Linearization in magnetoelasticity

References

[1] V. Agostiniani and A. DeSimone, Γ-convergence of energies for nematic elastomers in the small strain limit, Contin. Mech. Thermodyn. 23 (2011), no. 3, 257–274. 10.1007/s00161-011-0180-2Suche in Google Scholar

[2] V. Agostiniani and A. DeSimone, Dimension reduction via Γ-convergence for soft active materials, Meccanica 52 (2017), no. 14, 3457–3470. 10.1007/s11012-017-0630-4Suche in Google Scholar

[3] V. Agostiniani and A. DeSimone, Rigorous derivation of active plate models for thin sheets of nematic elastomers, Math. Mech. Solids 25 (2020), no. 10, 1804–1830. 10.1177/1081286517699991Suche in Google Scholar

[4] S. Almi, M. Caponi, M. Friedrich and F. Solombrino, Geometric rigidity on Sobolev spaces with variable exponent and applications, NoDEA Nonlinear Differential Equations Appl. 32 (2025), no. 1, Paper No. 12. 10.1007/s00030-024-01016-4Suche in Google Scholar

[5] M. Barchiesi and A. DeSimone, Frank energy for nematic elastomers: A nonlinear model, ESAIM Control Optim. Calc. Var. 21 (2015), no. 2, 372–377. 10.1051/cocv/2014022Suche in Google Scholar

[6] M. Barchiesi, D. Henao and C. Mora-Corral, Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 743–816. 10.1007/s00205-017-1088-1Suche in Google Scholar

[7] M. Barchiesi, D. Henao, C. Mora-Corral and R. Rodiac, Harmonic dipoles and the relaxation of the neo-Hookean energy in 3D elasticity, Arch. Ration. Mech. Anal. 247 (2023), no. 4, Paper No. 70. 10.1007/s00205-023-01897-2Suche in Google Scholar

[8] M. Barchiesi, D. Henao, C. Mora-Corral and R. Rodiac, On the lack of compactness in the axisymmetric neo-Hookean model, Forum Math. Sigma 12 (2024), Paper No. e26. 10.1017/fms.2024.9Suche in Google Scholar

[9] V. I. Bogachev, Measure Theory. Vol. I, II, Springer, Berlin, 2007. 10.1007/978-3-540-34514-5Suche in Google Scholar

[10] O. Bouchala, S. Hencl and A. Molchanova, Injectivity almost everywhere for weak limits of Sobolev homeomorphisms, J. Funct. Anal. 279 (2020), no. 7, Article ID 108658. 10.1016/j.jfa.2020.108658Suche in Google Scholar

[11] M. Bresciani, Linearized von Kármán theory for incompressible magnetoelastic plates, Math. Models Methods Appl. Sci. 31 (2021), no. 10, 1987–2037. 10.1142/S0218202521500445Suche in Google Scholar

[12] M. Bresciani, Quasistatic evolution in magnetoelasticity under subcritical coercivity assumptions, Calc. Var. Partial Differential Equations 62 (2023), no. 6, Paper No. 181. 10.1007/s00526-023-02521-7Suche in Google Scholar

[13] M. Bresciani, E. Davoli and M. Kružík, Existence results in large-strain magnetoelasticity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), no. 3, 557–592. 10.4171/aihpc/51Suche in Google Scholar

[14] M. Bresciani, M. Friedrich and C. Mora-Corral, Variational models with Eulerian–Lagrangian formulation allowing for material failure, Arch. Ration. Mech. Anal. 249 (2025), no. 1, Paper No. 4. 10.1007/s00205-024-02076-7Suche in Google Scholar

[15] M. Bresciani and M. Kružík, A reduced model for plates arising as low-energy Γ-limit in nonlinear magnetoelasticity, SIAM J. Math. Anal. 55 (2023), no. 4, 3108–3168. 10.1137/21M1446836Suche in Google Scholar

[16] W. F. Brown, Magnetoelastic Interactions, Springer, Berlin, 1966. 10.1007/978-3-642-87396-6Suche in Google Scholar

[17] P. G. Ciarlet, Mathematical Elasticity. Vol. I, Stud. Math. Appl. 20, North-Holland, Amsterdam, 1988. Suche in Google Scholar

[18] S. Conti and B. Schweizer, Rigidity and gamma convergence for solid-solid phase transitions with SO(2) invariance, Comm. Pure Appl. Math. 59 (2006), no. 6, 830–868. 10.1002/cpa.20115Suche in Google Scholar

[19] G. Dal Maso, M. Negri and D. Percivale, Linearized elasticity as Γ-limit of finite elasticity, Set-Valued Anal. 10 (2002), no. 2–3, 165–183. 10.1023/A:1016577431636Suche in Google Scholar

[20] E. Davoli, M. Kružík, P. Piovano and U. Stefanelli, Magnetoelastic thin films at large strains, Contin. Mech. Thermodyn. 33 (2021), no. 2, 327–341. 10.1007/s00161-020-00904-1Suche in Google Scholar PubMed PubMed Central

