Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation

Previous Article
Gamma-expansion for a 1D confined Lennard-Jones model with point defect
NHM Home
This Issue
Next Article
Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation

June 2013, 8(2): 481-499. doi: 10.3934/nhm.2013.8.481

Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation

Antonio DeSimone ^1, and Martin Kružík ^2,

1.	SISSA, International School of Advanced Studies, Via Bonomea 265, 34136 Trieste
2.	Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, 182 08 Prague

Received October 2012 Revised March 2013 Published May 2013

Abstract
Related Papers

We propose a sharp-interface model which describes rate-independent hysteresis in phase-transforming solids (such as shape memory alloys) by resolving explicitly domain patterns and their dissipative evolution. We show that the governing Gibbs' energy functional is the $\Gamma$-limit of a family of regularized Gibbs' energies obtained through a phase-field approximation. This leads to the convergence of the solution of the quasistatic evolution problem associated with the regularized energy to the one corresponding to the sharp interface model. Based on this convergence result, we propose a numerical scheme which allows us to simulate mechanical experiments for both spatially homogeneous and heterogeneous samples. We use the latter to assess the role that impurities and defects may have in determining the response exhibited by real samples. In particular, our numerical results indicate that small heterogeneities are essential in order to obtain spatially localized nucleation of a new martensitic variant from a pre-existing one in stress-controlled experiments.

Keywords: shape memory alloys, multi-well energies, phase-field models, rate-independent processes, Hysteresis, sharp-interface limits..

Mathematics Subject Classification: Primary: 74N30, 49J45, 74D10; Secondary: 74S0.

Citation: Antonio DeSimone, Martin Kružík. Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation. Networks and Heterogeneous Media, 2013, 8 (2) : 481-499. doi: 10.3934/nhm.2013.8.481

References:

