Scardia, Lucia Damage as \(\Gamma\)-limit of microfractures in anti-plane linearized elasticity. (English) Zbl 1157.74003 Math. Models Methods Appl. Sci. 18, No. 10, 1703-1740 (2008). Summary: We give a homogenization result for a material having brittle inclusions arranged in a periodic structure. According to the relation between the softness parameter and the size of microstructure, three different limit models are deduced via \(\Gamma\)-convergence. In particular, damage is obtained as limit of periodically distributed microfractures. Cited in 8 Documents MSC: 74A45 Theories of fracture and damage 74Q05 Homogenization in equilibrium problems of solid mechanics 74R05 Brittle damage 74B15 Equations linearized about a deformed state (small deformations superposed on large) Keywords:integral representation; homogenization; brittle inclusions PDFBibTeX XMLCite \textit{L. Scardia}, Math. Models Methods Appl. Sci. 18, No. 10, 1703--1740 (2008; Zbl 1157.74003) Full Text: DOI arXiv References: [1] DOI: 10.1016/0362-546X(92)90015-7 · Zbl 0779.35011 · doi:10.1016/0362-546X(92)90015-7 [2] Adams R. A., Sobolev Spaces (2003) [3] Alberti G., Boll. Un. Mat. Ital. B(7) 11 pp 375– [4] DOI: 10.1007/BF01190895 · Zbl 0837.49011 · doi:10.1007/BF01190895 [5] DOI: 10.1007/PL00011302 · doi:10.1007/PL00011302 [6] Ambrosio L., Functions of Bounded Variations and Free Discontinuity Problems (2000) · Zbl 0957.49001 [7] Attouch H., Variational Convergence for Functions and Operators (1984) · Zbl 0561.49012 [8] Braides A., Homogenization of Multiple Integrals (1998) · Zbl 0911.49010 [9] DOI: 10.1007/BF02198476 · Zbl 0924.35015 · doi:10.1007/BF02198476 [10] DOI: 10.1007/978-1-4612-0327-8 · doi:10.1007/978-1-4612-0327-8 [11] DOI: 10.1007/BF02392977 · Zbl 0772.49006 · doi:10.1007/BF02392977 [12] DOI: 10.1098/rsta.1921.0006 · doi:10.1098/rsta.1921.0006 [13] DOI: 10.1070/SM1979v035n02ABEH001474 · Zbl 0421.35019 · doi:10.1070/SM1979v035n02ABEH001474 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.