Incompatible deformation field and Riemann curvature tensor

doi:10.1007/s10483-017-2176-8

Abstract

Abstract:

Compatibility conditions of a deformation field in continuum mechanics have been revisited via two different routes. One is to use the deformation gradient, and the other is a pure geometric one. Variations of the displacement vector and the displacement density tensor are obtained explicitly in terms of the Riemannian curvature tensor. The explicit relations reconfirm that the compatibility condition is equivalent to the vanishing of the Riemann curvature tensor and reveals the non-Euclidean nature of the space in which the dislocated continuum is imbedded. Comparisons with the theory of Kröner and Le-Stumpf are provided.

Key words: Riemann curvature tensor, Burgers vector, dislocation density tensor, compatibility condition, deformation gradient

2010 MSC Number:

Bohua SUN. Incompatible deformation field and Riemann curvature tensor. Applied Mathematics and Mechanics (English Edition), 2017, 38(3): 311-332.

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References

[1] Chien, W. Z. and Ye, K. Y. Theory of Elasticity (in Chinese), China Scientific Press, Beijing (1956)
[2] Truesdell, C. and Toupin, R. Principles of Classical Mechanics and Field Theory, Springer-Verlag, Berlin (1960)
[3] Truesdell, C. and Noll, W. The Non-Linear Field Theories of Mechanics, Springer-Verlag, Berlin (1969)
[4] Eringen, A. C. Mechanics of continua. Journal of Applied Mechanics, 35, 846-846(1968)
[5] Taylor, G. I. The mechanism of plastic deformation of crystals I:theoretical. Proceedings of the Royal Society of London Series A, 145, 362-387(1934)
[6] Le, K. C. and Günther, C. Nonlinear continuum dislocation theory revisited. International Journal of Plasticity, 53, 164-178(2014)
[7] Le, K. C. Three-dimensional continuum dislocation theory. International Journal of Plasticity, 76, 213-230(2016)
[8] Nye, J. F. Some geometrical relations in dislocated crystals. Acta Metallurgica, 1, 153-162(1953)
[9] Kondo, K. Geometry of elastic deformation and incompatibility. Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, 1, 5-17(1955)
[10] 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, 1, 6-17(1955)
[11] Bilby, B. A., Bullough, R., and 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, 231, 263-273(1955)
[12] Kröner, E. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis, 4, 273-334(1960)
[13] Sedov, L. I. and Berdichevsky, V. L. A Dynamic Theory of Continual Dislocations, Springer-Verlag, Berlin (1967)
[14] Gairola, B. K. D. Nonlinear Elastic Problems, North-Holland Publishing Company, North-Holland (1979)
[15] Aifantis, E. C. The physics of plastic deformation. International Journal of Plasticity, 3, 211-247(1987)
[16] Guo, Z. H. and Liang, H. Y. Compatiable Deformation Field of Continue Media (in Chinese), Chongqing Publisher, Chongqing (1989)
[17] Sun, B. Incompatibel deformation field. Lecture Notes for Seminar at the National Technical University of Athens, Cape Peninsula University of Technology, Athens (2016)
[18] Le, K. C. and Stumpf, H. Finite Elastoplasticity with Microstructure, Ruhr-Universität Bochum, Germany (1994)
[19] Le, K. C. and Stumpf, H. Nonlinear continuum theory of dislocations. International Journal of Engineering Science, 34, 339-358(1995)
[20] Le, K. C. and Stumpf, H. On the determination of the crystal reference in nonlinear continuum theory of dislocations. Proceedings of the Royal Society of London Series A, 452, 359-371(1996)
[21] Le, K. C. and Stumpf, H. A model of elastoplastic bodies with continuously distributed dislocations. International Journal of Plasticity, 12, 611-627(1996)
[22] Stumpf, H., Makowski, J., and Hackl, K. Configurational force and couples in fracture mechanics accounting for microstructures and dissipation. International Journal of Solids and Structures, 47, 2380-2389(2010)
