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A hierarchy of theories for thin elastic bodies

A hierarchy of theories for thin elastic bodies . Stefan Müller MPI for Mathematics in the Sciences, Leipzig www.mis.mpg.de. B ath I nstitute for C omplex S ystems Multi-scale problems: Modelling, analysis and applications 12th – 14th September 2005.

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A hierarchy of theories for thin elastic bodies

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  1. A hierarchy of theoriesfor thin elastic bodies Stefan Müller MPI for Mathematics in the Sciences, Leipzig www.mis.mpg.de Bath Institute for Complex Systems Multi-scale problems: Modelling, analysis and applications 12th – 14th September 2005

  2. Nonlinear elasticity 3d  2d • Major question since the beginning of elasticity theory • Why ? • 2d simpler to understand, visualize • Important in engineering and biology • Qualitatively new behaviour: crumpling, collapse • Subtle influence of geometry (large rotations) • Very non-scalar behaviour `Zoo of theories´ First rigorous results: LeDret-Raoult (´93-´96) (membrane theory, -convergence) Acerbi-Buttazzo-Percivale (´91) (rods, -convergence) Mielke (´88) (rods, centre manifolds)

  3. Beyond membranes Key point: Low energy  close to rotation Classical result Need quantitative version

  4. Rigidity estimate/ Nonlinear Korn Thm. (Friesecke, James, M.) L2 distance from a point L2 distance from a set Remarks 1. F. John (1961) u BiLip, dist (u, SO(n)) <   Birth of BMO 2. Y.G. Reshetnyak Almost conformal maps: weak implies strong 3. Linearization  Korn´s inequality 4. Scaling is optimal (and this is crucial) 5. Ok for Lp, 1 < p < 

  5. Rigidity estimate – an application L2 distance from a point L2 distance from a set Thm. (DalMaso-Negri-Percivale) 3d nonlinear elasticity 3d geom. linear elasticity Gives rigorous status to singular solutions in linear elasticity Question: For which sets besides SO(n) does such an estimate hold ? Faraco-Zhong (quasiconformal), Chaudhuri-M. (2 wells), DeLellis-Szekelyhidi (abstract version)

  6. Idea of proof 1. Four-line proof for (Reshetnyak, Kinderlehrer) 2. First part of the real proof: perturb this argument This yields (interior) bound by , not

  7. Proof of rigidity estimate I Step 0: Wlog `truncation of gradients´ (Liu, Ziemer, Evans-Gariepy) Step1: Let Take divergence Compute

  8. Proof of rigidity estimate II Step 2: We know Linearize at F = Id Set Korn  interior estimate with optimal scaling • Step 3: Estimate up to the boundary. • Cover by cubes with boundary distance  size • Weighted Poincaré inequality (`Hardy ineq.´)

  9. 3d nonlinear elasticity

  10. 3d  2d Rem. Same for shells (FJM + M.G. Mora)

  11. Gamma-convergence (De Giorgi)

  12. The limit functional (Kirchhoff 1850) isometry curvature „bending energy“ Geometrically nonlinear, Stress-strain relation linear (only matters)

  13. Idea of proof • One key point: compactness • Unscale to S x (0,h), divide into cubes of size h • Apply rigidity estimate to each cube: •  good approximation of deformation gradient • by rotation • Apply rigidity estimate to union of two neighbouring • cubes: • difference quotient estimate •  compactness, higher differentiability of the limit

  14. in-plane displacement out-of plane displacement Different scaling limits (Modulo rigid motions) Given  such that find , ,  for which

  15. A hierarchy of theories(natural boundary conditions) For  > 2 assume that force points in a single direction (which can be assumed normal to the plate) and has zero moment

  16. A hierarchy of theories(clamped boundary conditions, normal load)

  17. Unified limit for  > 2 (natural bc)

  18. Constrained theory for 2 <  < 4 One crucial ingredient for upper bound: Rem. Hartmann-Nirenberg, Pogorelov, Vodopyanov-Goldstein

  19. A wide field The range is a no man‘s land where interesting things happen Two signposts: • = 1: Complex blistering patterns in thin films with Dirichlet boundary conditions Scaling known/ Gamma-limit open (depends on bdry cond. ?) BenBelgacem-Conti-DeSimone-M., Jin-Sternberg, Hornung = 5/3: Crumpling of paper ? T. Witten et al., Pomeau, Ben Amar, Audoly, Mahadevan et al., Sharon et al., Venkataramani, Conti-Maggi, ... More general: reduced theories which capture systematically both membrane and bending effects

  20. Beyond minimizers (2d  1d)

  21. Beyond minimizers (2d  1d) A. Mielke, Centre manifolds

  22. Conclusions Rigidity estimate/ Nonlinear Korn inequality Small energy  Close to rigid motion • Reduction 3d to 2d: • Key point is geometry/ understanding (large) rotations • (F. John) • Hierarchy of limiting theories ordered by scaling of the energy Interesting and largely unexplored scaling regimes where different limiting theories interact Beyond minimizers …

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