Reference shape of bodies with enhanced kinematics. Part I. Geometric methods

K. G. Koifman

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 29 (2023) no. 4, pp. 26-53

Résumé

The work develops differential-geometric methods for modeling finite incompatible deformations of hyperelastic solids. They are based on the representation of a body as a smooth manifold, on which a metric and a non-Euclidean connection are synthesized. The resulting geometric space is interpreted as global stress-free shape, and the physical response and material balance equations are formulated relative to it. Within the framework of the geometric approach, deformations are modeled as embeddings of a non-Euclidean shape in physical space. Measures of incompatibility are represented by invariants of the affine connection, namely, curvature, torsion and nonmetricity, and the connection itself is determined by the type of physical process. This article is the first part of the study. The proposed geometric approach is applied to bodies whose response depends on the first deformation gradient. Compatibility conditions are obtained and their geometric interpretation is proposed.

Détail

Keywords: hyperelasticity, body with enhanced kinematics, residual stresses, non-Euclidean geometry, material metric, material connection, curvature, non-metricity.
Mots-clés : second gradient, microstructure, incompatible deformations, torsion

Bibliographie

[1] Gurtin M.E., Fried E., Anand L., The Mechanics and Thermodynamics of Continua, Cambridge University Press, Cambridge, 2010, 718 pp. | DOI | MR

[2] Toupin R.A., “Theories of elasticity with couple-stress”, Archive for Rational Mechanics and Analysis, 17 (1964), 85–112 | DOI | MR | Zbl

[3] Lychev S.A., “On conservation laws of micromorphic nondissipative thermoelasticity”, Vestnik of Samara State University, 2007, no. 4(54), 225–262 (In Russ.) | MR | Zbl

[4] dell'Isola F., Andreaus U., Placidi L., “At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola”, Mathematics and Mechanics of Solids, 20:8 (2015), 887–928 | DOI | MR | Zbl

[5] dell'Isola F., Della Corte A., Giorgio I., “Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives”, Mathematics and Mechanics of Solids, 22:4 (2017), 852–872 | DOI | MR | Zbl

[6] Voigt W., “Theoretische Studien über die Elasticitätsverhältnisse der Krystalle. II.”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen, 34 (1887), 53–100 https://eudml.org/doc/135897

[7] Duhem P., “Le potentiel thermodynamique et la pression hydrostatique”, Annales scientifiques de l'École Normale Supérieure, Serie 3, 10 (1893), 183–230 | DOI | MR

[8] Cosserat E. Cosserat F., Théorie des corps déformables, A. Hermann et fils, Paris, 1909, 226 pp.

[9] Ericksen J.L., “Conservation Laws for Liquid Crystals”, Transactions of The Society of Rheology, 5:1 (1961), 23–34 | DOI | MR

[10] Ericksen J.L., Truesdell C., “Exact theory of stress and strain in rods and shells”, Archive for Rational Mechanics and Analysis, 1 (1957), 295–323 | DOI | MR

[11] Toupin R.A., “Elastic materials with couple-stresses”, Archive for Rational Mechanics and Analysis, 11 (1962), 385–414 | DOI | MR | Zbl

[12] Mindlin R.D., Tiersten H.F., “Effects of couple-stresses in linear elasticity”, Archive for Rational Mechanics and Analysis, 11 (1962), 415–448 | DOI | MR | Zbl

[13] Mindlin R.D., “Micro-structure in linear elasticity”, Archive for Rational Mechanics and Analysis, 16 (1964), 51–78 | DOI | MR | Zbl

[14] Cross J.J., “Mixtures of Fluids and Isotropic Solids”, Archives of Mechanics, 25 (1973), 1024–1039 https://rcin.org.pl/Content/167085/WA727_106772_P.262-Cross-Mixtures.pdf | MR

[15] Noll W., “A mathematical theory of the mechanical behavior of continuous media”, Archive for Rational Mechanics and Analysis, 2 (1958), 197–226 | DOI | MR | Zbl

[16] Wang C.-C., “On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations”, Archive for Rational Mechanics and Analysis, 27:1 (1967), 33–94 | DOI | MR | Zbl

[17] Aifantis E.C., “On the Microstructural Origin of Certain Inelastic Models”, Journal of Engineering Materials and Technology, 106:4 (1984), 326–330 | DOI

[18] Triantafyllidis N., Aifantis E.C., “A gradient approach to localization of deformation. I. Hyperelastic materials”, Journal of Elasticity, 16 (1986), 225–237 | DOI | MR | Zbl

[19] Altan S.B., Aifantis E.C., “On the structure of the mode III crack-tip in gradient elasticity”, Scripta Metallurgica et Materialia, 26:2 (1992), 319–324 | DOI

