# American Institute of Mathematical Sciences

• Previous Article
On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions
• DCDS Home
• This Issue
• Next Article
Control of crack propagation by shape-topological optimization
June  2015, 35(6): 2615-2623. doi: 10.3934/dcds.2015.35.2615

## Existence results for incompressible magnetoelasticity

 1 Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, CZ-182 08 Praha 8, Czech Republic 2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna 3 Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ-166 29 Praha 6, Czech Republic

Received  November 2013 Revised  April 2014 Published  December 2014

We investigate a variational theory for magnetoelastic solids under the incompressibility constraint. The state of the system is described by deformation and magnetization. While the former is classically related to the reference configuration, magnetization is defined in the deformed configuration instead. We discuss the existence of energy minimizers without relying on higher-order deformation gradient terms. Then, by introducing a suitable positively $1$-homogeneous dissipation, a quasistatic evolution model is proposed and analyzed within the frame of energetic solvability.
Citation: Martin Kružík, Ulisse Stefanelli, Jan Zeman. Existence results for incompressible magnetoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2615-2623. doi: 10.3934/dcds.2015.35.2615
##### References:
 [1] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Ration. Mech. Anal., 63 (): 337. doi: 10.1007/BF00279992. Google Scholar [2] M. Barchiesi and A. DeSimone, Frank energy for nematic elastomers: A nonlinear model,, Preprint CVGMT Pisa, (2013). Google Scholar [3] W. Bielski and B. Gambin, Relationship between existence of energy minimizers of incompressible and nearly incompressible magnetostrictive materials,, Rep. Math. Phys., 66 (2010), 147. doi: 10.1016/S0034-4877(10)00023-6. Google Scholar [4] A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape-memory single crystals,, Z. Angew. Math. Phys., 64 (2013), 343. 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. doi: 10.1142/S0218202511005246. Google Scholar [6] W. F. Brown, Jr., Magnetoelastic Interactions,, Springer, (1966). doi: 10.1007/978-3-642-87396-6. Google Scholar [7] S. Chikazumi, Physics of Magnetism,, J. Wiley, (1964). Google Scholar [8] P. G. Ciarlet, Mathematical Elasticity,, Vol. I: Three-dimensional Elasticity, (1988). Google Scholar [9] P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Ration. Mech. Anal., 97 (1987), 171. doi: 10.1007/BF00250807. Google Scholar [10] B. Dacorogna, Direct Methods in the Calculus of Variations,, Second edition. Springer, (2008). Google Scholar [11] A. DeSimone, Energy minimizers for large ferromagnetic bodies,, Arch. Ration. Mech. Anal., 125 (1993), 99. doi: 10.1007/BF00376811. Google Scholar [12] A. DeSimone and G. Dolzmann, Existence of minimizers for a variational problem in two-dimensional nonlinear magnetoelasticity,, Arch. Ration. Mech. Anal., 144 (1998), 107. doi: 10.1007/s002050050114. Google Scholar [13] 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. Google Scholar [14] G. Eisen, A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals,, Manuscripta Math. 27 (1979), 27 (1979), 73. doi: 10.1007/BF01297738. Google Scholar [15] 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. Google Scholar [16] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials,, Contin. Mech. Thermodyn., 2 (1990), 215. doi: 10.1007/BF01129598. Google Scholar [17] R. D. James and D. Kinderlehrer, Theory of magnetostriction with application to $Tb_xDy_{1-x}Fe_2$,, Phil. Mag. B, 68 (1993), 237. doi: 10.1080/01418639308226405. Google Scholar [18] J. Liakhova, A Theory of Magnetostrictive Thin Films with Applications,, PhD Thesis, (1999). Google Scholar [19] J. Liakhova, M. Luskin and T. Zhang, Computational modeling of ferromagnetic shape memory thin films,, Ferroelectrics, 342 (2005), 73. doi: 10.1080/00150190600946211. Google Scholar [20] M. Luskin and T. Zhang, Numerical analysis of a model for ferromagnetic shape memory thin films,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 37. doi: 10.1016/j.cma.2006.10.039. Google Scholar [21] A. Mielke, Evolution of rate-independent systems,, in Handbook of Differential Equations, 2 (2005), 461. Google Scholar [22] A. Mielke and F. Theil, Mathematical model for rate-independent phase transformations,, In: Models of Cont. Mechanics in Analysis and Engineering (Alber, (1999), 117. Google Scholar [23] A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151. doi: 10.1007/s00030-003-1052-7. Google Scholar [24] A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using extremum principle,, Arch. Ration. Mech. Anal., 162 (2002), 137. doi: 10.1007/s002050200194. Google Scholar [25] T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics,, Z. Angew. Math. Phys., 55 (2004), 159. doi: 10.1007/s00033-003-0110-7. Google Scholar [26] 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. Google Scholar [27] P. Rybka and M. Luskin, Existence of energy minimizers for magnetostrictive materials,, SIAM J. Math. Anal., 36 (2005), 2004. doi: 10.1137/S0036141004442021. Google Scholar

