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]

Ann. Global Anal. Geom., 26 (2004), 321-332. doi: 10.1023/B:AGAG.0000047509.63818.4f.  Google Scholar

[2]

Proc. Roy. Soc. A, 231 (1955), 263-273. doi: 10.1098/rspa.1955.0171.  Google Scholar

[3]

Proc. Roy. Soc. Edin. A, 236 (1956), 481-505. doi: 10.1098/rspa.1956.0150.  Google Scholar

[4]

Birkhäuser, 1992.  Google Scholar

[5]

Cambridge University Press, 2007. doi: 10.1017/CBO9780511755217.  Google Scholar

[6]

Eur. Phys. J. E, 36 (2013), p106. doi: 10.1140/epje/i2013-13106-0.  Google Scholar

[7]

Jyväskylän Yliopistopaino, 2005.  Google Scholar

[8]

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]

in Les Houches Summer School Proceedings (eds. R. Balian, M. Kleman and J.-P. Poirier), North-Holland, Amsterdam, 1981. Google Scholar

[10]

Arch. Rat. Mech. Anal., 216 (2015), 1009-1047. Google Scholar

[11]

Acta Met., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6.  Google Scholar

[12]

Math. Mech. Solids, 19 (2014), 299-307. doi: 10.1177/1081286512463720.  Google Scholar

[13]

2nd edition, Springer, 2006.  Google Scholar

[14]

Phys. Rev. A, 38 (1988), 1005-1018. doi: 10.1103/PhysRevA.38.1005.  Google Scholar

[15]

Ann. Sci. Ecole Norm. Sup. Paris 1907, 24 (1907), 401-518. Google Scholar

[16]

Arch. Rat. Mech. Anal., 27 (1967), 33-94. doi: 10.1007/BF00276434.  Google Scholar

show all references

References:
[1]

Ann. Global Anal. Geom., 26 (2004), 321-332. doi: 10.1023/B:AGAG.0000047509.63818.4f.  Google Scholar

[2]

Proc. Roy. Soc. A, 231 (1955), 263-273. doi: 10.1098/rspa.1955.0171.  Google Scholar

[3]

Proc. Roy. Soc. Edin. A, 236 (1956), 481-505. doi: 10.1098/rspa.1956.0150.  Google Scholar

[4]

Birkhäuser, 1992.  Google Scholar

[5]

Cambridge University Press, 2007. doi: 10.1017/CBO9780511755217.  Google Scholar

[6]

Eur. Phys. J. E, 36 (2013), p106. doi: 10.1140/epje/i2013-13106-0.  Google Scholar

[7]

Jyväskylän Yliopistopaino, 2005.  Google Scholar

[8]

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]

in Les Houches Summer School Proceedings (eds. R. Balian, M. Kleman and J.-P. Poirier), North-Holland, Amsterdam, 1981. Google Scholar

[10]

Arch. Rat. Mech. Anal., 216 (2015), 1009-1047. Google Scholar

[11]

Acta Met., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6.  Google Scholar

[12]

Math. Mech. Solids, 19 (2014), 299-307. doi: 10.1177/1081286512463720.  Google Scholar

[13]

2nd edition, Springer, 2006.  Google Scholar

[14]

Phys. Rev. A, 38 (1988), 1005-1018. doi: 10.1103/PhysRevA.38.1005.  Google Scholar

[15]

Ann. Sci. Ecole Norm. Sup. Paris 1907, 24 (1907), 401-518. Google Scholar

[16]

Arch. Rat. Mech. Anal., 27 (1967), 33-94. doi: 10.1007/BF00276434.  Google Scholar

[1]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020

[2]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2777-2808. doi: 10.3934/dcds.2020385

[3]

Yuan Gao, Jian-Guo Liu, Tao Luo, Yang Xiang. Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3177-3207. doi: 10.3934/dcdsb.2020224

[4]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3725-3757. doi: 10.3934/dcds.2021014

[5]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311

[6]

Micol Amar, Daniele Andreucci, Claudia Timofte. Homogenization of a modified bidomain model involving imperfect transmission. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021040

[7]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[8]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190

[9]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008

[10]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053

[11]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

[12]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[13]

Mayte Pérez-Llanos, Juan Pablo Pinasco, Nicolas Saintier. Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations. Networks & Heterogeneous Media, 2021, 16 (2) : 257-281. doi: 10.3934/nhm.2021006

[14]

Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258

[15]

Yaonan Ma, Li-Zhi Liao. The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1681-1711. doi: 10.3934/jimo.2020040

[16]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[17]

Zehui Jia, Xue Gao, Xingju Cai, Deren Han. The convergence rate analysis of the symmetric ADMM for the nonconvex separable optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1943-1971. doi: 10.3934/jimo.2020053

[18]

Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021065

[19]

Antonio De Rosa, Domenico Angelo La Manna. A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021059

[20]

Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]