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On certain convex compactifications for relaxation in evolution problems
1. | Mathematical Institute, Charles University, Sokolovská 83, CZ-186 75 Praha 8 |
References:
[1] |
P. Alexandroff, Untersuchungen über gestalt und lage abgeschlossener mengen beliebiger dimension,, Math. Anal., 30 (1929), 101.
|
[2] |
S. Aubri, M. Fago and M. Ortiz, A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials,, Comp. Meth. in Appl. Mech. Engr., 192 (2003), 2823.
|
[3] |
V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", D. Reidel Publ., (1986).
|
[4] |
S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143.
doi: 10.1016/j.cma.2003.12.065. |
[5] |
S. A. Belov and V. V. Chistyakov, A selection principle or mappings of bounded variation,, J. Math. Anal. Appl., 249 (2000), 351.
doi: 10.1006/jmaa.2000.6844. |
[6] |
F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measure approach,, SISSA, (). Google Scholar |
[7] |
C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity,, Proc. Royal Soc. London, 458 (2002), 299.
|
[8] |
V. V. Chistyakov, Mappings of bounded variations,, J. Dyn. Cont. Syst., 3 (1997), 261.
doi: 10.1007/BF02465896. |
[9] |
V. V. Chistyakov and O. E. Galkin, Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$,, J. Dyn. Cont. Syst., 4 (1998), 217.
doi: 10.1023/A:1022889902536. |
[10] |
G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures,, Netw. Heterog. Media, 2 (2007), 1.
|
[11] |
G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening,, Arch. Rational Mech. Anal., 189 (2007), 469.
doi: 10.1007/s00205-008-0117-5. |
[12] |
G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening,, Netw. Heterog. Media, 3 (2008), 567.
|
[13] |
R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations,, Comm. Math. Phys., 108 (1987), 667.
doi: 10.1007/BF01214424. |
[14] |
S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology,", Princeton, (1952).
|
[15] |
R. Engelking, "General Topology,", PWN, (1977).
|
[16] |
A. Fiaschi, A Young measure approach to a quasistatic evolution for a class of material models with nonconvex elastic energies,, ESAIM: Control, 15 (2009), 245.
doi: 10.1051/cocv:2008030. |
[17] |
A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy,, Ann. Inst. H. Poincaré, 26 (2009), 1055.
|
[18] |
A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, (). Google Scholar |
[19] |
S. Govindjee, A. Mielke and G. J. Hall, Free-energy of mixing for $n$-variant martensitic phase transformations using quasi-convex analysis,, J. Mech. Physics Solids, 50 (2002), 1897.
doi: 10.1016/S0022-5096(02)00009-1. |
[20] |
K. Hackl and D. M. Kochmann, Relaxed potentials and evolution equations for inelastic microstructures,, in, (2008), 27.
doi: 10.1007/978-1-4020-9090-5_3. |
[21] |
B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés., J. Mécanique, 14 (1975), 39.
|
[22] |
E. Helly, Über lineare funktionaloperationen,, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265. Google Scholar |
[23] |
M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi,, Meccanica, 40 (2005), 389.
doi: 10.1007/s11012-005-2106-1. |
[24] |
M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism,, SIAM Rev., 48 (2006), 439.
doi: 10.1137/S0036144504446187. |
[25] |
M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces,, (preprint no. 5/07, (2007).
doi: 10.1051/cocv:2008060. |
[26] |
M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity., (preprint no. 20/08, (2008). Google Scholar |
[27] |
M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures,, (preprint no. 19/08, (2008). Google Scholar |
[28] |
S. Lefschetz, On compact spaces,, Math. Anal., 32 (1931), 521.
doi: 10.2307/1968249. |
[29] |
A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. PDEs, 22 (2005), 73.
|
[30] |
A. Mielke, Deriving new evolution equations for microstructures via relaxation of variational incremental problems,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5095.
doi: 10.1016/j.cma.2004.07.003. |
[31] |
A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.
|
[32] |
A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case,, in, 28 (2006), 351.
