The mathematics of thin structures. (English) Zbl 1511.49011
The paper consists of five papers:
Section 1 gives a review of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case and rise to various models in plate theory (membrane, bending, von Kármán, etc.); Section 2 adds brittleness and delamination, Section 3 revisits the classical setting in a non-Euclidean elasticity induced by the presence of a prestrain in the models.
Thus the concern of this investigation is to study what happens to a thin three-dimensional elastic body when its thickness vanishes asymptotically?
Briefly, the first section contains discussions to the other sections and numerating and detailing the various topics that have been included or excluded from this volume. And also discusses the main historical steps that have led to this pursuits. It includes proven theorems with full proofs and remarks explaining and supporting those assumption and theories.
The second section presents fracture versus delamination of thin films. It starts with motivation and giving real examples of cracks that appear in some places. A thin vinyl sticker is bonded to a metal panel and exposed to atmospheric conditions. After exhibiting the work that had been carried out before, and all different models treated mathematical or otherwise and also criticizing some models for the lack of mathematical proofs, the paper states that the object of this note is that it is possible to rigorously derive a phenomenological model introduced in the literature before, starting from three-brittle fracture in the contest of linear elasticity, by letting the thickness of the film tend to zero. The description of the problem is given and proved through several theorems with some supporting remarks, pointing out some open questions and similar remarks.
Section 3: Geometry and morphogenesis of thin films. This section presents the authors’ choice of topics and results motivated by the mathematical study of curvature-driven morphogenises. The paper takes into account analytical results concerning the dimension reduction for prestrained materials only. While for a larger scope, review and a list of open problems and a reference for that is given. The article discusses: the set-up of non-Euclidean elasticity, thin prestrained films, some other energy scalings, the infinite hierarchy of \(\Gamma\) limits, the weak prestrain, classical linear elasticity (case of no prestrain), the infinite hierarchy of shell theories and the matching propertie, remarks for further investigations and follow up. The section is also equipped with many proved theorems, corollaries and mathematical analysis.
In Section 4, Micromagnetics of curved thin films, a definition of skymions in the mathematical framework of the vibrational theory of micromagnetism, a brief review of magnetic thin film in planar structures and the recent developments about curved thin films are presented. And analysis of magnetic skyrmions in spherical thin films with open questions in cylindrical surfaces are all discussed with concentration showing how simple geometries can be techniques for the analysis of more complex scenarios. Finally considered are: Magnetic skyrmions in curved geometries, the variation theory of micromagnetism, the planar thin-film regime, the curved thin-film regime, topologically protected states in spherical thin films, plus many figures, theorems, remarks explaining these ideas. It ends with conclusions and further outlook.
Section 5 deals with one-dimensional walls in thin film ferromagnetics. It starts with nicely written and clearly explained historical and existing remarks about magnetism. The aim of this paper is to give a brief overview and some open questions in the modeling and analysis of domain wall solutions in the ferromagnetic films with the magnetization lying mostly in the film plane. This has been achieved through a discussion of: micro magnetic energy functional, domain walls in bulk materials, micromagnetics of thin films, domain walls in thin films. The section ends with conclusions stating that recently a number of developments in modeling and analysis of the domain walls are more open questions than answers that will inspire the next generations of researchers in the calculus of variation and analysis of PDE to further advance this area of the intersection of mathematics and material science.
The whole volume ends with Acknowledgements and a list of (218) references.
- 1.
- The mathematics of thin structures – an introduction by G. Francfort and I. Fonseca,
- 2.
- Fracture versus delamination of thin films (by J. F. Babadjian),
- 3.
- Geometry and morphogenesis of thin films (by Marta Lewicka),
- 4.
- Micromagnetics of curved thin films (by Giovanni Di Fratta),
- 5.
- One-dimensional domain walls in thin film ferromagnets: an overview (by C. Muratov).
Section 1 gives a review of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case and rise to various models in plate theory (membrane, bending, von Kármán, etc.); Section 2 adds brittleness and delamination, Section 3 revisits the classical setting in a non-Euclidean elasticity induced by the presence of a prestrain in the models.
Thus the concern of this investigation is to study what happens to a thin three-dimensional elastic body when its thickness vanishes asymptotically?
Briefly, the first section contains discussions to the other sections and numerating and detailing the various topics that have been included or excluded from this volume. And also discusses the main historical steps that have led to this pursuits. It includes proven theorems with full proofs and remarks explaining and supporting those assumption and theories.
The second section presents fracture versus delamination of thin films. It starts with motivation and giving real examples of cracks that appear in some places. A thin vinyl sticker is bonded to a metal panel and exposed to atmospheric conditions. After exhibiting the work that had been carried out before, and all different models treated mathematical or otherwise and also criticizing some models for the lack of mathematical proofs, the paper states that the object of this note is that it is possible to rigorously derive a phenomenological model introduced in the literature before, starting from three-brittle fracture in the contest of linear elasticity, by letting the thickness of the film tend to zero. The description of the problem is given and proved through several theorems with some supporting remarks, pointing out some open questions and similar remarks.
Section 3: Geometry and morphogenesis of thin films. This section presents the authors’ choice of topics and results motivated by the mathematical study of curvature-driven morphogenises. The paper takes into account analytical results concerning the dimension reduction for prestrained materials only. While for a larger scope, review and a list of open problems and a reference for that is given. The article discusses: the set-up of non-Euclidean elasticity, thin prestrained films, some other energy scalings, the infinite hierarchy of \(\Gamma\) limits, the weak prestrain, classical linear elasticity (case of no prestrain), the infinite hierarchy of shell theories and the matching propertie, remarks for further investigations and follow up. The section is also equipped with many proved theorems, corollaries and mathematical analysis.
In Section 4, Micromagnetics of curved thin films, a definition of skymions in the mathematical framework of the vibrational theory of micromagnetism, a brief review of magnetic thin film in planar structures and the recent developments about curved thin films are presented. And analysis of magnetic skyrmions in spherical thin films with open questions in cylindrical surfaces are all discussed with concentration showing how simple geometries can be techniques for the analysis of more complex scenarios. Finally considered are: Magnetic skyrmions in curved geometries, the variation theory of micromagnetism, the planar thin-film regime, the curved thin-film regime, topologically protected states in spherical thin films, plus many figures, theorems, remarks explaining these ideas. It ends with conclusions and further outlook.
Section 5 deals with one-dimensional walls in thin film ferromagnetics. It starts with nicely written and clearly explained historical and existing remarks about magnetism. The aim of this paper is to give a brief overview and some open questions in the modeling and analysis of domain wall solutions in the ferromagnetic films with the magnetization lying mostly in the film plane. This has been achieved through a discussion of: micro magnetic energy functional, domain walls in bulk materials, micromagnetics of thin films, domain walls in thin films. The section ends with conclusions stating that recently a number of developments in modeling and analysis of the domain walls are more open questions than answers that will inspire the next generations of researchers in the calculus of variation and analysis of PDE to further advance this area of the intersection of mathematics and material science.
