# American Institute of Mathematical Sciences

February  2022, 16(1): 197-214. doi: 10.3934/ipi.2021046

## Identification and stability of small-sized dislocations using a direct algorithm

 1 URMIT University, School of Science, Discipline of Mathematical Sciences, Melbourne, 3000, Australia 2 Université de Technologie de Compiègne, LMAC, 60205 Compiègne Cedex, France

* Corresponding author: Abdellatif El Badia

Received  December 2020 Revised  April 2021 Published  February 2022 Early access  July 2021

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 [9], 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.

Citation: Batoul Abdelaziz, Abdellatif El Badia, Ahmad El Hajj. Identification and stability of small-sized dislocations using a direct algorithm. Inverse Problems and Imaging, 2022, 16 (1) : 197-214. doi: 10.3934/ipi.2021046
##### References:
 [1] B. Abdelaziz, A. E. Badia and 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 (2013), 797-801.  doi: 10.1016/j.crma.2013.10.010. [2] B. Abdelaziz, A. E. Badia and A. E. Hajj, Direct algorithm for multipolar sources reconstruction, Journal of Mathematical Analysis and Applications, 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013. [3] B. Abdelaziz, A. E. Badia and A. E. Hajj, Some remarks on the small electromagnetic inhomogeneities reconstruction problem, Inverse Problems & Imaging, 11 (2017), 1027-1046.  doi: 10.3934/ipi.2017047. [4] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004. [5] O. Alvarez, P. Hoch, Y. L. Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Archive for Rational Mechanics and Analysis, 181 (2006), 449-504.  doi: 10.1007/s00205-006-0418-5. [6] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245. [7] H. Ammari, H. Kang, G. Nakamura and 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 (2002), 97-129.  doi: 10.1023/A:1023940025757. [8] A. Aspri, E. Beretta, A. L. Mazzucato and M. V. De Hoop, Analysis of a model of elastic dislocations in geophysics, Arch. Ration. Mech. Anal., 236 (2020), 71-111.  doi: 10.1007/s00205-019-01462-w. [9] A. E. Badia and A. E. Hajj, Identification of dislocations in materials from boundary measurements, SIAM Journal on Applied Mathematics, 73 (2013), 84-103.  doi: 10.1137/110833920. [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. doi: 10.1088/0266-5611/29/12/125008. [11] A. E. Badia and T. H. -Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663.  doi: 10.1088/0266-5611/16/3/308. [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. doi: 10.1088/0266-5611/27/10/105001. [13] W. Bollmann, Interference effects in the electron microscopy of thin crystal foils, Phys. Rev., 103 (1956), 1588-1589. [14] G. Canova and L. Kubin, Dislocation microstructure and plastic flow: A three dimensional simulation, Continuum Models and Discrete Systems, 2 (1991), 93-101. [15] D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements, Continuous Dependence and Computational Reconstruction, Inverse Problems, 14 (1998), 553-595. [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 and M. Condat, Model validation of a 3d simulation of dislocation dynamics: discretization and line tension effects, Acta Metallurgica Et Materialia, 40 (1992), 2629-2637. [18] B. Devincre and L. Kubin, Mesoscopic simulations of dislocations and plasticity, Materials Science and Engineering: A, 234 (1997), 8-14. [19] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Rational Mech. Anal., 105 (1989), 299-326.  doi: 10.1007/BF00281494. [20] A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM Journal on Mathematical Analysis, 36 (2005), 1943-1964.  doi: 10.1137/S003614100343768X. [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 and M. J. Whelan, Direct observations of the arrangement. and motion of dislocations in aluminium, Phil. Mag., 1 (1956), 677-684. [23] A. G. Khachaturyan, Theory of Structural Transformations in Solids, Courier Corporation, 2013. [24] M. Koslowski, A. M. Cuitino and 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 (2002), 2597-2635.  doi: 10.1016/S0022-5096(02)00037-6. [25] L. P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis and Y. Bréchet, Dislocation microstructures and plastic flow: A 3d simulation, Solid State Phenomena, Trans Tech Publ, 23-24 (1992), 455-472.  doi: 10.4028/www.scientific.net/SSP.23-24.455. [26] J.-L. Lions and E. Magenes, Problemes aux limites non homogenes et applications, (1968)., [27] E. Orowan, Zur kristallplastizitat i-iii, Z. Phys., 89 (1934), 605-634. [28] M. Polanyi, Uber eine art gitterstorung, die einem kristall plastisch machen konnte, Z. Phys., 89 (1934), 660-664. [29] O. Politano and J. Salazar, A 3d mesoscopic approach for discrete dislocation dynamics, Materials Science and Engineering: A, 309 (2001), 261-264. [30] D. Rodney, Y. Le Bouar and A. Finel, Phase field methods and dislocations, Acta Materialia, 51 (2003), 17-30. [31] K. Schwarz, Simulation of dislocations on the mesoscopic scale. i. Methods and examples, Journal of Applied Physics, 85 (1999), 108-119. [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 (1934), 362-387. [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. doi: 10.24033/asens.583. [37] Y. U. Wang, Y. Jin, A. Cuitino and A. Khachaturyan, Nanoscale phase field microelasticity theory of dislocations: Model and 3d simulations, Acta Materialia, 49 (2001), 1847-1857. [38] Y. Xiang, L. -T. Cheng, D. J. Srolovitz and E. Weinan, A level set method for dislocation dynamics, Acta Materialia, 51 (2003), 5499-5518.

