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Compatibility conditions of continua using Riemann-Cartan geometry. (English) Zbl 07357414

Summary: The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann-Cartan geometry. We show that Vallée’s compatibility condition in linear elasticity theory is equivalent to the vanishing of the three-dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye’s tensor also arises from the three-dimensional Einstein tensor, which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions that can be obtained using our geometrical approach and apply it to the microcontinuum theories.

MSC:

74-XX Mechanics of deformable solids
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[1] Vallée, C. Compatibility equations for large deformations. Int J Eng Sci 1992; 30(12): 1753-1757. · Zbl 0825.73217
[2] Ciarlet, PG, Gratie, L, Iosifescu, O, et al. Another approach to the fundamental theorem of Riemannian geometry in \(\mathbb{R}^3\), by way of rotation fields. J Math Pures Appl 2007; 87(3): 237-252. · Zbl 1114.53033
[3] Nye, JF. Some geometrical relations in dislocated crystals. Acta Metall 1953; 1(2): 153-162.
[4] Cottrell, AH, Bilby, BA. Dislocation theory of yielding and strain ageing of iron. Proc Phys Soc London, Sect A 1949; 62(1): 49.
[5] Kröner, E. Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch Ration Mech Anal 1959; 4(1): 273. · Zbl 0090.17601
[6] Noll, W. A mathematical theory of the mechanical behavior of continuous media. Arch Ration Mech Anal 1958; 2(1): 197-226. · Zbl 0083.39303
[7] Kondo, K. On the analytical and physical foundations of the theory of dislocations and yielding by the differential geometry of continua. Int J Eng Sci 1964; 2(3): 219-251. · Zbl 0143.45203
[8] Bilby, BA . Geometry and continuum mechanics. In: Kröner, E . (ed.) Mechanics of generalized continua. Berlin: Springer, 1968, 180-199. · Zbl 0222.73117
[9] Kleinert, H. Gravity as a theory of defects in a crystal with only second gradient elasticity. Ann Phys 1987; 499(2): 117-119.
[10] Katanaev, MO, Volovich, IV. Theory of defects in solids and three-dimensional gravity. Ann Phys 1992; 216(1): 1-28. · Zbl 0875.53018
[11] Hehl, FW, Obukhov, YN. Élie Cartan’s torsion in geometry and in field theory, an essay. Ann Fond Louis de Broglie 2007; 32: 157-194. · Zbl 1329.53110
[12] Yavari, A, Goriely, A. Riemann-Cartan geometry of nonlinear dislocation mechanics. Arch Ration Mech Anal 2012; 205(1): 59-118. · Zbl 1281.74006
[13] Yavari, A, Goriely, A. Riemann-Cartan geometry of nonlinear disclination mechanics. Math Mech Solids 2013; 18(1): 91-102. · Zbl 1528.74014
[14] deWit, R. Relation between dislocations and disclinations. J Appl Phys 1971; 42(9): 3304-3308.
[15] Kléman, M. Relationship between Burgers circuit, Volterra process and homotopy groups. J Phys Lett - Paris 1977; 38(10): 199-202.
[16] Kléman, M. Defects in liquid crystals. Rep Prog Phys 1989; 52(5): 555-654.
[17] Hehl, FW, Von Der Heyde, P, Kerlick, GD, et al. General relativity with spin and torsion: foundations and prospects. Rev Mod Phys 1976; 48: 393-416. · Zbl 1371.83017
[18] Arcos, HI, Pereira, JG. Torsion gravity: a reappraisal. Int J Mod Phys D 2004; 13(10): 2193-2240. · Zbl 1082.83029
[19] Böhmer, CG, Downes, RJ. From continuum mechanics to general relativity. Int J Mod Phys D 2014; 23(12): 1442015. · Zbl 1305.83003
[20] Aldrovandi, R, Pereira, JG. Teleparallel gravity: an introduction (Theoretical, Mathematical & Computational Physics, vol. 173). Dordrecht, Springer, 2013. · Zbl 1259.83002
[21] Lazar, M, Hehl, FW. Cartan’s spiral staircase in physics and, in particular, in the gauge theory of dislocations. Found Phys 2010; 40(9-10): 1298-1325. · Zbl 1216.83043
[22] Kleinert, H . New gauge symmetry in gravity and the evanescent role of torsion. In: Proceedings of the Conference in Honour of Murray Gell-Mann’s 80th Birthday, 2010. · Zbl 1225.83021
[23] Blagojević, M, Hehl, FW. Gauge theories of gravitation: a reader with commentaries. London: Imperial College Press, 2013. · Zbl 1282.83046
[24] Kleinert, H, Zaanen, J. Nematic world crystal model of gravity explaining absence of torsion in spacetime. Phys Lett A 2004; 324(5-6): 361-365. · Zbl 1123.82368
[25] Beekman, AJ, Nissinen, J, Wu, K, et al. Dual gauge field theory of quantum liquid crystals in two dimensions. Phys Rep 2017; 683: 1-110. · Zbl 1366.82072
[26] Nissinen, J. Emergent spacetime and gravitational Nieh-Yan anomaly in chiral \(p + \operatorname{ip}\) Weyl superfluids and superconductors. Phys Rev Lett 2020; 124(11): 117002.
