrara 54, 297–318 (2008)
Tollenaere, H., Caillerie, D.: Continuous modeling of lattice structures by homogenization. Adv. Eng. Softw. 29, 699–705 (1998)
dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. Roy. Soc. A 472, 1–23 (2016)
dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2D lattices: a numerical analysis on the static response and wave propagation. Proc. Estonian Acad. Sci. 64, 219–225 (2015)
Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Continuum Mech. Thermodyn. 9, 241–257 (1997)
Caillerie, D., Mourad, A., Raoult, A.: Discrete homogenization in graphene sheet modeling. J. Elasticity 84, 33–68 (2006)
Abdoul-Anziz, H., Seppecher, P.: Strain gradient and generalized continua obtained by homogenizing frame lattices. Math. Mech. Complex 6, 213–250 (2018)
Abdoul-Anziz, H., Seppecher, P.: Homogenization of periodic graph-based elastic structures. Journal de l’École polytechnique — Mathématiques, 5, 259–288 (2018)
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Boutin, C., Contrafatto, L., Cuomo, M., Gazzo, S., Greco, L. (2020). Discrete Homogenization Procedure for Estimating the Mechanical Properties of Nets and Pantographic Structures. In: Carcaterra, A., Paolone, A., Graziani, G. (eds) Proceedings of XXIV AIMETA Conference 2019. AIMETA 2019. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-41057-5_58
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