[1]	G. Alberti and A. DeSimone, Quasistatic evolution of sessile drops and contact angle hysteresis, Arch. Rat. Mech. Anal., 202 (2011), 295-348. doi: 10.1007/s00205-011-0427-x.
[2]	L. Ambrosio, Metric space valued functions of bounded variations, Ann. Scuola Normale Sup. Pisa Cl. Sci. (4), 17 (1990), 439-478.
[3]	S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 67-90.
[4]	S. Baldo and G. Belletini, $\Gamma$-convergence and numerical analysis: An application to the minimal partition problem, Ricerche Mat., 40 (1991), 33-64.
[5]	H. Ben Belgacem, S. Conti, A. DeSimone and S. Müller, Rigorous bounds for the Föppl-von Kármán theory of isotropically compressed plates, Journal of Nonlinear Science, 10 (2000), 661-683. doi: 10.1007/s003320010007.
[6]	B. Benešová, Global optimization numerical strategies for rate-independent processes, J. Global Optim., 50 (2011), 197-220. doi: 10.1007/s10898-010-9560-6.
[7]	W. F. Brown, Virtues and weaknesses of the domain concept, Revs. Mod. Physics, 17 (1945), 15-19.
[8]	R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Scientific Computing, 16 (1995), 1190-1208. doi: 10.1137/0916069.
[9]	C. Collins, D. Kinderlehrer and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potential, SIAM J. Num. Anal., 28 (1991), 321-332. doi: 10.1137/0728018.
[10]	R. Conti, C. Tamagnini and A. DeSimone, Critical softening in Cam-Clay plasticity: Adaptive viscous regularization, dilated time and numerical integration across stress-strain jump discontinuities, Comput. Methods Appl. Mech. Engrg., 258 (2013), 118-133. doi: 10.1016/j.cma.2013.02.002.
[11]	J. Cooper, "Working Analysis," Elsevier Academic Press, 2005. doi: 10.1249/00005768-199205001-00495.
[12]	G. Dal Maso, "An Introduction to $\Gamma$-Convegence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.
[13]	G. Dal Maso and A. DeSimone, Quasistatic evolution for Cam-Clay plasticity: Examples of spatially homogeneous solutions, Math. Model. Meth. Appl. Sci., 19 (2009), 1643-1711. doi: 10.1142/S0218202509003942.
[14]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Rat. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5.
[15]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: A weak formulation via viscoplastic regularization and time rescaling, Calc. Var. PDE, 40 (2011), 125-181. doi: 10.1007/s00526-010-0336-0.
[16]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solutions, Calc. Var. PDE, 44 (2012), 495-541. doi: 10.1007/s00526-011-0443-6.
[17]	R. Delville, R. D. James, U. Salman, A. Finel and D. Schryvers, Transmission electron microscopy study of low-hysteresis shape memory alloys, in "Proceedings of ESOMAT 2009," 2009. doi: 10.1051/esomat/200902005.
[18]	A. DeSimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles, Meccanica, 30 (1995), 591-603. doi: 10.1007/BF01557087.
[19]	A. DeSimone, N. Grunewald and F. Otto, A new model for contact angle hysteresis, Netw. Heterog. Media, 2 (2007), 211-225. doi: 10.3934/nhm.2007.2.211.
[20]	A. DeSimone and L. Teresi, Elastic energies for nematic elastomers, Europ. Phys. J. E, 29 (2009), 191-204. doi: 10.1140/epje/i2009-10467-9.
[21]	L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[22]	H. Garcke, "On Mathematical Models for Phase Separation in Elastically Stressed Solids," Habilitation Thesis, University of Bonn, Bonn, 2000.
[23]	P. Germain, Q. Nguyen and P. Suquet, Continuum thermodynamics, J. Applied Mechanics, 50 (1983), 1010-1020. doi: 10.1115/1.3167184.
[24]	L. Fedeli, A. Turco and A. DeSimone, Metastable equilibria of capillary drops on solid surfaces: A phase field approach, Cont. Mech. Thermodyn., 23 (2011), 453-471. doi: 10.1007/s00161-011-0189-6.
[25]	G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91. doi: 10.1515/CRELLE.2006.044.
[26]	R. D. James, Hysteresis in phase transformations, in "ICIAM 95" (Hamburg, 1995), Math. Res., 87, Akademie Verlag, Berlin, (1996), 133-154.
[27]	L. Juhász, H. Andrä and O. Hesebeck, A simple model for shape memory alloys under multi-axial non-proportional loading, in "Smart Materials" (ed. K.-H. Hoffmann), Proceedings of the 1st Caesarium, Springer, Berlin, (2000), 51-66.
[28]	M. Kružík and M. Luskin, The computation of martensitic microstructure with piecewise laminates, Journal of Scientific Computing, 19 (2003), 293-308. doi: 10.1023/A:1025364227563.
[29]	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), 389-418. doi: 10.1007/s11012-005-2106-1.
[30]	M. Kružík and F. Otto, A phenomenological model for hysteresis in polycrystalline shape memory alloys, ZAMM Z. Angew. Math. Mech., 84 (2004), 835-842. doi: 10.1002/zamm.200310139.
[31]	S. Leclerq, G. Bourbon and C. Lexcellent, Plasticity like model of martensite phase transition in shape memory alloys, J. Physique IV France, 5 (1995), 513-518. doi: 10.1051/jp4:1995279.
[32]	S. Leclerq and C. Lexcellent, A general macroscopic description of thermomechanical behavior of shape memory alloys, J. Mech. Phys. Solids, 44 (1996), 953-980. doi: 10.1016/0022-5096(96)00013-0.
[33]	C. Lexcellent, S. Moyne, A. Ishida and S. Miyazaki, Deformation behavior associated with stress-induced martensitic transformation in Ti-Ni thin films and their thermodynamical modelling, Thin Solid Films, 324 (1998), 184-189. doi: 10.1016/S0040-6090(98)00352-6.
[34]	A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var., 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.
[35]	A. Mielke and F. Theil, Mathematical model for rate-independent phase transformations, in "Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering" (eds. H.-D. Alber, R. Balean and R. Farwig), Shaker-Verlag, Aachen, (1999), 117-129.
[36]	A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.
[37]	A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using extremum principle, Arch. Rat. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.
[38]	I. Müller, Modelling and simulation of phase transition in shape memory metals, in "Smart Materials" (ed. K.-H. Hoffmann), Proceedings of the 1st Caesarium, Springer, Berlin, (2000), 97-114.
[39]	F. Nishimura, T. Hayashi, C. Lexcellent and K. Tanaka, Phenomenological analysis of subloops and cyclic behavior in shape memory alloys under mechanical and/or thermal loads, Mech. of Mat., 19 (1995), 281-292.
[40]	T. Roubíček, Evolution model for martensitic phase transformation in shape-memory alloys, Interfaces and Free Boundaries, 4 (2002), 111-136. doi: 10.4171/IFB/55.
[41]	Y. C. Shu and J. H. Yen, Multivariant model of martensitic microstructure in thin films, Acta Materialia, 56 (2008), 3969-3981. doi: 10.1016/j.actamat.2008.04.018.
[42]	M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discr. Cont. Dyn. Syst. Ser. S, 6 (2013), 235-255. doi: 10.3934/dcdss.2013.6.235.
[43]	J. M. T. Thomson and G. W. Hunt, "Elastic Instability Phenomena," J. Wiley and Sons, Chichester, 1984.

