Identification and stability of small-sized dislocations using a direct algorithm. (English) Zbl 1481.35388

Summary: This paper considers the problem of identifying dislocation lines of curvilinear form in three-dimensional materials from boundary measurements, when the areas surrounded by the dislocation lines are assumed to be small-sized. The objective of this inverse problem is to reconstruct the number, the initial position and certain characteristics of these dislocations and establish, using certain test functions, a Hölder stability of the centers. This paper can be considered as a generalization of [A. El Badia and A. El Hajj, SIAM J. Appl. Math. 73, No. 1, 84–103 (2013; Zbl 1267.35252)], where instead of reconstructing point-wise dislocations, as done in the latter paper, our aim is to recover the parameters of line dislocations by employing a direct algebraic algorithm.


35R30 Inverse problems for PDEs
35Q70 PDEs in connection with mechanics of particles and systems of particles
74G75 Inverse problems in equilibrium solid mechanics
74B05 Classical linear elasticity
74B10 Linear elasticity with initial stresses
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)


Zbl 1267.35252
Full Text: DOI


[1] B. Abdelaziz; A. E. Badia; A. E. Hajj, Reconstruction of extended sources with small supports in the elliptic equation \(\Delta u + \mu u = F\) from a single Cauchy data, C. R. Math. Acad. Sci. Paris, 351, 797-801 (2013) · Zbl 1282.35405
[2] B. Abdelaziz; A. E. Badia; A. E. Hajj, Direct algorithm for multipolar sources reconstruction, Journal of Mathematical Analysis and Applications, 428, 306-336 (2015) · Zbl 1325.65149
[3] B. Abdelaziz; A. E. Badia; A. E. Hajj, Some remarks on the small electromagnetic inhomogeneities reconstruction problem, Inverse Problems & Imaging, 11, 1027-1046 (2017) · Zbl 1377.49038
[4] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. · Zbl 1190.35228
[5] O. Alvarez; P. Hoch; Y. L. Bouar; R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Archive for Rational Mechanics and Analysis, 181, 449-504 (2006) · Zbl 1158.74335
[6] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2004. · Zbl 1113.35148
[7] H. Ammari; H. Kang; G. Nakamura; K. Tanuma, Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elasticity, 67, 97-129 (2002) · Zbl 1089.74576
[8] A. Aspri; E. Beretta; A. L. Mazzucato; M. V. De Hoop, Analysis of a model of elastic dislocations in geophysics, Arch. Ration. Mech. Anal., 236, 71-111 (2020) · Zbl 1437.35251
[9] A. E. Badia; A. E. Hajj, Identification of dislocations in materials from boundary measurements, SIAM Journal on Applied Mathematics, 73, 84-103 (2013) · Zbl 1267.35252
[10] A. E. Badia and A. E. Hajj, Stability estimates for an inverse source problem of Helmholtz’s equation from single cauchy data at a fixed frequency, Inverse Problems, 29 (2013), 125008. · Zbl 1292.65114
[11] A. E. Badia; T. H. -Duong, An inverse source problem in potential analysis, Inverse Problems, 16, 651-663 (2000) · Zbl 0963.35194
[12] A. E. Badia and T. Nara, An inverse source problem for Helmholtz’s equation from the cauchy data with a single wave number, Inverse Problems, 27 (2011), 105001. · Zbl 1231.35299
[13] W. Bollmann, Interference effects in the electron microscopy of thin crystal foils, Phys. Rev., 103, 1588-1589 (1956)
[14] G. Canova; L. Kubin, Dislocation microstructure and plastic flow: A three dimensional simulation, Continuum Models and Discrete Systems, 2, 93-101 (1991)