The whole volume ends with Acknowledgements and a list of (218) references.
Reviewer: Faitori Omer Salem (Tripoli)
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49S05 | Variational principles of physics |
| 74B20 | Nonlinear elasticity |
| 74K20 | Plates |
| 74K35 | Thin films |
| 53A35 | Non-Euclidean differential geometry |
Keywords:
thin three-dimensional elastic bodyReferences:
| [1] | Acerbi, Emilio, Existence and regularity for mixtures of micromagnetic materials, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2225-2243 (2006) · Zbl 1149.74333 · doi:10.1098/rspa.2006.1655 |
| [2] | A. Aharoni, Introduction to the Theory of Ferromagnetism, volume 109 of International Series of Monographs on Physics, Oxford University Press, New York, 2nd ed., 2001. |
| [3] | Acharya, Amit, The metric-restricted inverse design problem, Nonlinearity, 1769-1797 (2016) · Zbl 1342.74099 · doi:10.1088/0951-7715/29/6/1769 |
| [4] | S. Almi and E. Tasso, Brittle fracture in linearly elastic plates, Preprint, 2020. · Zbl 1509.49009 |
| [5] | Alouges, Fran\c{c}ois, Homogenization of composite ferromagnetic materials, Proc. A., 20150365, 19 pp. (2015) · Zbl 1371.82148 · doi:10.1098/rspa.2015.0365 |
| [6] | Alouges, Fran\c{c}ois, Liouville type results for local minimizers of the micromagnetic energy, Calc. Var. Partial Differential Equations, 525-560 (2015) · Zbl 1325.35208 · doi:10.1007/s00526-014-0757-2 |
| [7] | Ambrosio, Luigi, Entire solutions of semilinear elliptic equations in \(\mathbf R^3\) and a conjecture of De Giorgi, J. Amer. Math. Soc., 725-739 (2000) · Zbl 0968.35041 · doi:10.1090/S0894-0347-00-00345-3 |
| [8] | Ambrosio, Luigi, Fine properties of functions with bounded deformation, Arch. Rational Mech. Anal., 201-238 (1997) · Zbl 0890.49019 · doi:10.1007/s002050050051 |
| [9] | Ansini, Nadia, The nonlinear sieve problem and applications to thin films, Asymptot. Anal., 113-145 (2004) · Zbl 1210.35117 |
| [10] | Ansini, Nadia, The Neumann sieve problem and dimensional reduction: a multiscale approach, Math. Models Methods Appl. Sci., 681-735 (2007) · Zbl 1120.74040 · doi:10.1142/S0218202507002078 |
| [11] | O. Anza Hafsa and J.-P. Mandallena, The nonlinear membrane energy: variational derivation under the constraint “\( \det \nabla u > 0\)”, Bull. Sci. Math. 132 (2008). 272-291. · Zbl 1148.35004 |
| [12] | Atiyah, M. F., Geometry and kinematics of two skyrmions, Comm. Math. Phys., 391-422 (1993) · Zbl 0778.53051 |
| [13] | Babadjian, Jean-Fran\c{c}ois, Quasistatic evolution of a brittle thin film, Calc. Var. Partial Differential Equations, 69-118 (2006) · Zbl 1089.74037 · doi:10.1007/s00526-005-0369-y |
| [14] | Babadjian, Jean-Fran\c{c}ois, Lower semicontinuity of quasi-convex bulk energies in \(\text{SBV}\) and integral representation in dimension reduction, SIAM J. Math. Anal., 1921-1950 (2008) · Zbl 1151.74025 · doi:10.1137/060676416 |
| [15] | Babadjian, Jean-Fran\c{c}ois, Traces of functions of bounded deformation, Indiana Univ. Math. J., 1271-1290 (2015) · Zbl 1339.26030 · doi:10.1512/iumj.2015.64.5601 |
| [16] | Babadjian, Jean-Fran\c{c}ois, Reduced models for linearly elastic thin films allowing for fracture, debonding or delamination, Interfaces Free Bound., 545-578 (2016) · Zbl 1381.74147 · doi:10.4171/IFB/373 |
| [17] | Ball, John M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 337-403 (1976/77) · Zbl 0368.73040 · doi:10.1007/BF00279992 |
| [18] | Barker, Blake, Existence and stability of viscoelastic shock profiles, Arch. Ration. Mech. Anal., 491-532 (2011) · Zbl 1233.35023 · doi:10.1007/s00205-010-0363-1 |
| [19] | R. G. Barrera, G. A. Estevez, and J. Giraldo, Vector spherical harmonics and their application to magnetostatics, Eur. J. Phys. 6 (1885), no. 4, 287-294. |
| [20] | Bellettini, G., Compactness and lower semicontinuity properties in \(\operatorname{SBD}(\Omega )\), Math. Z., 337-351 (1998) · Zbl 0914.46007 · doi:10.1007/PL00004617 |
| [21] | A. Berger and H. P. Oepen, Magnetic domain walls in ultrathin fcc cobalt films, Phys. Rev. B 45 (1992), 12596-12599. |
| [22] | Bernand-Mantel, Anne, A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree \(\pm 1\) harmonic maps from \(\mathbb{R}^2\) to \(\mathbb{S}^2\), Arch. Ration. Mech. Anal., 219-299 (2021) · Zbl 1466.78007 · doi:10.1007/s00205-020-01575-7 |
| [23] | Bhattacharya, Kaushik, An asymptotic study of the debonding of thin films, Arch. Ration. Mech. Anal., 205-229 (2002) · Zbl 0999.74079 · doi:10.1007/s002050100177 |
| [24] | Bhattacharya, Kaushik, Plates with incompatible prestrain, Arch. Ration. Mech. Anal., 143-181 (2016) · Zbl 1382.74081 · doi:10.1007/s00205-015-0958-7 |
| [25] | F. Bloch, Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika, Z. Physik 74 (1932), 295-335. · Zbl 0003.42303 |
| [26] | A. Bogdanov and A. Hubert, Thermodynamically stable magnetic vortex states in magnetic crystals, Journal of Magnetism and Magnetic Materials 138 (1994), no.3, 255-269, 1994. DOI 10.1016/0304-8853(94)90046-9. |
| [27] | Bouchitt\'{e}, Guy, A global method for relaxation in \(W^{1,p}\) and in \(\text{SBV}_p\), Arch. Ration. Mech. Anal., 187-242 (2002) · Zbl 1028.49009 · doi:10.1007/s00205-002-0220-y |
| [28] | Bouchitt\'{e}, Guy, The Cosserat vector in membrane theory: a variational approach, J. Convex Anal., 351-365 (2009) · Zbl 1179.35015 |
| [29] | Bourdin, Blaise, The variational approach to fracture, J. Elasticity, 5-148 (2008) · Zbl 1176.74018 · doi:10.1007/s10659-007-9107-3 |