show all references

References:

[1]	G. Alberti and A. DeSimone, Quasistatic evolution of sessile drops and contact angle hysteresis, Arch. Rat. Mech. Anal., 202 (2011), 295-348. doi: 10.1007/s00205-011-0427-x.
[2]	L. Ambrosio, Metric space valued functions of bounded variations, Ann. Scuola Normale Sup. Pisa Cl. Sci. (4), 17 (1990), 439-478.
[3]	S. Baldo, Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 67-90.
[4]	S. Baldo and G. Belletini, $\Gamma$-convergence and numerical analysis: An application to the minimal partition problem, Ricerche Mat., 40 (1991), 33-64.
[5]	H. Ben Belgacem, S. Conti, A. DeSimone and S. Müller, Rigorous bounds for the Föppl-von Kármán theory of isotropically compressed plates, Journal of Nonlinear Science, 10 (2000), 661-683. doi: 10.1007/s003320010007.
[6]	B. Benešová, Global optimization numerical strategies for rate-independent processes, J. Global Optim., 50 (2011), 197-220. doi: 10.1007/s10898-010-9560-6.
[7]	W. F. Brown, Virtues and weaknesses of the domain concept, Revs. Mod. Physics, 17 (1945), 15-19.
[8]	R. H. Byrd, P. Lu, J. Nocedal and C. Zhu, A limited memory algorithm for bound constrained optimization, SIAM J. Scientific Computing, 16 (1995), 1190-1208. doi: 10.1137/0916069.
[9]	C. Collins, D. Kinderlehrer and M. Luskin, Numerical approximation of the solution of a variational problem with a double well potential, SIAM J. Num. Anal., 28 (1991), 321-332. doi: 10.1137/0728018.
[10]	R. Conti, C. Tamagnini and A. DeSimone, Critical softening in Cam-Clay plasticity: Adaptive viscous regularization, dilated time and numerical integration across stress-strain jump discontinuities, Comput. Methods Appl. Mech. Engrg., 258 (2013), 118-133. doi: 10.1016/j.cma.2013.02.002.
[11]	J. Cooper, "Working Analysis," Elsevier Academic Press, 2005. doi: 10.1249/00005768-199205001-00495.
[12]	G. Dal Maso, "An Introduction to $\Gamma$-Convegence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.
[13]	G. Dal Maso and A. DeSimone, Quasistatic evolution for Cam-Clay plasticity: Examples of spatially homogeneous solutions, Math. Model. Meth. Appl. Sci., 19 (2009), 1643-1711. doi: 10.1142/S0218202509003942.
[14]	G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini, A vanishing viscosity approach to quasistatic evolution in plasticity with softening, Arch. Rat. Mech. Anal., 189 (2008), 469-544. doi: 10.1007/s00205-008-0117-5.
[15]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: A weak formulation via viscoplastic regularization and time rescaling, Calc. Var. PDE, 40 (2011), 125-181. doi: 10.1007/s00526-010-0336-0.
[16]	G. Dal Maso, A. DeSimone and F. Solombrino, Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solutions, Calc. Var. PDE, 44 (2012), 495-541. doi: 10.1007/s00526-011-0443-6.
[17]	R. Delville, R. D. James, U. Salman, A. Finel and D. Schryvers, Transmission electron microscopy study of low-hysteresis shape memory alloys, in "Proceedings of ESOMAT 2009," 2009. doi: 10.1051/esomat/200902005.
[18]	A. DeSimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles, Meccanica, 30 (1995), 591-603. doi: 10.1007/BF01557087.
[19]	A. DeSimone, N. Grunewald and F. Otto, A new model for contact angle hysteresis, Netw. Heterog. Media, 2 (2007), 211-225. doi: 10.3934/nhm.2007.2.211.
[20]	A. DeSimone and L. Teresi, Elastic energies for nematic elastomers, Europ. Phys. J. E, 29 (2009), 191-204. doi: 10.1140/epje/i2009-10467-9.
[21]	L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.
[22]	H. Garcke, "On Mathematical Models for Phase Separation in Elastically Stressed Solids," Habilitation Thesis, University of Bonn, Bonn, 2000.
