Effective Theories and Minimal Energy Configurations for Heterogeneous Multilayers

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1 Effective Theories and Minimal Energy Configurations for Universität Augsburg, Germany Minneapolis, May 16 th,

2 Overview 1 Motivation 2

3 Overview 1 Motivation 2 Effective Theories 2

4 Overview 1 Motivation 2 Effective Theories 3 Minimizing Configurations 2

5 Overview 1 Motivation 2 Effective Theories 3 Minimizing Configurations 3

6 Physical Experiments Self-induced fabrication of Nanotubes O. G. Schmidt, K. Eberl, Nature

7 Physical Experiments Self-induced fabrication of Nanotubes O. G. Schmidt, K. Eberl, Nature

8 Physical Experiments Experiments within the DFG project FOR 522 H. Paetzelt et al., PSS

9 Questions Main goal: Give a rigorous explanation of these observations. 6

10 Questions Main goal: Give a rigorous explanation of these observations. Work plan: Describe the system in terms of nonlinear elasticity theory. Note: Objects undergo large deformations. Derive effective energy functionals for thin multilayers. Investigate those functionals for minimal energy configurations. 6

11 Overview 1 Motivation 2 Effective Theories 3 Minimizing Configurations 7

12 Nonlinear elasticity theory For continuum deformations of a 3d bulk material in Ω: y 3d elasticity th. E = Ω W ( y). 8

13 Nonlinear elasticity theory For continuum deformations of a 3d bulk material in Ω: y 3d elasticity th. E = Ω W ( y). Here W is the stored energy function, satisfying in particular W 0 and W (F ) = 0 F SO(3), W (RF ) = W (F ) for all R SO(3), F R 3 3, W in C 2 near SO(3), W (F ) dist 2 (F, SO(3)) for all F R

14 Nonlinear elasticity theory For continuum deformations of a 3d bulk material in Ω: y 3d elasticity th. E = Ω W ( y). Here W is the stored energy function, satisfying in particular W 0 and W (F ) = 0 F SO(3), W (RF ) = W (F ) for all R SO(3), F R 3 3, W in C 2 near SO(3), W (F ) dist 2 (F, SO(3)) for all F R 3 3. For small strains ( y) T y Id with Q 3 = 2 W F (Id)( ) 2 ( ) E = W ( y) = W R ( y) T y 1 Q 3 ( ( y) 2 T y). Ω Ω Ω 8

15 Thin films Thin films: 1nm h = height lateral dimensions y - New phenomena: crumpling, large deformations at low energy - Interesting in many applications. 9

16 Thin films Thin films: 1nm h = height lateral dimensions y - New phenomena: crumpling, large deformations at low energy - Interesting in many applications. Classical problem in elasticity theory: Derive suitable energy functinals in the limit of singular geometries. This leads, in particular, to E h: membrane theory E h 3 : Kirchhoff s plate theory E h 5 : von Kármán s plate theory 9

17 Thin films Thin films: 1nm h = height lateral dimensions y - New phenomena: crumpling, large deformations at low energy - Interesting in many applications. Classical problem in elasticity theory: Derive suitable energy functinals in the limit of singular geometries. This leads, in particular, to E h: membrane theory E h 3 : Kirchhoff s plate theory E h 5 : von Kármán s plate theory For recent rigorous Γ-convergence results, see LeDret/Raoult 93: membranes Friesecke/James/Müller 02 & 05: Hierarchy of plate theories 9

18 Gamma-convergence Aim: Derive an effective functional E 0 for multilayers in the Kirchhoff scaling regime as a Γ-limit E h = h 3 E E 0. Definition. Let E h be functionals on a metric space X. (E h ) is said to Γ-converge to E 0, if (i) ( lim inf-inequality ) for every sequence y h y in X one has lim inf E h (y h ) E 0 (y), (ii) ( recovery sequence ) and if for every y X there is a sequence y h y such that lim E h (y h ) = E 0 (y). 10

19 Gamma-convergence Aim: Derive an effective functional E 0 for multilayers in the Kirchhoff scaling regime as a Γ-limit E h = h 3 E E 0. Definition. Let E h be functionals on a metric space X. (E h ) is said to Γ-converge to E 0, if (i) ( lim inf-inequality ) for every sequence y h y in X one has lim inf E h (y h ) E 0 (y), (ii) ( recovery sequence ) and if for every y X there is a sequence y h y such that lim E h (y h ) = E 0 (y). Fact: Γ-convergence & compactness convergence of minimizers. Consequence: Study the minimizers of E 0 instead of E h. 10

