Jiménez, Víctor Manuel; de León, Manuel; Epstein, Marcelo Material distributions. (English) Zbl 1487.53123 Math. Mech. Solids 25, No. 7, 1450-1458 (2020). Summary: The concept of material distribution is introduced as describing the geometric material structure of a general non-uniform body. Any smooth constitutive law is shown to give rise to a unique smooth integrable singular distribution. Ultimately, the material distribution and its associated singular foliation result in a rigorous and unique subdivision of the material body into strictly smoothly uniform components. Thus, the constitutive law induces a unique partition of the body into smoothly uniform sub-bodies, laminates, filaments and isolated points. Cited in 5 Documents MSC: 53Z30 Applications of differential geometry to engineering 74E05 Inhomogeneity in solid mechanics Keywords:singular distributions; material uniformity; differential geometry; Lie groupoids; Lie algebroids; Stefan-Sussman theorem PDFBibTeX XMLCite \textit{V. M. Jiménez} et al., Math. Mech. Solids 25, No. 7, 1450--1458 (2020; Zbl 1487.53123) Full Text: DOI References: [1] Wang, C-C. On the geometric structure of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations. Arch Ration Mech Anal 1967; 27: 33-94. · Zbl 0187.48802 [2] Elżanowski, M, Epstein, M, Śniatycki, J. G-structures and material homogeneity. J Elast 1990; 23: 167-180. · Zbl 0709.73002 [3] Epstein, M, Elżanowski, M. Material inhomogeneities and their evolution: a geometric approach. Berlin: Springer-Verlag, 2007. · Zbl 1130.74001 [4] Epstein, M, de León, M. Geometrical theory of uniform Cosserat media. J Geom Phys 1998; 26: 127-170. · Zbl 0928.74009 [5] Mackenzie, K. Lie groupoids and Lie algebroids in differential geometry (London Mathematical Society Lecture Note Series, vol. 124). Cambridge, UK: Cambridge University Press, 1987. · Zbl 0683.53029 [6] Jiménez, VM, de León, M, Epstein, M. Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies. arXiv:1607.04043v2, 2016. · Zbl 1431.74010 [7] Epstein, M, de León, M. Homogeneity without uniformity: towards a mathematical theory of functionally graded materials. Int J Solids Struct 2000; 37: 7577-7591. · Zbl 0991.74022 [8] Elżanowski, M, Epstein, M. Geometric characterization of hyperelastic uniformity. Arch Ration Mech Anal 1985; 88: 347-357. · Zbl 0574.73010 [9] Michor, PW. Topics in differential geometry (Graduate Studies in Mathematics, vol. 93). Providence, RI: American Mathematical Society, 2008. [10] Noll, W. Materially uniform bodies with inhomogeneities. Arch Ration Mech Anal. 1967; 27: 1-32. · Zbl 0168.45701 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.