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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

A hierarchy of theoriesfor thin elastic bodies

Stefan Müller

MPI for Mathematics in the Sciences, Leipzig


Bath Institute for Complex Systems

Multi-scale problems:

Modelling, analysis and applications

12th – 14th September 2005

Nonlinear elasticity 3d 2d
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
Beyond membranes

Key point: Low energy  close to rotation

Classical result

Need quantitative version

Rigidity estimate nonlinear korn
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


3. Linearization  Korn´s inequality

4. Scaling is optimal (and this is crucial)

5. Ok for Lp, 1 < p < 

Rigidity estimate an application
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
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
Proof of rigidity estimate I

Step 0: Wlog

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

Step1: Let

Take divergence


Proof of rigidity estimate ii
Proof of rigidity estimate II

Step 2: We know

Linearize at F = Id


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
3d  2d

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

The limit functional kirchhoff 1850
The limit functional (Kirchhoff 1850)



„bending energy“

Geometrically nonlinear,

Stress-strain relation linear (only matters)

Idea of proof1
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

Different scaling limits

in-plane displacement

out-of plane displacement

Different scaling limits

(Modulo rigid motions)

Given  such that

find , ,  for which

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

Constrained theory for 2 4
Constrained theory for 2 <  < 4

One crucial ingredient for upper bound:

Rem. Hartmann-Nirenberg, Pogorelov, Vodopyanov-Goldstein

A wide field
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,


= 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 1d1
Beyond minimizers (2d  1d)

A. Mielke, Centre manifolds


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


Interesting and largely unexplored scaling regimes

where different limiting theories interact

Beyond minimizers …