| [30] | Bourquin, Fr\'{e}d\'{e}ric, \( \Gamma \)-convergence et analyse asymptotique des plaques minces, C. R. Acad. Sci. Paris S\'{e}r. I Math., 1017-1024 (1992) · Zbl 0769.73043 |
| [31] | Braides, A., Brittle thin films, Appl. Math. Optim., 299-323 (2001) · Zbl 0999.49012 · doi:10.1007/s00245-001-0022-x |
| [32] | Braides, Andrea, Homogenization of multiple integrals, Oxford Lecture Series in Mathematics and its Applications, xiv+298 pp. (1998), The Clarendon Press, Oxford University Press, New York · Zbl 0911.49010 |
| [33] | A. Brataas, A. D. Kent, and H. Ohno, Current-induced torques in magnetic materials, Nature Mat. 11 (2012), 372-381. |
| [34] | W. F. Brown, Magnetostatic Principles in Ferromagnetism, North-Holland, Amsterdam, 1962. · Zbl 0116.23703 |
| [35] | W. F. Brown, Micromagnetics, Interscience Publishers, London, 1963. |
| [36] | W. F. Brown, Thermal fluctuations of a single-domain particle, Phys. Rev. 130 (1963), 1677-1686. |
| [37] | W. F. Brown, The fundamental theorem of the theory of fine ferromagnetic particles, Annals of the New York Academy of Sciences 147 (1969), no.12, 463-488. DOI 10.1111/j.1749-6632.1969.tb41269.x. |
| [38] | Cao, Wentao, Very weak solutions to the two-dimensional Monge-Amp\'{e}re equation, Sci. China Math., 1041-1056 (2019) · Zbl 1421.35185 · doi:10.1007/s11425-018-9516-7 |
| [39] | A. Capella, H. Kn\"upfer, and C. B. Muratov, Existence and structure of \(360^\circ\) walls in thin uniaxial ferromagnetic films, Preprint, 2021. |
| [40] | Capella, Antonio, Wave-type dynamics in ferromagnetic thin films and the motion of N\'{e}el walls, Nonlinearity, 2519-2537 (2007) · Zbl 1135.82037 · doi:10.1088/0951-7715/20/11/004 |
| [41] | Carbou, G., Thin layers in micromagnetism, Math. Models Methods Appl. Sci., 1529-1546 (2001) · Zbl 1012.82031 · doi:10.1142/S0218202501001458 |
| [42] | Chermisi, Milena, One-dimensional N\'{e}el walls under applied external fields, Nonlinearity, 2935-2950 (2013) · Zbl 1276.78003 · doi:10.1088/0951-7715/26/11/2935 |
| [43] | H. S. Cho, C. Hou, M. Sun, and H. Fujiwara, Characteristics of \(360^\circ \)-domain walls observed by magnetic force microscope in exchange-biased NiFe films, J. Appl. Phys. 85 (1999), 5160-5162. |
| [44] | Choksi, Rustum, Bounds on the micromagnetic energy of a uniaxial ferromagnet, Comm. Pure Appl. Math., 259-289 (1998) · Zbl 0909.49004 · doi:10.1002/(SICI)1097-0312(199803)51:3<259::AID-CPA3>3.0.CO;2-9 |
| [45] | Choksi, Rustum, Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy, Comm. Math. Phys., 61-79 (1999) · Zbl 1023.82011 · doi:10.1007/s002200050549 |
| [46] | Ciarlet, Philippe G., Mathematical elasticity. Vol. II, Studies in Mathematics and its Applications, lxiv+497 pp. (1997), North-Holland Publishing Co., Amsterdam · Zbl 0888.73001 |
| [47] | Ciarlet, P. G., A justification of the two-dimensional linear plate model, J. M\'{e}canique, 315-344 (1979) · Zbl 0415.73072 |
| [48] | Conti, Sergio, Nonlinear partial differential equations. \(h\)-principle and rigidity for \(C^{1,\alpha }\) isometric embeddings, Abel Symp., 83-116 (2012), Springer, Heidelberg · Zbl 1255.53038 · doi:10.1007/978-3-642-25361-4\_5 |
| [49] | R. Collette, Shape and energy of n[e-acute]el walls in very thin ferromagnetic films, J. Appl. Phys. 35 (1964), 3294-3301. |
| [50] | Conti, Sergio, Branched microstructures: scaling and asymptotic self-similarity, Comm. Pure Appl. Math., 1448-1474 (2000) · Zbl 1032.74044 · doi:10.1002/1097-0312(200011)53:11<1448::AID-CPA6>3.0.CO;2-C |
| [51] | Conti, Sergio, Analysis, modeling and simulation of multiscale problems. Derivation of elastic theories for thin sheets and the constraint of incompressibility, 225-247 (2006), Springer, Berlin · Zbl 1366.74043 · doi:10.1007/3-540-35657-6\_9 |
| [52] | Conti, S., Phase field approximation of cohesive fracture models, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 1033-1067 (2016) · Zbl 1345.49012 · doi:10.1016/j.anihpc.2015.02.001 |
| [53] | Conti, Sergio, Confining thin elastic sheets and folding paper, Arch. Ration. Mech. Anal., 1-48 (2008) · Zbl 1127.74005 · doi:10.1007/s00205-007-0076-2 |
| [54] | Conti, Sergio, Rigorous derivation of F\"{o}ppl’s theory for clamped elastic membranes leads to relaxation, SIAM J. Math. Anal., 657-680 (2006) · Zbl 1146.74025 · doi:10.1137/050632567 |
| [55] | Dal Maso, Gianni, An introduction to \(\Gamma \)-convergence, Progress in Nonlinear Differential Equations and their Applications, xiv+340 pp. (1993), Birkh\"{a}user Boston, Inc., Boston, MA · Zbl 0816.49001 · doi:10.1007/978-1-4612-0327-8 |
| [56] | Dal Maso, Gianni, Generalised functions of bounded deformation, J. Eur. Math. Soc. (JEMS), 1943-1997 (2013) · Zbl 1271.49029 · doi:10.4171/JEMS/410 |
| [57] | Dal Maso, Gianni, Fracture models as \(\Gamma \)-limits of damage models, Commun. Pure Appl. Anal., 1657-1686 (2013) · Zbl 1345.74097 · doi:10.3934/cpaa.2013.12.1657 |
| [58] | Davoli, E., Homogenization of chiral magnetic materials: a mathematical evidence of Dzyaloshinskii’s predictions on helical structures, J. Nonlinear Sci., 1229-1262 (2020) · Zbl 1439.35037 · doi:10.1007/s00332-019-09606-8 |
| [59] | Davoli, Elisa, Micromagnetics of thin films in the presence of Dzyaloshinskii-Moriya interaction, Math. Models Methods Appl. Sci., 911-939 (2022) · Zbl 1492.78005 · doi:10.1142/S0218202522500208 |
| [60] | De Giorgi, Ennio, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis. Convergence problems for functionals and operators, 131-188 (1978), Pitagora, Bologna · Zbl 0405.49001 |
