# 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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] W. Bollmann, Interference effects in the electron microscopy of thin crystal foils, Phys. Rev., 103 (1956), 1588-1589.   Google Scholar [14] G. Canova and L. Kubin, Dislocation microstructure and plastic flow: A three dimensional simulation, Continuum Models and Discrete Systems, 2 (1991), 93-101.   Google Scholar [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.   Google Scholar [16] B. Devincre, Simulations dynamiques des dislocations a une echelle mesoscopique: Une etude de la deformation plastique, Ph.D thesis, Paris, 1993. Google Scholar [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.   Google Scholar [18] B. Devincre and L. Kubin, Mesoscopic simulations of dislocations and plasticity, Materials Science and Engineering: A, 234 (1997), 8-14.   Google Scholar [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.  Google Scholar [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.  Google Scholar [21] M. Haataja and F. Léonard, Influence of mobile dislocations on phase separation in binary alloys, Physical Review B, 69 (2004), 081201. Google Scholar [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.   Google Scholar [23] A. G. Khachaturyan, Theory of Structural Transformations in Solids, Courier Corporation, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [26] J.-L. Lions and E. Magenes, Problemes aux limites non homogenes et applications, (1968)., Google Scholar [27] E. Orowan, Zur kristallplastizitat i-iii, Z. Phys., 89 (1934), 605-634.   Google Scholar [28] M. Polanyi, Uber eine art gitterstorung, die einem kristall plastisch machen konnte, Z. Phys., 89 (1934), 660-664.   Google Scholar [29] O. Politano and J. Salazar, A 3d mesoscopic approach for discrete dislocation dynamics, Materials Science and Engineering: A, 309 (2001), 261-264.   Google Scholar [30] D. Rodney, Y. Le Bouar and A. Finel, Phase field methods and dislocations, Acta Materialia, 51 (2003), 17-30.   Google Scholar [31] K. Schwarz, Simulation of dislocations on the mesoscopic scale. i. Methods and examples, Journal of Applied Physics, 85 (1999), 108-119.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] W. Bollmann, Interference effects in the electron microscopy of thin crystal foils, Phys. Rev., 103 (1956), 1588-1589.   Google Scholar [14] G. Canova and L. Kubin, Dislocation microstructure and plastic flow: A three dimensional simulation, Continuum Models and Discrete Systems, 2 (1991), 93-101.   Google Scholar [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.   Google Scholar [16] B. Devincre, Simulations dynamiques des dislocations a une echelle mesoscopique: Une etude de la deformation plastique, Ph.D thesis, Paris, 1993. Google Scholar [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.   Google Scholar [18] B. Devincre and L. Kubin, Mesoscopic simulations of dislocations and plasticity, Materials Science and Engineering: A, 234 (1997), 8-14.   Google Scholar [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.  Google Scholar [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.  Google Scholar [21] M. Haataja and F. Léonard, Influence of mobile dislocations on phase separation in binary alloys, Physical Review B, 69 (2004), 081201. Google Scholar [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.   Google Scholar [23] A. G. Khachaturyan, Theory of Structural Transformations in Solids, Courier Corporation, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [26] J.-L. Lions and E. Magenes, Problemes aux limites non homogenes et applications, (1968)., Google Scholar [27] E. Orowan, Zur kristallplastizitat i-iii, Z. Phys., 89 (1934), 605-634.   Google Scholar [28] M. Polanyi, Uber eine art gitterstorung, die einem kristall plastisch machen konnte, Z. Phys., 89 (1934), 660-664.   Google Scholar [29] O. Politano and J. Salazar, A 3d mesoscopic approach for discrete dislocation dynamics, Materials Science and Engineering: A, 309 (2001), 261-264.   Google Scholar [30] D. Rodney, Y. Le Bouar and A. Finel, Phase field methods and dislocations, Acta Materialia, 51 (2003), 17-30.   Google Scholar [31] K. Schwarz, Simulation of dislocations on the mesoscopic scale. i. Methods and examples, Journal of Applied Physics, 85 (1999), 108-119.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.   Google Scholar
Edge dislocation.
The form of the dislocations
The form of the dislocations.
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