Abstract
We present two case studies how analysis can be used to derive a hierarchy of models to capture multiscale behavior of materials. The determination, via Γ-convergence, of the thin film limit of micromagnetism delivers a reduced two-dimensional model for soft ferromagnetic films which justifies previously known theories for small fields and extends them to the regime of field penetration. The analytic evaluation of the quasiconvex envelope of the microscopic energy density of nematic elastomers allows efficient numerical computations with finite elements and shows the existence of a new “smectic” phase. In both cases, the numerical solution of the coarse-grained model is complemented by a reconstruction of the microscopic pattern associated with the reduced field.
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References
G. Alberti and S. Müller, A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 (2001), 761–825.
L. Ambrosio, C. De Lellis, and C. Mantegazza, Line energies for gradient vector fields in the plane, Calc. Var. Partial Diff. Eqs. 9 (1999), 327–355.
G. Anzellotti, S. Baldo, and A. Visintin, Asymptotic behavior of the LandauLifshitz model of ferromagnetism, Appl. Math. Optim. 23 (1991), 171–192.
J. Ball and R. D. James, Fine phase mixtures as minimizers of the energy, Arch. Rat. Mech. Anal. 100 (1987), 13–52.
5. H. Ben Belgacem, S. Conti, A. DeSimone, and S. Müller, Energy scaling of
compressed elastic films,Arch. Rat. Mech. Anal. 164 (2002), 1–37.
Rigorous bounds for the Föppl-von Kc rmó,n theory of isotropically com-
pressed plates, J. Nonlinear Sci. 10 (2000), 661–683.
H. A. M. van den Berg, Self-consistent domain theory in soft ferromagnetic media. ii. basic domain structures in thin film objects, J. Appl. Phys. 60 (1986), 1104–1113.
G. Bertotti, Hysteresis in magnetism, Academic Press, San Diego, 1998.
P. Bladon, E. M. Terentjev, and M. Warner, Transitions and instabilities in liquid-crystal elastomers, Phys. Rev. E 47 (1993), R3838 - R3840.
A. Braides and A. Defranceschi, Homogeneization of multiple integrals, Clare-don Press, Oxford, 1998.
W. F. Brown, Micromagnetics, Wiley, 1963.
P. Bryant and H. Suhl, Thin-film magnetic patterns in an external field, Appl. Phys. Lett. 54 (1989), 2224.
M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals, Arch. Rat. Mech. Anal. 103 (1988), 237–277.
S. Conti, G. Dolzmann, and A. DeSimone, Soft elastic response of stretched sheets of nematic elastomers: a numerical study, J. Mech. Phys. Solids 50 (2002), 1431–1451.
B. Dacorogna, A relaxation theorem and its application to the equilibrium of gases, Arch. Rat. Mech. Anal. 77 (1981), 359–386.
B. Dacorogna, Direct methods in the calculus of variations, Springer, Berlin, 1989.
B. Dacorogna and C. Tanteri, Implicit partial differential equations and the constraints of non linear elasticity, J. Math. Pure Appl. 81 (2002), 311–341.
G. Dal Maso, An introduction to F-convergence, Birkhäuser, Boston, 1993.
E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell’area, Rend. Mat. 8 (1975), 277–294.
E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Mat. 58 (1975), 842–850.
A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Arch. Rat. Mech. Anal. 161 (2002), 181–204.
A. DeSimone and R.D. James, A constrained theory of magnetoelasticity, J. Mech. Phys. Solids 50 (2002), 283–320.
A. DeSimone, R.V. Kohn, S. Müller, and F. Otto, A compactness result in the gradient theory of phase transitions, Proc. Roy. Soc. Edin. A 131 (2001), 833–844.
A. DeSimone, R.V. Kohn, S. Müller, and F. Otto, Magnetic microstructures - a paradigm of multiscale problems,ICIAM
J.M. Ball and J.C.R. Hunt, eds. ), Oxford Univ. Press, 2000, pp. 175–190.
A reduced theory for thin-film micromagnetics,Comm. Pure Appl. Math. (to appear).
Repulsive interaction of Néel wall tails,Mult. Model. and Simul. (in press).
A. DeSimone, R.V. Kohn, S. Müller, F. Otto, and R. Schäfer, Two-dimensional modeling of soft ferromagnetic films,Proc. Roy. Soc. Lond. A 457 (2001), 29832992.
L.C. Evans, Partial differential equations, American Mathematical Society, Providence, 1998.
H. Finkelmann, I. Kundler, E.M. Terentjev, and M. Warner, Critical stripe-domain instability of nematic elastomers,J. Phys. II France 7 (1997), 10591069.
G. Friesecke, R. James, and S. Müller, Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Acad. Sci. Paris Série I 334 (2002), 173–178.
G. Friesecke and F. Theil, Validity and failure of the Cauchy-Born hypothesis in a 2D mass-spring lattice, preprint (2001).
C.J. Garcia-Cervera and W.E, Effective dynamics for ferromagnetic thin films, J. Appl. Phys. 90 (2001), 370–374.
