Abstract
We study the homogenization of a linear elastic solid reinforced by a small volume fraction of very stiff fibers. The effective material is characterized by the emergence of concentrations of strain energies around and within the fibers. The former is expressed in terms of the discrepancy appearing between the averaged effective displacements in the fibers and in the matrix, by means of a tensor-valued capacity of the cross-sections of the fibers. The latter corresponds to a combination of stretching, bending and torsional energies, where the torsional contribution is specific to anisotropic constitutive components.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I: abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. (1991). https://doi.org/10.1007/BF00375065
Bellieud, M.: Torsion effects in elastic composites with high contrast. SIAM J. Math. Anal. 41(6), 2514–2553 (2010). https://doi.org/10.1137/07069362X
Bellieud, M.: A notion of capacity related to linear elasticity. Applications to homogenization. Arch. Ration. Mech. Anal. (2012). https://doi.org/10.1007/s00205-011-0448-5
Bellieud, M.: Homogenization of Norton-Hoff fibered composites with high viscosity contrast. SIAM J. Math. Anal. (2022). https://doi.org/10.1137/20M1345785
Bellieud, M., Bouchitté, G.: Homogenization of elliptic problems in a fiber reinforced structure. Non local effects. Ann. Sc. Norm. Super. Pisa, Cl. Sci. IV 26, 407–436 (1998)
Bellieud, M., Gruais, I.: Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non local effects. Memory effects. J. Math. Pures Appl. (2005). https://doi.org/10.1016/j.matpur.2004.02.003
Bellieud, M., Licht, C., Orankitjaroen, S.: Non linear capacitary problems for a general distribution of fibers. Appl. Math. Res. Express (2014). https://doi.org/10.1093/amrx/abt002
Bouchitte, G., Felbacq, D.: Homogenization of a wire photonic crystal: the case of small volume fraction. SIAM J. Appl. Math. (2006). https://doi.org/10.1137/050633147
Briane, M.: Homogenization of the Stokes equations with high-contrast viscosity. J. Math. Pures Appl. (2003). https://doi.org/10.1016/S0021-7824(03)00012-6
Caillerie, D., Dinari, B.: A Perturbation Problem with Two Small Parameters in the Framework of the Heat Conduction of a Fiber Reinforced Body. Partial Differential Equations. Banach Center Publications, Warsaw (1987)
Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In: I. Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. II. Res. Notes in Math., vol. 60, pp. 98–138. Pitman, Boston (1982)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Birkhäuser, Basel (1993)
El Jarroudi, M., Brillard, A.: Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibres. IMA J. Appl. Math. (2001). https://doi.org/10.1093/imamat/66.6.567
Khruslov, E.Y.: Homogenized Models of Composite Media. Progress in Nonlinear Differential Equations and Their Application. Birkhäuser, Basel (1991). https://doi.org/10.1007/978-1-4684-6787-1
Murat, F., Sili, A.: Effets non locaux dans le passage 3d-1d en élasticité linéarisée anisotrope hétérogène. (French) [Nonlocal effects in the passage from 3d to 1d in linearized, anisotropic, heterogeneous elasticity]. C. R. Acad. Sci., Sér. 1 Math. 330(8), 745–750 (2000). https://doi.org/10.1016/S0764-4442(00)00232-9
Paroni, R., Sili, A.: Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers. J. Differ. Equ. 260(3), 2026–2059 (2016). https://doi.org/10.1016/j.jde.2015.09.055
Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9, 241–257 (1997). https://doi.org/10.1007/s001610050069
Author information
Authors and Affiliations
Contributions
Author 1 wrote the manuscript, prepared the figure, and reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bellieud, M. Homogenization of an Anisotropic Elastic Material Reinforced by a Small Volume Fraction of Very Stiff Anisotropic Fibers. Non Local Effects. Bending Effects. Torsional Effects. J Elast 153, 245–274 (2023). https://doi.org/10.1007/s10659-023-09986-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-023-09986-9