Quasistatic evolution of magnetoelastic plates via dimension reduction

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December 2015, 35(12): 5999-6013. doi: 10.3934/dcds.2015.35.5999

Quasistatic evolution of magnetoelastic plates via dimension reduction

Martin Kružík ^1, , Ulisse Stefanelli ^2, and Chiara Zanini ^3,

1.	Institute of Information Theory and Automation of the ASCR, Pod vodárenskou věží 4, 182 08 Prague
2.	Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna
3.	Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Received May 2014 Published May 2015

Abstract
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A rate-independent model for the quasistatic evolution of a magnetoelastic plate is advanced and analyzed. Starting from the three-dimensional setting, we present an evolutionary $\Gamma$-convergence argument in order to pass to the limit in one of the material dimensions. By taking into account both conservative and dissipative actions, a nonlinear evolution system of rate-independent type is obtained. The existence of so-called energetic solutions to such system is proved via approximation.

Keywords: Magnetoelasticity, energetic solution, $\Gamma$-convergence for rate-independent processes., existence, dimension reduction.

Mathematics Subject Classification: Primary: 74F15, 74N30, 35K5.

Citation: Martin Kružík, Ulisse Stefanelli, Chiara Zanini. Quasistatic evolution of magnetoelastic plates via dimension reduction. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5999-6013. doi: 10.3934/dcds.2015.35.5999

References:

[1]	N. Anuniwat, M. Ding, S. J. Poon, S. A. Wolf and J. Lu, Strain-induced enhancement of coercivity in amorphous TbFeCo films, J. Appl. Phys., 113 (2013), 043905. doi: 10.1063/1.4788807.
[2]	J.-F. Babadjian, Quasistatic evolution of a brittle thin film, Calc. Var. Partial Differential Equations, 26 (2006), 69-118. doi: 10.1007/s00526-005-0369-y.
[3]	B. Benešová, M Kružík and G. Pathó, A mesoscopic thermomechanically coupled model for thin-film shape-memory alloys by dimension reduction and scale transition, Contin. Mech. Thermodyn., 26 (2014), 683-713. doi: 10.1007/s00161-013-0323-8.
[4]	A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys, Z. Angew. Math. Phys., 64 (2013), 343-359. doi: 10.1007/s00033-012-0223-y.
[5]	A.-L. Bessoud and U. Stefanelli, Magnetic shape memory alloys: Three-dimensional modeling and analysis, Math. Models Meth. Appl. Sci., 21 (2011), 1043-1069. doi: 10.1142/S0218202511005246.
[6]	A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.
[7]	W. F. Brown, Micromagnetics, {Wiley, New-York, 1963.}
[8]	C. Chappert and P. Bruno, Magnetic anisotropy in metallic ultrathin films and related experiments on cobalt films, J. Appl. Phys., 64 (1988), p5736. doi: 10.1063/1.342243.
[9]	G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.
[10]	D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mat. Struct., 22 (2013), 095009.
[11]	E. Davoli, Linearized plastic plate models as $\Gamma$-limits of 3D finite elastoplasticity, ESAIM Control Optim. Calc. Var., 20 (2014), 725-747. doi: 10.1051/cocv/2013081.
[12]	E. Davoli, Quasistatic evolution models for thin plates arising as low energy $\Gamma$-limits of finite plasticity, Math. Models Methods Appl. Sci., 24 (2014), 2085-2153. doi: 10.1142/S021820251450016X.
[13]	E. Davoli and M. G. Mora, A quasistatic evolution model for perfectly plastic plates derived by Gamma-convergence, Ann. Inst. H. Poincaré Anal. Nonlin., 30 (2013), 615-660. doi: 10.1016/j.anihpc.2012.11.001.
[14]	A. DeSimone and G. Dolzmann, Existence of minimizers for a variational problem in two-dimensional nonlinear magnetoelasticity, Arch. Rational Mech. Anal., 144 (1998), 107-120. doi: 10.1007/s002050050114.
[15]	A. DeSimone and R. D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids, 50 (2002), 283-320. doi: 10.1016/S0022-5096(01)00050-3.
[16]	A. Dorfmann and R. W. Ogden, Some problems in nonlinear magnetoelasticity, Z. Angew. Math. Phys., 56 (2005), 718-745. doi: 10.1007/s00033-004-4066-z.
[17]	L. Freddi, R. Paroni and C. Zanini, Dimension reduction of a crack evolution problem in a linearly elastic plate, Asymptotic Anal., 70 (2010), 101-123.
[18]	L. Freddi, T. Roubíček, R. Paroni and C. Zanini, Quasistatic delamination models for Kirchhoff-Love plates, Z. Angew. Math. Mech., 91 (2011), 845-865. doi: 10.1002/zamm.201000171.
[19]	L. Freddi, T. Roubíček and C. Zanini, Quasistatic delamination of sandwich-like Kirchhoff-Love plates, J. Elasticity, 113 (2013), 219-250. doi: 10.1007/s10659-012-9419-9.
[20]	V. Gehanno, A. Marty, B. Gilles and Y. Samson, Magnetic domains in epitaxial ordered FePd(001) thin films with perpendicular magnetic anisotropy, Phys. Rev. B, 55 (1997), 12552-12555. doi: 10.1103/PhysRevB.55.12552.
[21]	G. Gioia and R. D. James, Micromagnetics of very thin films, Proc. Roy. Soc. Lond. A, 453 (1997), 213-223. doi: 10.1098/rspa.1997.0013.
[22]	M. L. Hodgdon, Applications of a theory of ferromagnetic hysteresis, IEEE Trans. Mag., 24 (1988), 218-221. doi: 10.1109/20.43893.
[23]	A. Hubert and R. Schäfer, Magnetic Domains, Springer, New York, 1998.
[24]	R. D. James, Configurationsl forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever, Contin. Mech. Thermodyn., 14 (2002), 55-86. doi: 10.1007/s001610100072.
[25]	R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Continuum Mech. Thermodyn., 2 (1990), 215-239. doi: 10.1007/BF01129598.