[20] Ru C.Q., Aifantis E.C., “A simple approach to solve boundary-value problems in gradient elasticity”, Acta Mechanica, 101 (1993), 59–68 | DOI | MR | Zbl

[21] Gutkin M.Yu., Aifantis E.C., “Screw dislocation in gradient elasticity”, Scripta Materialia, 35:11 (1996), 1353–1358 | DOI

[22] Gutkin M.Yu., Aifantis E.C., “Edge dislocation in gradient elasticity”, Scripta Materialia, 36:1 (1997), 129–135 | DOI

[23] Gutkin M.Y., Aifantis E.C., “Dislocations and disclinations in the gradient theory of elasticity”, Physics of the Solid State, 41:12 (1999), 1980–1988 | DOI

[24] Kalpakides V., Agiasofitou E., “On Material Equations in Second Gradient Electroelasticity”, Journal of elasticity and the physical science of solids, 67 (2002), 205–227 | DOI | MR | Zbl

[25] Lurie S., Solyaev Y., “Anti-plane inclusion problem in the second gradient electroelasticity theory”, International Journal of Engineering Science, 144 (2019), 103129 | DOI | MR | Zbl

[26] Volkov-Bogorodskii D.B., Lurie S.A., “Solution of the Eshelby problem in gradient elasticity for multilayer spherical inclusions”, Mechanics of Solids, 51:2 (2016), 161–176 (in English; original in Russian) | DOI

[27] Ma H., Hu G., Wei Y., Liang L., “Inclusion problem in second gradient elasticity”, International Journal of Engineering Science, 132 (2018), 60–78 | DOI | MR | Zbl

[28] Solyaev Y.O., Lurie S.A., “Eshelby integral formulas in second gradient elasticity”, Nanoscience and Technology: An International Journal, 11:2 (2020), 99–107 | DOI

[29] Eremeyev V.A., “Local material symmetry group for first- and second-order strain gradient fluids”, Mathematics and Mechanics of Solids, 26:8 (2021), 1173–1190 | DOI | MR

[30] Eremeyev V.A., “Strong Ellipticity and Infinitesimal Stability within Nth-Order Gradient Elasticity”, Mathematics, 11:4 (2023), 1024 | DOI

[31] Kondo K., “On the analytical and physical foundations of the theory of dislocations and yielding by the differential geometry of continua”, International Journal of Engineering Science, 2:3 (1964), 219–251 | DOI | MR | Zbl

[32] Yavari A., Goriely A., “Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics”, Archive for Rational Mechanics and Analysis, 205:1 (2012), 59–118 | DOI | MR | Zbl

[33] 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 | DOI | MR | Zbl

[34] Lychev S.A., Koifman K.G., “Reference shape of bodies with finite incompatible deformations”, Vestnik of Samara University. Natural Science Series, 28:3–4 (2022), 53–87 (In Russ.) | DOI | MR

[35] Epstein M., El$\dot{z}$anowski M., Material inhomogeneities and their evolution: A geometric approach, Springer Science Business Media, Berlin–Heidelberg, 2007, 261 pp. | DOI | MR

[36] Epstein M., The Geometrical Language of Continuum Mechanics, Cambridge University Press, Cambridge, 2010, 312 pp. | DOI | MR | Zbl

[37] Lychev S., Koifman K., Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics, De Gruyter, Berlin, 2019, 388 pp. | DOI | MR

[38] Kröner E., “Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen”, Archive for Rational Mechanics and Analysis, 4:4 (1959), 18–334 https://libcats.org/book/789336 | MR

[39] Anthony K.H., “Die theorie der disklinationen”, Archive for Rational Mechanics and Analysis, 39 (1970), 43–88 | DOI | MR | Zbl

[40] Anthony K.H., “Die theorie der nichtmetrischen Spannungen in Kristallen”, Archive for Rational Mechanics and Analysis, 40 (1971), 50–78 | DOI | MR | Zbl

[41] Lycheva T.N., Lychev S.A., “Evolution of the field of distributed defects in a crystal during contact interaction with a system of rigid stamps”, Vestnik of Samara University. Natural Science Series, 28:1-2 (2022), 55–73 (In Russ.) | DOI | MR

[42] Yavari A., “A Geometric Theory of Growth Mechanics”, Journal of Nonlinear Science, 20:6 (2010), 781–830 | DOI | MR | Zbl

[43] Lychev S.A., Manzhirov A.V., “The mathematical theory of growing bodies. Finite deformations”, Journal of Applied Mathematics and Mechanics, 77:4 (2013), 421–432 (in English; original in Russian) | DOI | MR | Zbl

[44] Lychev S., “Equilibrium equations for transversely accreted shells”, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 94:1–2 (2014), 118–129 | DOI | MR | Zbl