[27] Böhmer, CG, Downes, RJ, Vassiliev, D. Rotational elasticity. Q J Mech Appl Math 2011; 64(4): 415-439. · Zbl 1248.74007
[28] Böhmer, CG, Obukhov, YN. A gauge-theoretic approach to elasticity with microrotations. Proc R Soc London, Ser A 2012; 468(2141): 1391-1407. · Zbl 1364.74017
[29] Böhmer, CG, Tamanini, N. Rotational elasticity and couplings to linear elasticity. Math Mech Solids 2013; 20(8): 959-974. · Zbl 1330.74010
[30] Peshkov, I, Romenski, E, Dumbser, M. Continuum mechanics with torsion. Continuum Mech Thermodyn 2019; 31(5): 1517-1541.
[31] Nissinen, J, Volovik, GE. Tetrads in solids: from elasticity theory to topological quantum Hall systems and Weyl fermions. J Exp Theor Phys 2018; 127(5): 948-957.
[32] Nissinen, J, Volovik, GE. Elasticity tetrads, mixed axial-gravitational anomalies, and (3+1)-d quantum Hall effect. Phys Rev Res 2019; 1: 023007.
[33] Lankeit, J, Neff, P, Osterbrink, F. Integrability conditions between the first and second Cosserat deformation tensor in geometrically nonlinear micropolar models and existence of minimizers. Z Angew Math Mech 2016; 68(1): 11. · Zbl 1437.74002
[34] Edelen, DGB. A new look at the compatability [sic] problem of elasticity theory. Int J Eng Sci 1990; 28(1): 23-27. · Zbl 0707.73016
[35] Skyrme, THR. A non-linear theory of strong interactions. Proc R Soc London, Ser A 1958; 247: 260-278.
[36] Skyrme, THR. A unified model of K- and \(\pi \)-mesons. Proc R Soc London, Ser A 1959; 252: 236-245. · Zbl 0086.43503
[37] Skyrme, THR. A non-linear field theory. Proc R Soc London, Ser A 1961; 260(1300): 127-138. · Zbl 0102.22605
[38] Trebin, HR. Elastic energies of a directional medium. J Phys 1981; 42(11): 1573-1576.
[39] Trebin, HR. The topology of non-uniform media in condensed matter physics. Adv Phys 1982; 31(3): 195-254.
[40] Esteban, MJ. A direct variational approach to Skyrme’s model for meson fields. Commun Math Phys 1986; 105(4): 571-591. · Zbl 0621.58035
[41] Esteban, MJ, Müller, S. Sobolev maps with integer degree and applications to Skyrme’s problem. Proc R Soc London, Ser A 1992; 436(1896): 197-201. · Zbl 0757.49010
[42] Unzicker, A. Topological defects in an elastic medium: a valid model particle physics. Structured media, Proceedings of an International Symposium in Memory of E. Kröner, 2001.
[43] Randono, A, Hughes, TL. Torsional monopoles and torqued geometries in gravity and condensed matter. Phys Rev Lett 2011; 106(16): 161102.
[44] Neff, P, Lankeit, J, Madeo, A. On Grioli’s minimum property and its relation to Cauchy’s polar decomposition. Int J Eng Sci 2014; 80: 209-217. · Zbl 1423.74041
[45] Fischle, A, Neff, P. Grioli’s theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations. Rend Lincei - Mat Appl 2017; 28(3): 573-600. · Zbl 1386.74009
[46] Neff, P, Fischle, A, Borisov, L. Explicit global minimization of the symmetrized Euclidean distance by a characterization of real matrices with symmetric square. SIAM J Appl Algebra Geom 2019; 3(1): 31-43. · Zbl 1416.15023
[47] Borisov, L, Fischle, A, Neff, P. Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices. Z Angew Math Mech 2019; 99(6): e201800120.