[23] Love, A. E. H. A treatise on the mathematical theory of elasticity. nature, 74, 74-75(1906)
[24] Kirchhoff, G. Vorlesungen über Mechanik, Birkhäuser Basel, Leipzig (1876)
[25] Beltrami, E. Sull' interpretazione meccanica delle formole di Maxwell. Mem. dell' Accad. di Bologna, 7, 1-38(1886)
[26] Michell, J. H. On the direct determination of stress in an elastic solid, with application to the theory of plates. Proceedings of the London Mathematical Society, 31, 100-124(1899)
[27] Weingarten, G. Sulle superficie di discontinuitá nella teoria della elasticitá dei corpi solidi. Atti della Reale Accademia dei Lincei, 5, 57-60(1901)
[28] Cesàro, E. Sulle formole del volterra, fondamentali nella teoria delle distorsioni elastiche. Il Nuovo Cimento, 12, 311-321(1906)
[29] Volterra, V. Sulle equazioni di erenziali lineari. Rend. Acad. Lincei, 3, 143-154(1887)
[30] Volterra, V. Sur léquilibre des corps élastiques multiplement connexes. Annales Scientifiques de École Normale Supérieure, 24, 401-518(1907)
[31] Krutkov, Y. A. The Stress Tensor and the Solution of General Problems in the Static Theory of Elasticity (in Russian), Nauka, Moscow (1949)
[32] Beltrami, E. Sur la théorie de la déformation infiniment petite d'un milieu. Comptes Rendus, 108, 502-505(1889)
[33] Green, A. E. and Zerna, W. Theory of elasticity in general coordinates. Philosophical Magazine, 41, 313-336(1950)
[34] Seugling, W. R. Equations of compatibility for finite deformation of a continuous medium. American Mathematical Monthly, 57, 679-681(1950)
[35] Pietraszkiewicz, W. Determination of displacements from given strains in the nonlinear continuum mechanics. ZAMM, 62, 154-156(1982)
[36] Pietraszkiewicz, W. and Badur, J. Finite rotations in the description of continuum deformation. International Journal of Engineering Science, 21, 1097-1115(1983)
[37] Blume, J. A. Compatibility conditions for a left Cauchy-Green strain field. Journal of Elasticity, 21, 271-308(1989)
[38] Acharya, A. On compatibility conditions for the left Cauchy-Green deformation field in three dimensions. Journal of Elasticity, 56, 95-105(1999)
[39] Yavari, A. and Goriely, A. Riemann-Cartan geometry of nonlinear dislocation mechanics. Archive for Rational Mechanics and Analysis, 205, 59-118(2012)
[40] Yavari, A. Compatibility equations of nonlinear elasticity for non-simply-connected bodies. Archive for Rational Mechanics and Analysis, 209, 237-253(2013)
[41] Frankel, T. The Geometry of Physics, Cambridge University, Cambridge (1997)
[42] Huang, K. C. Tensor Analysis (in Chinese), Tsinghua University Press, Beijing (2003)
[43] Kröner, E. Der fundamentale zusammenhang zwischen versetzungsdichte und spannungsfunktionen. Zeitschrift für Physik, 142, 463-475(1955)
[44] Arnold, V. I. Mathematical Methods of Classical Mechanics, Springer-Verlag, New York (1989)
[45] Wu, J. K. and Huang, K. F. Differential Geometry and Application in Mechanics (in Chinese), Peking Unviversity Press, Beijing (2011)
[46] Cartan, É. Sur certaines expressions différentielles et le probléme de Pfaff. Annales Scientifiques de École Normale Supérieure, 16, 239-332(1899)
[47] Cartan, H. Differential Forms, Dover, New York (2006)
[48] Poincaré, H. Analysis situs. École Polytech, 2, 1-123(1895)
[49] Suhubi, E. S. Exterior Analysis, Academic Press, Waltham (2013)
[50] Sun, B. Dimensional Analysis and Lie Group (in Chinese), Higher Education Press, Beijing (2016)
[51] Chern, S. S., Chen, W. H., and Lam, K. S. Lectures on Differential Geometry, World Scientific, Singapore (1999)
[52] De Rham, G. Sur l'analysis situs des variétésán dimensions. Journal de Mathématiques Pures et Appliquées, 9, 115-200(1931)
[53] Hassani, S. Mathematical Physics, Springer-Verlag, Berlin (2006)