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

Quasistatic evolution of magnetoelastic plates via dimension reduction

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

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.
Citation: Martin Kružík, Ulisse Stefanelli, Chiara Zanini. Quasistatic evolution of magnetoelastic plates via dimension reduction. Discrete & Continuous Dynamical Systems - A, 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). doi: 10.1063/1.4788807.

[2]

J.-F. Babadjian, Quasistatic evolution of a brittle thin film,, Calc. Var. Partial Differential Equations, 26 (2006), 69. 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. 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. 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. doi: 10.1142/S0218202511005246.

[6]

A. Braides, $\Gamma$-Convergence for Beginners,, Oxford University Press, (2002). doi: 10.1093/acprof:oso/9780198507840.001.0001.

[7]

W. F. Brown, Micromagnetics,, {Wiley, (1963).

[8]

C. Chappert and P. Bruno, Magnetic anisotropy in metallic ultrathin films and related experiments on cobalt films,, J. Appl. Phys., 64 (1988). doi: 10.1063/1.342243.

[9]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäser, (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).

[11]

E. Davoli, Linearized plastic plate models as $\Gamma$-limits of 3D finite elastoplasticity,, ESAIM Control Optim. Calc. Var., 20 (2014), 725. 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. 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. 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. doi: 10.1007/s002050050114.

[15]

A. DeSimone and R. D. James, A constrained theory of magnetoelasticity,, J. Mech. Phys. Solids, 50 (2002), 283. 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. 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.

[18]

L. Freddi, T. Roubíček, R. Paroni and C. Zanini, Quasistatic delamination models for Kirchhoff-Love plates,, Z. Angew. Math. Mech., 91 (2011), 845. 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. 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. 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. doi: 10.1098/rspa.1997.0013.

[22]

M. L. Hodgdon, Applications of a theory of ferromagnetic hysteresis,, IEEE Trans. Mag., 24 (1988), 218. doi: 10.1109/20.43893.

[23]

A. Hubert and R. Schäfer, Magnetic Domains,, Springer, (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. doi: 10.1007/s001610100072.

[25]

R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials,, Continuum Mech. Thermodyn., 2 (1990), 215. doi: 10.1007/BF01129598.

[26]

M. Kaltenbacher, M. Meiler and M. Ertl, Physical modeling and numerical computation of magnetostriction,, COMPEL, 28 (2009), 819. doi: 10.1108/03321640910958946.

[27]

M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism,, SIAM Review, 48 (2006), 439. 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. 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. doi: 10.1007/s00030-011-0137-y.

[30]

A. Mielke, Evolution in rate-independent systems (ch. 6),, in Handbook of Differential Equations, (2005), 461.

[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, (2006), 399. 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. 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. 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. 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. doi: 10.1007/s00030-003-1052-7.

[36]

L. Néel, L'approche a la saturation de la magnétostriction,, J. Phys. Radium, 15 (1954), 376.

[37]

H. J. Richter, The transition from longitudinal to perpendicular recording,, J. Phys. D: Appl. Phys., 40 (2007). 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], (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. 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. 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. 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. 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. 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).

[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).

[46]

A. Visintin, Modified Landau-Lifshitz equation for ferromagnetism,, Phys. B, 233 (1997), 365. doi: 10.1016/S0921-4526(97)00322-0.

[47]

A. Visintin, Maxwell's equations with vector hysteresis,, Arch. Ration. Mech. Anal., 175 (2005), 1. 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. 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). doi: 10.1063/1.4788807.

[2]

J.-F. Babadjian, Quasistatic evolution of a brittle thin film,, Calc. Var. Partial Differential Equations, 26 (2006), 69. 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. 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. 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. doi: 10.1142/S0218202511005246.

[6]

A. Braides, $\Gamma$-Convergence for Beginners,, Oxford University Press, (2002). doi: 10.1093/acprof:oso/9780198507840.001.0001.

[7]

W. F. Brown, Micromagnetics,, {Wiley, (1963).

[8]

C. Chappert and P. Bruno, Magnetic anisotropy in metallic ultrathin films and related experiments on cobalt films,, J. Appl. Phys., 64 (1988). doi: 10.1063/1.342243.

[9]

G. Dal Maso, An Introduction to $\Gamma$-Convergence,, Birkhäser, (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).

[11]

E. Davoli, Linearized plastic plate models as $\Gamma$-limits of 3D finite elastoplasticity,, ESAIM Control Optim. Calc. Var., 20 (2014), 725. 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. 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. 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. doi: 10.1007/s002050050114.

[15]

A. DeSimone and R. D. James, A constrained theory of magnetoelasticity,, J. Mech. Phys. Solids, 50 (2002), 283. 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. 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.

[18]

L. Freddi, T. Roubíček, R. Paroni and C. Zanini, Quasistatic delamination models for Kirchhoff-Love plates,, Z. Angew. Math. Mech., 91 (2011), 845. 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. 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. 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. doi: 10.1098/rspa.1997.0013.

[22]

M. L. Hodgdon, Applications of a theory of ferromagnetic hysteresis,, IEEE Trans. Mag., 24 (1988), 218. doi: 10.1109/20.43893.

[23]

A. Hubert and R. Schäfer, Magnetic Domains,, Springer, (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. doi: 10.1007/s001610100072.

[25]

R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials,, Continuum Mech. Thermodyn., 2 (1990), 215. doi: 10.1007/BF01129598.

[26]

M. Kaltenbacher, M. Meiler and M. Ertl, Physical modeling and numerical computation of magnetostriction,, COMPEL, 28 (2009), 819. doi: 10.1108/03321640910958946.

[27]

M. Kružík and A. Prohl, Recent developments in modeling, analysis and numerics of ferromagnetism,, SIAM Review, 48 (2006), 439. 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. 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. doi: 10.1007/s00030-011-0137-y.

[30]

A. Mielke, Evolution in rate-independent systems (ch. 6),, in Handbook of Differential Equations, (2005), 461.

[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, (2006), 399. 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. 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. 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. 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. doi: 10.1007/s00030-003-1052-7.

[36]

L. Néel, L'approche a la saturation de la magnétostriction,, J. Phys. Radium, 15 (1954), 376.

[37]

H. J. Richter, The transition from longitudinal to perpendicular recording,, J. Phys. D: Appl. Phys., 40 (2007). 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], (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. 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. 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. 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. 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. 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).

[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).

[46]

A. Visintin, Modified Landau-Lifshitz equation for ferromagnetism,, Phys. B, 233 (1997), 365. doi: 10.1016/S0921-4526(97)00322-0.

[47]

A. Visintin, Maxwell's equations with vector hysteresis,, Arch. Ration. Mech. Anal., 175 (2005), 1. 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. doi: 10.1016/j.jmps.2012.02.003.

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