a hierarchy of theories for thin elastic bodies stefan müller mpi for mathematics in the sciences,...
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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 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