Frustrating geometry: Non Euclidean plates

Transcription

1 Frustrating geometry: Non Euclidean plates Efi Efrati In collaboration with Eran Sharon & Raz Kupferman Racah Institute of Physics The Hebrew University of Jerusalem September 009

2 Outline: Growth and geometric frustration. Incompatible 3D hyperelasticity. Non Euclidean plates: Examples. Reduced D elastic theory of non Euclidean plates. Non Euclidean plates: Analysis. An application: Almost minimal surfaces.

3 Outline: Growth and geometric frustration. Incompatible 3D hyperelasticity. Non Euclidean plates: Examples. Reduced D elastic theory of non Euclidean plates. Non Euclidean plates: Analysis. An application: Almost minimal surfaces.

4 Growth Arabidopsis mutant Peach leaf curl CINCINNATA mutant, [Nath et al 003] Plant growth mechanics Residual stress due to local growth. Biological response to mechanical stress. Manipulating growth regulation causes morphological changes.

5 Time scale separation τ << τ Elastic Shaping Plant growth (Arabidopsis): - Cell division hours - % area growth by expansion hours - Acoustic time 1 μs - Cantilever mode typical time 100 μs

6 Spontaneous growth and geometric frustration Why local growth is likely to result in residual stress? Given 8 points 8 rods of arbitrary lengths, you will (probably) not be able to connect every pair once. think in rod length rather than points

7 Spontaneous growth and geometric frustration Why local growth is likely to result in residual stress? Example: Isotropic non-homogeneous growth. Every point initially at rest expands isotropically by a factor. λ(r r )

8 Spontaneous growth and geometric frustration Why local growth is likely to result in residual stress? Example: Isotropic non-homogeneous growth. Every point initially at rest expands isotropically by a factor. λ(r r ) The only expansion factor which does not result in residual stress : λ( r) r r r C r r 0

9 Soft active deformations Residual stress must be addressed when describing motion by auto-deformation. Making use of unused residual work.

10 Signature of geometric frustration Tempered glass fragments cannot be re-joined When tempered glass shatters, each of its fragments deforms to relax internal stresses. The relaxed fragments do not fit one another.

11 Signature of geometric frustration The helicoidal form of a Bauhinia pod is residually stressed. Slicing along its length generate helices of a higher pitch.

12 Outline: Growth and geometric frustration. Incompatible 3D hyperelasticity. Non Euclidean plates: Examples. Reduced D elastic theory of non Euclidean plates. Non Euclidean plates: Analysis. An application: Almost minimal surfaces.

13 Hyper-elasticity [Truesdell 195] + A measure of local deformation: Strain Local elastic energy density as a function of the deformation Hyperelastic description E W(ε ) dv Stress is obtained as a Frechet derivative of the elastic energy density. S W ε

14 A measure of deformation: The Strain Stress free configuration x Deformed configuration r r(x) x r u The displacement field ( ) ( ) I r r u u u u T T T + + ε ) ( ) ( ) ( 1 1 Cauchy St-Venant strain

15 A measure of deformation: The Strain The displacement field u r x x Stress free configuration r(x) r Deformed configuration Rely on the existence of a stress free configuration

16 Differential geometry I : Three dimensions The metric tensor: g ij r x i r x j r(x) Length element: ds g ij dx i dx j A 3 by 3 matrix is the metric of a body in Euclidean space. If two embeddable metrics are identical, the bodies they describe differ by a rigid motion. All components of the Riemann curvature tensor vanish. Such metrics are called: Euclidean Compatible Embeddable

17 3D incompatible hyperelasticity 3D metric determine a configuration uniquely. Our adaptation of Truesdell s hyperelasticity principle is formulated in terms of metric. The energy stored within a deformed elastic body is a volume integral of an elastic energy density which depends only on the local metric tensor and on tensors that characterize the body, but are independent of the configuration.

18 3D incompatible hyperelasticity The energy density depends on the configuration through the metric. x For every there exists a unique SPD tensor g(x) for which the energy density vanish, i.e. g(x) is called the reference metric. The Cauchy-St. Venant strain W ( g, x) W ( g( x), x) 1 ε ( g g ) 0 current configuration metric reference metric W ( ε, x) 0 ε 0

19 When a stress free configuration exists, we may set. In such coordinates the Cauchy-St. Venant strain adopts the familiar form 3D incompatible hyperelasticity g I ( ) ( ) ( ) u u u u I r r I g T T T + + ε ) ( ) ( ) ( 1 1 1

