Wrinkling, Microstructure, and Energy Scaling Laws

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1 , Microstructure, and Energy Scaling Laws Courant Institute, NYU IMA Workshop on Strain Induced Shape Formation, May 211 First half: an overview Second half: recent progress with Hoai-Minh Nguyen

2 Plan (1) A long introduction (2) Some mathematical results (still a survey) (3) The wrinkling cascade seen in a floating elastic sheet (new)

3 as microstructure Focus: a (flat) sheet with thickness h and shape R 2 Nonlinear elastic viewpoint: u : R 3 E h = membrane + bending h Du T Du I 2 + h 3 D 2 u 2 + loads Membrane term is nonconvex; it prefers isometric immersion. Bending term is regularizing singular perturbation, penalizing curvature. Von Karman viewpoint: u = (x + h β/2 w 1, y + h β/2 w 2, h β/4 u 3 ) E h h β+1 e(w) + 1 u 2 3 u h β 2 +3 D 2 u loads Same essential character as nonlinear elastic model.

4 as microstructure, cont d Overall idea: 1 E h h Du T Du I 2 + h 2 D 2 u 2 (1) Find lim h 1 h E h = E, the minimum membrane energy, allowing for the presence of wrinkling if necessary. (2) Estimate correction due to h >, e.g. by showing that Notes: E + C 1 h γ 1 h E h E + C 2 h γ E is predicted by tension field theory (mechanics) or by a relaxed variational problem (mathematics). It vanishes if loads are compressive. In task (2), upper bounds come from constructions (conceptually easy) but lower bounds must be ansatz-independent (usually harder). If γ < 2 then deformation develops microstructure.

5 An elementary mathematical analogue 1 min v()=v(1)= (v 2 x 1) 2 + ε 2 v 2 xx + αv 2 dx When ε =, α >, min value is, not attained. Min sequence has v x = ±1 with prob 1/2 each. When ε >, min scales like ε 2/3 α 1/3, since for sawtooth with N teeth, value is about εn + αn 2. Best N (α/ε) 1/3. Case ε >, α = is different: just one tooth; min value is c ε. Optimal profile of tooth (which determines c ) found by minimizing ε 1 (w 2 1) 2 + εw 2 x subject to w ±1 as x ±. Exactly solvable. Singular perturbation induces defects and organizes microstructure.

2D is richer than 1D Simple model of martensite twins near an austenite interface Z min v = at x= (vy2 1)2 + vx2 + ε2 v 2 Like 1D example (in y ), but microstructure is required by bdry cond rather

6 2D is richer than 1D Simple model of martensite twins near an austenite interface Z min v = at x= (vy2 1)2 + vx2 + ε2 v 2 Like 1D example (in y ), but microstructure is required by bdry cond rather than a lower-order term Microstructural length scale depends on x (finer near x = ) Much is known (Kohn-Müller, Conti, Schreiber), especially for a sharp-interface analogue. Energy scaling law, and more: microstructure is self-similar near x =.

Elastic sheets are richer still Depending on loads and bdry conds, limit as h may involve: formation of defects (e.g. point defects or ridges) formation of microstructure (e.

7 Elastic sheets are richer still Depending on loads and bdry conds, limit as h may involve: formation of defects (e.g. point defects or ridges) formation of microstructure (e.g. wrinkles) or both (e.g. crumpled paper) or neither (e.g. paper held at edge)

8 Plan (1) A long introduction (2) Some mathematical results (still a survey) (3) The wrinkling cascade seen in a floating elastic sheet (new)

9 The relaxed problem 1 h E h = W (Du)+h 2 (bending); Du is a 3 2 matrix; W Du T Du I 2 Limiting value as h is min QW (Du), where QW (F ) = minimal membrane energy/unit area (allowing for wrinkling) if the macroscopic deformation gradient is F. QW worked out explicitly by Jack Pipkin; there are 3 cases: - biaxial tension no wrinkles, QW (F) = F - uniaxial tension 1D wrinkles avoid compression - unstressed complex wrinkling pattern, QW (F) =. QW is typically convex, but very degenerate. If solution of relaxed problem is unique it tells us a lot. But often it isn t unique.

10 Nonlinear bending theory 1 h E h = W (Du)+h 2 (bending); Du is a 3 2 matrix; W Du T Du I 2 If 1 h E h = C h 2 then the prefactor C minimizes the bending energy among isometries. Similar result holds starting from 3D elasticity in a thin domain (Friesecke-James-Müller; see talk of Lewicka). Must know absence of microstructure to apply this result.