[48] Cosserat, E, Cosserat, F. Théorie des corps déformables. Paris: Librairie Scientifique A. Hermann et Fils, 1909. · JFM 40.0862.02
[49] Eringen, AC. Microcontinuum field theories: I. Foundations and solids. New York: Springer, 1999. · Zbl 0953.74002
[50] Neff, P. Existence of minimizers for a finite-strain micromorphic elastic solid. Proc R Soc Edinburgh A 2006; 136: 997-1012. · Zbl 1106.74010
[51] Neff, P, Münch, I. Curl bounds Grad on SO(3). ESAIM: Control Optim Calc Var 2008; 14(1): 148-159. · Zbl 1139.74008
[52] Neff, P . Existence of minimizers in nonlinear elastostatics of micromorphic solids. In: Iesan, D (ed.) Encyclopedia of thermal stresses. Heidelberg: Springer, 2013, 1475-1485.
[53] Neff, P, Ghiba, ID, Lazar, M, et al. The relaxed linear micromorphic continuum: well-posedness of the static problem and relations to the gauge theory of dislocations. Q J Mech Appl Math 2015; 68(1): 53-84. · Zbl 1310.74037
[54] Neff, P, Bîrsan, M, Osterbrink, F. Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements. J Elast 2015; 121(1): 119-141. · Zbl 1327.74035
[55] Bîrsan, M, Neff, P. On the dislocation density tensor in the Cosserat theory of elastic shells. In: Naumenko, K, Aßmus, M (eds) Advanced methods of continuum mechanics for materials and structures (Advanced Structured Materials, vol. 60). Singapore: Springer, 2016, 391-413.
[56] Ciarlet, PG. An introduction to differential geometry with applications to elasticity. J Elast 2005; 78(1-3): 1-125. · Zbl 1086.74001
[57] Finkelstein, D. Kinks . J Math Phys 1966; 7(7): 1218-1225. · Zbl 0151.44203
[58] Shankar, R. Applications of topology to the study of ordered systems. J Phys 1977; 38(11): 1405-1412.
[59] Nakahara, M. Toy-skyrmions in superfluid \(^3 He-A\). Prog Theor Phys 1987; 77(5): 1011-1013.
[60] Schouten, JA. Ricci-Calculus. Berlin: Springer, 1954. · Zbl 0057.37803
[61] Lubarda, MV, Lubarda, VA. A note on the compatibility equations for three-dimensional axisymmetric problems. Math Mech Solids 2020; 25(2): 160-165. · Zbl 1446.74070
[62] Romenskii, EI. Hyperbolic equations of Maxwell’s nonlinear model of elastoplastic heat-conducting media. Sib Math J 1989; 30(4): 606-625. · Zbl 0741.73022
[63] Zubov, LM. Nonlinear theory of dislocations and disclinations in elastic bodies (Lecture Notes in Physics Monographs, vol. 47). Berlin: Springer, 1997.
[64] Derezin, S, Zubov, L. Disclinations in nonlinear elasticity. Z Angew Math Mech 2011; 91(6): 433-442. · Zbl 1316.74007
[65] Zelenina, AA, Zubov, LM. Spherically symmetric deformations of micropolar elastic medium with distributed dislocations and disclinations. In: dell’Isola, F et al. (eds) Advances in mechanics of microstructured media and structures (Advanced Structured Materials, vol. 87). Cham: Springer, 2018, 357-369.
[66] Goloveshkina, EG, Zubov, LM. Universal spherically symmetric solution of nonlinear dislocation theory for incompressible isotropic elastic medium. Arch Appl Mech 2019; 89(3): 409-424.
[67] Bilby, BA, Bullough, R, Smith, E. Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry. Proc R Soc London, Ser A 1955; 231(1185): 263-273.
[68] Kondo, K. Non-Riemannian and Finslerian approaches to the theory of yielding. Int J Eng Sci 1963; 1(1): 71-88. · Zbl 0137.20205
[69] Godunov, SK, Romenskii, EI. Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates. J Appl Mech Tech Phys 1972; 13(6): 868-884.
[70] Kröner, E . Continuum theory of defects. In: Balian, R, Kléman, M, Poirier, JP (eds.) Physics of defects (Les Houches, vol. 35). Amsterdam: North-Holland, 1980, 215-315.
[71] Kleinert, H. Gauge fields in condensed matter vol. 2: Stresses and defects (differential geometry, crystal melting). Singapore: World Scientific, 1989. · Zbl 0785.53061
[72] Godunov, SK, Romenskii, E. Elements of continuum mechanics and conservation laws. New York: Kluwer Academic, 2003. · Zbl 1031.74004
[73] Yavari, A, Goriely, A. Weyl geometry and the nonlinear mechanics of distributed point defects. Proc R Soc London, Ser A 2012; 468(2148): 3902-3922. · Zbl 1371.74046
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