[23]	P. Germain, Q. Nguyen and P. Suquet, Continuum thermodynamics, J. Applied Mechanics, 50 (1983), 1010-1020. doi: 10.1115/1.3167184.
[24]	L. Fedeli, A. Turco and A. DeSimone, Metastable equilibria of capillary drops on solid surfaces: A phase field approach, Cont. Mech. Thermodyn., 23 (2011), 453-471. doi: 10.1007/s00161-011-0189-6.
[25]	G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math., 595 (2006), 55-91. doi: 10.1515/CRELLE.2006.044.
[26]	R. D. James, Hysteresis in phase transformations, in "ICIAM 95" (Hamburg, 1995), Math. Res., 87, Akademie Verlag, Berlin, (1996), 133-154.
[27]	L. Juhász, H. Andrä and O. Hesebeck, A simple model for shape memory alloys under multi-axial non-proportional loading, in "Smart Materials" (ed. K.-H. Hoffmann), Proceedings of the 1st Caesarium, Springer, Berlin, (2000), 51-66.
[28]	M. Kružík and M. Luskin, The computation of martensitic microstructure with piecewise laminates, Journal of Scientific Computing, 19 (2003), 293-308. doi: 10.1023/A:1025364227563.
[29]	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), 389-418. doi: 10.1007/s11012-005-2106-1.
[30]	M. Kružík and F. Otto, A phenomenological model for hysteresis in polycrystalline shape memory alloys, ZAMM Z. Angew. Math. Mech., 84 (2004), 835-842. doi: 10.1002/zamm.200310139.
[31]	S. Leclerq, G. Bourbon and C. Lexcellent, Plasticity like model of martensite phase transition in shape memory alloys, J. Physique IV France, 5 (1995), 513-518. doi: 10.1051/jp4:1995279.
[32]	S. Leclerq and C. Lexcellent, A general macroscopic description of thermomechanical behavior of shape memory alloys, J. Mech. Phys. Solids, 44 (1996), 953-980. doi: 10.1016/0022-5096(96)00013-0.
[33]	C. Lexcellent, S. Moyne, A. Ishida and S. Miyazaki, Deformation behavior associated with stress-induced martensitic transformation in Ti-Ni thin films and their thermodynamical modelling, Thin Solid Films, 324 (1998), 184-189. doi: 10.1016/S0040-6090(98)00352-6.
[34]	A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-limits and relaxations for rate-independent evolutionary problems, Calc. Var., 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.
[35]	A. Mielke and F. Theil, Mathematical model for rate-independent phase transformations, in "Proceedings of the Workshop on Models of Continuum Mechanics in Analysis and Engineering" (eds. H.-D. Alber, R. Balean and R. Farwig), Shaker-Verlag, Aachen, (1999), 117-129.
[36]	A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlin. Diff. Eq. Appl., 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.
[37]	A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using extremum principle, Arch. Rat. Mech. Anal., 162 (2002), 137-177. doi: 10.1007/s002050200194.
[38]	I. Müller, Modelling and simulation of phase transition in shape memory metals, in "Smart Materials" (ed. K.-H. Hoffmann), Proceedings of the 1st Caesarium, Springer, Berlin, (2000), 97-114.
[39]	F. Nishimura, T. Hayashi, C. Lexcellent and K. Tanaka, Phenomenological analysis of subloops and cyclic behavior in shape memory alloys under mechanical and/or thermal loads, Mech. of Mat., 19 (1995), 281-292.
[40]	T. Roubíček, Evolution model for martensitic phase transformation in shape-memory alloys, Interfaces and Free Boundaries, 4 (2002), 111-136. doi: 10.4171/IFB/55.
[41]	Y. C. Shu and J. H. Yen, Multivariant model of martensitic microstructure in thin films, Acta Materialia, 56 (2008), 3969-3981. doi: 10.1016/j.actamat.2008.04.018.
[42]	M. Thomas, Quasistatic damage evolution with spatial BV-regularization, Discr. Cont. Dyn. Syst. Ser. S, 6 (2013), 235-255. doi: 10.3934/dcdss.2013.6.235.
[43]	J. M. T. Thomson and G. W. Hunt, "Elastic Instability Phenomena," J. Wiley and Sons, Chichester, 1984.