| [61] | De Lellis, Camillo, A Nash-Kuiper theorem for \(C^{1,1/5-\delta }\) immersions of surfaces in 3 dimensions, Rev. Mat. Iberoam., 1119-1152 (2018) · Zbl 1421.53045 · doi:10.4171/RMI/1019 |
| [62] | Del Pino, Manuel, On De Giorgi’s conjecture and beyond, Proc. Natl. Acad. Sci. USA, 6845-6850 (2012) · Zbl 1309.35035 · doi:10.1073/pnas.1202687109 |
| [63] | De Simone, Antonio, Energy minimizers for large ferromagnetic bodies, Arch. Rational Mech. Anal., 99-143 (1993) · Zbl 0811.49030 · doi:10.1007/BF00376811 |
| [64] | De Simone, Antonio, Hysteresis and imperfection sensitivity in small ferromagnetic particles, Meccanica, 591-603 (1995) · Zbl 0836.73060 · doi:10.1007/BF01557087 |
| [65] | DeSimone, Antonio, 2-d stability of the N\'{e}el wall, Calc. Var. Partial Differential Equations, 233-253 (2006) · Zbl 1158.78300 · doi:10.1007/s00526-006-0019-z |
| [66] | DeSimone, Antonio, ICIAM 99 (Edinburgh). Magnetic microstructures-a paradigm of multiscale problems, 175-190 (2000), Oxford Univ. Press, Oxford · Zbl 0991.82038 |
| [67] | Desimone, Antonio, A reduced theory for thin-film micromagnetics, Comm. Pure Appl. Math., 1408-1460 (2002) · Zbl 1027.82042 · doi:10.1002/cpa.3028 |
| [68] | Desimone, Antonio, Repulsive interaction of N\'{e}el walls, and the internal length scale of the cross-tie wall, Multiscale Model. Simul., 57-104 (2003) · Zbl 1059.82046 · doi:10.1137/S1540345902402734 |
| [69] | The science of hysteresis. Vol. II, xii+648 pp. (2006), Elsevier/Academic Press, Amsterdam · Zbl 1117.34046 |
| [70] | H.-D. Dietze and H. Thomas, Bloch- und N\'eel-W\"ande in d\"unnen ferromagnetischen Schichten, Z. Physik 163 (1961), 523-534. |
| [71] | G. Di Fratta, C. Serpico, and M. d’Aquino, A generalization of the fundamental theorem of Brown for fine ferromagnetic particles, Physica B: Condensed Matter 407 (2012), no.9, 1368-1371. DOI 10.1016/j.physb.2011.10.010. |
| [72] | Di Fratta, Giovanni, The Newtonian potential and the demagnetizing factors of the general ellipsoid, Proc. A., 20160197, 7 pp. (2016) · Zbl 1371.78010 · doi:10.1098/rspa.2016.0197 |
| [73] | Di Fratta, Giovanni, Micromagnetics of curved thin films, Z. Angew. Math. Phys., Paper No. 111, 19 pp. (2020) · Zbl 1446.82087 · doi:10.1007/s00033-020-01336-2 |
| [74] | G. Di Fratta, A. Fiorenza, and V. Slastikov, On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces, 2110.08755, 2021. · Zbl 1465.46031 |
| [75] | Di Fratta, Giovanni, Symmetry properties of minimizers of a perturbed Dirichlet energy with a boundary penalization, SIAM J. Math. Anal., 3636-3653 (2022) · Zbl 1494.35011 · doi:10.1137/21M143011X |
| [76] | Di Fratta, Giovanni, Variational principles of micromagnetics revisited, SIAM J. Math. Anal., 3580-3599 (2020) · Zbl 1446.35197 · doi:10.1137/19M1261365 |
| [77] | Di Fratta, Giovanni, On a sharp Poincar\'{e}-type inequality on the 2-sphere and its application in micromagnetics, SIAM J. Math. Anal., 3373-3387 (2019) · Zbl 1420.35012 · doi:10.1137/19M1238757 |
| [78] | P. A. M. Dirac, On the theory of quantum mechanics, Proc. R. Soc. Lond. Ser. A 112 (1926), 661-677. · JFM 52.0975.01 |
| [79] | P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. Ser. A 117 (1928), 610-624. · JFM 54.0973.01 |
| [80] | do Carmo, Manfredo P., Differential geometry of curves and surfaces, viii+503 pp. (1976), Prentice-Hall, Inc., Englewood Cliffs, N.J. · Zbl 0816.53001 |
| [81] | D\"{o}ring, Lukas, A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types, J. Eur. Math. Soc. (JEMS), 1377-1422 (2014) · Zbl 1301.49124 · doi:10.4171/JEMS/464 |
| [82] | I. Dzyaloshinsky, A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics, Journal of Physics and Chemistry of Solids 4 (1958), no.4, 241-255. DOI 10.1016/0022-3697(58)90076-3. |
| [83] | Esteban, Maria J., Existence of 3D skyrmions. Erratum to: “A direct variational approach to Skyrme”s model for meson fields” [Comm. Math. Phys. {\bf 105} (1986), no. 4, 571-591; MR0852091] and “A new setting for Skyrme”s problem” [in Variational methods (Paris, 1988), 77-93, Birkh\"{a}user Boston, Boston, MA, 1990; MR1205147], Comm. Math. Phys., 209-210 (2004) · Zbl 1085.58501 · doi:10.1007/s00220-004-1139-y |
| [84] | Esteban, Maria J., Skyrmions and symmetry, Asymptotic Anal., 187-192 (1988) · Zbl 0825.58008 |
| [85] | E. Feldtkeller and W. Liesk, \(360^\circ -W\)\"ande in magnetischen Schichten, Z. Angew. Phys. 14 (1962), 195-199. |
| [86] | A. Fert, V. Cros, and J. Sampaio, Skyrmions on the track, Nature Nanotechnology 8 (2013), no.3, 152-156. DOI 10.1038/nnano.2013.29. |
| [87] | J. Fidler and T. Schrefl, Micromagnetic modelling-the current state of the art, J. Phys. D: Appl. Phys. 33 (2000), R135-R156. |
| [88] | Focardi, M., Asymptotic analysis of Ambrosio-Tortorelli energies in linearized elasticity, SIAM J. Math. Anal., 2936-2955 (2014) · Zbl 1301.49030 · doi:10.1137/130947180 |
| [89] | Francfort, G. A., Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 1319-1342 (1998) · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9 |
| [90] | Freddi, Lorenzo, Dimension reduction of a crack evolution problem in a linearly elastic plate, Asymptot. Anal., 101-123 (2010) · Zbl 1219.35306 |
| [91] | Friesecke, Gero, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris, 697-702 (2003) · Zbl 1140.74481 · doi:10.1016/S1631-073X(03)00028-1 |
| [92] | Friesecke, Gero, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 1461-1506 (2002) · Zbl 1021.74024 · doi:10.1002/cpa.10048 |
