Thin-limit behavior of engineered non-Euclidean plates, a ... · Thin-limit behavior of engineered non-Euclidean plates, a theoretical analysis Thin non-Euclidean structures Γ-convergence:

Thin-limit behavior of engineered non-Euclidean plates, a theoretical analysis

Thin-limit behavior of engineered non-Euclideanplates, a theoretical analysis

Reza Pakzad

University of Pittsburgh

IMA, Hot Topic Workshop on Strain Induced Shape FormationMay 17, 2011


3d Nonlinear Elasticity

Standard variational theory

Standard variational theory

! Total energy functional:

J(u) =Z

!W (!u)!

Z

!fu

u := !" R3 deformation, x # !$ R3 ref configuration,f : !" R3 external body force.

! Elastic energy: E(u) =R! W (!u)

Assumptions on energy density W : R3%3 " R+:• W (RF) = W (F) &R # SO(3) frame indifference• W (Id) = minW = 0 normalization• W (F)' c dist2(F ,SO(3)), c > 0 non-degeneracy• C2 close to SO(3)



Elasticity and growth

Elasticity and growthObservation: residual stress at free equilibria in leaves.

The shape of a long leafHaiyi Liang, L. Mahadevan ! †

!School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138

Submitted to Proceedings of the National Academy of Sciences of the United States of America

Long leaves in terrestrial plants and their submarine counterparts,algal blades have a typical saddle-like mid-surface and rippled edges.To understand the origin of these morphologies, we dissect leavesand di!erentially stretch foam ribbons to show that these shapesarise from a simple cause, the elastic relaxation via bending thatfollows either di!erential growth (in leaves) or di!erential stretchingpast the yield point (in ribbons). We quantify these di!erent modal-ities in terms of a mathematical model for the shape of an initiallyflat elastic sheet with lateral gradients in longitudinal growth. Usinga combination of scaling concepts, stability analysis, and numericalsimulations we map out the shape space for these growing ribbonsand find that as the relative growth strain is increased, a long flatlamina may switch to a saddle shape and/or developing undulationsthat may lead to strongly localized ripples as the growth strain is lo-calized to the edge of the leaf. Our theory delineates the geometricand growth control parameters that determine the shape space offinite laminae and thus allows for a comparative study of elongatedleaf morphology.

growing surfaces | edge actuation | leaves | buckling | rippling

Laminae or leaf-like structures are thin, i.e. they have onedimension much smaller than the other two. They arise

in biology in a variety of situations, ranging from the grace-fully undulating submarine avascular algal blades [1] to thesaddle-shaped, coiled or edge-rippled leaves of many terrestrialplants. The variety of their planforms and three-dimensionalshapes reflects both their growth history and their mechanicalproperties, and poses many physico-chemical questions thatmay be broadly classified into two kinds: (i) how does inho-mogeneous growth at the molecular and cellular level lead tothe observed complex shapes at the mesoscopic/macroscopiclevel ? (ii) how does the resulting mesoscopic shape influencethe underlying molecular growth processes ? At the molecularlevel, mutants responsible for di!erential cell proliferation [2]lead to a range of leaf shapes.

At the macroscopic level, stresses induced by externalloads lead to phenotypic plasticity in algal blades that switchbetween long, narrow blade-like shapes in rapid flow tobroader undulating shapes in slow flow [1]. Understandingthe origin of these morphological variants requires a mathe-matical theory that accounts for the process by which shapeis generated by inhomogeneous growth in a tissue. Recentwork has focused on some of these questions by highlight-ing the self-similar structures that form near the edge due tovariations in a prescribed intrinsic metric of a surface that isasymptotically flat at infinity [3, 5], and also on the case ofa circular disk with edge-localized growth [6], but does notconsider the subtle role of the boundary conditions at the freeedge, nor the the role of finite size of a leaf on the stability,and the phase space of di!erent shapes.

Here, in contrast, motivated by our experimental obser-vations of long leaves and artificial mimics thereof, we ad-dress the question of the morphology of a long leaf or laminaof finite dimensions (length L, width W , thickness H , withH ! W < L). In particular, by coupling growth to shape,

! = 5%

! = 20%

(a)

(b)

(c)Saddle Rippled

"(y)

0.00

0.05

0.10

0.15

0.20

y (cm)-2 -1 0 1 2

y

x

Fig. 1. (a)Shape of a Corn lily leaf showing the saddle-like shape of the mid-surface and the rippled edges. Dissection along the midrib leads to a relief of theincompatible strain induced by di!erential lateral growth and causes the midrib tostraighten, except near the tip, consistent with the notion that the shape is a result ofelastic interactions of a growing plate. The dashed red line is the original position ofthe midrib. (b) A foam ribbon that is stretched beyond the elastic limit relaxes intoa saddle shape when the edge strain is ! ! 5% (see text), but relaxes into a rippledshape when the edge strain is ! ! 20%. (c) The observed lateral strain "(y)(seetext) is approximately parabolic for the saddle-shaped ribbon but is localized morestrongly to the edge for the rippled ribbon.

