# American Institute of Mathematical Sciences

March  2012, 17(2): 527-552. doi: 10.3934/dcdsb.2012.17.527

## On the energetic formulation of the Gurtin and Anand model in strain gradient plasticity

 1 Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy

Received  July 2010 Revised  September 2011 Published  December 2011

We discuss the energetic formulation of the Gurtin and Anand model (J. Mech. Phys. Solids, 2005) in strain gradient plasticity, and illustrate the related mathematical analysis concerning the existence of quasi-static evolutions.
Citation: Alessandro Giacomini. On the energetic formulation of the Gurtin and Anand model in strain gradient plasticity. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 527-552. doi: 10.3934/dcdsb.2012.17.527
##### References:
 [1] E. C. Aifantis, On the microstructural origin of certain inelastic models, Trans. ASME J. Eng. Mater. Technol., 106 (1984), 326-330. doi: 10.1115/1.3225725. [2] M. F. Ashby, The deformation of plastically non-homogeneous alloys, Philos. Mag., 21 (1970), 399-424. doi: 10.1080/14786437008238426. [3] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [4] G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291. doi: 10.1007/s00205-005-0407-0. [5] G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5. [6] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4. [7] I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," Translated from the French, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. [8] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [9] N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech., 33 (1997), 295-361. doi: 10.1016/S0065-2156(08)70388-0. [10] N. A. Fleck and J. W. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids, 49 (2001), 2245-2271. doi: 10.1016/S0022-5096(01)00049-7. [11] G. A. Francfort and C. J. Larsen, Existence and convergence for quasistatic evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), 1465-1500. doi: 10.1002/cpa.3039. [12] G. A. Francfort and J.-J. Marigo, Revisiting brittle fractures as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9. [13] A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc. (JEMS), 12 (2010), 1231-1266. doi: 10.4171/JEMS/228. [14] A. Giacomini and L. Lussardi, Quasi-static evolution for a model in strain gradient plasticity, SIAM J. Math. Anal., 40 (2008), 1201-1245. doi: 10.1137/070708202. [15] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J., 31 (1964), 159-178. doi: 10.1215/S0012-7094-64-03115-1. [16] P. Gudmundson, A unified treatment of strain gradient plasticity, J. Mech. Phys. Solids, 52 (2004), 1379-1406. doi: 10.1016/j.jmps.2003.11.002. [17] M. E. Gurtin and L. Anand, A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. I. Small deformations, J. Mech. Phys. Solids, 53 (2005), 1624-1649. doi: 10.1016/j.jmps.2004.12.008. [18] M. E. Gurtin and B. D. Reddy, Alternative formulations of isotropic hardening for Mises materials, and associated variational inequalities, Contin. Mech. Thermodyn., 21 (2009), 237-250. doi: 10.1007/s00161-009-0107-3. [19] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999. [20] J. Kratochvíl, M. Kružík and R. Sedláček, Energetic approach to gradient plasticity, ZAMM Z. Angew. Math. Mech., 90 (2010), 122-135. doi: 10.1002/zamm.200900227. [21] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99. [22] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009), 221-248. doi: 10.1007/s00332-008-9033-y. [23] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," Vol. II, 461-559, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005. [24] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. [25] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. [26] H.-B. Mühlhaus and E. C. Aifantis, A variational principle for gradient plasticity, Int. J. Solids Structures, 28 (1991), 845-857. doi: 10.1016/0020-7683(91)90004-Y. [27] P. Neff, K. Chelmiński and H.-D. Alber, Notes on strain gradient plasticity: Finite strain covariant modelling and global existence in the infinitesimal rate-independent case, Math. Models Methods Appl. Sci., 19 (2009), 307-346. [28] J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metall., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6. [29] B. D. Reddy, F. Ebobisse and A. McBride, Well-posedness of a model of strain gradient plasticity for plastically irrotational materials, Int. J. Plasticity, 24 (2008), 55-73. doi: 10.1016/j.ijplas.2007.01.013. [30] B. D. Reddy, F. Ebobisse and A. McBride, On the mathematical formulations of a model of strain gradient plasticity, in "Theoretical, Modelling and Computational Aspects of Inelastic Media" (Proceedings of IUTAM Symposium, ed. B. D. Reddy), Springer, Berlin, (2008), 117-128. [31] P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, (French) [On the equations of plasticity: Existence and regularity of solutions], J. Mécanique, 20 (1981), 3-39.