20 Multilayers Multilayers: W = W (h 1 x 3, F ) and W (t, F ) = W 0 (t, F (Id + hb (h) (t))) [ ( )] 1 = W hf (h) (t) F. with W 0 (t, ) as above ( unformly in t ). Suppose B (h) B. 11

21 Multilayers Multilayers: W = W (h 1 x 3, F ) and W (t, F ) = W 0 (t, F (Id + hb (h) (t))) [ ( )] 1 = W hf (h) (t) F. with W 0 (t, ) as above ( unformly in t ). Suppose B (h) B. Let Ω h = S ( h 2, h 2 ), S R2 rather general. Rescale: y(x, x 3 ) = v(x, hx 3 ) and set E h (v) = h Ω 3 W (h 1 x 3, v(x)) dx h = h 2 Ω 1 W (x 3, y(x), h 1 y,3 (x)) dx =: I h (y) to get functionals on a fixed domain. 11

22 Effective theory for h 0 Theorem. (Compactness) If (y h ) W 1,2 (Ω 1 ), I h (y h ) C, then ( y, h 1 y,3 ) L2 ( y, b) SO(3) a.e. with ( y, b) H 1 (Ω 1 ) independent of x 3. Proof. Follows directly from the homogeneous case. (up to subseq.) 12

23 Effective theory for h 0 Theorem. (Compactness) If (y h ) W 1,2 (Ω 1 ), I h (y h ) C, then ( y, h 1 y,3 ) L2 ( y, b) SO(3) a.e. with ( y, b) H 1 (Ω 1 ) independent of x 3. Proof. Follows directly from the homogeneous case. (up to subseq.) Theorem. I h Γ-converges to I 0 with { 1 I 0 (y) = Q 2 S 2 (II) for y A, otherwise, where A = {u W 2,2 (S; R 3 ) : u O(2; 3) a.e.} (the set of isometric immersions), II is the second fundamental form of y and Q 2 is a polynomial of degree 2. 12

24 The quadratic form For F R 2 2 : let ˆF R 3 3 (fill up with 0) and for F R 3 3 : let ˇF R 2 2 (omit last row and column). 13

25 The quadratic form For F R 2 2 : let ˆF R 3 3 (fill up with 0) and for F R 3 3 : let ˇF R 2 2 (omit last row and column). Set Q 2 (t, F ) := min c R 3 Q 3(t, ˆF + c e 3 ) (= The quadratic form of Kirchhoff s homogeneous plate theory.) 13

26 The quadratic form For F R 2 2 : let ˆF R 3 3 (fill up with 0) and for F R 3 3 : let ˇF R 2 2 (omit last row and column). Set Q 2 (t, F ) := min c R 3 Q 3(t, ˆF + c e 3 ) (= The quadratic form of Kirchhoff s homogeneous plate theory.) Then 1 2 Q 2 (F ) = min Q 2 (t, A + tf + ˇB(t)) dt. A R

27 The quadratic form For F R 2 2 : let ˆF R 3 3 (fill up with 0) and for F R 3 3 : let ˇF R 2 2 (omit last row and column). Set Q 2 (t, F ) := min c R 3 Q 3(t, ˆF + c e 3 ) (= The quadratic form of Kirchhoff s homogeneous plate theory.) Then 1 2 Q 2 (F ) = min Q 2 (t, A + tf + ˇB(t)) dt. A R In fact: For a quadratic form Q (pos. def. on symm. matrices) Q 2 (F ) = Q(F F 0 ) + α. Q can be calculated explicitly from the first moments t k Q 2 (t, ) dt, k = 0, 1, 2, of the quadratic forms Q 2 (t, ) of the homogeneous Kirchhoff theory. 13

28 On the proof I liminf-ineq.: Adapt the Friesecke/James/Müller approach for homogeneous plates: ( ) I h (y h ) = h Ω 2 W 0 x 3, ( y, h 1 y,3 )(Id + hb (h) ) Ω 1 Q 3 ( x 3, R(x ) T ( y, h 1 y,3 )(Id + hb (h) ) ) Id. h 14

29 On the proof I liminf-ineq.: Adapt the Friesecke/James/Müller approach for homogeneous plates: ( ) I h (y h ) = h Ω 2 W 0 x 3, ( y, h 1 y,3 )(Id + hb (h) ) Ω 1 Q 3 ( x 3, R(x ) T ( y, h 1 y,3 )(Id + hb (h) ) ) Id. h Taking lim inf h, using lower semicontinuity and minimizing out the last row and column: lim inf I h (y h ) 1 ( Q 2 x3, A(x ) + x 3 II(x ) + h 2 ˇB(x 3 ) ). Ω 1 14

