Far Eastern Mathematical Journal

To content of the issue

Geometric aspects of the theory of incompatible deformations for simple structurally inhomogeneous solids with variable composition

Lychev S.A., Koifman K.G.

2017, issue 2, P.

The present paper is intended to formalize strain measures on non-Euclidean spaces for simple body. Use of non-Euclidean geometry methods allows one: i) to identify a global materially uniform reference shape for bodies with structural inhomogeneity, which caused by layer-by-layer formation of a solid during an additive manufacturing process; ii) to identify a global actual shape for bodies immersed into non-Euclidean physical space, in particular, for 2-dimensional solids on material surfaces. In present paper the expressions for strain measures are derived. The latter are generated by embeddings of Riemannian manifold, which represents a simple body, into Riemannian manifold, which represents a physical space. A method for description of solids with variable composition is suggested. Such a solid is considered as a family of Riemannian manifolds. Operations of partitioning and joining are defined over them. These operations characterize structural features of inhomogeneities, which are defined by a scenario of an additive manufacturing process. Specific cases for discrete and continuous structural inhomogeneity are considered in detail. A procedure for material metric synthesizing is suggested. Inclusion map is introduced. It allows one to establish relationship between the classical deformation gradient and tangent map, which is defined over smooth manifold representing a shape of the body. Essential features of suggested method for description of deformation incompatibility are demonstrated by example of hollow structurally inhomogeneous cylinder with incompressible material.

incompatible deformations, strain measures, residual stresses, material manifold, non-Euclidian geometry

Download the article (PDF-file)