| [93] | Friesecke, Gero, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 183-236 (2006) · Zbl 1100.74039 · doi:10.1007/s00205-005-0400-7 |
| [94] | P. Gambardella and I. M. Miron. Current-induced spin-orbit torques, Phil. Trans. Roy. Soc. London, Ser. A 369 (2011), 3175-3197. |
| [95] | Y. Gaididei, V. P. Kravchuk, and D. D. Sheka, Curvature effects in thin magnetic shells, Physical Review Letters 112 (2014), no.25, 257203. DOI 10.1103/PhysRevLett.112.257203. |
| [96] | Garcia Cervera, Carlos Javier, Magnetic domains and magnetic domain walls, 132 pp. (1999), ProQuest LLC, Ann Arbor, MI |
| [97] | Garc\'{\i }a-Cervera, Carlos J., One-dimensional magnetic domain walls, European J. Appl. Math., 451-486 (2004) · Zbl 1094.82019 · doi:10.1017/S0956792504005595 |
| [98] | Garc\'{\i }a-Cervera, Carlos J., Numerical micromagnetics: a review, Bol. Soc. Esp. Mat. Apl. SeMA, 103-135 (2007) · Zbl 1242.82045 |
| [99] | Ghoussoub, N., On a conjecture of De Giorgi and some related problems, Math. Ann., 481-491 (1998) · Zbl 0918.35046 · doi:10.1007/s002080050196 |
| [100] | T. L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetisation field, Phys. Rev. 100 (1955). DOI 10.1103/PhysRev.100.1243. |
| [101] | T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn. 40 (2004), 3443-3449. |
| [102] | G. Gioia and R. D. James. Micromagnetics of very thin films. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 453(1956):213-223, 1997. DOI 10.1098/rspa.1997.0013. |
| [103] | A. A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond. 221A (1920), 163-198. · Zbl 1454.74137 |
| [104] | Gurtin, Morton E., Thermomechanics of evolving phase boundaries in the plane, Oxford Mathematical Monographs, xi+148 pp. (1993), The Clarendon Press, Oxford University Press, New York · Zbl 0787.73004 |
| [105] | Q. Han and M. Lewicka, Convex integration for the Monge-Amp\`ere systems, in preparation (2021). |
| [106] | Q. Han, M. Lewicka, and L. Mahadevan, Geodesics and isometric immersions in kirigami, Bull. Amer. Math. Soc. 59 (2022), 331-369. · Zbl 1492.35370 |
| [107] | Harutyunyan, Davit, Gaussian curvature as an identifier of shell rigidity, Arch. Ration. Mech. Anal., 743-766 (2017) · Zbl 1374.35400 · doi:10.1007/s00205-017-1143-y |
| [108] | B. Heinrich and J. F. Cochran, Ultrathin metallic magnetic films: magnetic anisotropies and exchange interactions, Adv. Phys. 42 (1993), 523-639. |
| [109] | W. Heisenberg, Mehrk\"orperproblem und Resonanz in der Quantenmechanik, Z. Physik 38 (1926), 411-426. · JFM 52.0962.02 |
| [110] | W. Heisenberg, Zur Theorie des Ferromagnetismus, Z. Physik 49 (1928), 619-636. · JFM 54.0979.02 |
| [111] | R. M. Hornreich, \(90^\) magnetization curling in thin films, J. Appl. Phys. 34 (1963), 1071-1072. |
| [112] | Hornung, Peter, Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells, J. Elasticity, 1-19 (2013) · Zbl 1322.74049 · doi:10.1007/s10659-012-9391-4 |
| [113] | A. Hubert and R. Sch\"afer, Magnetic Domains, Springer, Berlin, 1998. |
| [114] | A. Hubert and R. Sch\"afer, Magnetic Domains: The Analysis of Magnetic Microstructures, Springer Science & Business Media, 2008. |
| [115] | Ignat, Radu, A \(\Gamma \)-convergence result for N\'{e}el walls in micromagnetics, Calc. Var. Partial Differential Equations, 285-316 (2009) · Zbl 1175.49014 · doi:10.1007/s00526-009-0229-2 |
| [116] | Ignat, R., Renormalized energy between vortices in some Ginzburg-Landau models on 2-dimensional Riemannian manifolds, Arch. Ration. Mech. Anal., 1577-1666 (2021) · Zbl 1462.35375 · doi:10.1007/s00205-020-01598-0 |
| [117] | Ignat, Radu, Vortex energy and \(360^\circ N\)\'{e}el walls in thin-film micromagnetics, Comm. Pure Appl. Math., 1677-1724 (2010) · Zbl 1200.49046 · doi:10.1002/cpa.20322 |
| [118] | Ignat, Radu, Global Jacobian and \(\Gamma \)-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices, J. Funct. Anal., Paper No. 108928, 66 pp. (2021) · Zbl 1458.35396 · doi:10.1016/j.jfa.2021.108928 |
| [119] | Ignat, Radu, Entropy method for line-energies, Calc. Var. Partial Differential Equations, 375-418 (2012) · Zbl 1241.49010 · doi:10.1007/s00526-011-0438-3 |
| [120] | Ignat, Radu, A zigzag pattern in micromagnetics, J. Math. Pures Appl. (9), 139-159 (2012) · Zbl 1248.49064 · doi:10.1016/j.matpur.2012.01.005 |
| [121] | Ignat, Radu, N\'{e}el walls with prescribed winding number and how a nonlocal term can change the energy landscape, J. Differential Equations, 5846-5901 (2017) · Zbl 1402.35096 · doi:10.1016/j.jde.2017.07.006 |
| [122] | Bola\~{n}os, Silvia Jim\'{e}nez, Dimension reduction for thin films prestrained by shallow curvature, Proc. A., Paper No. 20200854, 24 pp. (2021) |
| [123] | C. Kittel, Physical theory of ferromagnetic domains, Rev. Mod. Phys. 21 (1949), 541-583. |
| [124] | Klein, Yael, Shaping of elastic sheets by prescription of non-Euclidean metrics, Science, 1116-1120 (2007) · Zbl 1226.74013 · doi:10.1126/science.1135994 |
| [125] | Kohn, Robert V., Another thin-film limit of micromagnetics, Arch. Ration. Mech. Anal., 227-245 (2005) · Zbl 1074.78012 · doi:10.1007/s00205-005-0372-7 |
| [126] | Kn\"{u}pfer, Hans, Domain structure of bulk ferromagnetic crystals in applied fields near saturation, J. Nonlinear Sci., 921-962 (2011) · Zbl 1298.35014 · doi:10.1007/s00332-011-9105-2 |
| [127] | Kn\"{u}pfer, Hans, Magnetic domains in thin ferromagnetic films with strong perpendicular anisotropy, Arch. Ration. Mech. Anal., 727-761 (2019) · Zbl 1412.78005 · doi:10.1007/s00205-018-1332-3 |
