# American Institute of Mathematical Sciences

September  2015, 7(3): 361-387. doi: 10.3934/jgm.2015.7.361

## The emergence of torsion in the continuum limit of distributed edge-dislocations

 1 Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel, Israel

Received  September 2014 Revised  May 2015 Published  July 2015

We present a rigorous homogenization theorem for distributed edge-dislocations. We construct a sequence of locally-flat 2D Riemannian manifolds with dislocation-type singularities. We show that this sequence converges, as the dislocations become denser, to a flat non-singular Weitzenböck manifold, i.e. a flat manifold endowed with a metrically-consistent connection with zero curvature and non-zero torsion. In the process, we introduce a new notion of convergence of Weitzenböck manifolds, which is relevant to this class of homogenization problems.
Citation: Raz Kupferman, Cy Maor. The emergence of torsion in the continuum limit of distributed edge-dislocations. Journal of Geometric Mechanics, 2015, 7 (3) : 361-387. doi: 10.3934/jgm.2015.7.361
##### References:
 [1] I. Agricola and C. Thier, The geodesics of metric connections with vectorial torsion, Ann. Global Anal. Geom., 26 (2004), 321-332. doi: 10.1023/B:AGAG.0000047509.63818.4f. [2] B. Bilby, R. Bullough and E. Smith, Continuous distributions of dislocations: A new application of the methods of Non-Riemannian geometry, Proc. Roy. Soc. A, 231 (1955), 263-273. doi: 10.1098/rspa.1955.0171. [3] B. Bilby and E. Smith, Continuous distributions of dislocations. III, Proc. Roy. Soc. Edin. A, 236 (1956), 481-505. doi: 10.1098/rspa.1956.0150. [4] M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992. [5] D. J. H. Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, 2007. doi: 10.1017/CBO9780511755217. [6] J. Guven, J. Hanna, O. Kahraman and M. Müller, Dipoles in thin sheets, Eur. Phys. J. E, 36 (2013), p106. doi: 10.1140/epje/i2013-13106-0. [7] J. Heinonen, Lectures on Lipschitz Analysis, Jyväskylän Yliopistopaino, 2005. [8] K. Kondo, Geometry of elastic deformation and incompatibility, in Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry (ed. K. Kondo), 1 (1955), 5-17. [9] E. Kroner, The Physics of Defects, in Les Houches Summer School Proceedings (eds. R. Balian, M. Kleman and J.-P. Poirier), North-Holland, Amsterdam, 1981. [10] R. Kupferman, M. Moshe and J. Solomon, Metric description of defects in amorphous materials, Arch. Rat. Mech. Anal., 216 (2015), 1009-1047. [11] J. Nye, Some geometrical relations in dislocated crystals, Acta Met., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6. [12] A. Ozakin and A. Yavari, Affine development of closed curves in Weitzenböck manifolds and the burgers vector of dislocation mechanics, Math. Mech. Solids, 19 (2014), 299-307. doi: 10.1177/1081286512463720. [13] P. Petersen, Riemannian Geometry, 2nd edition, Springer, 2006. [14] H. Seung and D. Nelson, Defects in flexible membranes with crystalline order, Phys. Rev. A, 38 (1988), 1005-1018. doi: 10.1103/PhysRevA.38.1005. [15] V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. Ecole Norm. Sup. Paris 1907, 24 (1907), 401-518. [16] C.-C. Wang, On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rat. Mech. Anal., 27 (1967), 33-94. doi: 10.1007/BF00276434.

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##### References:
 [1] I. Agricola and C. Thier, The geodesics of metric connections with vectorial torsion, Ann. Global Anal. Geom., 26 (2004), 321-332. doi: 10.1023/B:AGAG.0000047509.63818.4f. [2] B. Bilby, R. Bullough and E. Smith, Continuous distributions of dislocations: A new application of the methods of Non-Riemannian geometry, Proc. Roy. Soc. A, 231 (1955), 263-273. doi: 10.1098/rspa.1955.0171. [3] B. Bilby and E. Smith, Continuous distributions of dislocations. III, Proc. Roy. Soc. Edin. A, 236 (1956), 481-505. doi: 10.1098/rspa.1956.0150. [4] M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992. [5] D. J. H. Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, 2007. doi: 10.1017/CBO9780511755217. [6] J. Guven, J. Hanna, O. Kahraman and M. Müller, Dipoles in thin sheets, Eur. Phys. J. E, 36 (2013), p106. doi: 10.1140/epje/i2013-13106-0. [7] J. Heinonen, Lectures on Lipschitz Analysis, Jyväskylän Yliopistopaino, 2005. [8] K. Kondo, Geometry of elastic deformation and incompatibility, in Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry (ed. K. Kondo), 1 (1955), 5-17. [9] E. Kroner, The Physics of Defects, in Les Houches Summer School Proceedings (eds. R. Balian, M. Kleman and J.-P. Poirier), North-Holland, Amsterdam, 1981. [10] R. Kupferman, M. Moshe and J. Solomon, Metric description of defects in amorphous materials, Arch. Rat. Mech. Anal., 216 (2015), 1009-1047. [11] J. Nye, Some geometrical relations in dislocated crystals, Acta Met., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6. [12] A. Ozakin and A. Yavari, Affine development of closed curves in Weitzenböck manifolds and the burgers vector of dislocation mechanics, Math. Mech. Solids, 19 (2014), 299-307. doi: 10.1177/1081286512463720. [13] P. Petersen, Riemannian Geometry, 2nd edition, Springer, 2006. [14] H. Seung and D. Nelson, Defects in flexible membranes with crystalline order, Phys. Rev. A, 38 (1988), 1005-1018. doi: 10.1103/PhysRevA.38.1005. [15] V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. Ecole Norm. Sup. Paris 1907, 24 (1907), 401-518. [16] C.-C. Wang, On the geometric structures of simple bodies, a mathematical foundation for the theory of continuous distributions of dislocations, Arch. Rat. Mech. Anal., 27 (1967), 33-94. doi: 10.1007/BF00276434.
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