• Previous Article
    Control of crack propagation by shape-topological optimization
  • DCDS Home
  • This Issue
  • Next Article
    On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions
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 and 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.

[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.

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.

[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.

[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 and 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 and Continuous Dynamical Systems, 2010, 27 (3) : 1189-1217. doi: 10.3934/dcds.2010.27.1189

[3]

Marita Thomas. Quasistatic damage evolution with spatial $\mathrm{BV}$-regularization. Discrete and 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 and 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 and Continuous Dynamical Systems, 2015, 35 (12) : 5999-6013. doi: 10.3934/dcds.2015.35.5999

[6]

Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058

[7]

Gianni Dal Maso, Francesco Solombrino. Quasistatic evolution for Cam-Clay plasticity: The spatially homogeneous case. Networks and Heterogeneous Media, 2010, 5 (1) : 97-132. doi: 10.3934/nhm.2010.5.97

[8]

Virginia Agostiniani. Second order approximations of quasistatic evolution problems in finite dimension. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1125-1167. doi: 10.3934/dcds.2012.32.1125

[9]

Mirelson M. Freitas, Anderson J. A. Ramos, Baowei Feng, Mauro L. Santos, Helen C. M. Rodrigues. Existence and continuity of global attractors for ternary mixtures of solids. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3563-3583. doi: 10.3934/dcdsb.2021196

[10]

Anna Maria Candela, Addolorata Salvatore. Existence of minimizers for some quasilinear elliptic problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3335-3345. doi: 10.3934/dcdss.2020241

[11]

Federica Mennuni, Addolorata Salvatore. Existence of minimizers for a quasilinear elliptic system of gradient type. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022013

[12]

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 179-195. doi: 10.3934/dcdss.2021028

[13]

Duvan Henao, Rémy Rodiac. On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4509-4536. doi: 10.3934/dcds.2018197

[14]

Pierpaolo Soravia. Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation. Mathematical Control and Related Fields, 2012, 2 (4) : 399-427. doi: 10.3934/mcrf.2012.2.399

[15]

Luís Balsa Bicho, António Ornelas. Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 439-451. doi: 10.3934/dcds.2011.29.439

[16]

Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687

[17]

Sandro Zagatti. Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 2005-2025. doi: 10.3934/dcds.2021181

[18]

Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925

[19]

Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure and Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039

[20]

Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]