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A hierarchy of theories for thin elastic bodies

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### 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

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)

Beyond membranes

Key point: Low energy close to rotation

Classical result

Need quantitative version

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 <

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)

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

Proof of rigidity estimate I

Step 0: Wlog

`truncation of gradients´ (Liu, Ziemer, Evans-Gariepy)

Step1: Let

Take divergence

Compute

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.´)

3d 2d

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

The limit functional (Kirchhoff 1850)

isometry

curvature

„bending energy“

Geometrically nonlinear,

Stress-strain relation linear (only matters)

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

out-of plane displacement

Different scaling limits(Modulo rigid motions)

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) and

has zero moment

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

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 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

Beyond minimizers (2d 1d)

A. Mielke, Centre manifolds

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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