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##### References:
 [1] E. C. Aifantis, On the microstructural origin of certain inelastic models, Trans. ASME J. Eng. Mater. Technol., 106 (1984), 326-330. doi: 10.1115/1.3225725. [2] M. F. Ashby, The deformation of plastically non-homogeneous alloys, Philos. Mag., 21 (1970), 399-424. doi: 10.1080/14786437008238426. [3] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [4] G. Dal Maso, A. DeSimone and M. G. Mora, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal., 180 (2006), 237-291. doi: 10.1007/s00205-005-0407-0. [5] G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Ration. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5. [6] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal., 176 (2005), 165-225. doi: 10.1007/s00205-004-0351-4. [7] I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," Translated from the French, Corrected reprint of the 1976 English edition, Classics in Applied Mathematics, 28, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. [8] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [9] N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech., 33 (1997), 295-361. doi: 10.1016/S0065-2156(08)70388-0. [10] N. A. Fleck and J. W. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids, 49 (2001), 2245-2271. doi: 10.1016/S0022-5096(01)00049-7. [11] G. A. Francfort and C. J. Larsen, Existence and convergence for quasistatic evolution in brittle fracture, Comm. Pure Appl. Math., 56 (2003), 1465-1500. doi: 10.1002/cpa.3039. [12] G. A. Francfort and J.-J. Marigo, Revisiting brittle fractures as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), 1319-1342. doi: 10.1016/S0022-5096(98)00034-9. [13] A. Garroni, G. Leoni and M. Ponsiglione, Gradient theory for plasticity via homogenization of discrete dislocations, J. Eur. Math. Soc. (JEMS), 12 (2010), 1231-1266. doi: 10.4171/JEMS/228. [14] A. Giacomini and L. Lussardi, Quasi-static evolution for a model in strain gradient plasticity, SIAM J. Math. Anal., 40 (2008), 1201-1245. doi: 10.1137/070708202. [15] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals, Duke Math. J., 31 (1964), 159-178. doi: 10.1215/S0012-7094-64-03115-1. [16] P. Gudmundson, A unified treatment of strain gradient plasticity, J. Mech. Phys. Solids, 52 (2004), 1379-1406. doi: 10.1016/j.jmps.2003.11.002. [17] M. E. Gurtin and L. Anand, A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. I. Small deformations, J. Mech. Phys. Solids, 53 (2005), 1624-1649. doi: 10.1016/j.jmps.2004.12.008. [18] M. E. Gurtin and B. D. Reddy, Alternative formulations of isotropic hardening for Mises materials, and associated variational inequalities, Contin. Mech. Thermodyn., 21 (2009), 237-250. doi: 10.1007/s00161-009-0107-3. [19] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis," Interdisciplinary Applied Mathematics, 9, Springer-Verlag, New York, 1999. [20] J. Kratochvíl, M. Kružík and R. Sedláček, Energetic approach to gradient plasticity, ZAMM Z. Angew. Math. Mech., 90 (2010), 122-135. doi: 10.1002/zamm.200900227. [21] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations, 22 (2005), 73-99. [22] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci., 19 (2009), 221-248. doi: 10.1007/s00332-008-9033-y. [23] A. Mielke, Evolution of rate-independent systems, in "Evolutionary Equations," Vol. II, 461-559, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2005. [24] A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. [25] A. Mielke and F. Theil, On rate-independent hysteresis models, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151-189. [26] H.-B. Mühlhaus and E. C. Aifantis, A variational principle for gradient plasticity, Int. J. Solids Structures, 28 (1991), 845-857. doi: 10.1016/0020-7683(91)90004-Y. [27] P. Neff, K. Chelmiński and H.-D. Alber, Notes on strain gradient plasticity: Finite strain covariant modelling and global existence in the infinitesimal rate-independent case, Math. Models Methods Appl. Sci., 19 (2009), 307-346. [28] J. F. Nye, Some geometrical relations in dislocated crystals, Acta Metall., 1 (1953), 153-162. doi: 10.1016/0001-6160(53)90054-6. [29] B. D. Reddy, F. Ebobisse and A. McBride, Well-posedness of a model of strain gradient plasticity for plastically irrotational materials, Int. J. Plasticity, 24 (2008), 55-73. doi: 10.1016/j.ijplas.2007.01.013. [30] B. D. Reddy, F. Ebobisse and A. McBride, On the mathematical formulations of a model of strain gradient plasticity, in "Theoretical, Modelling and Computational Aspects of Inelastic Media" (Proceedings of IUTAM Symposium, ed. B. D. Reddy), Springer, Berlin, (2008), 117-128. [31] P.-M. Suquet, Sur les équations de la plasticité: Existence et régularité des solutions, (French) [On the equations of plasticity: Existence and regularity of solutions], J. Mécanique, 20 (1981), 3-39.
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