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Volume 27, 2021
Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
Article Number S16
Number of page(s) 35
Published online 01 March 2021
  1. H. Abels, M.G. Mora and S. Müller, The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity. Calc. Var. Partial Differ. Equ. 41 (2011) 241–259. [Google Scholar]
  2. E. Acerbi, G. Buttazzo and D. Percivale, Thin inclusions in linear elasticity: A variational approach. J. Reine Angew. Math. 386 (1988) 99–115. [Google Scholar]
  3. E. Acerbi, G. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string. J. Elast. 25 (1991) 137–148. [Google Scholar]
  4. G. Anzellotti, S. Baldo and D. Percivale, Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity. Asymptot. Anal. 9 (1994) 61–100. [Google Scholar]
  5. H. B. Belgacem, S. Conti, A. DeSimone and S. Müller, Energy scaling of compressed elastic films – three-dimensional elasticity and reduced theories. Arch. Ration. Mech. Anal. 164 (2002) 1–37. [Google Scholar]
  6. K. Bhattacharya, M. Lewicka and M. Schäffner, Plates with incompatible prestrain. Arch. Ration. Mech. Anal. 221 (2016) 143–181. [Google Scholar]
  7. A. Braides, A handbook of Γ-convergence, in Stationary Partial Differential Equations, edited by M. Chipot and P. Quittner, Vol. 3. Handbook of Differential Equations. Elsevier (2006) 101–213. [Google Scholar]
  8. J. Braun and B. Schmidt, An atomistic derivation of von-Kármán plate theory. Preprint (2019). [Google Scholar]
  9. P.G. Ciarlet, Mathematical Elasticity. Vol. II: Theory of Plates, Vol. 27. Studies in Mathematics and Its Applications. North-Holland Publishing Co., Amsterdam (1997). [Google Scholar]
  10. P.G. Ciarlet, Mathematical Elasticity. Vol. III: Theory of Shells, Vol. 29. Studies in Mathematics and Its Applications. North-Holland Publishing Co., Amsterdam (2000). [Google Scholar]
  11. S. Conti, Low-Energy Deformations of Thin Elastic Plates: Isometric Embeddings and Branching Patterns. Habilitations-schreiben, Universität Leipzig (2004). [Google Scholar]
  12. S. Conti and G. Dolzmann, Γ-convergence for incompressible elastic plates. Calc. Var. Partial Diff. Equ. 34 (2009) 531–551. [Google Scholar]
  13. S. Conti and F. Maggi, Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. 187 (2008) 1–48. [Google Scholar]
  14. M. de Benito Delgado, Effective two dimensional theories for multi-layered plates. Doctoral dissertation, Universität Augsburg (2019). [Google Scholar]
  15. M. de Benito Delgado and B. Schmidt, Energy minimizing configurations of pre-strained multilayers. J. Elast. 140 (2020) 303–335. [Google Scholar]
  16. E. Efrati, E. Sharon and R. Kupferman, Elastic theory of unconstrained non-Euclidean plates. J. Mech. Phys. Solids 57 (2009) 762–775. [Google Scholar]
  17. A.I. Egunov, J.G. Korvink and V.A. Luchnikov, Polydimethylsiloxane bilayer films with an embedded spontaneous curvature. Soft Matter 12 (2016) 45–52. [PubMed] [Google Scholar]
  18. L. Euler, Methodus inveniendi lineas curvas, additamentum I: De curvis elasticis (1744), in Opera Omnia Ser. Prima, Vol. XXIV. Orell Füssli, Bern (1952) 231–297. [Google Scholar]
  19. M. Finot and S. Suresh, Small and large deformation of thick and thin-film multi-layers: Effects of layer geometry, plasticity and compositional gradients. Mechanics and Physics of Layered and Graded Materials. J. Mech. Phys. Solids 44 (1996) 683–721. [Google Scholar]
  20. L.B. Freund, Substrate curvature due to thin film mismatch strain in the nonlinear deformation range. The J. R. Willis 60th anniversary volume. J. Mech. Phys. Solids 48 (2000) 1159–1174. [Google Scholar]
  21. G. Friesecke, R.D. James, M.G. Mora and S. Müller. Derivation of nonlinear bending theory for shells from three-dimensional nonlinearelasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris 336 (2003) 697–702. [CrossRef] [MathSciNet] [Google Scholar]
  22. G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55 (2002) 1461–1506. [Google Scholar]
  23. G. Friesecke, R.D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by Γ-convergence. Arch. Ration. Mech. Anal. 180 (2006) 183–236. [Google Scholar]
  24. M. Grundmann, Nanoscroll formation from strained layer heterostructures. Appl. Phys. Lett. 83 (2003) 2444–2446. [Google Scholar]
  25. P. Hornung, Approximation of flat W2,2 isometric immersions by smooth ones. Arch. Ration. Mech. Anal. 199 (2011) 1015–1067. [Google Scholar]
  26. P. Hornung, Fine level set structure of flat isometric immersions. Arch. Ration. Mech. Anal. 199 (2011) 943–1014. [Google Scholar]