30 On the proof I liminf-ineq.: Adapt the Friesecke/James/Müller approach for homogeneous plates: ( ) I h (y h ) = h Ω 2 W 0 x 3, ( y, h 1 y,3 )(Id + hb (h) ) Ω 1 Q 3 ( x 3, R(x ) T ( y, h 1 y,3 )(Id + hb (h) ) ) Id. h Taking lim inf h, using lower semicontinuity and minimizing out the last row and column: lim inf I h (y h ) 1 ( Q 2 x3, A(x ) + x 3 II(x ) + h 2 ˇB(x 3 ) ). Ω 1 So by construction of Q 2 : lim inf I h (y h ) 1 h 2 S Q 2 (II(x )). 14

31 On the proof II Problem: This shifts the main problem to the construction of the Recovery sequences: For y A with b = y,1 y,2 set ( ) y h (x, x 3 ) = y(x ) + h (x 3 λ(x ))b(x ) + y(x ) f (x ) + h 2 D(x, x 3 ). 15

32 On the proof II Problem: This shifts the main problem to the construction of the Recovery sequences: For y A with b = y,1 y,2 set ( ) y h (x, x 3 ) = y(x ) + h (x 3 λ(x ))b(x ) + y(x ) f (x ) + h 2 D(x, x 3 ). Technical subtlety: Need density (cf. Pakzad/Hornung) of smooth isometric immersions and a representation result: Lemma. Let y be a well-behaved isometric immersion. Then any symmetric tensor field A supported on {II 0}) is of the form A = λii ( f + ( f )T ). Note: For y A, det II = 0. Thus, the PDE A λii = symmetrized gradient for λ is non-elliptic. 15

33 Overview 1 Motivation 2 Effective Theories 3 Minimizing Configurations 16

34 Cylinders Observation. Certain cylinders are minimizers. Proof. Note II N = {F : F = F T, det F = 0} = {a a} To minimize S Q(II F 0) choose a.e. II F argmin{q(f F 0 ) : F N }. 17

35 Cylinders Observation. Certain cylinders are minimizers. Proof. Note II N = {F : F = F T, det F = 0} = {a a} To minimize S Q(II F 0) choose a.e. II F argmin{q(f F 0 ) : F N }. Proposition. Every minimizer is a cylinder. 17

36 Cylinders Observation. Certain cylinders are minimizers. Proof. Note II N = {F : F = F T, det F = 0} = {a a} To minimize S Q(II F 0) choose a.e. II F argmin{q(f F 0 ) : F N }. Proposition. Every minimizer is a cylinder. Proof. a a b b N = a a, b b are lin. indep. 17

37 Cylinders Observation. Certain cylinders are minimizers. Proof. Note II N = {F : F = F T, det F = 0} = {a a} To minimize S Q(II F 0) choose a.e. II F argmin{q(f F 0 ) : F N }. Proposition. Every minimizer is a cylinder. Proof. a a b b N = a a, b b are lin. indep. y(s) is a ruled surface, locally paramtrized by (t, s) y(γ(t) + sν(t)), γ : I S the leading curve with normal ν and curvature κ s.t. II(γ(t) + sν(t)) = λ(t) 1 sκ(t) γ (t) γ (t). y minimizer = II N a.e. = κ 0 = γ, λ const. = y cylinder. 17

38 A 1D model In 1D: Film becomes a u curve γ, E(γ) = L 0 (κ(t) κ 0 ) 2 dt. For large films (curves of length L h 1 ) one should require a non-interpenetration condition. 18

39 A 1D model In 1D: Film becomes a u curve γ, E(γ) = L 0 (κ(t) κ 0 ) 2 dt. For large films (curves of length L h 1 ) one should require a non-interpenetration condition. Theorem. Under these conditions the energy minimizers are double spirals 18

40 A 1D model In 1D: Film becomes a u curve γ, E(γ) = L 0 (κ(t) κ 0 ) 2 dt. For large films (curves of length L h 1 ) one should require a non-interpenetration condition. Theorem. Under these conditions the energy minimizers are double spirals Etch from both sides: 18

41 Thanks Thank you for your attention! References: Minimal energy configurations of strained multi-layers. Calc. Var. Partial Diff. Eq. 30 (2007) pp Plate theory for stressed heterogeneous multilayers of finite bending energy. J. Math. Pures Appl. 88 (2007) pp