show all references

##### References:
 [1] B. Abdelaziz, A. E. Badia and 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 (2013), 797-801.  doi: 10.1016/j.crma.2013.10.010. [2] B. Abdelaziz, A. E. Badia and A. E. Hajj, Direct algorithm for multipolar sources reconstruction, Journal of Mathematical Analysis and Applications, 428 (2015), 306-336.  doi: 10.1016/j.jmaa.2015.03.013. [3] B. Abdelaziz, A. E. Badia and A. E. Hajj, Some remarks on the small electromagnetic inhomogeneities reconstruction problem, Inverse Problems & Imaging, 11 (2017), 1027-1046.  doi: 10.3934/ipi.2017047. [4] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004. [5] O. Alvarez, P. Hoch, Y. L. Bouar and R. Monneau, Dislocation dynamics: Short-time existence and uniqueness of the solution, Archive for Rational Mechanics and Analysis, 181 (2006), 449-504.  doi: 10.1007/s00205-006-0418-5. [6] H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2004. doi: 10.1007/b98245. [7] H. Ammari, H. Kang, G. Nakamura and 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 (2002), 97-129.  doi: 10.1023/A:1023940025757. [8] A. Aspri, E. Beretta, A. L. Mazzucato and M. V. De Hoop, Analysis of a model of elastic dislocations in geophysics, Arch. Ration. Mech. Anal., 236 (2020), 71-111.  doi: 10.1007/s00205-019-01462-w. [9] A. E. Badia and A. E. Hajj, Identification of dislocations in materials from boundary measurements, SIAM Journal on Applied Mathematics, 73 (2013), 84-103.  doi: 10.1137/110833920. [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. doi: 10.1088/0266-5611/29/12/125008. [11] A. E. Badia and T. H. -Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663.  doi: 10.1088/0266-5611/16/3/308. [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. doi: 10.1088/0266-5611/27/10/105001. [13] W. Bollmann, Interference effects in the electron microscopy of thin crystal foils, Phys. Rev., 103 (1956), 1588-1589. [14] G. Canova and L. Kubin, Dislocation microstructure and plastic flow: A three dimensional simulation, Continuum Models and Discrete Systems, 2 (1991), 93-101. [15] D. J. Cedio-Fengya, S. Moskow and M. S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements, Continuous Dependence and Computational Reconstruction, Inverse Problems, 14 (1998), 553-595. [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 and M. Condat, Model validation of a 3d simulation of dislocation dynamics: discretization and line tension effects, Acta Metallurgica Et Materialia, 40 (1992), 2629-2637. [18] B. Devincre and L. Kubin, Mesoscopic simulations of dislocations and plasticity, Materials Science and Engineering: A, 234 (1997), 8-14. [19] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Rational Mech. Anal., 105 (1989), 299-326.  doi: 10.1007/BF00281494. [20] A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIAM Journal on Mathematical Analysis, 36 (2005), 1943-1964.  doi: 10.1137/S003614100343768X. [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 and M. J. Whelan, Direct observations of the arrangement. and motion of dislocations in aluminium, Phil. Mag., 1 (1956), 677-684. [23] A. G. Khachaturyan, Theory of Structural Transformations in Solids, Courier Corporation, 2013. [24] M. Koslowski, A. M. Cuitino and 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 (2002), 2597-2635.  doi: 10.1016/S0022-5096(02)00037-6. [25] L. P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis and Y. Bréchet, Dislocation microstructures and plastic flow: A 3d simulation, Solid State Phenomena, Trans Tech Publ, 23-24 (1992), 455-472.  doi: 10.4028/www.scientific.net/SSP.23-24.455. [26] J.-L. Lions and E. Magenes, Problemes aux limites non homogenes et applications, (1968)., [27] E. Orowan, Zur kristallplastizitat i-iii, Z. Phys., 89 (1934), 605-634. [28] M. Polanyi, Uber eine art gitterstorung, die einem kristall plastisch machen konnte, Z. Phys., 89 (1934), 660-664. [29] O. Politano and J. Salazar, A 3d mesoscopic approach for discrete dislocation dynamics, Materials Science and Engineering: A, 309 (2001), 261-264. [30] D. Rodney, Y. Le Bouar and A. Finel, Phase field methods and dislocations, Acta Materialia, 51 (2003), 17-30. [31] K. Schwarz, Simulation of dislocations on the mesoscopic scale. i. Methods and examples, Journal of Applied Physics, 85 (1999), 108-119. [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 (1934), 362-387. [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. doi: 10.24033/asens.583. [37] Y. U. Wang, Y. Jin, A. Cuitino and A. Khachaturyan, Nanoscale phase field microelasticity theory of dislocations: Model and 3d simulations, Acta Materialia, 49 (2001), 1847-1857. [38] Y. Xiang, L. -T. Cheng, D. J. Srolovitz and E. Weinan, A level set method for dislocation dynamics, Acta Materialia, 51 (2003), 5499-5518.
Edge dislocation.
The form of the dislocations
The form of the dislocations.
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