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

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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
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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 & 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. 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.
[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.
[4]	S. Baldo and G. Belletini, $\Gamma$-convergence and numerical analysis: An application to the minimal partition problem,, Ricerche Mat., 40 (1991), 33.
[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. doi: 10.1007/s003320010007.
[6]	B. Benešová, Global optimization numerical strategies for rate-independent processes,, J. Global Optim., 50 (2011), 197. doi: 10.1007/s10898-010-9560-6.
[7]	W. F. Brown, Virtues and weaknesses of the domain concept,, Revs. Mod. Physics, 17 (1945), 15.
[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. 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. 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. 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 (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. 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. 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. 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. 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, (2009). doi: 10.1051/esomat/200902005.
[18]	A. DeSimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles,, Meccanica, 30 (1995), 591. 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. 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. 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, (1992).
[22]	H. Garcke, "On Mathematical Models for Phase Separation in Elastically Stressed Solids,", Habilitation Thesis, (2000).
[23]	P. Germain, Q. Nguyen and P. Suquet, Continuum thermodynamics,, J. Applied Mechanics, 50 (1983), 1010. 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. 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. doi: 10.1515/CRELLE.2006.044.
[26]	R. D. James, Hysteresis in phase transformations,, in, 87 (1996), 133.
[27]	L. Juhász, H. Andrä and O. Hesebeck, A simple model for shape memory alloys under multi-axial non-proportional loading,, in, (2000), 51.
[28]	M. Kružík and M. Luskin, The computation of martensitic microstructure with piecewise laminates,, Journal of Scientific Computing, 19 (2003), 293. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00526-007-0119-4.
[35]	A. Mielke and F. Theil, Mathematical model for rate-independent phase transformations,, in, (1999), 117.
[36]	A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151. 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. doi: 10.1007/s002050200194.
[38]	I. Müller, Modelling and simulation of phase transition in shape memory metals,, in, (2000), 97.
[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.
[40]	T. Roubíček, Evolution model for martensitic phase transformation in shape-memory alloys,, Interfaces and Free Boundaries, 4 (2002), 111. 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. 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. doi: 10.3934/dcdss.2013.6.235.
[43]	J. M. T. Thomson and G. W. Hunt, "Elastic Instability Phenomena,", J. Wiley and Sons, (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. 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.
[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.
[4]	S. Baldo and G. Belletini, $\Gamma$-convergence and numerical analysis: An application to the minimal partition problem,, Ricerche Mat., 40 (1991), 33.
[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. doi: 10.1007/s003320010007.
[6]	B. Benešová, Global optimization numerical strategies for rate-independent processes,, J. Global Optim., 50 (2011), 197. doi: 10.1007/s10898-010-9560-6.
[7]	W. F. Brown, Virtues and weaknesses of the domain concept,, Revs. Mod. Physics, 17 (1945), 15.
[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. 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. 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. 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 (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. 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. 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. 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. 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, (2009). doi: 10.1051/esomat/200902005.
[18]	A. DeSimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles,, Meccanica, 30 (1995), 591. 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. 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. 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, (1992).
[22]	H. Garcke, "On Mathematical Models for Phase Separation in Elastically Stressed Solids,", Habilitation Thesis, (2000).
[23]	P. Germain, Q. Nguyen and P. Suquet, Continuum thermodynamics,, J. Applied Mechanics, 50 (1983), 1010. 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. 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. doi: 10.1515/CRELLE.2006.044.
[26]	R. D. James, Hysteresis in phase transformations,, in, 87 (1996), 133.
[27]	L. Juhász, H. Andrä and O. Hesebeck, A simple model for shape memory alloys under multi-axial non-proportional loading,, in, (2000), 51.
[28]	M. Kružík and M. Luskin, The computation of martensitic microstructure with piecewise laminates,, Journal of Scientific Computing, 19 (2003), 293. 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. 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. 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. 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. 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. 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. doi: 10.1007/s00526-007-0119-4.
[35]	A. Mielke and F. Theil, Mathematical model for rate-independent phase transformations,, in, (1999), 117.
[36]	A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151. 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. doi: 10.1007/s002050200194.
[38]	I. Müller, Modelling and simulation of phase transition in shape memory metals,, in, (2000), 97.
[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.
[40]	T. Roubíček, Evolution model for martensitic phase transformation in shape-memory alloys,, Interfaces and Free Boundaries, 4 (2002), 111. 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. 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. doi: 10.3934/dcdss.2013.6.235.
[43]	J. M. T. Thomson and G. W. Hunt, "Elastic Instability Phenomena,", J. Wiley and Sons, (1984).

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