11 Blisters in compressed thin films min W (Du) + h 2 (bending) u =(1 η)x Boundary condition consistent with uniform compression (using stress-free state of film as reference state). Film buckles to reduce membrane energy. Wrinkles accumulate at boundary. Studied by Jin-Sternberg (von Karman version) and Ben Belgacem-Conti-DeSimone-Müller (von Karman and nonlinear). Main result: 1 h E h scales like hη 3/2. (Linear not quadratic in h, due to accumulation of wrinkles near boundary.) Lower bound is relatively easy in this case (it focuses on a thin strip near the boundary).

12 Metric-induced wrinkling Do leaves and flowers buckle due to elastic energy minimization? Model problem: clamped at 3 sides RHS is free g = Du T Du g 2 + h 2 D 2 u 2 ( 1 m 2 (x) ), m() = 1, m x >. Image of x = x wants to be longer than left edge. Bdry condition accommodates this only by buckling. Energy scaling law? See poster by Peter Bella.

An annulus in tension Recall the wrinkled sheet loaded in tension (Cerda-Mahadevan, PRL, 23): No wrinkling at extremes; lots in middle (local length scale as sheet thickness ).

13 An annulus in tension Recall the wrinkled sheet loaded in tension (Cerda-Mahadevan, PRL, 23): No wrinkling at extremes; lots in middle (local length scale as sheet thickness ). A free bdry separates the wrinkled and unwrinkled regions. Awkward for analysis (what are loads? where is free bdry?). More accessible: annulus-shaped sheet, loaded by uniform tension at both boundaries (cf preprint by Davidovitch, Schroll, Vella, Adda-Bedia, Cerda). No wrinkling at larger radii; lots of lot of wrinkling at smaller radii. Free boundary at r = r. Loads and geometry would force circles r = const to shrink, for r < r, if deformation were planar. But sheet prefers not to stretch or shrink. So the circles buckle out of plane. Hence the wrinkling. Energy scaling law? See talks by Bella and Davidovitch.

14 Plan (1) A long introduction (2) Some mathematical results (still a survey) (3) The wrinkling cascade seen in a floating elastic sheet

15 The floating elastic sheet sheet floats on water confined on 2 sides surface tension pulls free edges wrinkles form, refining at edges Experiment and theory: Huang, Davidovitch, Santangelo, Russell, Menon (PRL 21); also Davidovitch (PRE 29) Mathematical analysis: joint work with Hoai-Minh Nguyen. Key conclusions: 1 Excess energy due to wrinkling scales like h 3/2 log h 2 Local length scale is ch 3/4 x + h 3/2 near x =.

16 Floating elastic sheet the model Von Karman theory: displacement (w 1, w 2, u 3 ) Confinement: w 2 (x, ) =, w 2 (x, 1) = /2 Domain = [, L] [, 1], with L 1 For simplicity: u 3 is periodic in y E h = α mh e(w) + 1 u 2 3 u h 3 u 3 2 +α g u αs u 3 2 (α c α s) xw 1 + α c u 3 2 H 1/2 (Γ) Key hypothesis: α c > α s. Notation: Γ = free edges. Notes: Extra surf energy due to u 3 is α s 1 2 u α c u 3 2 H 1/2 (Γ) Extra surf energy due to in-plane def is (α s α c) div w Since div w = xw 1 + const, surf tension is tensile when α c > α s. H 1/2 term is energy of capillary fringe field: u 3 2 H 1/2 (Γ) = min water u 3 2

17 Floating elastic sheet the model Von Karman theory: displacement (w 1, w 2, u 3 ) Confinement: w 2 (x, ) =, w 2 (x, 1) = /2 Domain = [, L] [, 1], with L 1 For simplicity: u 3 is periodic in y E h = α mh e(w) + 1 u 2 3 u h 3 u 3 2 +α g u αs u 3 2 (α c α s) xw 1 + α c u 3 2 H 1/2 (Γ) Key hypothesis: α c > α s. Notation: Γ = free edges. Notes: Extra surf energy due to u 3 is α s 1 2 u α c u 3 2 H 1/2 (Γ) Extra surf energy due to in-plane def is (α s α c) div w Since div w = xw 1 + const, surf tension is tensile when α c > α s. H 1/2 term is energy of capillary fringe field: u 3 2 H 1/2 (Γ) = min water u 3 2