[26]	M. Kaltenbacher, M. Meiler and M. Ertl, Physical modeling and numerical computation of magnetostriction, COMPEL, 28 (2009), 819-832. doi: 10.1108/03321640910958946.
[27]	M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism, SIAM Review, 48 (2006), 439-483. doi: 10.1137/S0036144504446187.
[28]	M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via $\Gamma$-convergence, Math. Models Meth. Appl. Sci., 21 (2011), 1961-1986. doi: 10.1142/S0218202511005611.
[29]	M. Liero and T. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $\Gamma$-convergence, NoDEA Nonlinear Differential Eq. Applications, 19 (2012), 437-457. doi: 10.1007/s00030-011-0137-y.
[30]	A. Mielke, Evolution in rate-independent systems (ch. 6), in Handbook of Differential Equations, Evolutionary Equations (eds. C. Dafermos and E. Feireisl), Vol. 2, Elsevier B. V., 2005, 461-559.
[31]	A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case, in Multifield Problems in Solid and Fluid Mechanics (eds. R. Helmig, A. Mielke and B. Wohlmuth), Series Lecture Notes in Applied and Computational Mechanics, 28, Springer, 2006, 399-428. doi: 10.1007/978-3-540-34961-7_12.
[32]	A. Mielke, Generalized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction, Phys. B, 407 (2012), 1330-1335. doi: 10.1016/j.physb.2011.10.013.
[33]	A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, M2AN Math. Model. Numer. Anal., 43 (2009), 399-428. doi: 10.1051/m2an/2009009.
[34]	A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-imits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.
[35]	A. Mielke and F. Theil, On rate-independent hysteresis model, NoDEA Nonlinear Diff. Equations Applications, 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.
[36]	L. Néel, L'approche a la saturation de la magnétostriction, J. Phys. Radium, 15 (1954), 376-378.
[37]	H. J. Richter, The transition from longitudinal to perpendicular recording, J. Phys. D: Appl. Phys., 40 (2007), R149-R177. doi: 10.1088/0022-3727/40/9/R01.
[38]	R. T. Rockafellar and R. J.-B Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02431-3.
[39]	T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics, Z. Angew. Math. Phys., 55 (2004), 159-182. doi: 10.1007/s00033-003-0110-7.
[40]	T. Roubíček and M. Kružík, Mesoscopic model for ferromagnets with isotropic hardening, Z. Angew. Math. Phys., 56 (2005), 107-135. doi: 10.1007/s00033-003-2108-6.
[41]	T. Roubíček and U. Stefanelli, Magnetic shape-memory alloys: Thermomechanical modeling and analysis, Contin. Mech. Thermodyn., 26 (2014), 783-810. doi: 10.1007/s00161-014-0339-8.
[42]	T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Arch. Ration. Mech. Anal., 210 (2013), 1-43. doi: 10.1007/s00205-013-0648-2.
[43]	P. Rybka and M. Luskin, Existence of energy minimizers for magnetostrictive materials, SIAM J. Math. Anal., 36 (2005), 2004-2019. doi: 10.1137/S0036141004442021.
[44]	B. Schulz and K. Baberschke, Crossover form in-plane to perpendicular magnetization in ultrathin Ni/Cu(001) films, Phys. Rev. B, 50 (1994), 13467.
[45]	A. D. C. Viegas, M. A. Correa, L. Santi, R. B. da Silva, F. Bohn, M. Carara and R. L. Somme, Thickness dependence of the high-frequency magnetic permeability in amorphous $Fe_{73.5}Cu_1Nb_3Si_{13.5}B_9$ thin films, J. Appl. Phys., 101 (2007), 033908.
[46]	A. Visintin, Modified Landau-Lifshitz equation for ferromagnetism, Phys. B, 233 (1997), 365-369. doi: 10.1016/S0921-4526(97)00322-0.
[47]	A. Visintin, Maxwell's equations with vector hysteresis, Arch. Ration. Mech. Anal., 175 (2005), 1-37. doi: 10.1007/s00205-004-0333-6.
[48]	J. Wang and P. Steinmann, A variational approach towards the modeling of magnetic field-induced strains in magnetic shape memory alloys, J. Mech. Phys. Solids, 60 (2012), 1179-1200. doi: 10.1016/j.jmps.2012.02.003.