we are able to pose a nonlinear boundary value problem. Weanalyze the resulting equations using a combination of scal-ing concepts, asymptotics and stability analysis to deduce thevarious morphologies that arise and show that the finite width

†To whom correspondence should be addressed. E-mail: [email protected]

c"2007 by The National Academy of Sciences of the USA1 In the case of the stretched ribbon, there is in general some thinning of the material due to thePoisson e!ect as well, but we ignore this in our qualitative description of the phenomenon

www.pnas.org — — PNAS Issue Date Volume Issue Number 1–7

Torn plastic sheet are reminiscent of patterns encountered in plants. The outer edge undergoes a cascade of spontaneous buckling transitions.

beet leaves

torn plastic

30 mm

9.6 mm

2.8 mm

0.84 mm

0.25 mm

0.08 mm

The rippled structure exhibits a fractal!like behavior. One can identify up to 6 generations of self!similar patterns. The cuto" seems to be associated with the thickness of the sheet.

! A model incorporating growth by Rodriguez-Hoger-McCulloch(J. Biomech 1994)

decomposition !u = FA

A : !" R3%3 = growth tensor; change in mass. det A > 0.F = elastic part; reorganization of body (= (!u)A!1)

! A(x): Growth, Plasticity, Thermal expansion or contraction, ...

! EnergyZ

W (F) =Z

!W ((!u)A!1) dx



3d non-Euclidean elasticity

3d non-Euclidean elasticity

! E(u) =Z

!W ((!u)A(x)!1)dx

! E(u) = 0() !u = RA a.e.() (!u)T !u = AT A and det!u > 0

! G := AT A is a Riemannian metric, (!u)T !u: pull-back metric.! Lemma (Lewicka-P.) R iem *+ 0) infu#W 1,2 E(u) > 0.

! Comparable Energy: E(u) =Z

|(!u)T !u!G|2dvolG.

A constraint (e.g. det!u > 0) must be imposed, otherwiseinf E = 0.

! Given target metric G, study E with a suitable constraint;or let A =

,G and study E .


Thin non-Euclidean structures

"-convergence: a model problem

An experiment

Shaping of elastic sheets by prescription of Non-EuclideanmetricsKlein, Efrati, Sharon, Science (2007)Reprinted with permission of AAAS




Modeling by non-Euclidean Elasticity

Thin plates: ! = Sh = S% (!h/2,h/2), S $ R2

S h

[g"#] metric on S.

! Assume that G(x ,x3) =

!

" [g"#(x)]00

0 0 1

#

$ ,

Eh(u) =1h

Z

ShW ((!u)A!1)

! R iem(G)+ 0- Kg + 0 (If Kg *+ 0) inf Eh > 0)! Questions: What is the limit of minimizers as h" 0?

How do inf Eh scale?




!-convergence

Metatheorem "-convergence- conv. of minimizers.

! X a metric space, Fn : X " [!$,+$]. Fn"!" F whenever

(i) (The "-liminf inequality)&xn " x in X , F(x). lim infn"$ Fn(xn).

(ii) (The "-limsup inequality)&x # X , /xn # X , s. t. xn " x and limsupn"$ Fn(xn). F(x).

! (Compactness) A sequence of approximate minimizers {xn} hasa converging subsequence in X .




A !-limit theorem! Assume Kg *+ 0 and that the class of isometric immersions

A = {y : (S,g)" R3 ; (!y)T !y = g ,Z

S|!2y |2 < $} *= /0.

! Theorem (Lewicka, P. (10))(1) infEh 0 h2.(2) The "-limit of 1

h2 Eh is minimizing a bending functional I(y) fory # A .

!I(y) =

124

Z

SQ2(A!1

"##A!1"# )

Normal!n = %1y%%2y|%1y%%2y | , 2nd form # =( !y)T !!n, A"# =

%[g"#].

Q2 is a quadratic form and can be calculated from D2W (Id)! det(A!1

"##A!1"# ) = Kg , tr(A!1

"##A!1"# ) = 2H.

! I|A admits at least one minimizer.




Theoretically, W 2,2 Isometric Immersions kill wrinkles! Key Compactness Ingredient:

Eh(uh). Ch2 ) uh(x ,hz3) =: yh " y # A .! K *+ 0, then A *= /0() infEh 0 h2.! “Minimizers” of Eh converge to minimizers of I on A .