18 Floating elastic sheet the model Von Karman theory: displacement (w 1, w 2, u 3 ) Confinement: w 2 (x, ) =, w 2 (x, 1) = /2 Domain = [, L] [, 1], with L 1 For simplicity: u 3 is periodic in y E h = α mh e(w) + 1 u 2 3 u h 3 u 3 2 +α g u αs u 3 2 (α c α s) xw 1 + α c u 3 2 H 1/2 (Γ) Key hypothesis: α c > α s. Notation: Γ = free edges. Notes: Extra surf energy due to u 3 is α s 1 2 u α c u 3 2 H 1/2 (Γ) Extra surf energy due to in-plane def is (α s α c) div w Since div w = xw 1 + const, surf tension is tensile when α c > α s. H 1/2 term is energy of capillary fringe field: u 3 2 H 1/2 (Γ) = min water u 3 2

19 Floating elastic sheet energy scaling E h = membrane + bending + gravitational + surface = α mh e(w) + 1 u 2 3 u h 3 u 3 2 +α g u αs u 3 2 (α c α s) xw 1 + α c u 3 2 H 1/2 (Γ) min E h = (value assoc tensile forces) + (correction due to bending energy) value assoc tensile forces = (αc αs)2 4α mh L αmh 2 L [αmh αs]2 + L 4α mh correction α 1/2 c h 3/2 log h [αmh αs]2 + α mh Comment: wrinkles form only if α mh > α s.

20 A convenient reorganization E 1 = α mh E 2 = α mh E 3 = h 3 E h = E 1 + E 2 + E 3 + E 4 ( xw xu ) 2 (α c α s) ( xw xu ) ( yw yu ) 2 + α s E 4 = 1 2 αmh u αc xu α g xw 2 + yw 1 + xu 3 yu 3 2 yu 3 2 u α c u 3 2 H 1/2 (Γ) E 1 captures stretching due to surface tension. It slaves 1 w 1 to xu 3. E 2 captures effect of confining bdry conditions. It determines 1 ( yu 3) 2. Minimization of E 1 and E 2 gives the value assoc to tensile forces. E 3 captures effect of bending resistance. Its min value is the correction due to bending energy. E 4 is unimportant due to symmetry of bdry conditions.

21 Energy due to tensile forces E 1 = α mh E 2 = α mh ( xw xu ) 2 (α c α s) ( xw xu ) ( yw yu ) 2 + α s yu 3 2 E 1 captures stretching due to surface tension. To minimize integrand, E 1 prefers xw xu 3 2 to be a special value. So xw 1 is slaved to xu 3. E 2 captures effect of the confining bdry condition. Bdry condn gives 1 yw 2 dy = /2. Using Jensen s ineq, integral in y is minimized (at each x) when z = 1 yu 3 2 dy achieves α mh min z> 4 ( z) αsz. No wrinkling if best z is (this occurs when α mh < α s).

22 The key term The picture so far: E 1 (αc αs)2 4α mh L, E αmh 2 L [αmh αs]2 + L 4α mh with equality in the latter only when 1 ( yu 3) 2 dy = δ for all x, defining [ ] αmh α s δ = α mh Moving forward: E 3 controls the wrinkling. Recall: E 3 = h 3 u αc xu α g The capillary term forces u 3 to be small at free bdry. + u α c u 3 2 H 1/2 (Γ). If u 3 is not uniformly small, then xu 3 must be large. Expensive wrt surface energy. If u 3 is uniformly small, then length scale of wrinkling must be small (since slope yu 3 was fixed by E 2 ). Expensive wrt bending energy.

23 A heuristic argument Recall (taking α g = α c = 1 for simplicity): E 3 = h 3 u xu u u 3 2 H 1/2 (Γ) Let s assume u 3 (x, y) = a(x) sin(θ(x)y). Since 1 ( yu 3) 2 dy = δ, we require I claim that a 2 (x)θ 2 (x) = 2δ. E 3 Ch 3/2 log h [αmh αs]2 +, α mh and minimization requires θ 1 (x) δ 1/2 h 3/4 x + h 3/2 near x =. STEP 1. If a(x) is unif small then θ(x) is unif large, so u 3 2 is large. So it suffices to consider the case when for some x (, 1/2), a(x ) is large: a 2 (x ) Cδh 3/2 log h 1. STEP 2. If a() is large then u 3 2 is large. So sufft to consider H 1/2 (Γ) a 2 () Cδh 3 log h 2.