[45] Lychev S.A., Koifman K.G., “Geometric Aspects of the Theory of Incompatible Deformations. Part I. Uniform Configurations”, Nanomechanics Science and Technology: An International Journal, 7:3 (2016), 177–233 | DOI | MR

[46] Lychev S., Koifman K., “Nonlinear evolutionary problem for a laminated inhomogeneous spherical shell”, Acta Mechanica, 230:11 (2019), 3989–4020 | DOI | MR | Zbl

[47] Lychev S.A., Kostin G.V., Lycheva T.N., Koifman K.G., “Non-Euclidean Geometry and Defected Structure for Bodies with Variable Material Composition”, Journal of Physics: Conference Series, 1250 (2019), 012035 | DOI

[48] Sozio F., Yavari A., “Nonlinear mechanics of accretion”, Journal of Nonlinear Science, 29:4 (2019), 1813–1863 | DOI | MR | Zbl

[49] Bout D.K., Bychkov P.S., Lychev S.A., “The theoretical and experimental study of the bending of a thin substrate during electrolytic deposition”, PNRPU Mechanics Bulletin, 2020, no. 1, 17–31 (In Russ.) | DOI

[50] Lychev S.A., Kostin G.V., Koifman K.G., Lycheva T.N., “Modeling and Optimization of Layer-by-Layer Structures”, Journal of Physics: Conference Series, 1009 (2018), 012014 | DOI

[51] Lychev S.A., Fekry M., “Residual stresses in a thermoelastic cylinder resulting from layer-by-layer surfacing”, Vestnik of Samara University. Natural Science Series, 26:3 (2020), 63–90 (In Russ.) | DOI | MR

[52] Lychev S.A., Fekry M., “Evaluation of residual stresses in additively produced thermoelastic cylinder. Part I. Thermal fields”, Mechanics of Advanced Materials and Structures, 30:10 (2023), 1975–1990 | DOI

[53] Lychev S.A., Fekry M., “Evaluation of residual stresses in additively produced thermoelastic cylinder. Part II. Residual stresses”, Mechanics of Advanced Materials and Structures, 30:10 (2023), 1991–2000 | DOI

[54] Fekry M., “Thermal stresses in growing thermoviscoelastic cylinder and their evolution in the course of selective laser melting processing”, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 103:2 (2023), e202100519 | DOI | MR

[55] Epstein M., Burton D.A., Tucker R., “Relativistic anelasticity”, Classical and Quantum Gravity, 23:10 (2006), 3545–3571 | DOI | MR | Zbl

[56] Lychev S., Koifman K., Bout D., “Finite Incompatible Deformations in Elastic Solids: Relativistic Approach”, Lobachevskii Journal of Mathematics, 43 (2022), 1908–1933 | DOI | MR | Zbl

[57] Lychev S., Koifman K., Pivovaroff N., “Incompatible Deformations in Relativistic Elasticity”, Lobachevskii Journal of Mathematics, 44:6 (2023), 2352–2397 | DOI | MR

[58] Lychev S.A., Koifman K.G., “Material Affine Connections for Growing Solids”, Lobachevskii Journal of Mathematics, 41:10 (2020), 2034–2052 | DOI | MR | Zbl

[59] Lychev S.A., Koifman K.G., “Geometric Aspects of the Theory of Incompatible Deformations. Part II. Strain and Stress Measures”, Nanomechanics Science and Technology: An International Journal, 10:2 (2019), 97–121 | DOI | MR

[60] Lychev S.A., Lycheva T.N., Koifman K.G., “The Nonlinear evolutionary problem for self-stressed multilayered hyperelastic spherical bodies”, PNRPU Mechanics Bulletin, 2020, no. 1, 43–59 (In Russ.) | DOI

[61] Lychev S., Koifman K., Djuzhev N., “Incompatible Deformations in Additively Fabricated Solids: Discrete and Continuous Approaches”, Symmetry, 13:12 (2021), 2331 | DOI

[62] Morgan A.J.A., “Inhomogeneous materially uniform higher order gross bodies”, Archive for Rational Mechanics and Analysis, 57 (1975), 189–253 | DOI | MR | Zbl

[63] El$\dot{z}$anowski M., Epstein M., “The symmetry group of second-grade materials”, International Journal of Non-Linear Mechanics, 27:4 (1992), 635–638 | DOI | MR

[64] de León M., Epstein M., “On the integrability of second-order G-structures with applications to continuous theories of dislocations”, Reports on Mathematical Physics, 33:3 (1993), 419–436 | DOI | MR | Zbl

[65] de León M., Epstein M., “The geometry of uniformity in second-grade elasticity”, Acta Mechanica, 114 (1996), 217–224 | DOI | MR | Zbl

[66] Epstein M., de León M., “Geometrical theory of uniform Cosserat media”, Journal of Geometry and Physics, 26 (1998), 127–170 | DOI | MR | Zbl

[67] Bucataru I., Epstein M., “Geometrical theory of dislocations in bodies with microstructure”, Journal of Geometry and Physics, 52:1 (2004), 57–73 | DOI | MR | Zbl

[68] Mac Lane S., Categories for the Working Mathematician, Fizmatlit, M., 2004, 154 pp. (In Russ.)

