American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

W. F. Brown, Micromagnetics, {Wiley, New-York, 1963.} Google Scholar

[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.  Google Scholar

[9]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[10]

D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mat. Struct., 22 (2013), 095009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. L. Hodgdon, Applications of a theory of ferromagnetic hysteresis, IEEE Trans. Mag., 24 (1988), 218-221. doi: 10.1109/20.43893.  Google Scholar

[23]

A. Hubert and R. Schäfer, Magnetic Domains, Springer, New York, 1998. Google Scholar

[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.  Google Scholar

[25]

R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Continuum Mech. Thermodyn., 2 (1990), 215-239. doi: 10.1007/BF01129598.  Google Scholar

[26]

M. Kaltenbacher, M. Meiler and M. Ertl, Physical modeling and numerical computation of magnetostriction, COMPEL, 28 (2009), 819-832. doi: 10.1108/03321640910958946.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

L. Néel, L'approche a la saturation de la magnétostriction, J. Phys. Radium, 15 (1954), 376-378. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

A. Visintin, Modified Landau-Lifshitz equation for ferromagnetism, Phys. B, 233 (1997), 365-369. doi: 10.1016/S0921-4526(97)00322-0.  Google Scholar

[47]

A. Visintin, Maxwell's equations with vector hysteresis, Arch. Ration. Mech. Anal., 175 (2005), 1-37. doi: 10.1007/s00205-004-0333-6.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

W. F. Brown, Micromagnetics, {Wiley, New-York, 1963.} Google Scholar

[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.  Google Scholar

[9]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[10]

D. Davino, P. Krejčí and C. Visone, Fully coupled modeling of magnetomechanical hysteresis through thermodynamic compatibility, Smart Mat. Struct., 22 (2013), 095009. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. L. Hodgdon, Applications of a theory of ferromagnetic hysteresis, IEEE Trans. Mag., 24 (1988), 218-221. doi: 10.1109/20.43893.  Google Scholar

[23]

A. Hubert and R. Schäfer, Magnetic Domains, Springer, New York, 1998. Google Scholar

[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.  Google Scholar

[25]

R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Continuum Mech. Thermodyn., 2 (1990), 215-239. doi: 10.1007/BF01129598.  Google Scholar

[26]

M. Kaltenbacher, M. Meiler and M. Ertl, Physical modeling and numerical computation of magnetostriction, COMPEL, 28 (2009), 819-832. doi: 10.1108/03321640910958946.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

L. Néel, L'approche a la saturation de la magnétostriction, J. Phys. Radium, 15 (1954), 376-378. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[46]

A. Visintin, Modified Landau-Lifshitz equation for ferromagnetism, Phys. B, 233 (1997), 365-369. doi: 10.1016/S0921-4526(97)00322-0.  Google Scholar

[47]

A. Visintin, Maxwell's equations with vector hysteresis, Arch. Ration. Mech. Anal., 175 (2005), 1-37. doi: 10.1007/s00205-004-0333-6.  Google Scholar

[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.  Google Scholar

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