show all references

References:

[1]	N. Anuniwat, M. Ding, S. J. Poon, S. A. Wolf and J. Lu, Strain-induced enhancement of coercivity in amorphous TbFeCo films, J. Appl. Phys., 113 (2013), 043905. doi: 10.1063/1.4788807.
[2]	J.-F. Babadjian, Quasistatic evolution of a brittle thin film, Calc. Var. Partial Differential Equations, 26 (2006), 69-118. doi: 10.1007/s00526-005-0369-y.
[3]	B. Benešová, M Kružík and G. Pathó, A mesoscopic thermomechanically coupled model for thin-film shape-memory alloys by dimension reduction and scale transition, Contin. Mech. Thermodyn., 26 (2014), 683-713. doi: 10.1007/s00161-013-0323-8.
[4]	A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape memory alloys, Z. Angew. Math. Phys., 64 (2013), 343-359. doi: 10.1007/s00033-012-0223-y.
[5]	A.-L. Bessoud and U. Stefanelli, Magnetic shape memory alloys: Three-dimensional modeling and analysis, Math. Models Meth. Appl. Sci., 21 (2011), 1043-1069. doi: 10.1142/S0218202511005246.
[6]	A. Braides, $\Gamma$-Convergence for Beginners, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.
[7]	W. F. Brown, Micromagnetics, {Wiley, New-York, 1963.}
[8]	C. Chappert and P. Bruno, Magnetic anisotropy in metallic ultrathin films and related experiments on cobalt films, J. Appl. Phys., 64 (1988), p5736. doi: 10.1063/1.342243.
[9]	G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.
[10]	D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mat. Struct., 22 (2013), 095009.
[11]	E. Davoli, Linearized plastic plate models as $\Gamma$-limits of 3D finite elastoplasticity, ESAIM Control Optim. Calc. Var., 20 (2014), 725-747. doi: 10.1051/cocv/2013081.
[12]	E. Davoli, Quasistatic evolution models for thin plates arising as low energy $\Gamma$-limits of finite plasticity, Math. Models Methods Appl. Sci., 24 (2014), 2085-2153. doi: 10.1142/S021820251450016X.
[13]	E. Davoli and M. G. Mora, A quasistatic evolution model for perfectly plastic plates derived by Gamma-convergence, Ann. Inst. H. Poincaré Anal. Nonlin., 30 (2013), 615-660. doi: 10.1016/j.anihpc.2012.11.001.
[14]	A. DeSimone and G. Dolzmann, Existence of minimizers for a variational problem in two-dimensional nonlinear magnetoelasticity, Arch. Rational Mech. Anal., 144 (1998), 107-120. doi: 10.1007/s002050050114.
[15]	A. DeSimone and R. D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids, 50 (2002), 283-320. doi: 10.1016/S0022-5096(01)00050-3.
[16]	A. Dorfmann and R. W. Ogden, Some problems in nonlinear magnetoelasticity, Z. Angew. Math. Phys., 56 (2005), 718-745. doi: 10.1007/s00033-004-4066-z.
[17]	L. Freddi, R. Paroni and C. Zanini, Dimension reduction of a crack evolution problem in a linearly elastic plate, Asymptotic Anal., 70 (2010), 101-123.
[18]	L. Freddi, T. Roubíček, R. Paroni and C. Zanini, Quasistatic delamination models for Kirchhoff-Love plates, Z. Angew. Math. Mech., 91 (2011), 845-865. doi: 10.1002/zamm.201000171.
[19]	L. Freddi, T. Roubíček and C. Zanini, Quasistatic delamination of sandwich-like Kirchhoff-Love plates, J. Elasticity, 113 (2013), 219-250. doi: 10.1007/s10659-012-9419-9.
[20]	V. Gehanno, A. Marty, B. Gilles and Y. Samson, Magnetic domains in epitaxial ordered FePd(001) thin films with perpendicular magnetic anisotropy, Phys. Rev. B, 55 (1997), 12552-12555. doi: 10.1103/PhysRevB.55.12552.
[21]	G. Gioia and R. D. James, Micromagnetics of very thin films, Proc. Roy. Soc. Lond. A, 453 (1997), 213-223. doi: 10.1098/rspa.1997.0013.
[22]	M. L. Hodgdon, Applications of a theory of ferromagnetic hysteresis, IEEE Trans. Mag., 24 (1988), 218-221. doi: 10.1109/20.43893.
[23]	A. Hubert and R. Schäfer, Magnetic Domains, Springer, New York, 1998.
[24]	R. D. James, Configurationsl forces in magnetism with application to the dynamics of a small-scale ferromagnetic shape memory cantilever, Contin. Mech. Thermodyn., 14 (2002), 55-86. doi: 10.1007/s001610100072.
[25]	R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Continuum Mech. Thermodyn., 2 (1990), 215-239. doi: 10.1007/BF01129598.
