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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 |
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. |
[2] |
M. F. Ashby, The deformation of plastically non-homogeneous alloys,, Philos. Mag., 21 (1970), 399.
doi: 10.1080/14786437008238426. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems,", Oxford Mathematical Monographs, (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.
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.
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.
doi: 10.1007/s00205-004-0351-4. |
[7] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems,", Translated from the French, 28 (1976).
|
[8] |
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Interdisciplinary Applied Mathematics, 9 (1999).
|
[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. |
[21] |
A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73.
|
[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. |
[23] |
A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.
|
[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.
|
[25] |
A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.
|
[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. |
[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.
|
[28] |
J. F. Nye, Some geometrical relations in dislocated crystals,, Acta Metall., 1 (1953), 153.
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.
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, (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.
|
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. |
[2] |
M. F. Ashby, The deformation of plastically non-homogeneous alloys,, Philos. Mag., 21 (1970), 399.
doi: 10.1080/14786437008238426. |
[3] |
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variations and Free Discontinuity Problems,", Oxford Mathematical Monographs, (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.
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.
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.
doi: 10.1007/s00205-004-0351-4. |
[7] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems,", Translated from the French, 28 (1976).
|
[8] |
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
W. Han and B. D. Reddy, "Plasticity. Mathematical Theory and Numerical Analysis,", Interdisciplinary Applied Mathematics, 9 (1999).
|
[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. |
[21] |
A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. Partial Differential Equations, 22 (2005), 73.
|
[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. |
[23] |
A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.
|
[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.
|
[25] |
A. Mielke and F. Theil, On rate-independent hysteresis models,, NoDEA Nonlinear Differential Equations Appl., 11 (2004), 151.
|
[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. |
[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.
|
[28] |
J. F. Nye, Some geometrical relations in dislocated crystals,, Acta Metall., 1 (1953), 153.
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.
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, (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.
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