|
[33] |
A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys,, Multiscale Model. Simul., 1 (2003), 571.
doi: 10.1137/S1540345903422860. |
[34] |
A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151.
|
[35] |
A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Archive Rat. Mech. Anal., 162 (2002), 137.
doi: 10.1007/s002050200194. |
[36] |
T. Roubíček, Convex compactifications and special extensions of optimization problems,, Nonlinear Anal., 16 (1991), 1117.
doi: 10.1016/0362-546X(91)90199-B. |
[37] |
T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus,", W. de Gruyter, (1997).
|
[38] |
T. Roubíček, Convex locally compact extensions of Lebesgue spaces and their applications,, in:, (1999), 237.
|
[39] |
T. Roubíček and K.-H. Hoffmann, About the concept of measure-valued solutions to distributed parameter systems,, Math. Methods Appl. Sci., 18 (1995), 671.
doi: 10.1002/mma.1670180902. |
[40] |
T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics,, Z. angew. Math. Physik, 55 (2004), 159.
|
[41] |
A. Tychonoff, Über die topologische Erweiterung von Räumen,, Math. Annalen, 102 (1930), 544.
doi: 10.1007/BF01782364. |
show all references
References:
[1] |
P. Alexandroff, Untersuchungen über gestalt und lage abgeschlossener mengen beliebiger dimension,, Math. Anal., 30 (1929), 101.
|
[2] |
S. Aubri, M. Fago and M. Ortiz, A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials,, Comp. Meth. in Appl. Mech. Engr., 192 (2003), 2823.
|
[3] |
V. Barbu and T. Precupanu, "Convexity and Optimization in Banach Spaces,", D. Reidel Publ., (1986).
|
[4] |
S. Bartels, C. Carstensen, K. Hackl and U. Hoppe, Effective relaxation for microstructure simulations: Algorithms and applications,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5143.
doi: 10.1016/j.cma.2003.12.065. |
[5] |
S. A. Belov and V. V. Chistyakov, A selection principle or mappings of bounded variation,, J. Math. Anal. Appl., 249 (2000), 351.
doi: 10.1006/jmaa.2000.6844. |
[6] |
F. Cagnetti and R. Toader, Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: A Young measure approach,, SISSA, (). Google Scholar |
[7] |
C. Carstensen, K. Hackl and A. Mielke, Non-convex potentials and microstructures in finite-strain plasticity,, Proc. Royal Soc. London, 458 (2002), 299.
|
[8] |
V. V. Chistyakov, Mappings of bounded variations,, J. Dyn. Cont. Syst., 3 (1997), 261.
doi: 10.1007/BF02465896. |
[9] |
V. V. Chistyakov and O. E. Galkin, Mappings of bounded $\Phi$-variation with arbitrary function $\Phi$,, J. Dyn. Cont. Syst., 4 (1998), 217.
doi: 10.1023/A:1022889902536. |
[10] |
G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Time-dependent systems of generalized Young measures,, Netw. Heterog. Media, 2 (2007), 1.
|
[11] |
G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening,, Arch. Rational Mech. Anal., 189 (2007), 469.
doi: 10.1007/s00205-008-0117-5. |
[12] |
G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, Globally stable quasistatic evolution in plasticity with softening,, Netw. Heterog. Media, 3 (2008), 567.
|
[13] |
R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations,, Comm. Math. Phys., 108 (1987), 667.
doi: 10.1007/BF01214424. |
[14] |
S. Eilenberg and N. Steenrod, "Foundation of Algebraic Topology,", Princeton, (1952).
|
[15] |
R. Engelking, "General Topology,", PWN, (1977).
|
[16] |
A. Fiaschi, A Young measure approach to a quasistatic evolution for a class of material models with nonconvex elastic energies,, ESAIM: Control, 15 (2009), 245.
doi: 10.1051/cocv:2008030. |
[17] |
A. Fiaschi, A vanishing viscosity approach to a quasistatic evolution problem with nonconvex energy,, Ann. Inst. H. Poincaré, 26 (2009), 1055.