[15] D. J. Cedio-Fengya; S. Moskow; M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements, Continuous Dependence and Computational Reconstruction, Inverse Problems, 14, 553-595 (1998) · Zbl 0916.35132
[16] B. Devincre, Simulations dynamiques des dislocations a une echelle mesoscopique: Une etude de la deformation plastique, Ph.D thesis, Paris, 1993.
[17] B. Devincre; M. Condat, Model validation of a 3d simulation of dislocation dynamics: discretization and line tension effects, Acta Metallurgica Et Materialia, 40, 2629-2637 (1992)
[18] B. Devincre; L. Kubin, Mesoscopic simulations of dislocations and plasticity, Materials Science and Engineering: A, 234, 8-14 (1997)
[19] A. Friedman; M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Rational Mech. Anal., 105, 299-326 (1989) · Zbl 0684.35087
[20] A. Garroni; S. Müller, \( \Gamma \)-limit of a phase-field model of dislocations, SIAM Journal on Mathematical Analysis, 36, 1943-1964 (2005) · Zbl 1094.82008
[21] M. Haataja and F. Léonard, Influence of mobile dislocations on phase separation in binary alloys, Physical Review B, 69 (2004), 081201.
[22] P. B. Hirsch; R. W. Horne; M. J. Whelan, Direct observations of the arrangement. and motion of dislocations in aluminium, Phil. Mag., 1, 677-684 (1956)
[23] A. G. Khachaturyan, Theory of Structural Transformations in Solids, Courier Corporation, 2013.
[24] M. Koslowski; A. M. Cuitino; M. Ortiz, A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals, Journal of the Mechanics and Physics of Solids, 50, 2597-2635 (2002) · Zbl 1094.74563
[25] L. P. Kubin; G. Canova; M. Condat; B. Devincre; V. Pontikis; Y. Bréchet, Dislocation microstructures and plastic flow: A 3d simulation, Solid State Phenomena, Trans Tech Publ, 23-24, 455-472 (1992)
[26] J.-L. Lions and E. Magenes, Problemes aux limites non homogenes et applications, (1968)., · Zbl 0165.10801
[27] E. Orowan, Zur kristallplastizitat i-iii, Z. Phys., 89, 605-634 (1934)
[28] M. Polanyi, Uber eine art gitterstorung, die einem kristall plastisch machen konnte, Z. Phys., 89, 660-664 (1934)
[29] O. Politano; J. Salazar, A 3d mesoscopic approach for discrete dislocation dynamics, Materials Science and Engineering: A, 309, 261-264 (2001)
[30] D. Rodney; Y. Le Bouar; A. Finel, Phase field methods and dislocations, Acta Materialia, 51, 17-30 (2003)
[31] K. Schwarz, Simulation of dislocations on the mesoscopic scale. i. Methods and examples, Journal of Applied Physics, 85, 108-119 (1999)
[32] V. Shenoy, R. Kukta and R. Phillips, Mesoscopic analysis of structure and strength of dislocation junctions in fcc metals, Physical Review Letters, 84 (2000), 1491.
[33] G. W. Stewart, Introduction to Matrix Computations, Computer Science and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973.
[34] G. I. Taylor, The mechanism of plastic deformation of crystals. part I. Theoretical, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 145, 362-387 (1934) · JFM 60.0712.02
[35] M. Verdier, M. Fivel and I. Groma, Mesoscopic scale simulation of dislocation dynamics in fcc metals: Principles and applications, Modelling and Simulation in Materials Science and Engineering, 6 (1998), 755.
[36] V. Volterra, Sur l’équilibre des corps élastiques multiplement connexes, Ann. Sci. École Norm. Sup. (3), 24 (1907), 401-517. · JFM 38.0814.01
[37] Y. U. Wang; Y. Jin; A. Cuitino; A. Khachaturyan, Nanoscale phase field microelasticity theory of dislocations: Model and 3d simulations, Acta Materialia, 49, 1847-1857 (2001)
[38] Y. Xiang; L. -T. Cheng; D. J. Srolovitz; E. Weinan, A level set method for dislocation dynamics, Acta Materialia, 51, 5499-5518 (2003)
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