| [128] | Kn\"{u}pfer, Hans, \( \Gamma \)-limit for two-dimensional charged magnetic zigzag domain walls, Arch. Ration. Mech. Anal., 1875-1923 (2021) · Zbl 1462.82083 · doi:10.1007/s00205-021-01606-x |
| [129] | Kohn, Robert V., International Congress of Mathematicians. Vol. I. Energy-driven pattern formation, 359-383 (2007), Eur. Math. Soc., Z\"{u}rich · Zbl 1140.49030 · doi:10.4171/022-1/15 |
| [130] | Kohn, Robert V., Another thin-film limit of micromagnetics, Arch. Ration. Mech. Anal., 227-245 (2005) · Zbl 1074.78012 · doi:10.1007/s00205-005-0372-7 |
| [131] | V. P. Kravchuk, D. D. Sheka, R. Streubel, D. Makarov, O. G. Schmidt, and Y. Gaididei, Out-of-surface vortices in spherical shells, Physical Review B 85 (2012), no.14, 1-6. DOI 10.1103/PhysRevB.85.144433. |
| [132] | Kupferman, Raz, A Riemannian approach to reduced plate, shell, and rod theories, J. Funct. Anal., 2989-3039 (2014) · Zbl 1305.74022 · doi:10.1016/j.jfa.2013.09.003 |
| [133] | Kurzke, Matthias, Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation, Arch. Ration. Mech. Anal., 843-888 (2011) · Zbl 1229.35283 · doi:10.1007/s00205-010-0356-0 |
| [134] | A. E. LaBonte, Two dimensional Bloch-type domain walls in ferromagnetic films, J. Appl. Phys. 40 (1969), 2450-2458. |
| [135] | L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Perspectives in Theoretical Physics 8 (2012), no.153, 51-65. DOI: 10.1016/b978-0-08-036364-6.50008-9. · Zbl 0012.28501 |
| [136] | Landau, L. D., Course of theoretical physics. Vol. 7, viii+187 pp. (1986), Pergamon Press, Oxford |
| [137] | Le Dret, H., The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci., 59-84 (1996) · Zbl 0844.73045 · doi:10.1007/s003329900003 |
| [138] | Le Dret, Herv\'{e}, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 549-578 (1995) · Zbl 0847.73025 |
| [139] | J. Leliaert, M. Dvornik, J. Mulkers, J. De Clercq, M. V. Milosevi\'c, and B. Van Waeyenberge, Fast micromagnetic simulations on GPUrecent advances made with Mumax3, J. Phys. D: Appl. Phys. 51 (2018), 123002. |
| [140] | Lewicka, Marta, Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models, Arch. Ration. Mech. Anal., 1677-1707 (2020) · Zbl 1433.74078 · doi:10.1007/s00205-020-01500-y |
| [141] | Lewicka, Marta, Dimension reduction for thin films with transversally varying prestrain: oscillatory and nonoscillatory cases, Comm. Pure Appl. Math., 1880-1932 (2020) · Zbl 1452.49026 · doi:10.1002/cpa.21871 |
| [142] | Lewicka, Marta, Geometry, analysis, and morphogenesis: problems and prospects, Bull. Amer. Math. Soc. (N.S.), 331-369 (2022) · Zbl 1492.35370 · doi:10.1090/bull/1765 |
| [143] | Lewicka, Marta, The F\"{o}ppl-von K\'{a}rm\'{a}n equations for plates with incompatible strains, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 402-426 (2011) · Zbl 1219.74027 · doi:10.1098/rspa.2010.0138 |
| [144] | M. Lewicka, L. Mahadevan, and R. Pakzad, Models for elastic shells with incompatible strains, Proceedings of the Royal Society A 470 (2014), 20130604. · Zbl 1219.74027 |
| [145] | Lewicka, Marta, The Monge-Amp\`ere constraint: matching of isometries, density and regularity, and elastic theories of shallow shells, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 45-67 (2017) · Zbl 1359.35085 · doi:10.1016/j.anihpc.2015.08.005 |
| [146] | Lewicka, Marta, A nonlinear theory for shells with slowly varying thickness, C. R. Math. Acad. Sci. Paris, 211-216 (2009) · Zbl 1168.74036 · doi:10.1016/j.crma.2008.12.017 |
| [147] | Lewicka, Marta, Shell theories arising as low energy \(\Gamma \)-limit of 3d nonlinear elasticity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 253-295 (2010) · Zbl 1425.74298 |
| [148] | Lewicka, Marta, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Ration. Mech. Anal., 1023-1050 (2011) · Zbl 1291.74130 · doi:10.1007/s00205-010-0387-6 |
| [149] | Lewicka, Marta, A local and global well-posedness results for the general stress-assisted diffusion systems, J. Elasticity, 19-41 (2016) · Zbl 1382.35296 · doi:10.1007/s10659-015-9545-2 |
| [150] | Lewicka, Marta, The uniform Korn-Poincar\'{e} inequality in thin domains, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 443-469 (2011) · Zbl 1253.74055 · doi:10.1016/j.anihpc.2011.03.003 |
| [151] | Lewicka, Marta, On the optimal constants in Korn’s and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions, Indiana Univ. Math. J., 377-397 (2016) · Zbl 1405.74010 · doi:10.1512/iumj.2016.65.5797 |
| [152] | Lewicka, Marta, Variational models for prestrained plates with Monge-Amp\`ere constraint, Differential Integral Equations, 861-898 (2015) · Zbl 1363.74063 |
| [153] | Lewicka, Marta, Scaling laws for non-Euclidean plates and the \(W^{2,2}\) isometric immersions of Riemannian metrics, ESAIM Control Optim. Calc. Var., 1158-1173 (2011) · Zbl 1300.74028 · doi:10.1051/cocv/2010039 |
| [154] | Lewicka, Marta, Infinite dimensional dynamical systems. The infinite hierarchy of elastic shell models: some recent results and a conjecture, Fields Inst. Commun., 407-420 (2013), Springer, New York · Zbl 1263.74035 · doi:10.1007/978-1-4614-4523-4\_16 |
| [155] | Lewicka, Marta, Convex integration for the Monge-Amp\`ere equation in two dimensions, Anal. PDE, 695-727 (2017) · Zbl 1370.35151 · doi:10.2140/apde.2017.10.695 |
| [156] | Lewicka, Marta, Plates with incompatible prestrain of high order, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 1883-1912 (2017) · Zbl 1457.74121 · doi:10.1016/j.anihpc.2017.01.003 |