24 A heuristic argument Recall (taking α g = α c = 1 for simplicity): E 3 = h 3 u xu u u 3 2 H 1/2 (Γ) Let s assume u 3 (x, y) = a(x) sin(θ(x)y). Since 1 ( yu 3) 2 dy = δ, we require I claim that a 2 (x)θ 2 (x) = 2δ. E 3 Ch 3/2 log h [αmh αs]2 +, α mh and minimization requires θ 1 (x) δ 1/2 h 3/4 x + h 3/2 near x =. STEP 1. If a(x) is unif small then θ(x) is unif large, so u 3 2 is large. So it suffices to consider the case when for some x (, 1/2), a(x ) is large: a 2 (x ) Cδh 3/2 log h 1. STEP 2. If a() is large then u 3 2 is large. So sufft to consider H 1/2 (Γ) a 2 () Cδh 3 log h 2.

25 A heuristic argument Recall (taking α g = α c = 1 for simplicity): E 3 = h 3 u xu u u 3 2 H 1/2 (Γ) Let s assume u 3 (x, y) = a(x) sin(θ(x)y). Since 1 ( yu 3) 2 dy = δ, we require I claim that a 2 (x)θ 2 (x) = 2δ. E 3 Ch 3/2 log h [αmh αs]2 +, α mh and minimization requires θ 1 (x) δ 1/2 h 3/4 x + h 3/2 near x =. STEP 1. If a(x) is unif small then θ(x) is unif large, so u 3 2 is large. So it suffices to consider the case when for some x (, 1/2), a(x ) is large: a 2 (x ) Cδh 3/2 log h 1. STEP 2. If a() is large then u 3 2 is large. So sufft to consider H 1/2 (Γ) a 2 () Cδh 3 log h 2.

26 A heuristic argument cont d STEP 3. Consider tradeoff between surface energy and bending terms: if u 3 = a(x) sin(θ(x)y) with a 2 θ 2 = 2δ then and x 1 ( xu 3 ) 2 = h 3 x 1 x 1 yyu 3 2 h 3 x δ 2 a 2 (x) dx, [a x sin(θy) + aθ x cos(θy)] 2 1 x ax 2 dx. 2 So (using z 2 + w 2 2zw with equality when z = w) sum x h 3 δ 2 a x 2 a2 x dx h 3/2 δ dx. a Using steps 1 and 2, this is at least Ch 3/2 δ log h ; moreover, the good choice of a(x) has h 3 δ 2 /a 2 (x) = 1 2 a2 x(x) (an ODE for a(x)). ax

27 How is the rigorous version different? The heuristic calculation has (almost) all the main ideas. All that s left: (1) For the upper bound, cannot really use u 3 = a(x) sin(θ(x)y) because the term we ignored ( x a 2 θ 2 x dx) is too large. Easy to fix. (2) For the lower bound, need an ansatz-independent version of the heuristic calculation. Its essence is this Lemma: if g(x, y) : [, 1] 2 R is periodic in y and 1 g2 (, y) dy a (sufficiently small) 1 g2 (1, y) dy b (sufficiently large) 1 ( yg) 2 (x, y) dy 1 for most x (all but measure ε) then provided ε t 1 1 ( yyg) 2 + α α ab 2 αt log(b/a). 1 1 ( xg) 2 C αt log(b/a)

28 Stepping back can be microstructure (when its length scale as h ). Advantage of focus on energy scaling law: permits ansatz-free analysis. Some successes, but still many open problems. For the floating elastic sheet: the scaling law is rigorous, but the ODE for local length scale of wrinkling is (thus far) just heuristic. Different tools have different strengths: - Numerics predicts patterns, but finds local minima - Minimization within an ansatz suggests what to expect - Ansatz-independent lower bounds confirm (or refute) adequacy of a particular ansatz

Credits Images are from: E. Cerda and L. Mahadevan, Phys Rev Lett 9 (23) 7432 J. Huang, B. Davidovitch, C. Santangelo, T. Russell, and N. Menon, Phys Rev Lett 15 (21) 3832 B.-K. Lai, K. Kerman, and S.

29 Credits Images are from: E. Cerda and L. Mahadevan, Phys Rev Lett 9 (23) 7432 J. Huang, B. Davidovitch, C. Santangelo, T. Russell, and N. Menon, Phys Rev Lett 15 (21) 3832 B.-K. Lai, K. Kerman, and S. Ramanathan, J Power Sources 195 (21) S. Conti and F. Maggi, Arch Rational Mech Anal 187 (28) 1-48 E. Cerda and L. Mahadevan, Proc R Soc A 461 (25) Funding from NSF is gratefully acknowledged.

Pattern formation in compressed elastic films on compliant substrates: an explanation of the herringbone structure

Pattern formation in compressed elastic films on compliant substrates: an explanation of the herringbone structure Robert V. Kohn Courant Institute, NYU SIAM Annual Meeting, July 2012 Joint work with Hoai-Minh