# 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 & 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. doi: 10.1115/1.3225725. Google Scholar [2] M. F. Ashby, The deformation of plastically non-homogeneous alloys,, Philos. Mag., 21 (1970), 399. doi: 10.1080/14786437008238426. Google Scholar [3] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). Google Scholar [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. doi: 10.1007/s00205-005-0407-0. Google Scholar [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. doi: 10.1007/s00205-008-0117-5. Google Scholar [6] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165. doi: 10.1007/s00205-004-0351-4. Google Scholar [7] I. Ekeland and R. Témam, "Convex Analysis and Variational Problems,", Translated from the French, 28 (1976). Google Scholar [8] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992). Google Scholar [9] N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity,, Adv. Appl. Mech., 33 (1997), 295. doi: 10.1016/S0065-2156(08)70388-0. Google Scholar [10] N. A. Fleck and J. W. Hutchinson, A reformulation of strain gradient plasticity,, J. Mech. Phys. Solids, 49 (2001), 2245. doi: 10.1016/S0022-5096(01)00049-7. Google Scholar [11] G. A. Francfort and C. J. Larsen, Existence and convergence for quasistatic evolution in brittle fracture,, Comm. Pure Appl. Math., 56 (2003), 1465. doi: 10.1002/cpa.3039. Google Scholar [12] G. A. Francfort and J.-J. Marigo, Revisiting brittle fractures as an energy minimization problem,, J. Mech. Phys. Solids, 46 (1998), 1319. doi: 10.1016/S0022-5096(98)00034-9. Google Scholar [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. doi: 10.4171/JEMS/228. Google Scholar [14] A. Giacomini and L. Lussardi, Quasi-static evolution for a model in strain gradient plasticity,, SIAM J. Math. Anal., 40 (2008), 1201. doi: 10.1137/070708202. Google Scholar [15] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals,, Duke Math. J., 31 (1964), 159. doi: 10.1215/S0012-7094-64-03115-1. Google Scholar [16] P. Gudmundson, A unified treatment of strain gradient plasticity,, J. Mech. Phys. Solids, 52 (2004), 1379. doi: 10.1016/j.jmps.2003.11.002. Google Scholar [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. doi: 10.1016/j.jmps.2004.12.008. Google Scholar [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. doi: 10.1007/s00161-009-0107-3. Google Scholar [19] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Interdisciplinary Applied Mathematics, 9 (1999). Google Scholar [20] J. Kratochvíl, M. Kružík and R. Sedláček, Energetic approach to gradient plasticity,, ZAMM Z. Angew. Math. Mech., 90 (2010), 122. doi: 10.1002/zamm.200900227. Google Scholar [21] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73. Google Scholar [22] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain,, J. Nonlinear Sci., 19 (2009), 221. doi: 10.1007/s00332-008-9033-y. Google Scholar [23] A. Mielke, Evolution of rate-independent systems,, in, (2005), 461. Google Scholar [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. Google Scholar [25] A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151. Google Scholar [26] H.-B. Mühlhaus and E. C. Aifantis, A variational principle for gradient plasticity,, Int. J. Solids Structures, 28 (1991), 845. doi: 10.1016/0020-7683(91)90004-Y. Google Scholar [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. Google Scholar [28] J. F. Nye, Some geometrical relations in dislocated crystals,, Acta Metall., 1 (1953), 153. doi: 10.1016/0001-6160(53)90054-6. Google Scholar [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. doi: 10.1016/j.ijplas.2007.01.013. Google Scholar [30] B. D. Reddy, F. Ebobisse and A. McBride, On the mathematical formulations of a model of strain gradient plasticity,, in, (2008), 117. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] E. C. Aifantis, On the microstructural origin of certain inelastic models,, Trans. ASME J. Eng. Mater. Technol., 106 (1984), 326. doi: 10.1115/1.3225725. Google Scholar [2] M. F. Ashby, The deformation of plastically non-homogeneous alloys,, Philos. Mag., 21 (1970), 399. doi: 10.1080/14786437008238426. Google Scholar [3] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). Google Scholar [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. doi: 10.1007/s00205-005-0407-0. Google Scholar [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. doi: 10.1007/s00205-008-0117-5. Google Scholar [6] G. Dal Maso, G. A. Francfort and R. Toader, Quasistatic crack growth in nonlinear elasticity,, Arch. Ration. Mech. Anal., 176 (2005), 165. doi: 10.1007/s00205-004-0351-4. Google Scholar [7] I. Ekeland and R. Témam, "Convex Analysis and Variational Problems,", Translated from the French, 28 (1976). Google Scholar [8] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992). Google Scholar [9] N. A. Fleck and J. W. Hutchinson, Strain gradient plasticity,, Adv. Appl. Mech., 33 (1997), 295. doi: 10.1016/S0065-2156(08)70388-0. Google Scholar [10] N. A. Fleck and J. W. Hutchinson, A reformulation of strain gradient plasticity,, J. Mech. Phys. Solids, 49 (2001), 2245. doi: 10.1016/S0022-5096(01)00049-7. Google Scholar [11] G. A. Francfort and C. J. Larsen, Existence and convergence for quasistatic evolution in brittle fracture,, Comm. Pure Appl. Math., 56 (2003), 1465. doi: 10.1002/cpa.3039. Google Scholar [12] G. A. Francfort and J.-J. Marigo, Revisiting brittle fractures as an energy minimization problem,, J. Mech. Phys. Solids, 46 (1998), 1319. doi: 10.1016/S0022-5096(98)00034-9. Google Scholar [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. doi: 10.4171/JEMS/228. Google Scholar [14] A. Giacomini and L. Lussardi, Quasi-static evolution for a model in strain gradient plasticity,, SIAM J. Math. Anal., 40 (2008), 1201. doi: 10.1137/070708202. Google Scholar [15] C. Goffman and J. Serrin, Sublinear functions of measures and variational integrals,, Duke Math. J., 31 (1964), 159. doi: 10.1215/S0012-7094-64-03115-1. Google Scholar [16] P. Gudmundson, A unified treatment of strain gradient plasticity,, J. Mech. Phys. Solids, 52 (2004), 1379. doi: 10.1016/j.jmps.2003.11.002. Google Scholar [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. doi: 10.1016/j.jmps.2004.12.008. Google Scholar [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. doi: 10.1007/s00161-009-0107-3. Google Scholar [19] W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Interdisciplinary Applied Mathematics, 9 (1999). Google Scholar [20] J. Kratochvíl, M. Kružík and R. Sedláček, Energetic approach to gradient plasticity,, ZAMM Z. Angew. Math. Mech., 90 (2010), 122. doi: 10.1002/zamm.200900227. Google Scholar [21] A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73. Google Scholar [22] A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain,, J. Nonlinear Sci., 19 (2009), 221. doi: 10.1007/s00332-008-9033-y. Google Scholar [23] A. Mielke, Evolution of rate-independent systems,, in, (2005), 461. Google Scholar [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. Google Scholar [25] A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151. Google Scholar [26] H.-B. Mühlhaus and E. C. Aifantis, A variational principle for gradient plasticity,, Int. J. Solids Structures, 28 (1991), 845. doi: 10.1016/0020-7683(91)90004-Y. Google Scholar [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. Google Scholar [28] J. F. Nye, Some geometrical relations in dislocated crystals,, Acta Metall., 1 (1953), 153. doi: 10.1016/0001-6160(53)90054-6. Google Scholar [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. doi: 10.1016/j.ijplas.2007.01.013. Google Scholar [30] B. D. Reddy, F. Ebobisse and A. McBride, On the mathematical formulations of a model of strain gradient plasticity,, in, (2008), 117. Google Scholar [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. Google Scholar
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