A hierarchy of theoriesfor thin elastic bodies

Stefan Müller

MPI for Mathematics in the Sciences, Leipzig

www.mis.mpg.de

Bath Institute for Complex SystemsMulti-scale problems:

Modelling, analysis and applications12th – 14th September 2005

Nonlinear elasticity 3d 2dMajor 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)

Beyond membranes

Key point: Low energy close to rotation

Classical result

Need quantitative version

Rigidity estimate/ Nonlinear Korn

Thm. (Friesecke, James, M.)

Remarks 1. F. John (1961) u BiLip, dist (u, SO(n)) < Birth of BMO2. Y.G. Reshetnyak Almost conformal maps: weak implies strong3. Linearization Korn´s inequality4. Scaling is optimal (and this is crucial)5. Ok for Lp, 1 < p <

L2 distance from a point L2 distance from a set

Rigidity estimate – an application

Thm. (DalMaso-Negri-Percivale) 3d nonlinear elasticity 3d geom. linear elasticity

L2 distance from a point L2 distance from a set

Gives rigorous status to singular solutions in linear elasticity

Question: For which sets besides SO(n) does such an estimatehold ? Faraco-Zhong (quasiconformal), Chaudhuri-M. (2 wells), DeLellis-Szekelyhidi (abstract version)

Idea of proof

1. Four-line proof for(Reshetnyak, Kinderlehrer)

2. First part of the real proof: perturb this argumentThis yields (interior) bound by , not

Proof of rigidity estimate I

Step 0: Wlog `truncation of gradients´ (Liu, Ziemer, Evans-Gariepy)

Step1: Let

Compute

Take divergence

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 sizeb) Weighted Poincaré inequality (`Hardy ineq.´)

3d nonlinear elasticity

3d 2d

Rem. Same for shells (FJM + M.G. Mora)

Gamma-convergence (De Giorgi)

The limit functional (Kirchhoff 1850)

Geometrically nonlinear, Stress-strain relation linear (only matters)

isometry

„bending energy“

curvature

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

Different scaling limits

(Modulo rigid motions)

in-plane displacement out-of plane displacement

Given such that

find , , for which

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) andhas zero moment

A hierarchy of theories(clamped boundary conditions,

normal load)

Unified limit for > 2 (natural bc)

Constrained theory for 2 < < 4

One crucial ingredient for upper bound:

Rem. Hartmann-Nirenberg, Pogorelov, Vodopyanov-Goldstein

A wide field

The range is a no man‘s landwhere interesting things happenTwo 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 capturesystematically both membrane and bending effects

Beyond minimizers (2d 1d)

Beyond minimizers (2d 1d)

A. Mielke, Centre manifolds

Conclusions

Rigidity estimate/ Nonlinear Korn inequalitySmall 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 regimeswhere different limiting theories interact