Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the dynamics of continuous distributions of dislocations

Authors: Ernst Binz, Günter Schwarz and Jan Wenzelburger
Journal: Quart. Appl. Math. 59 (2001), 225-239
MSC: Primary 53C80; Secondary 58A10, 74A99
DOI: https://doi.org/10.1090/qam/1827812
MathSciNet review: MR1827812
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Abstract | References | Similar Articles | Additional Information

Abstract: For materials with continuous distributions of dislocations a configuration space which unifies the continuum theory of defects with classical elasticity is given. Weak equations of motion are derived from the principle of virtual work. Using the Helmholtz decomposition theorem, this yields a coupled system of equations for the dynamics of dislocations and classical elasticity.

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  • [1] Stuart S. Antman and John E. Osborn, The principle of virtual work and integral laws of motion, Arch. Rational Mech. Anal. 69 (1979), no. 3, 231–262. MR 522525, https://doi.org/10.1007/BF00248135
  • [2] Ernst Binz, Jędrzej Śniatycki, and Hans Fischer, Geometry of classical fields, North-Holland Mathematics Studies, vol. 154, North-Holland Publishing Co., Amsterdam, 1988. Notas de Matemática [Mathematical Notes], 123. MR 972499
  • [3] E. Binz, Symmetry, constitutive laws of bounded smoothly deformable media and Neumann problems, Symmetries in science, V (Lochau, 1990) Plenum, New York, 1991, pp. 31–65. MR 1143586
  • [4] E. Binz, Global differential geometric methods in elasticity and hydrodynamics, Differential geometry, group representations, and quantization, Lecture Notes in Phys., vol. 379, Springer, Berlin, 1991, pp. 3–29. MR 1180079
  • [5] E. Binz, On the irredundant part of the first Piola-Kirchhoff stress tensor, Rep. Math. Phys. 32 (1993), no. 2, 175–210. MR 1276471, https://doi.org/10.1016/0034-4877(93)90013-5
  • [6] E. Binz and G. Schwarz, The principle of virtual work and a symplectic reduction of nonlocal continuum mechanics, Rep. Math. Phys. 32 (1993), no. 1, 49–69. MR 1247163, https://doi.org/10.1016/0034-4877(93)90071-L
  • [7] Marcelo Epstein and Reuven Segev, Differentiable manifolds and the principle of virtual work in continuum mechanics, J. Math. Phys. 21 (1980), no. 5, 1243–1245. MR 574898, https://doi.org/10.1063/1.524516
  • [8] E. Hellinger, Die allgemeinen Ansätze der Mechanik der Kontinua, Enzykl. Math. Wiss. 4/4 (1914)
  • [9] A. M. Kosevich, Crystal dislocations and the theory of elasticity, In: Dislocations in Solids, ed. F. R. N. Nabarro, North Holland, Amsterdam, 1979, pp. 33-165
  • [10] Rainer Kreß, Potentialtheoretische Randwertprobleme bei Tensorfeldern beliebiger Dimension und beliebigen Ranges, Arch. Rational Mech. Anal. 47 (1972), 59–80 (German). MR 0361131, https://doi.org/10.1007/BF00252189
  • [11] E. Kröner, Continuum Theory of Defects, Les Houches, Session XXXV, 1980-Physics of Defects, R. Balian et al., eds., North-Holland, 1981
  • [12] E. Kröner, Stress space and strain space in continuum mechanics, Phys. Stat. Solids (b) 144, 39-44 (1987)
  • [13] E. Kröner, A Variational Principle in Nonlinear Dislocation Theory. In: Proc. 2nd Internat. Conf. Nonlinear Mechanics, ed. Chien Wei-zang, Peking University Press, Beijing, 1993, pp. 59-64
  • [14] E. Kröner, Dislocation theory as a physical field theory, Continuum models and discrete systems (Varna, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 522–537. MR 1442987
  • [15] L. D. Landau and E. M. Lifshitz, Course of theoretical physics. Vol. 7, 3rd ed., Pergamon Press, Oxford, 1986. Theory of elasticity; Translated from the Russian by J. B. Sykes and W. H. Reid. MR 884707
  • [16] J. E. Marsden and J. R. Hughes, Mathematical Foundations of Elasticity, Prentice Hall, Englewood Cliffs, New Jersey, 1983
  • [17] Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1994. A basic exposition of classical mechanical systems. MR 1304682
  • [18] G. A. Maugin, The method of virtual power in continuum mechanics: application to coupled fields, Acta Mech. 35 (1980), no. 1-2, 1–70 (English, with German summary). MR 563729, https://doi.org/10.1007/BF01190057
  • [19] Gérard A. Maugin, The thermomechanics of plasticity and fracture, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992. MR 1173212
  • [20] Walter Noll, Materially uniform simple bodies with inhomogeneities, Arch. Rational Mech. Anal. 27 (1967/1968), 1–32. MR 0225530, https://doi.org/10.1007/BF00276433
  • [21] G. Schwarz, The Euclidean group in global models of continuum mechanics and the existence of a symmetric stress tensor, Rep. Math. Phys. 33, 397-412 (1993)
  • [22] Günter Schwarz, Hodge decomposition—a method for solving boundary value problems, Lecture Notes in Mathematics, vol. 1607, Springer-Verlag, Berlin, 1995. MR 1367287
  • [23] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. II, Publish or Perish, Boston, 1970
  • [24] G. I. Taylor, The mechanism of plastic deformation of crystals, parts I and II, Proc. Roy. Soc. London A 145, 362-387, 388-404 (1934)
  • [25] C.-C. Wang, On the geometric structures of simple bodies. A mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rational Mech. Anal. 27 (1967/1968), 33–94. MR 0229420, https://doi.org/10.1007/BF00276434
  • [26] J. Wenzelburger, Ein globales differentialgeometrisches Modell dünner Schalen, Diplomarbeit Universität Karlsruhe, Karlsruhe, 1989
  • [27] J. Wenzelburger, Die Hodge-Zerlegung in der Kontinuumstheorie von Defekten, Dissertation Universität Mannheim, Verlag Shaker, Aachen, 1994
  • [28] Jan Wenzelburger, A kinematic model for continuous distributions of dislocations, J. Geom. Phys. 24 (1998), no. 4, 334–352. MR 1610787, https://doi.org/10.1016/S0393-0440(97)00016-8

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DOI: https://doi.org/10.1090/qam/1827812
Article copyright: © Copyright 2001 American Mathematical Society

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