show all references

##### References:
 [1] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity,, Arch. Ration. Mech. Anal., 63 (): 337. doi: 10.1007/BF00279992. Google Scholar [2] M. Barchiesi and A. DeSimone, Frank energy for nematic elastomers: A nonlinear model,, Preprint CVGMT Pisa, (2013). Google Scholar [3] W. Bielski and B. Gambin, Relationship between existence of energy minimizers of incompressible and nearly incompressible magnetostrictive materials,, Rep. Math. Phys., 66 (2010), 147. doi: 10.1016/S0034-4877(10)00023-6. Google Scholar [4] A.-L. Bessoud, M. Kružík and U. Stefanelli, A macroscopic model for magnetic shape-memory single crystals,, Z. Angew. Math. Phys., 64 (2013), 343. 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. doi: 10.1142/S0218202511005246. Google Scholar [6] W. F. Brown, Jr., Magnetoelastic Interactions,, Springer, (1966). doi: 10.1007/978-3-642-87396-6. Google Scholar [7] S. Chikazumi, Physics of Magnetism,, J. Wiley, (1964). Google Scholar [8] P. G. Ciarlet, Mathematical Elasticity,, Vol. I: Three-dimensional Elasticity, (1988). Google Scholar [9] P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity,, Arch. Ration. Mech. Anal., 97 (1987), 171. doi: 10.1007/BF00250807. Google Scholar [10] B. Dacorogna, Direct Methods in the Calculus of Variations,, Second edition. Springer, (2008). Google Scholar [11] A. DeSimone, Energy minimizers for large ferromagnetic bodies,, Arch. Ration. Mech. Anal., 125 (1993), 99. doi: 10.1007/BF00376811. Google Scholar [12] A. DeSimone and G. Dolzmann, Existence of minimizers for a variational problem in two-dimensional nonlinear magnetoelasticity,, Arch. Ration. Mech. Anal., 144 (1998), 107. doi: 10.1007/s002050050114. Google Scholar [13] 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. Google Scholar [14] G. Eisen, A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals,, Manuscripta Math. 27 (1979), 27 (1979), 73. doi: 10.1007/BF01297738. Google Scholar [15] 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. Google Scholar [16] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials,, Contin. Mech. Thermodyn., 2 (1990), 215. doi: 10.1007/BF01129598. Google Scholar [17] R. D. James and D. Kinderlehrer, Theory of magnetostriction with application to $Tb_xDy_{1-x}Fe_2$,, Phil. Mag. B, 68 (1993), 237. doi: 10.1080/01418639308226405. Google Scholar [18] J. Liakhova, A Theory of Magnetostrictive Thin Films with Applications,, PhD Thesis, (1999). Google Scholar [19] J. Liakhova, M. Luskin and T. Zhang, Computational modeling of ferromagnetic shape memory thin films,, Ferroelectrics, 342 (2005), 73. doi: 10.1080/00150190600946211. Google Scholar [20] M. Luskin and T. Zhang, Numerical analysis of a model for ferromagnetic shape memory thin films,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 37. doi: 10.1016/j.cma.2006.10.039. Google Scholar [21] A. Mielke, Evolution of rate-independent systems,, in Handbook of Differential Equations, 2 (2005), 461. Google Scholar [22] A. Mielke and F. Theil, Mathematical model for rate-independent phase transformations,, In: Models of Cont. Mechanics in Analysis and Engineering (Alber, (1999), 117. Google Scholar [23] A. Mielke and F. Theil, On rate-independent hysteresis models,, Nonlin. Diff. Eq. Appl., 11 (2004), 151. doi: 10.1007/s00030-003-1052-7. Google Scholar [24] A. Mielke, F. Theil and V. Levitas, A variational formulation of rate-independent phase transformations using extremum principle,, Arch. Ration. Mech. Anal., 162 (2002), 137. doi: 10.1007/s002050200194. Google Scholar [25] T. Roubíček and M. Kružík, Microstructure evolution model in micromagnetics,, Z. Angew. Math. Phys., 55 (2004), 159. doi: 10.1007/s00033-003-0110-7. Google Scholar [26] 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. Google Scholar [27] P. Rybka and M. Luskin, Existence of energy minimizers for magnetostrictive materials,, SIAM J. Math. Anal., 36 (2005), 2004. doi: 10.1137/S0036141004442021. Google Scholar
 [1] Marco Cicalese, Antonio DeSimone, Caterina Ida Zeppieri. Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers. Networks & Heterogeneous Media, 2009, 4 (4) : 667-708. doi: 10.3934/nhm.2009.4.667 [2] Francesco Solombrino. Quasistatic evolution for plasticity with softening: The spatially homogeneous case. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1189-1217. doi: 10.3934/dcds.2010.27.1189 [3] Marita Thomas. Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 235-255. doi: 10.3934/dcdss.2013.6.235 [4] G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Globally stable quasistatic evolution in plasticity with softening. Networks & Heterogeneous Media, 2008, 3 (3) : 567-614. doi: 10.3934/nhm.2008.3.567 [5] 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 [6] Gianni Dal Maso, Francesco Solombrino. Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case. Networks & Heterogeneous Media, 2010, 5 (1) : 97-132. doi: 10.3934/nhm.2010.5.97 [7] Virginia Agostiniani. Second order approximations of quasistatic evolution problems in finite dimension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125 [8] Duvan Henao, Rémy Rodiac. On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4509-4536. doi: 10.3934/dcds.2018197 [9] Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control & Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399 [10] Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925 [11] Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039 [12] Luís Balsa Bicho, António Ornelas. Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 439-451. doi: 10.3934/dcds.2011.29.439 [13] Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687 [14] Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895 [15] Alain Haraux. On the fast solution of evolution equations with a rapidly decaying source term. Mathematical Control & Related Fields, 2011, 1 (1) : 1-20. doi: 10.3934/mcrf.2011.1.1 [16] Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022 [17] Md. Abul Kalam Azad, Edite M.G.P. Fernandes. A modified differential evolution based solution technique for economic dispatch problems. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1017-1038. doi: 10.3934/jimo.2012.8.1017 [18] Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136 [19] Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213 [20] Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Small solids in an inviscid fluid. Networks & Heterogeneous Media, 2010, 5 (3) : 385-404. doi: 10.3934/nhm.2010.5.385

2018 Impact Factor: 1.143