[69] Postnikov M.M., Lections in geometry. Semester I. Analytic geometry, URSS, M., 2017, 416 pp. (In Russ.)

[70] Postnikov M.M., Lections in geometry. Semester II. Linear algebra, URSS, M., 2017, 400 pp. (In Russ.)

[71] Lee J.M., Introduction to Smooth Manifolds, Springer, New York, 2012, 708 pp. | DOI | MR

[72] Koifman K.G., “Reference shape of bodies with enhanced kinematics. Part II. Second gradient and microstructure”, Vestnik Samarskogo universiteta. Estestvennonauchnaya seriia – Vestnik of Samara University. Natural Science Series, 29:4 (2023), 54–76 | MR

[73] Kellogg O.D., Foundations of Potential Theory, Springer Nature, Berlin–Heidelberg, 1967, 386 pp. | DOI | MR | Zbl

[74] Truesdell C., Noll W., The Non-Linear Field Theories of Mechanics, Springer Science Business Media, New York, 2004, 602 pp. | DOI | MR

[75] Truesdell C., Toupin R., “The Classical Field Theories”, Principles of Classical Mechanics and Field Theory / Prinzipien der Klassischen Mechanik und Feldtheorie, Encyclopedia of Physics / Handbuch der Physik, 2 / 3 / 1, ed. Flugge S., Springer, Berlin–Heidelberg | DOI | MR | Zbl

[76] Noll W., “Materially uniform simple bodies with inhomogeneities”, Archive for Rational Mechanics and Analysis, 27:1 (1967), 1–32 | DOI | MR

[77] Eckart C., “The thermodynamics of irreversible processes. IV. The theory of elasticity and anelasticity”, Physical Review, 73:4 (1948), 373–382 | DOI | MR | Zbl

[78] Bilby B., Bullough R., Smith E., “Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 231:1185 (1955), 263–273 | DOI | MR

[79] Kondo K., “Non-Riemannian geometry of imperfect crystals from a macroscopic viewpoint”, Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, v. 1, ed. Kondo K., Division D-I, Gakujutsu Bunken Fukyo-Kai, 1955, 6–17 | MR

[80] Kondo K., “Non-Riemannian and Finslerian approaches to the theory of yielding”, International Journal of Engineering Science, 1:1 (1963), 71–88 | DOI | Zbl

[81] Marsden J.E., Hughes T.J., Mathematical foundations of elasticity, Courier Corporation, New York, 1994, 576 pp. https://archive.org/details/mathematicalfoun00mars | MR

[82] Bourbaki N., Theory of Sets, Mir, M., 1965, 456 pp. (In Russ.)

[83] Schwartz L., Analysis, v. 1, Mir, M., 1972, 824 pp. (In Russ.)

[84] Lychev S.A., Koifman K.G., “Contorsion of Material Connection in Growing Solids”, Lobachevskii Journal of Mathematics, 42:8 (2021), 1852–1875 | DOI | MR | Zbl

[85] Miri M., Rivier N., “Continuum elasticity with topological defects, including dislocations and extra-matter”, Journal of Physics A: Mathematical and General, 35:7 (2002), 1727–1739 | DOI | MR | Zbl

[86] Postnikov M.M., Lections in geometry. Semester V: Riemannian geometry, Faktorial, M., 1998, 496 pp. (In Russ.)

[87] Postnikov M.M., Lections in geometry. Semester IV. Differential geometry, URSS, M., 2017, 504 pp. (In Russ.)

[88] Fernandez O.E., Bloch A.M., “The Weitzenböck Connection and Time Reparameterization in Nonholonomic Mechanics”, Journal of Mathematical Physics, 52:1 (2011), 012901 | DOI | MR | Zbl

[89] Kartan E., Spaces of affine, projective and conformal connection, Izd-vo Kazanskogo universiteta, Kazan, 1962, 210 pp. (In Russ.) | MR

[90] Kartan E., Riemannian geometry in an orthogonal frame, Nauka, M., 1960, 207 pp. (In Russ.)

[91] Kartan E., Geometry of Riemannian spaces, Knizhnyi dom «Librokom», M., 2010, 248 pp. (In Russ.)

[92] Dhas B., Srinivasa A., Roy D., A Weyl geometric model for thermo-mechanics of solids with metrical defects, 2019, arXiv: 1904.06956 | DOI