[1] Kadomtsev S.B., Geometriia Lobachevskogo i fizika, Izd.stereotip., URSS, M., 2015.
[2] Mizner Ch., Torn K., Uiler Dzh., Gravitatsiia, T.1–3, Mir, M., 1977.
[3] Epstein M., The geometrical language of continuum mechanics, Cambridge University Press, 2010.
[4] Marsden J.E., Hughes T.J., Mathematical foundations of elasticity, Courier Corporation, 1994.
[5] Frankel T., The geometry of physics: an introduction, Cambridge University Press, 2003.
[6] Lychev S., Koifman K., “Geometric aspects of the theory of incompatible deformations. Part I. Uniform configurations”, Nanomechanics Science and Technology: An International Journal, 7:3, (2016), 177–233.
[7] Maugin G.A., Material inhomogeneities in elasticity, CRC Press, 1993.
[8] Lychev S.A., Manzhirov A.V., “Matematicheskaia teoriia rastushchikh tel. Konechnye deformatsii”, PMM, 77, (2013), 585–604.
[9] Yavari A., “A geometric theory of growth mechanics”, Journal of Nonlinear Science, 20:6, (2010), 781–830.
[10] Maugin G.A., Configurational Forces: Thermomechanics, Physics, Mathematics, and Numerics, CRC Press, 2010.
[11] Ciarletta P., Destrade M., Gower A.L., Taffetani M., “Morphology of residually stressed tubular tissues: Beyond the elastic multiplicative decomposition”, Journal of the Mechanics and Physics of Solids, 90, (2016), 242–253.
[12] Guzev M.A., Miasnikov V.P., “Neevklidova struktura polia vnutrennikh napriazhenii sploshnoi sredy”, Dal'nevost. matem. zhurn., 2:2, (2001), 29–44.
[13] Guzev M.A., Shepelov M.A., “Porogovoe povedenie mekhanicheskikh kharakteristik v neevklidovoi modeli sploshnoi sredy”, Dal'nevost. matem. zhurn., 10:1, (2010), 20–30.
[14] Choquet-Bruhat Y., General relativity and the Einstein equations, Oxford University Press, 2008.
[15] Yavari A., Goriely A., “Riemann–Cartan geometry of nonlinear disclination mechanics”, Mathematics and Mechanics of Solids, 2012, 1081286511436137.
[16] Yavari A., Goriely A., “Weyl geometry and the nonlinear mechanics of distributed point defects”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 468:2148, (2012), 3902–3922.
[17] Yavari A., Marsden J.E., Ortiz M., “On spatial and material covariant balance laws in elasticity”, Journal of Mathematical Physics, 47:4, (2006), 042903.
[18] Yavari A., Ozakin A., “Covariance in linearized elasticity”, Zeitschrift fur angewandte Mathematik und Physik, 59:6, (2008), 1081–1110.
[19] Youssef N.L., Sid-Ahmed A.M., “Linear connections and curvature tensors in the geometry of parallelizable manifolds”, Reports on mathematical physics, 60:1, (2007), 39–53.
[20] Guzev M.A., “Spektral'nye kharakteristiki polia samouravnoveshennykh napriazhenii”, Dal'nevost. matem. zhurn., 14:1, (2014), 41–47.
[21] Truesdell C., Noll W., The non-linear field theories of mechanics, ed. Stuart S. Antman, Springer Berlin Heidelberg, 2004.
[22] Xu Ma, Gonzalo R. Arce., Computational lithography, Wiley VCH Verlag GmbH, 2010.
[23] Multilayer thin films, eds. Gero Decher, Joe B. Schlenoff, Wiley VCH Verlag GmbH, 2012.
[24] Mechanical self-assembly, ed. Xi Chen, Springer Nature, 2013.
[25] Iu.G. Borisovich, N.M. Blizniakov, Ia.A. Izrailevich, T.N. Fomenko, Vvedenie v topologiiu, 2-e izd., dop., Nauka. Fizmatlit, M., 1995, 416 s.
[26] Burbaki N., Teoriia mnozhestv, Mir, M., 1965, 456 s.
[27] Lee J.M., Introduction to topological manifolds, Springer, New York, 2011.
[28] Lee J.M., Introduction to smooth manifolds, Springer New York, 2012.
[29] Schield R.T., “Inverse deformation results in finite elasticity”, Zeitschrift f’ur angewandte Mathematik und Physik ZAMP, 18:4, (1967), 490–500.
[30] Chadwick P., “Applications of an energy-momentum tensor in non-linear elastostatics”, Journal of Elasticity, 5:3-4, (1975), 249–258.
[31] Noll W., “Materially uniform simple bodies with inhomogeneities”, Archive for Rational Mechanics and Analysis, 27:1, (1967), 1–32.
[32] Shvarts L., Analiz, Tom 1, Mir, M., 1972, 824 s.
[33] Lur'e A.I., Nelineinaia teoriia uprugosti, Nauka, M., 1980, 512 s.
[34] Michal A.D., “Matrix and tensor calculus with applications to mechanics, elasticity and aeronautics”, 1947.
[35] Lychev S.A., “Geometric aspects of the theory of incompatible deformations in growing solids”, Advanced Structured Materials, Springer International Publishing, 2017, 327–347 pp.
[36] Choy K., “Chemical vapour deposition of coatings”, Progress in materials science, 48:2, (2003), 57–170.
[37] Gibson Ian, Rosen David W., Stucker Brent et al., Additive manufacturing technologies, Springer, 2010.
[38] Nastasi M.A., Mayer J.W., Ion implantation and synthesis of materials, 80, Springer, 2006.
[39] Lychev S.A., Mark A.V., “Osesimmetrichnoe narashchivanie pologo giperuprugogo tsilindra”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14, (2014), 209–226.
[40] Trusdell K., Pervonachal'nyi kurs ratsional'noi mekhaniki sploshnoi sredy, Mir, M., 1975, 592 s.
[41] Manzhirov A.V., Lychev S.A., “Matematicheskaia teoriia rastushchikh tel”, Aktual'nye problemy mekhaniki. 50 let Institutu problem mekhaniki im. A.Iu. Ishlinskogo RAN, Nauka, M., 2015.
[42] Abraham R., Marsden J.E., Ratiu T., Manifolds, tensor analysis, and applications, 75, Springer Science & Business Media, 1988.

To content of the issue