1. A hierarchy of theories for 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.) 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 L p, 1 < p <  L 2 distance from a pointL 2 distance from a set

5. Rigidity estimate – an application Thm. (DalMaso-Negri-Percivale) 3d nonlinear elasticity 3d geom. linear elasticity L 2 distance from a pointL 2 distance from a set 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 Compute Take divergence

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. a)Cover by cubes with boundary distance  size b)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) Geometrically nonlinear, Stress-strain relation linear (only matters) isometry „bending energy“ curvature

13. Idea of proof One key point: compactness 1.Unscale to S x (0,h), divide into cubes of size h 2.Apply rigidity estimate to each cube:  good approximation of deformation gradient by rotation 3.Apply rigidity estimate to union of two neighbouring cubes:  difference quotient estimate  compactness, higher differentiability of the limit

14. Different scaling limits (Modulo rigid motions) in-plane displacement out-of plane displacement 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. A. Mielke, Centre manifolds

22. Conclusions Rigidity estimate/ Nonlinear Korn inequality Small energy  Close to rigid motion Beyond minimizers … 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