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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, 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. Accepted in ESAIM Control Optim. Calc. Var. 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), 147-157. 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), 343-359. 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] W. F. Brown, Jr., Magnetoelastic Interactions, Springer, Berlin, 1966. doi: 10.1007/978-3-642-87396-6.  Google Scholar [7] S. Chikazumi, Physics of Magnetism, J. Wiley, New York, 1964. Google Scholar [8] P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988.  Google Scholar [9] P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 97 (1987), 171-188. doi: 10.1007/BF00250807.  Google Scholar [10] B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition. Springer, New York, 2008.  Google Scholar [11] A. DeSimone, Energy minimizers for large ferromagnetic bodies, Arch. Ration. Mech. Anal., 125 (1993), 99-143. 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), 107-120. doi: 10.1007/s002050050114.  Google Scholar [13] A. DeSimone and R. D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids, 50 (2002), 283-320. 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), 73-79. 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), 55-91. doi: 10.1515/CRELLE.2006.044.  Google Scholar [16] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Contin. Mech. Thermodyn., 2 (1990), 215-239. 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), 237-274. doi: 10.1080/01418639308226405.  Google Scholar [18] J. Liakhova, A Theory of Magnetostrictive Thin Films with Applications, PhD Thesis, University of Minnesota, 1999.  Google Scholar [19] J. Liakhova, M. Luskin and T. Zhang, Computational modeling of ferromagnetic shape memory thin films, Ferroelectrics, 342 (2005), 73-82. 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), 37-40. doi: 10.1016/j.cma.2006.10.039.  Google Scholar [21] A. Mielke, Evolution of rate-independent systems, in Handbook of Differential Equations, Evolutionary Equations (eds., C. Dafermos and E. Feireisl), Elsevier, 2 (2005), 461-559.  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, H.-D., Balean, R., Farwig, R. eds.) Shaker-Verlag, Aachen, 1999, pp. 117-129. Google Scholar [23] 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 [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), 137-177. doi: 10.1007/s002050200194.  Google Scholar [25] 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 [26] 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 [27] 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

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. Accepted in ESAIM Control Optim. Calc. Var. 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), 147-157. 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), 343-359. 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] W. F. Brown, Jr., Magnetoelastic Interactions, Springer, Berlin, 1966. doi: 10.1007/978-3-642-87396-6.  Google Scholar [7] S. Chikazumi, Physics of Magnetism, J. Wiley, New York, 1964. Google Scholar [8] P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988.  Google Scholar [9] P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Ration. Mech. Anal., 97 (1987), 171-188. doi: 10.1007/BF00250807.  Google Scholar [10] B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition. Springer, New York, 2008.  Google Scholar [11] A. DeSimone, Energy minimizers for large ferromagnetic bodies, Arch. Ration. Mech. Anal., 125 (1993), 99-143. 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), 107-120. doi: 10.1007/s002050050114.  Google Scholar [13] A. DeSimone and R. D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids, 50 (2002), 283-320. 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), 73-79. 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), 55-91. doi: 10.1515/CRELLE.2006.044.  Google Scholar [16] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Contin. Mech. Thermodyn., 2 (1990), 215-239. 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), 237-274. doi: 10.1080/01418639308226405.  Google Scholar [18] J. Liakhova, A Theory of Magnetostrictive Thin Films with Applications, PhD Thesis, University of Minnesota, 1999.  Google Scholar [19] J. Liakhova, M. Luskin and T. Zhang, Computational modeling of ferromagnetic shape memory thin films, Ferroelectrics, 342 (2005), 73-82. 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), 37-40. doi: 10.1016/j.cma.2006.10.039.  Google Scholar [21] A. Mielke, Evolution of rate-independent systems, in Handbook of Differential Equations, Evolutionary Equations (eds., C. Dafermos and E. Feireisl), Elsevier, 2 (2005), 461-559.  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, H.-D., Balean, R., Farwig, R. eds.) Shaker-Verlag, Aachen, 1999, pp. 117-129. Google Scholar [23] 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 [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), 137-177. doi: 10.1007/s002050200194.  Google Scholar [25] 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 [26] 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 [27] 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
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