P. Gérard, Microlocal defect measures, Comm. PDE 16 (1991), 1761–1794.
G. Gioia and M. Ortiz, Delamination of compressed thin films, Adv. Appl. Mech. 33 (1997), 119–192.
L. Golubovié and T. C. Lubensky, Nonlinear elasticity of amorphous solids, Phys. Rev. Lett. 63 (1989), 1082–1085.
A. Hubert and R. Schäfer, Magnetic domains, Springer, Berlin, 1998.
P. E. Jabin, F. Otto, and B. Perthame, Line-energy Ginzburg-Landau models: zero-energy states,Ann. Sc. Normale Pisa (in press).
P.E. Jabin and B. Perthame, Compactness in Ginzburg-Landau energy by kinetic averaging, Comm. Pure Appl. Math. 54 (2001), 1096–1109.
V.V. Jikov, S.M. Kozlov, and O.A. Oleinik, Homogeneization of differential operators and integral functionals, Springer, Berlin, 1994.
W. Jin and R.V. Kohn, Singular perturbation and the energy of folds, J. Nonlinear Sci. 10 (2000), 355–390.
W. Jin and P. Sternberg, Energy estimates of the von Kârmân model of thin-film blistering, J. Math. Phys. 42 (2001), 192–199.
J. Kundler and H. Finkelmann, Strain-induced director reorientation in nematic liquid single crystal elastomers, Macromol. Chem. Rapid Comm. 16 (1995), 679–686.
L.D. Landau and E.M. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys Z. Sowjetunion 8 (1935), 153–169.
C. Le Bris and X. Blanc amd P.-L. Lions, Convergence de modèles moléculaires vers des modèles de mécanique des milieux continus, C. R. Acad. Sci. Paris Série I 332 (2001), 949–956.
C. Melcher, The logarithmic tail of Néel walls in thin films, Preprint MPI-MIS 61 (2001).
C.B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin, 1966.
S. Müller, Variational models for microstructure and phase transitions, in Calculus of Variations and Geometric Evolution Problems, Le ctures given at the 2nd Session of the Centro Internazionale Matematico Estivo, Cetaro 1996 ( F. Bethuel, G. Huisken, S. Müller, K. Steffen, S. Hildebrandt, and M. Struwe, eds.), Springer, Berlin, 1999.
S. Müller and V. Sverâk, Convex integration with constraints and applications to phase transitions and partial differential equations, J. Eur. Math. Soc. (JEMS) 1 (1999), 393–442.
F. Murat and L. Tartar, Calcul des variations et homogénéisation,Les Méthodes de l’Homogénéisation: Théorie et Applications en Physique (D. Bergman et al., ed.), Collect. Dir. Etudes Rech. Electricité de France, vol. 57, Eyrolles, Paris, 1985, pp. 319–369, (translated in [501).
F. Murat and L. Tartar, Calculus of variations and homogenization,Topics in the Mathematical Modelling of Composite Materials (A. Cherkaev and R. Kohn, eds.), Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, 1997, pp. 139–173, (see also the other contributions in this volume).
O. Pantz, Une justification partielle du modèle de plaque en flexion par F-convergence, C. R. Acad. Sci. Paris Série I 332 (2001), 587–592.
T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm Pure Appl. Math. 54 (2001), 294–338.
J. A. Sethian, Level set methods, Cambridge University Press, 1996.
M. Silhavÿ, Relaxation in a class of SO(n)-invariant energies related to nematic elastomers, preprint (2001).
L. Tartar, Compensated compactness and partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium (R. Knops, ed.), vol. IV, Pitman, 1979, pp. 136–212.
L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edin. A 115 (1990), 193–230.
L. Tartar, Beyond Young measures, Meccanica 30 (1995), 505–526.
R. Tickle, R.D. James, T. Shield, M. Wuttig, and V.V. Kokorin, Ferromagnetic shape memory in the NiMnGa system,IEEE Trans. Magn. 35 (1999), 43014310.
G.C. Verwey, M. Warner, and E. M. Terentjev, Elastic instability and stripe domain in liquid crystalline elastomers, J. Phys. II France 6 (1996), 1273–1290.
M. Warner, New elastic behaviour arising from the unusual constitutive relation of nematic solids, J. Mech. Phys. Sol. 47 (1999), 1355–1377.
M. Warner and E.M. Terentjev, Nematic elastomers–a new state of matter?, Prog. Polym. Sci. 21 (1996), 853–891.
J. Weilepp and H.R. Brand, Director reorientation in nematic-liquid-singlecrystal elastomers by external mechanical stress, Europhys. Lett. 34 (1996), 495–500.
L.C. Young, Lectures on the calculus of variations and optimal control theory, Saunders, 1969, reprinted by Chelsea, 1980.
E.R. Zubarev, S.A. Kuptsov, T.I. Yuranova, R.V. Talroze, and H. Finkelmann, Monodomain liquid crystalline networks: reorientation mechanism from uniform to stripe domains, Liquid crystals 26 (1999), 1531–1540.
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Conti, S., DeSimone, A., Dolzmann, G., Müller, S., Otto, F. (2003). Multiscale Modeling of Materials — the Role of Analysis. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds) Trends in Nonlinear Analysis. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05281-5_11
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