Wrinkling effects must disappear as h" 0.! Existence of isometric immersions: Longstanding problem in

differential geometry(Nash-Kuiper) Every n dim Riemannian manifold has a C 1

isometric embedding in Rn+1

(Nirenberg) Every smooth [g"#] on S $ R2 with Kg > c > 0 has asmooth isometric immersion in R3

(Han-Hong) Same true for Kg <!c < 0 (bounded domain!)(Pogorelov) Example of g with no local C 2 isom. imm.Admits C1,1 immersion.




Mismatch: experiment and theory (Crime does not pay)

! A fundamental experimental differencebetween Kg > 0 and Kg < 0. (Klein,Venkataramani and Sharon)

! Possible explanations:

1. Metric g not smooth, but piece-wiseconstant on concentric annuli.

2. The evolution problem ’notwell-defined’ for Kg < 0.

3. Boundary layer effect?

4. Loss of radial symmetry for thehyperbolic case.




A one parameter family of minimizers! W isotropic,

S = [0,2&]% [!1,1],g = cosh2(x2)Id,Kg =!cosh(x2)!4.

! Under a negative determinantconstraint on a symmetricmatrix F , Q2(F) is minimizedwhen tr(F) = 0.

! There exists a one parameternon-trivial family yt # A suchthat Ht + 0 (tr(#yt )+ 0).

! I(yt) = min I > 0.! Are there other examples?

What happens for Kg > 0?

Smooth deformation of (VMC)Helicoids into Catenoid



Limit models for perturbations of Euclidean metric

Limit models for perturbations of Euclidean metricWrite equilibrium eqns. of a thin elastic body! = Sh, subject to growth tensor A = Id +perturbation:

G := AT A = Id +perturbation

Objectives:

(i) Derive a hierarchy/tree of pre-strained limit theories. Some ofthese theories are more easily accessible and can give insightinto the fully nonlinear bending problem.

(ii) A small step in the dynamic problem: perturbation as a first smalltime step.

(iii) Compare the infimum of energies with the magnitude of thecurvature tensor and dimensions of the domain throughestablishing scaling laws.




Von Karman - like morphogenesis model

! Gh = Id +h2'g +hx3(g metric on Sh, 'g,(g : S " R3%3sym

! Theorem (Lewicka-Mahadevan-P. 10) Assume curl((g)tan *= 0 orcurlT curl('g)tan + 1

2 det((g)tan *= 0. Then: inf Eh 0 h4.

! Theorem (Lewicka-Mahadevan-P.) 1h4 Eh "!" Ig , where

Ig(v ,u) = 124

RS Q2(!!2v! 1

2((g)tan) bending+1

2

RS Q2(sym!u + 1

2 !v1!v! 12('g)tan) stretching

Euler-Lagrange equations: (for Q2 = Id)(Mahadevan and Liang, PNAS 2009)

&$2% =!1

2 [v ,v ]! 12 curlT curl('g)tan

$2v = 12[%,v ]! 12 divT div((g)tan

+Free (natural) B.C.




Non-immersibility of Gh and scaling of the energy! Heuristically Gh = Id +h2'g +hx3(g prescribes

gh := Id+ h2

2 ('g)tan and #h = h2((g)tan on S.

! The corresponding Gauss-Codazzi equations:&

h curl((g)tan +O(h2) = 0h2 (curlT curl('g)tan + 1

2 det((g)tan)+O(h3) = 0

! Assume Gh does not depend on x3.R1212(Gh) = O(1) =) infEh ' Ch2

R1212(Gh) = O(h2) =) infEh ' Ch4

! Conjecture: 1h2R iem22

H!2 . C infEh.Result by Kupferman and Shamai for E in n = 2 and forR iem(G) > 0.

! A tree of models depending on “immersibility” of Gh, measured bythe scaling properties of Rh or by the degree of deficiency inGauss-Codazzi equations.




A useful intermediary model

! Gh = Id +h)'g metric on Sh, 'g : S " R3%3sym

Gauss curvature of Gh|S =!1

2 h) curlT curl('g)tan +O(h2))

! Theorem (Lewicka-P. 10) Assumef =!1

2 curlT curl('g)tan *+ 0, then:1h# infEh " $ &# > max{2),)+2}.

! 0 < ) < 2 =) 2 < # = )+2 < 4.! The "-limit of 1

h# Eh is given by the functional

J(v) =Z

SQ2(!2v),

over the space of scalar functions v satisfying det(!2v) = f .! Sample Question: Given f : S " R, is there a non-trivial one

parameter family of minimizers?


Thank You

Thank You!

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