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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

Raz Kupferman 1, and  Cy Maor 1,

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.
Keywords: torsion, homogenization, Dislocations, Gromov-Hausdorff convergence..
Mathematics Subject Classification: Primary: 53Z05; Secondary: 58K9.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992.  Google Scholar

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D. J. H. Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, 2007. doi: 10.1017/CBO9780511755217.  Google Scholar

[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.  Google Scholar

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J. Heinonen, Lectures on Lipschitz Analysis, Jyväskylän Yliopistopaino, 2005.  Google Scholar

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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. Google Scholar

[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. Google Scholar

[10]

R. Kupferman, M. Moshe and J. Solomon, Metric description of defects in amorphous materials, Arch. Rat. Mech. Anal., 216 (2015), 1009-1047. Google Scholar

[11]

J. Nye, Some geometrical relations in dislocated crystals, Acta Met., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6.  Google Scholar

[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.  Google Scholar

[13]

P. Petersen, Riemannian Geometry, 2nd edition, Springer, 2006.  Google Scholar

[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.  Google Scholar

[15]

V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. Ecole Norm. Sup. Paris 1907, 24 (1907), 401-518. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Do Carmo, Riemannian Geometry, Birkhäuser, 1992.  Google Scholar

[5]

D. J. H. Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, 2007. doi: 10.1017/CBO9780511755217.  Google Scholar

[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.  Google Scholar

[7]

J. Heinonen, Lectures on Lipschitz Analysis, Jyväskylän Yliopistopaino, 2005.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

R. Kupferman, M. Moshe and J. Solomon, Metric description of defects in amorphous materials, Arch. Rat. Mech. Anal., 216 (2015), 1009-1047. Google Scholar

[11]

J. Nye, Some geometrical relations in dislocated crystals, Acta Met., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6.  Google Scholar

[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.  Google Scholar

[13]

P. Petersen, Riemannian Geometry, 2nd edition, Springer, 2006.  Google Scholar

[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.  Google Scholar

[15]

V. Volterra, Sur l'équilibre des corps élastiques multiplement connexes, Ann. Sci. Ecole Norm. Sup. Paris 1907, 24 (1907), 401-518. Google Scholar

[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.  Google Scholar

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