[1]	Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks and Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145
[2]	Martin Kružík, Johannes Zimmer. Rate-independent processes with linear growth energies and time-dependent boundary conditions. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 591-604. doi: 10.3934/dcdss.2012.5.591
[3]	Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations and Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411
[4]	Claudio Giorgi. Phase-field models for transition phenomena in materials with hysteresis. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 693-722. doi: 10.3934/dcdss.2015.8.693
[5]	T. J. Sullivan, M. Koslowski, F. Theil, Michael Ortiz. Thermalization of rate-independent processes by entropic regularization. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 215-233. doi: 10.3934/dcdss.2013.6.215
[6]	Stefano Bosia, Michela Eleuteri, Elisabetta Rocca, Enrico Valdinoci. Preface: Special issue on rate-independent evolutions and hysteresis modelling. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : i-i. doi: 10.3934/dcdss.2015.8.4i
[7]	Alessia Berti, Claudio Giorgi, Elena Vuk. Free energies and pseudo-elastic transitions for shape memory alloys. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 293-316. doi: 10.3934/dcdss.2013.6.293
[8]	Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070
[9]	Daniele Davino, Ciro Visone. Rate-independent memory in magneto-elastic materials. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 649-691. doi: 10.3934/dcdss.2015.8.649
[10]	Federico Mario Vegni. Dissipativity of a conserved phase-field system with memory. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 949-968. doi: 10.3934/dcds.2003.9.949
[11]	Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[12]	Gianni Dal Maso, Alexander Mielke, Ulisse Stefanelli. Preface: Rate-independent evolutions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : i-ii. doi: 10.3934/dcdss.2013.6.1i
[13]	Alice Fiaschi. Rate-independent phase transitions in elastic materials: A Young-measure approach. Networks and Heterogeneous Media, 2010, 5 (2) : 257-298. doi: 10.3934/nhm.2010.5.257
[14]	Leonid Berlyand, Mykhailo Potomkin, Volodymyr Rybalko. Sharp interface limit in a phase field model of cell motility. Networks and Heterogeneous Media, 2017, 12 (4) : 551-590. doi: 10.3934/nhm.2017023
[15]	Sergiu Aizicovici, Hana Petzeltová. Convergence to equilibria of solutions to a conserved Phase-Field system with memory. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 1-16. doi: 10.3934/dcdss.2009.2.1
[16]	S. Gatti, M. Grasselli, V. Pata, M. Squassina. Robust exponential attractors for a family of nonconserved phase-field systems with memory. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 1019-1029. doi: 10.3934/dcds.2005.12.1019
[17]	Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete and Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257
[18]	Takashi Suzuki, Shuji Yoshikawa. Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 209-217. doi: 10.3934/dcdss.2012.5.209
[19]	Nobuyuki Kenmochi, Jürgen Sprekels. Phase-field systems with vectorial order parameters including diffusional hysteresis effects. Communications on Pure and Applied Analysis, 2002, 1 (4) : 495-511. doi: 10.3934/cpaa.2002.1.495
[20]	Michel Frémond, Elisabetta Rocca. A model for shape memory alloys with the possibility of voids. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1633-1659. doi: 10.3934/dcds.2010.27.1633

2020 Impact Factor: 1.213

Tools

Metrics

Other articles
by authors

on AIMS
Antonio DeSimone Martin Kružík
on Google Scholar
Antonio DeSimone Martin Kružík

[Back to Top]