| [157] | Li, Xinye, Lattice solutions in a Ginzburg-Landau model for a chiral magnet, J. Nonlinear Sci., 3389-3420 (2020) · Zbl 1471.35266 · doi:10.1007/s00332-020-09654-5 |
| [158] | Lin, Fanghua, Perspectives in nonlinear partial differential equations. Analysis on Faddeev knots and Skyrme solitons: recent progress and open problems, Contemp. Math., 319-344 (2007), Amer. Math. Soc., Providence, RI · Zbl 1200.81106 · doi:10.1090/conm/446/08639 |
| [159] | Lund, Ross G., One-dimensional domain walls in thin ferromagnetic films with fourfold anisotropy, Nonlinearity, 1716-1734 (2016) · Zbl 1342.78012 · doi:10.1088/0951-7715/29/6/1716 |
| [160] | Lund, Ross G., Edge domain walls in ultrathin exchange-biased films, J. Nonlinear Sci., 1165-1205 (2020) · Zbl 1434.82089 · doi:10.1007/s00332-019-09604-w |
| [161] | Lund, Ross G., One-dimensional in-plane edge domain walls in ultrathin ferromagnetic films, Nonlinearity, 728-754 (2018) · Zbl 1388.78007 · doi:10.1088/1361-6544/aa96c8 |
| [162] | Iurlano, Flaviana, Fracture and plastic models as \(\Gamma \)-limits of damage models under different regimes, Adv. Calc. Var., 165-189 (2013) · Zbl 1379.74003 · doi:10.1515/acv-2011-0011 |
| [163] | J. W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials, Adv. Appl. Mech. 29 (1991), 63-191. · Zbl 0790.73056 |
| [164] | Le\'{o}n Baldelli, Andr\'{e}s Alessandro, Fracture and debonding of a thin film on a stiff substrate: analytical and numerical solutions of a one-dimensional variational model, Contin. Mech. Thermodyn., 243-268 (2013) · Zbl 1343.74032 · doi:10.1007/s00161-012-0245-x |
| [165] | Le\'{o}n Baldelli, A. A., A variational model for fracture and debonding of thin films under in-plane loadings, J. Mech. Phys. Solids, 320-348 (2014) · Zbl 1328.74059 · doi:10.1016/j.jmps.2014.05.020 |
| [166] | Le\'{o}n Baldelli, Andr\'{e}s A., On the asymptotic derivation of Winkler-type energies from 3D elasticity, J. Elasticity, 275-301 (2015) · Zbl 1330.35439 · doi:10.1007/s10659-015-9528-3 |
| [167] | Manton, N. S., Geometry of skyrmions, Comm. Math. Phys., 469-478 (1987) · Zbl 0638.58029 |
| [168] | R. Mattheis, K. Ramst\"ock, and J. McCord, Formation and annihilation of edge walls in thin-film permalloy strips, IEEE Trans. Magn. 33 (1997), 3993-3995. |
| [169] | Melcher, Christof, The logarithmic tail of N\'{e}el walls, Arch. Ration. Mech. Anal., 83-113 (2003) · Zbl 1151.82437 · doi:10.1007/s00205-003-0248-7 |
| [170] | Melcher, Christof, Chiral skyrmions in the plane, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 20140394, 17 pp. (2014) · Zbl 1371.81320 · doi:10.1098/rspa.2014.0394 |
| [171] | Melcher, Christof, Curvature-stabilized skyrmions with angular momentum, Lett. Math. Phys., 2291-2304 (2019) · Zbl 1426.49051 · doi:10.1007/s11005-019-01188-6 |
| [172] | Milnor, John W., Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, xii+64 pp. (1997), Princeton University Press, Princeton, NJ · Zbl 1025.57002 |
| [173] | Modica, Luciano, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., 123-142 (1987) · Zbl 0616.76004 · doi:10.1007/BF00251230 |
| [174] | M. Morini, C. B. Muratov, M. Novaga, and V. V. Slastikov, Transverse domain walls in thin ferromagnetic strips, arXiv:2106.01338, 2021. · Zbl 1517.82045 |
| [175] | Morini, M., Geometrically constrained walls in two dimensions, Arch. Ration. Mech. Anal., 621-692 (2012) · Zbl 1287.74012 · doi:10.1007/s00205-011-0458-3 |
| [176] | T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism, Physical review 120 (1960), no.1, 91. |
| [177] | Moser, Roger, Boundary vortices for thin ferromagnetic films, Arch. Ration. Mech. Anal., 267-300 (2004) · Zbl 1099.82025 · doi:10.1007/s00205-004-0329-2 |
| [178] | Muratov, C. B., Optimal grid-based methods for thin film micromagnetics simulations, J. Comput. Phys., 637-653 (2006) · Zbl 1136.78329 · doi:10.1016/j.jcp.2005.12.018 |
| [179] | C. B. Muratov and V. V. Osipov, Theory of \(360^\circ\) domain walls in thin ferromagnetic films, J. Appl. Phys. 104 (2008), 053908. |
| [180] | Muratov, Cyrill B., Domain structure of ultrathin ferromagnetic elements in the presence of Dzyaloshinskii-Moriya interaction, Proc. A., 20160666, 19 pp. (2017) · Zbl 1404.82101 · doi:10.1098/rspa.2016.0666 |
| [181] | Muratov, Cyrill B., Uniqueness of one-dimensional N\'{e}el wall profiles, Proc. A., 20150762, 15 pp. (2016) · Zbl 1371.82151 · doi:10.1098/rspa.2015.0762 |
| [182] | N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nature Nanotechnology 8 (2013), no.12, 899-911. DOI 10.1038/nnano.2013.243. |
| [183] | J. C. N\'ed\'elec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Applied Mathematical Sciences Book, volume 144, Springer Science & Business Media, 2001. DOI DNLBO01000933132. · Zbl 0981.35002 |
| [184] | L. N\'eel, Les lois de l’aimantation et de la subdivision en domaines \'el\'ementaires d’un monocristal de fer, J. Phys. Radium 5 (1944), 241-251. |
| [185] | L. N\'eel, Les lois de l’aimantation et de la subdivision en domaines \'el\'ementaires d’un monocristal de fer, J. Phys. Radium 5 (1944), 265-276. |
| [186] | L. N\'eel, Quelques propri\'et\'es des parois des domaines \'el\'ementaires ferromagn\'etiques, Cah. Phys. 25 (1944), 1-20. |
| [187] | L. N\'eel, Energie des parois de Bloch dans les couches minces, C. R. Hebd. Seances Acad. Sci. 241 (1955), 533-537. |
| [188] | NIST, The object oriented micromagnetic framework (OOMMF) project at ITL/NIST, https://math.nist.gov/oommf/. |
