Article Contents
Article Contents

# Existence results for incompressible magnetoelasticity

• 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.
Mathematics Subject Classification: Primary: 74F15, 74K35; Secondary: 35Q74.

 Citation:

•  [1] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal., 63 (1976/77), 337-403. doi: 10.1007/BF00279992. [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. [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. [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. [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] W. F. Brown, Jr., Magnetoelastic Interactions, Springer, Berlin, 1966.doi: 10.1007/978-3-642-87396-6. [7] S. Chikazumi, Physics of Magnetism, J. Wiley, New York, 1964. [8] P. G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity, North-Holland, Amsterdam, 1988. [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. [10] B. Dacorogna, Direct Methods in the Calculus of Variations, Second edition. Springer, New York, 2008. [11] A. DeSimone, Energy minimizers for large ferromagnetic bodies, Arch. Ration. Mech. Anal., 125 (1993), 99-143.doi: 10.1007/BF00376811. [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. [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. [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. [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. [16] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Contin. Mech. Thermodyn., 2 (1990), 215-239.doi: 10.1007/BF01129598. [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. [18] J. Liakhova, A Theory of Magnetostrictive Thin Films with Applications, PhD Thesis, University of Minnesota, 1999. [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. [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. [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. [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. [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. [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. [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. [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. [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.