## Homogenization of an elastic medium having three phases.(English)Zbl 1320.35036

Summary: We study an elastostatic problem in an $$\epsilon$$-periodic medium having three phases: matrix, fibers, and fiber coatings. The rigidity is of order one along the fibers and is scaled by $$\epsilon^2$$ (the so-called double porosity scaling) in both the transverse directions and the fiber coatings. Using the homogenization process, we show that both the effective transverse traction and the longitudinal stress in the fibers are mainly influenced by the elastic properties of the fiber coatings.

### MSC:

 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 74B05 Classical linear elasticity 74Q05 Homogenization in equilibrium problems of solid mechanics
Full Text:

### References:

 [1] Allaire, G, Homogenization and two-scale convergence, SIAM J. Math. Anal., 6, 1482-1518, (1992) · Zbl 0770.35005 [2] Bellieud, M; Bouchité, G, Homogenization of a soft elastic material reinforced by fibers, Asymptotic Anal., 32, 153-183, (2002) · Zbl 1020.74032 [3] Boughammoura, A, A high-contrast Poincaré-type inequality for linear elasticity, Appl. Math. Lett., 25, 2203-2208, (2012) · Zbl 1386.35393 [4] Boughammoura, A, Homogenization of a degenerate parabolic problem in a highly heterogeneous medium with highly anisotropic fibers, Math. Computer Model., 49, 66-79, (2009) · Zbl 1165.80302 [5] Brillard, A; Jarroudi, M, Asymptotic behaviour of a cylindrical elastic structure periodically reinforced along identical fibres, IMA J. Appl. Math., 66, 567-598, (2001) · Zbl 1016.74053 [6] Caillerie, D, Homogénéisation d’un corps élastique renforcé par des fibres minces de grande rigidité et réparties périodiquement, C. R. Acad. Sci. Paris, Sér. II, 292, 477-480, (1981) · Zbl 0467.73022 [7] Gustafsson, B; Mossino, J, Compensated compactness for homogenization and reduction of dimension: the case of elastic laminates, Asymptotic Anal., 47, 139-169, (2006) · Zbl 1113.35018 [8] Hashin, Z, Thin interphase/imperfect interface in elasticity with application to coated fiber composites, J. Mech. Phys. Solids., 50, 2509-2537, (2002) · Zbl 1080.74006 [9] Kari, S; Berger, H; Gabbert, U; Guinovart-Diaz, R; Bravo-Castillero, J; Rodrigues-Ramos, R, Evaluation of influence of interphase material parameters on effective material properties of three phase composites, Compos. Sci. Technol., 68, 684-691, (2008) [10] Kari, S; Berger, H; Gabbert, U; Guinovart-Diaz, R; Bravo-Castillero, J; Rodrigues-Ramos, R, Numerical and analytical approaches for calculating the effective thermo-mechanical properties of three-phase composites, J. Thermal Stress., 30, 801-817, (2007) [11] Mikata, Y; Taya, M, Stress field in and around a coated short fiber in an infinite matrix subjected to uniaxial and biaxial loadings, ASME J. Appl. Mech., 52, 19-24, (1985) [12] Mikata, Y; Taya, M, Stress field in a coated continuous fiber composite subjected to thermal-mechanical loadings, J. Compos. Mater., 19, 554-578, (1985) [13] Nguetseng, G, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20, 608-623, (1989) · Zbl 0688.35007 [14] Pagano, NJ; Tandon, GP, Elastic response of multidirectional coated-fiber composites, Compos. Sci. Technol., 31, 273-293, (1988) [15] Russell, P St J, Photonic crystal fibers, Science, 299, 358-362, (2003) [16] Sili, A, Homogenization of an elastic medium reinforced by anisotropic fibers, Asymptotic Anal., 42, 133-171, (2005) · Zbl 1211.35035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.