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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

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), 295-348. doi: 10.1007/s00205-011-0427-x.  Google Scholar

[2]

L. Ambrosio, Metric space valued functions of bounded variations, Ann. Scuola Normale Sup. Pisa Cl. Sci. (4), 17 (1990), 439-478.  Google Scholar

[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.  Google Scholar

[4]

S. Baldo and G. Belletini, $\Gamma$-convergence and numerical analysis: An application to the minimal partition problem, Ricerche Mat., 40 (1991), 33-64.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

W. F. Brown, Virtues and weaknesses of the domain concept, Revs. Mod. Physics, 17 (1945), 15-19. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Cooper, "Working Analysis," Elsevier Academic Press, 2005. doi: 10.1249/00005768-199205001-00495.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. DeSimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles, Meccanica, 30 (1995), 591-603. doi: 10.1007/BF01557087.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H. Garcke, "On Mathematical Models for Phase Separation in Elastically Stressed Solids," Habilitation Thesis, University of Bonn, Bonn, 2000. Google Scholar

[23]

P. Germain, Q. Nguyen and P. Suquet, Continuum thermodynamics, J. Applied Mechanics, 50 (1983), 1010-1020. doi: 10.1115/1.3167184.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. D. James, Hysteresis in phase transformations, in "ICIAM 95" (Hamburg, 1995), Math. Res., 87, Akademie Verlag, Berlin, (1996), 133-154.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

J. M. T. Thomson and G. W. Hunt, "Elastic Instability Phenomena," J. Wiley and Sons, Chichester, 1984. Google Scholar

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.  Google Scholar

[2]

L. Ambrosio, Metric space valued functions of bounded variations, Ann. Scuola Normale Sup. Pisa Cl. Sci. (4), 17 (1990), 439-478.  Google Scholar

[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.  Google Scholar

[4]

S. Baldo and G. Belletini, $\Gamma$-convergence and numerical analysis: An application to the minimal partition problem, Ricerche Mat., 40 (1991), 33-64.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

W. F. Brown, Virtues and weaknesses of the domain concept, Revs. Mod. Physics, 17 (1945), 15-19. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Cooper, "Working Analysis," Elsevier Academic Press, 2005. doi: 10.1249/00005768-199205001-00495.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. DeSimone, Hysteresis and imperfection sensitivity in small ferromagnetic particles, Meccanica, 30 (1995), 591-603. doi: 10.1007/BF01557087.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

H. Garcke, "On Mathematical Models for Phase Separation in Elastically Stressed Solids," Habilitation Thesis, University of Bonn, Bonn, 2000. Google Scholar

[23]

P. Germain, Q. Nguyen and P. Suquet, Continuum thermodynamics, J. Applied Mechanics, 50 (1983), 1010-1020. doi: 10.1115/1.3167184.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

R. D. James, Hysteresis in phase transformations, in "ICIAM 95" (Hamburg, 1995), Math. Res., 87, Akademie Verlag, Berlin, (1996), 133-154.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

J. M. T. Thomson and G. W. Hunt, "Elastic Instability Phenomena," J. Wiley and Sons, Chichester, 1984. Google Scholar

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