On an Equation From the Theory of Field Dislocation Mechanics
Acharya, Amit ; Tartar, Luc
Bollettino dell'Unione Matematica Italiana, Tome 4 (2011), p. 409-444 / Harvested from Biblioteca Digitale Italiana di Matematica
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Global existence and uniqueness results for a quasilinear system of partial differential equations in one space dimension and time representing the transport of dislocation density are obtained. Stationary solutions of the system are also studied, and an infinite dimensional class of equilibria is derived. These time (in)dependent solutions include both periodic and aperiodic spatial distributions of smooth fronts of plastic distortion representing dislocation twist boundary microstructure. Dominated by hyperbolic transport-like features and at the same time containing a large class of equilibria, our system differs qualitatively from regularized systems of hyperbolic conservation laws and neither does it fit into a gradient flow structure.

Publié le : 2011-10-01
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     author = {Amit Acharya and Luc Tartar},
     title = {On an Equation From the Theory of Field Dislocation Mechanics},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {4},
     year = {2011},
     pages = {409-444},
     zbl = {1233.35186},
     mrnumber = {2906769},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2011_9_4_3_409_0}
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Acharya, Amit; Tartar, Luc. On an Equation From the Theory of Field Dislocation Mechanics. Bollettino dell'Unione Matematica Italiana, Tome 4 (2011) pp. 409-444. http://gdmltest.u-ga.fr/item/BUMI_2011_9_4_3_409_0/

[Ac1] Acharya, A., A model of crystal plasticity based on the theory of continuously distributed dislocations. J. Mech. Phys. Solids, 49 (2001), 761-784. | Zbl 1017.74010

[Ac2] Acharya, A., Driving forces and boundary conditions in continuum dislocation mechanics, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459, no. 2034 (2003), 1343-1363. Proceedings of the Royal Society. A, 459 (2003): 1343-1363. | MR 1994263 | Zbl 1041.74014

[Ac3] Acharya, A., Constitutive analysis of finite deformation field dislocation mechanics. J. Mech. Phys. Solids, 52, no. 2 (2004), 301-316. | MR 2033975 | Zbl 1106.74315

[Ac4] Acharya, A., New inroads in an old subject: Plasticity, from around the atomic to the macroscopic scale. J. Mech. Phys. Solids, 58, no. 5 (2010), 766- 778. | MR 2642309 | Zbl 1244.74026

[Ac5] Acharya, A., Microcanonical Entropy and mesoscale dislocation mechanics and plasticity. to appear in Journal of Elasticity. | MR 2806903 | Zbl 1320.74010

[Ac&Ma&Zi] Acharya, A. - Matthies, K. - Zimmer, J., Traveling wave solutions for a quasilinear model of field dislocation mechanics. J. Mech. Phys. Solids, 58 (2010), 2043-2053. | MR 2757690 | Zbl 1225.74012

[Ac&Ro] Acharya, A. - Roy, A., Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I. J. Mech. Phys. Solids, 54 (2006), 1687-1710. | MR 2232335 | Zbl 1120.74328

[Ag] Agmon, S., Lectures on Elliptic Boundary Value Problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Revised edition of the 1965 original. AMS Chelsea Publishing, Providence, RI, 2010. x+216 pp. ISBN: 978-0-8218-4910-1. MR2589244. Van Nostrand Mathematical Studies, No. 2 D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1965 v+291 pp. MR0178246 (31 #2504) | MR 178246

[Bi] Birkhoff, G., A Source Book in Classical Analysis, Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. xii+470 pp. | MR 729933

[Da] Dafermos, C., Hyperbolic Conservation Laws in Continuum Physics. Third edition. Grundlehren der Mathematischen Wissenschaften, 325. Springer-Verlag, Berlin, 2010. xxxvi+708 pp. [Second edition. 2005. xx+626 pp. First edition. 2000. xvi+443 pp.]. | MR 2574377 | Zbl 1078.35001

[Fo] Fox, N., A continuum theory of dislocations for single crystals. IMA Journal of Applied Mathematics, 2 (1966), 285-298.

[Kr] Kröner, E., Continuum theory of defects. Physics of Defects, Les Houches Summer School (North-Holland, 1981), 217-315.

[Li&Se1] Limkumnerd, S. - Sethna, J. P., Mesoscale theory of grains and cells: crystal plasticity and coarsening. Phys. Rev. Lett., (2006): 96, 095503.

[Li&;Se2] Limkumnerd, S. - Sethna, J. P., Shocks and slip systems: predictions from a mesoscale theory of continuum dislocation dynamics. J. Mech. Phys. Solids, 56, no. 4 (2008), 1450-1459. | MR 2404020 | Zbl 1171.74315

[Mu] Mura, T., Continuous distribution of moving dislocations. Phil. Mag., 8 (1963), 843-857.

[Na] Nabarro, F. R. N., Theory of Crystal Dislocations, Dover Books on Physics and Chemistry, 1987, 821 pp.

[Po1] Poincaré, H., Sur les équations aux dérivées partielles de la physique mathématique. Amer. J. Math., 12, no. 3 (1890), 211-294. | MR 1505534 | Zbl 22.0977.03

[Po2] Poincaré, H. , Sur les équations de la physique mathématique. Rend. Circ. Mat. Palermo, 8 (1894), 57-156. | Zbl 25.1526.01

[St] Stampacchia, G., Équations Elliptiques du Second Ordre à Coefficients Discontinus, Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que. 1966 326 pp. | MR 192177

[Ta1] Tartar, L., Imbedding theorems of Sobolev spaces into Lorentz spaces. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1, no. 3 (1998), 479-500. | MR 1662313 | Zbl 0929.46028

[Ta2] Tartar, L., Compensation effects in partial differential equations. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), 29, no. 1 (2005), 395-453. | MR 2305083

[Ta3] Tartar, L., An Introduction to Navier-Stokes Equation and Oceanography, Lecture Notes of the Unione Matematica Italiana, 1. Springer-Verlag, Berlin; UMI, Bologna, 2006. xxviii+245 pp. | MR 2258988

[Ta4] Tartar, L., An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, 3. Springer, Berlin; UMI, Bologna, 2007. xxvi+218 pp. | MR 2328004

[Ta5] Tartar, L., From Hyperbolic Systems to Kinetic Theory: A Personalized Quest, Lecture Notes of the Unione Matematica Italiana, 6. Springer-Verlag, Berlin; UMI, Bologna, 2008. xxviii+279 pp. | MR 2397052 | Zbl 1147.35004

[Ta6] Tartar, L., The General Theory of Homogenization: A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009. xxii+470 pp. | MR 2582099

[Wi] Willis, J. R., Second-order effects of dislocations in anisotropic crystals. Internat. J. Engrg. Sci., 5 (1967): 171-190. | Zbl 0147.43506