| [189] | Otto, Felix, The concertina pattern: from micromagnetics to domain theory, Calc. Var. Partial Differential Equations, 139-181 (2010) · Zbl 1237.78036 · doi:10.1007/s00526-009-0305-7 |
| [190] | Otto, Felix, Domain branching in uniaxial ferromagnets: asymptotic behavior of the energy, Calc. Var. Partial Differential Equations, 135-181 (2010) · Zbl 1193.49046 · doi:10.1007/s00526-009-0281-y |
| [191] | Outerelo, Enrique, Mapping degree theory, Graduate Studies in Mathematics, x+244 pp. (2009), American Mathematical Society, Providence, RI; Real Sociedad Matem\'{a}tica Espa\~{n}ola, Madrid · Zbl 1183.47056 · doi:10.1090/gsm/108 |
| [192] | W. Pauli \"Uber den Einfluder Geschwindigkeitsabh\"angigkeit der Elektronenmasse auf den Zeemaneffekt, Z. Physik 31 (1925), 373-385. · JFM 51.0742.02 |
| [193] | X. Portier and A. K. Petford-Long, The formation of \(360^\circ\) domain walls in magnetic tunnel junction elements, Appl. Phys. Lett. 76 (2000), 754-756. |
| [194] | R. Riedel and A. Seeger, Micromagnetic treatment of N\'eel walls, Phys. Stat. Sol. B 46 (1971), 377-384. |
| [195] | Roub\'{\i }\v{c}ek, Tom\'{a}\v{s}, Quasistatic delamination problem, Contin. Mech. Thermodyn., 223-235 (2009) · Zbl 1179.74130 · doi:10.1007/s00161-009-0106-4 |
| [196] | M. R\"uhrig, W. Rave, and A. Hubert, Investigation of micromagnetic edge structures of double-layer permalloy films, J. Magn. Magn. Mater. 84 (1990), 102-108. |
| [197] | M. R. Scheinfein, J. Unguris, J. L. Blue, K. J. Coakley, D. T. Pierce, R. J. Celotta, and P. J. Ryan, Micromagnetics of domain walls at surfaces, Phys. Rev. B 43 (1991), 3395-3422. |
| [198] | C. J. Serna, S. Veintemillas-Verdaguer, T. Gonz\'alez-Carre\~no, A. G. Roca, P. Tartaj, A. F. Rebolledo, R. Costo, and M. P. Morales, Progress in the preparation of magnetic nanoparticles for applications in biomedicine, Journal of Physics D: Applied Physics 42 (2009), no.22, 224002. DOI 10.1088/0022-3727/42/22/224002. |
| [199] | D. D. Sheka, D. Makarov, H. Fangohr, O. M. Volkov, H. Fuchs, J. van den Brink, Y. Gaididei, U. K. R\"oler, and V. P. Kravchuk, Topologically stable magnetization states on a spherical shell: Curvature-stabilized skyrmions, Physical Review B 94 (2016), no.14, 1-10. DOI 10.1103/physrevb.94.144402. |
| [200] | Skyrme, T. H. R., A unified field theory of mesons and baryons, Nuclear Phys., 556-569 (1962) |
| [201] | Slastikov, Valeriy, Micromagnetics of thin shells, Math. Models Methods Appl. Sci., 1469-1487 (2005) · Zbl 1111.82063 · doi:10.1142/S021820250500087X |
| [202] | M. I. Sloika, D. D. Sheka, V. P. Kravchuk, O. V. Pylypovskyi, and Y. Gaididei, Geometry induced phase transitions in magnetic spherical shell, Journal of Magnetism and Magnetic Materials 443 (2017), 404-412. DOI 10.1016/j.jmmm.2017.07.036. |
| [203] | E. C. Stoner and E. P. Wohlfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 240 (1948), no.826, 599-642. · Zbl 0031.38003 |
| [204] | Temam, Roger, Functions of bounded deformation, Arch. Rational Mech. Anal., 7-21 (1980/81) · Zbl 0472.73031 · doi:10.1007/BF00284617 |
| [205] | R. Streubel, D. Makarov, Y. Gaididei, O. G. Schmidt, V. P. Kravchuk, D. D. Sheka, F. Kronast, and P. Fischer, Magnetism in curved geometries, Journal of Physics D: Applied Physics 49 (2016), no.36, 363001. DOI 10.1088/0022-3727/49/36/363001. |
| [206] | R. Streubel, E. Y. Tsymbal, and P. Fischer, Magnetism in curved geometries, Journal of Applied Physics 129 (2021), no.21, 210902. |
| [207] | Suquet, Pierre-M., Un espace fonctionnel pour les \'{e}quations de la plasticit\'{e}, Ann. Fac. Sci. Toulouse Math. (5), 77-87 (1979) · Zbl 0405.46027 |
| [208] | Trabelsi, Karim, Modeling of a membrane for nonlinearly elastic incompressible materials via gamma-convergence, Anal. Appl. (Singap.), 31-60 (2006) · Zbl 1086.74025 · doi:10.1142/S0219530506000693 |
| [209] | Truesdell, C., Contemporary developments in continuum mechanics and partial differential equations. Some challenges offered to analysis by rational thermomechanics, North-Holland Math. Stud., 495-603 (1977), North-Holland, Amsterdam-New York · Zbl 0409.73097 |
| [210] | T. Trunk, M. Redjdal, A. K\'akay, M. F. Ruane, and F. B. Humphrey, Domain wall structure in permalloy films with decreasing thickness at the Bloch to N\'eel transition, J. Appl. Phys. 89 (2001), 7606-7608. |
| [211] | R. H. Wade, Some factors in easy axis magnetization of permalloy films, Phil. Mag. 10 (1964), 49-66. |
| [212] | P. Weiss, L’hypoth\`ese du champ mol\'eculaire et la propri\'et\'e ferromagn\'etique, J. Phys. Theor. Appl. 6 (1907), 661-690. · JFM 38.0901.02 |
| [213] | P. Weiss and G. Fo\"ex, Magn\'etisme, Armand Colin, Paris, 1926. · JFM 52.0914.04 |
| [214] | E. Winkler, Die Lehre Von der Elasticit\"at und Festigkeit, Dominicus, Prague, 1867, 388. · JFM 01.0350.01 |
| [215] | J. M. Winter, Bloch wall excitation. Application to nuclear resonance in a Bloch wall, Phys. Rev. 124 (1961), 452-459. · Zbl 0098.44705 |
| [216] | Z. C. Xia and J. W. Hutchinson, Crack patterns in thin films, J. Mech. Phys. Solids 48 (2000), 1107-1131. · Zbl 0966.74063 |
| [217] | Yao, Peng-Fei, Optimal exponentials of thickness in Korn’s inequalities for parabolic and elliptic shells, Ann. Mat. Pura Appl. (4), 379-401 (2021) · Zbl 1460.74059 · doi:10.1007/s10231-020-01000-6 |
| [218] | Zhang, Kewei, Quasiconvex functions, \( \operatorname{SO}(n)\) and two elastic wells, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 759-785 (1997) · Zbl 0918.49014 · doi:10.1016/S0294-1449(97)80132-1 |
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