[26]	M. Kaltenbacher, M. Meiler and M. Ertl, Physical modeling and numerical computation of magnetostriction, COMPEL, 28 (2009), 819-832. doi: 10.1108/03321640910958946.
[27]	M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism, SIAM Review, 48 (2006), 439-483. doi: 10.1137/S0036144504446187.
[28]	M. Liero and A. Mielke, An evolutionary elastoplastic plate model derived via $\Gamma$-convergence, Math. Models Meth. Appl. Sci., 21 (2011), 1961-1986. doi: 10.1142/S0218202511005611.
[29]	M. Liero and T. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $\Gamma$-convergence, NoDEA Nonlinear Differential Eq. Applications, 19 (2012), 437-457. doi: 10.1007/s00030-011-0137-y.
[30]	A. Mielke, Evolution in rate-independent systems (ch. 6), in Handbook of Differential Equations, Evolutionary Equations (eds. C. Dafermos and E. Feireisl), Vol. 2, Elsevier B. V., 2005, 461-559.
[31]	A. Mielke, A mathematical framework for generalized standard materials in the rate-independent case, in Multifield Problems in Solid and Fluid Mechanics (eds. R. Helmig, A. Mielke and B. Wohlmuth), Series Lecture Notes in Applied and Computational Mechanics, 28, Springer, 2006, 399-428. doi: 10.1007/978-3-540-34961-7_12.
[32]	A. Mielke, Generalized Prandtl-Ishlinskii operators arising from homogenization and dimension reduction, Phys. B, 407 (2012), 1330-1335. doi: 10.1016/j.physb.2011.10.013.
[33]	A. Mielke and T. Roubíček, Numerical approaches to rate-independent processes and applications in inelasticity, M2AN Math. Model. Numer. Anal., 43 (2009), 399-428. doi: 10.1051/m2an/2009009.
[34]	A. Mielke, T. Roubíček and U. Stefanelli, $\Gamma$-imits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differential Equations, 31 (2008), 387-416. doi: 10.1007/s00526-007-0119-4.
[35]	A. Mielke and F. Theil, On rate-independent hysteresis model, NoDEA Nonlinear Diff. Equations Applications, 11 (2004), 151-189. doi: 10.1007/s00030-003-1052-7.
[36]	L. Néel, L'approche a la saturation de la magnétostriction, J. Phys. Radium, 15 (1954), 376-378.
[37]	H. J. Richter, The transition from longitudinal to perpendicular recording, J. Phys. D: Appl. Phys., 40 (2007), R149-R177. doi: 10.1088/0022-3727/40/9/R01.
[38]	R. T. Rockafellar and R. J.-B Wets, Variational Analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 317, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02431-3.
[39]	T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics, Z. Angew. Math. Phys., 55 (2004), 159-182. doi: 10.1007/s00033-003-0110-7.
[40]	T. Roubíček and M. Kružík, Mesoscopic model for ferromagnets with isotropic hardening, Z. Angew. Math. Phys., 56 (2005), 107-135. doi: 10.1007/s00033-003-2108-6.
[41]	T. Roubíček and U. Stefanelli, Magnetic shape-memory alloys: Thermomechanical modeling and analysis, Contin. Mech. Thermodyn., 26 (2014), 783-810. doi: 10.1007/s00161-014-0339-8.
[42]	T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Arch. Ration. Mech. Anal., 210 (2013), 1-43. doi: 10.1007/s00205-013-0648-2.
[43]	P. Rybka and M. Luskin, Existence of energy minimizers for magnetostrictive materials, SIAM J. Math. Anal., 36 (2005), 2004-2019. doi: 10.1137/S0036141004442021.
[44]	B. Schulz and K. Baberschke, Crossover form in-plane to perpendicular magnetization in ultrathin Ni/Cu(001) films, Phys. Rev. B, 50 (1994), 13467.
[45]	A. D. C. Viegas, M. A. Correa, L. Santi, R. B. da Silva, F. Bohn, M. Carara and R. L. Somme, Thickness dependence of the high-frequency magnetic permeability in amorphous $Fe_{73.5}Cu_1Nb_3Si_{13.5}B_9$ thin films, J. Appl. Phys., 101 (2007), 033908.
[46]	A. Visintin, Modified Landau-Lifshitz equation for ferromagnetism, Phys. B, 233 (1997), 365-369. doi: 10.1016/S0921-4526(97)00322-0.
[47]	A. Visintin, Maxwell's equations with vector hysteresis, Arch. Ration. Mech. Anal., 175 (2005), 1-37. doi: 10.1007/s00205-004-0333-6.
[48]	J. Wang and P. Steinmann, A variational approach towards the modeling of magnetic field-induced strains in magnetic shape memory alloys, J. Mech. Phys. Solids, 60 (2012), 1179-1200. doi: 10.1016/j.jmps.2012.02.003.

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