[21] A. DeSimone and R. D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids 50 (2002), no. 2, 283–320. 10.1016/S0022-5096(01)00050-3Suche in Google Scholar

[22] A. DeSimone and L. Teresi, Elastic energies for nematic elastomers, Eur. Phys. J. E 29 (2009), 191–204. 10.1140/epje/i2009-10467-9Suche in Google Scholar PubMed

[23] A. Doležalová, S. Hencl and J. Malý, Weak limit of homeomorphisms in W 1 , n − 1 and (INV) condition, Arch. Ration. Mech. Anal. 247 (2023), no. 5, Paper No. 80. 10.1007/s00205-023-01911-7Suche in Google Scholar

[24] A. Doležalová, S. Hencl and A. Molchanova, Weak limit of homeomorphisms in W 1 , n − 1 : invertibility and lower semicontinuity of energy, ESAIM Control Optim. Calc. Var. 30 (2024), Paper No. 37. 10.1051/cocv/2024006Suche in Google Scholar

[25] L. Dorfmann and R. W. Ogden, Nonlinear Theory of Electroelastic and Magnetoelastic Interactions, Springer, New York, 2014. 10.1007/978-1-4614-9596-3Suche in Google Scholar

[26] G. Ethiraj and C. Miehe, Multiplicative magneto-elasticity of magnetosensitive polymers incorporating micromechanically-based network kernels, Internat. J. Engrg. Sci. 102 (2016), 93–119. 10.1016/j.ijengsci.2015.08.007Suche in Google Scholar

[27] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Oxford Lecture Ser. Math. Appl. 2, Oxford University, New York, 1995. 10.1093/oso/9780198511960.001.0001Suche in Google Scholar

[28] G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), no. 11, 1461–1506. 10.1002/cpa.10048Suche in Google Scholar

[29] G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal. 180 (2006), no. 2, 183–236. 10.1007/s00205-005-0400-7Suche in Google Scholar

[30] C. Gao, Z. Zeng, S. Peng and C. Shuai, Magnetostrictive alloys: Promising materials for biomedical applications, Bioactive Mater. 8 (2022), 177–195. 10.1016/j.bioactmat.2021.06.025Suche in Google Scholar PubMed PubMed Central

[31] P. Hajłasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), no. 1, 93–101. 10.4064/cm-64-1-93-101Suche in Google Scholar

[32] S. Hencl and P. Koskela, Lectures on Mappings of Finite Distortion, Lecture Notes in Math. 2096, Springer, Cham, 2014. 10.1007/978-3-319-03173-6Suche in Google Scholar

[33] A. Hubert and R. Schäfer, Magnetic Domains, Springer, Berlin, 1998. Suche in Google Scholar

[34] R. D. James and D. Kinderlehrer, Theory of magnetostriction with application to T ⁢ b x ⁢ D ⁢ y 1 − x ⁢ F ⁢ e 2 , Phil. Mag. B 68 (1993), 237–274. 10.1080/01418639308226405Suche in Google Scholar

[35] M. Kružík, U. Stefanelli and C. Zanini, Quasistatic evolution of magnetoelastic plates via dimension reduction, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5999–6013. 10.3934/dcds.2015.35.5999Suche in Google Scholar

[36] V. Maz’ya, Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Grundlehren Math. Wiss. 342, Springer, Heidelberg, 2011. 10.1007/978-3-642-15564-2Suche in Google Scholar

[37] C. Mudivarthi, S. Datta, J. Atulasimha, P. G. Evans, M. J. Dapino and A. B. Flatau, Anisotropy of constrained magnetostrictive materials, J. Magnetism Magn. Mat. 322 (2010), no. 20, 3028–3034. 10.1016/j.jmmm.2010.05.024Suche in Google Scholar

[38] S. Müller and S. J. Spector, An existence theory for nonlinear elasticity that allows for cavitation, Arch. Ration. Mech. Anal. 131 (1995), no. 1, 1–66. 10.1007/BF00386070Suche in Google Scholar

[39] T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis, Arch. Ration. Mech. Anal. 210 (2013), no. 1, 1–43. 10.1007/s00205-013-0648-2Suche in Google Scholar

[40] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. Suche in Google Scholar

[41] P. Rybka and M. Luskin, Existence of energy minimizers for magnetostrictive materials, SIAM J. Math. Anal. 36 (2005), no. 6, 2004–2019. 10.1137/S0036141004442021Suche in Google Scholar

[42] 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), no. 6, 1179–1200. 10.1016/j.jmps.2012.02.003Suche in Google Scholar

[43] A. Yavari and F. Sozio, On the direct and reverse multiplicative decompositions of deformation gradient in nonlinear anisotropic anelasticity, J. Mech. Phys. Solids 170 (2023), Article ID 105101. 10.1016/j.jmps.2022.105101Suche in Google Scholar