|
[18] |
A. Fiaschi, Rate-independent phase transitions in elastic materials: A Young-measure approach,, (preprint SISSA, (). Google Scholar |
[19] |
S. Govindjee, A. Mielke and G. J. Hall, Free-energy of mixing for $n$-variant martensitic phase transformations using quasi-convex analysis,, J. Mech. Physics Solids, 50 (2002), 1897.
doi: 10.1016/S0022-5096(02)00009-1. |
[20] |
K. Hackl and D. M. Kochmann, Relaxed potentials and evolution equations for inelastic microstructures,, in, (2008), 27.
doi: 10.1007/978-1-4020-9090-5_3. |
[21] |
B. Halphen and Q. S. Nguyen, Sur les matériaux standards généralisés., J. Mécanique, 14 (1975), 39.
|
[22] |
E. Helly, Über lineare funktionaloperationen,, Sitzungsberichte der Math.-Natur. Klasse der Kaiserlichen Akademie der Wissenschaften, 121 (1912), 265. Google Scholar |
[23] |
M. Kružík, A. Mielke and T. Roubíček, Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi,, Meccanica, 40 (2005), 389.
doi: 10.1007/s11012-005-2106-1. |
[24] |
M. Kružík and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism,, SIAM Rev., 48 (2006), 439.
doi: 10.1137/S0036144504446187. |
[25] |
M. Kružík and J. Zimmer, Evolutionary problems in non-reflexive spaces,, (preprint no. 5/07, (2007).
doi: 10.1051/cocv:2008060. |
[26] |
M. Kružík and J. Zimmer, A model of shape-memory alloys accounting for plasiticity., (preprint no. 20/08, (2008). Google Scholar |
[27] |
M. Kružík and J. Zimmer, A note on time-dependent DiPerna-Majda measures,, (preprint no. 19/08, (2008). Google Scholar |
[28] |
S. Lefschetz, On compact spaces,, Math. Anal., 32 (1931), 521.
doi: 10.2307/1968249. |
[29] |
A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems,, Calc. Var. PDEs, 22 (2005), 73.
|
[30] |
A. Mielke, Deriving new evolution equations for microstructures via relaxation of variational incremental problems,, Comput. Methods Appl. Mech. Engrg., 193 (2004), 5095.
doi: 10.1016/j.cma.2004.07.003. |
[31] |
A. Mielke, Evolution of rate-independent systems,, in, (2005), 461.
|
[32] |
A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case,, in, 28 (2006), 351.
|
[33] |
A. Mielke and T. Roubíček, A rate-independent model for inelastic behavior of shape-memory alloys,, Multiscale Model. Simul., 1 (2003), 571.
doi: 10.1137/S1540345903422860. |
[34] |
A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151.
|
[35] |
A. Mielke, F. Theil and V. I. Levitas, A variational formulation of rate-independent phase transformations using an extremum principle,, Archive Rat. Mech. Anal., 162 (2002), 137.
doi: 10.1007/s002050200194. |
[36] |
T. Roubíček, Convex compactifications and special extensions of optimization problems,, Nonlinear Anal., 16 (1991), 1117.
doi: 10.1016/0362-546X(91)90199-B. |
[37] |
T. Roubíček, "Relaxation in Optimization Theory and Variational Calculus,", W. de Gruyter, (1997).
|
[38] |
T. Roubíček, Convex locally compact extensions of Lebesgue spaces and their applications,, in:, (1999), 237.
|
[39] |
T. Roubíček and K.-H. Hoffmann, About the concept of measure-valued solutions to distributed parameter systems,, Math. Methods Appl. Sci., 18 (1995), 671.
doi: 10.1002/mma.1670180902. |
[40] |
T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics,, Z. angew. Math. Physik, 55 (2004), 159.
|
[41] |
A. Tychonoff, Über die topologische Erweiterung von Räumen,, Math. Annalen, 102 (1930), 544.
doi: 10.1007/BF01782364. |
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