  27. P. Hornung, S. Neukamm, and I. Velčić, Derivation of a homogenized nonlinear plate theory from 3d elasticity. Calc. Var. Partial Diff. Equ. 51 (2014) 677–699. [Google Scholar]
  28. P. Hornung, M. Pawelczyk and I. Velčić, Stochastic homogenization of the bending plate model. J. Math. Anal. Appl. 458 (2018) 1236–1273. [Google Scholar]
  29. C.S. Kim and S.J. Lombardo, Curvature and bifurcation of MgO-Al2O3 bilayer ceramic structures. J. Ceram. Process. Res. 9 (2008) 93–96. [Google Scholar]
  30. G. Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40 (1850) 51–88. [Google Scholar]
  31. Y. Klein, E. Efrati and E. Sharon, Shaping of elastic sheets by prescription of non-euclidean metrics. Science 315 (2007) 1116–1120. [Google Scholar]
  32. R. Kupferman and J.P. Solomon, A Riemannian approach to reduced plate, shell, and rod theories. J. Funct. Anal. 266 (2014) 2989–3039. [Google Scholar]
  33. H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549–578. [Google Scholar]
  34. M. Lewicka and D. Lučić, Dimension reduction for thin films with transversally varying prestrian: the oscillatory and the non-oscillatory case. Preprint (2018). [Google Scholar]
  35. M. Lewicka, L. Mahadevan and M.R. Pakzad, The Föppl-von Kármán equations for plates with incompatible strains. Proc. Roy. Soc. London Ser. A. Math. Phys. Eng. Sci. 467 (2011) 402–426. [Google Scholar]
  36. M. Lewicka, M.G. Mora and M.R. Pakzad, Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity. Annali della Scuola normale superiore di Pisa, Classe di scienze 9 (2008) 253–295. [Google Scholar]
  37. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover Publications, New York (1944). [Google Scholar]
  38. C. Maor and A. Shachar, On the role of curvature in the elastic energy of non-Euclidean thin bodies. J. Elast. 134 (2019) 149–173. [Google Scholar]
  39. C.B. Masters and N. Salamon, Geometrically nonlinear stress-deflection relations for thin film/substrate systems. Int. J. Eng. Sci. 31 (1993) 915–925. [Google Scholar]
  40. M.G. Mora, S. Müller and M.G. Schultz, Convergence of equilibria of planar thin elastic beams. Indiana Univ. Math. J. 56 (2007) 2413–2438. [CrossRef] [MathSciNet] [Google Scholar]
  41. S. Müller, Mathematical problems in thin elastic sheets: scaling limits, packing, crumpling and singularities, in Vector-Valued Partial Differential Equations and Applications, Vol. 2179. Lecture Notes Math. Springer, Cham (2017) 125–193. [Google Scholar]
  42. S. Müller and M.R. Pakzad, Regularity properties of isometric immersions. Mathematische Zeitschrift 251 (2005) 313–331. [Google Scholar]
  43. S. Müller and M.R. Pakzad, Convergence of equilibria of thin elastic plates—the von Kármán case. Comm. Partial Differ. Equ. 33 (2008) 1018–1032. [Google Scholar]
  44. S. Neukamm and I. Velčić, Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity. Math. Models Methods Appl. Sci. 23 (2013) 2701–2748. [Google Scholar]
  45. H. Paetzelt, V. Gottschalch, J. Bauer, H. Herrnberger and G. Wagner. Fabrication of III–V nano- and microtubes using MOVPE grown materials. Physica Status Solidi (A) 203 (2006) 817–824. [Google Scholar]
  46. M.R. Pakzad, On the Sobolev space of isometric immersions. J. Differ. Geom. 66 (2004) 47–69. [Google Scholar]
  47. V.Y. Prinz, D. Grützmacher, A. Beyer, C. David, B. Ketterer and E. Deckardt, A new technique for fabricating three-dimensional micro- and nanostructures of various shapes. Nanotechnology 12 (2001) 399–402. [Google Scholar]
  48. J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press (2003). [Google Scholar]
  49. N. Salamon and C.B. Masters, Bifurcation in isotropic thinfilm/substrate plates. Special topics in the theory of elastic: A volume in honour of Professor John Dundurs. Int. J. Solids Struct. 32 (1995) 473–481. [Google Scholar]
  50. B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Model. Simul. 5 (2006) 664–694. [Google Scholar]
  51. B. Schmidt, Minimal energy configurations of strained multi-layers. Cal. Var. Partial Diff. Equ. 30 (2007) 477–497. [Google Scholar]
  52. B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy. J. Math. Pures Appl. 88 (2007) 107–122. [Google Scholar]
  53. B. Schmidt, A Griffith-Euler-Bernoulli theory for thin brittle beams derived from nonlinear models in variational fracture mechanics. Math. Models Methods Appl. Sci. 27 (2017) 1685–1726. [Google Scholar]
  54. O.G. Schmidt and K. Eberl, Thin solid films roll up into nanotubes. Nature 410 (2001) 168. [CrossRef] [PubMed] [Google Scholar]
  55. T. von Kármán Festigkeitsprobleme im Maschinenbau, in Encyclopädie der Mathematischen Wissenschaften, Vol. IV/4. Teubner, Leipzig (1910) 311–385. [Google Scholar]

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