20 3D incompatible hyperelasticity g I When a stress free configuration exists, we may set. In such coordinates the Cauchy-St. Venant strain adopts the familiar form ε 1 ( ) ( T ) ( T T g I r) r I u + ( u) + ( u) u) 1 1 ( For such a case the elastic energy density of an isotropic material is of the form A' ijkl W A' ijkl ε ij ε kl + O ( 3 ε ) ( ik jl il jk δ δ δ ) ij kl λδ δ + μ δ + Lame coefficients

21 3D incompatible hyperelasticity g I When a stress free configuration exists, we may set. In such coordinates the Cauchy-St. Venant strain adopts the familiar form ε 1 ( ) ( T ) ( T T g I r) r I u + ( u) + ( u) u) 1 1 ( For such a case the elastic energy density of an isotropic material is of the form A' ijkl W A' ijkl ε ij ε kl + O ( 3 ε ) ( ik jl il jk δ δ δ ) ij kl λδ δ + μ δ + Lame coefficients A stress free configuration may not exist.

22 3D incompatible hyperelasticity Locally (only at a point) we may set new coordinates in which g ' I A' W ε 1 A' ijkl ε' ij 1 ' ij g ij δij ijkl ε' kl ( ' ) ij kl ik jl il jk δ + μ( δ δ δ δ ) λδ +

23 3D incompatible hyperelasticity Locally (only at a point) we may set new coordinates in which g ' I A' W ε 1 A' ijkl ε' ij 1 ' ij g ij δij ijkl ε' kl ( ' ) ij kl ik jl il jk δ + μ( δ δ δ δ ) λδ + Tensor transformation rules (Covariance becomes valuable) (Λ -1 ) Λ / k i k i k x' x k x / x' i i

24 3D incompatible hyperelasticity Locally (only at a point) we may set new coordinates in which g ' I A' W ε 1 A' ijkl ε' ij 1 ' ij g ij δij ijkl ε' kl ( ' ) ij kl ik jl il jk δ + μ( δ δ δ δ ) λδ + Tensor transformation rules (Covariance becomes valuable) (Λ -1 ) Λ / k i k i k x' x k x / x' i i W ε ij 1 1 A ijkl ε ij ε kl ( gij gij ) ij kl ik jl il jk g g + ( g g g g ) ijkl A λ μ +

25 3D incompatible hyperelasticity Locally (only at a point) we may set new coordinates in which g ' I A' W ε 1 A' ijkl ε' ij 1 ' ij g ij δij ijkl ε' kl ( ' ) ij kl ik jl il jk δ + μ( δ δ δ δ ) λδ + gij Λ k i Λ l j δ kl Tensor transformation rules (Covariance becomes valuable) (Λ -1 ) Λ / k i k i k x' x k x / x' i i W ε ij 1 1 A ijkl ε ij ε kl ( gij gij ) ij kl ik jl il jk g g + ( g g g g ) ijkl A λ μ +

26 3D incompatible hyperelasticity Recapitulation E W g 1 dx dx dx 3 ε ij 1 ( g g ) ij ij Strain is measured with respect to a reference metric. W 1 ijkl A ijε kl λ ε ε ε + με i i k k i k ε k i [Efrati Sharon & Kupferman, JMPS 009]

27 3D incompatible hyperelasticity Recapitulation E W g 1 dx dx dx 3 ε ij 1 ( g g ) ij ij Strain is measured with respect to a reference metric. W 1 ijkl A ijε kl λ ε ε ε + με i i k k i k ε k i [Efrati Sharon & Kupferman, JMPS 009] The reference metric need not be immersible.

28 Outline: Growth and geometric frustration. Incompatible 3D hyperelasticity. Non Euclidean plates: Examples. Reduced D elastic theory of non Euclidean plates. Non Euclidean plates: Analysis. An application: Almost minimal surfaces.

29 Experimental realization of Non Euclidean plates Environmentally responsive gels. N-Isopropylacrylamide undergoes a large volume reduction when heated above 33 c o. The volume reduction ratio depends strongly on monomer concentration. When injected cold with a catalyst, the gel polymerizes in a few minutes, freezing the monomer concentration gradient. [Klein Efrati & Sharon. Science 007]

30 Experimental realization of Non Euclidean plates [Klein Efrati & Sharon. Science 007]

31 Experimental realization of Non Euclidean plates Does not shrink Shrinks considerably Shrinkage homogeneous across the thickness Large thickness: Flat and strained. Small thickness: Buckled.

32 Experimental realization of Non Euclidean plates Shrinks considerably Does not shrink Shrinkage homogeneous across the thickness Large thickness: Flat and strained. Small thickness: Buckled.

33 Experimental realization of Non Euclidean plates Hyperbolic metric (saddle like everywhere)

34 Experimental realization of Non Euclidean plates The only shaping mechanism is the Prescription of a Non-Euclidean Metrics

35 Shaping by metric prescription Leaf shaping Plastic deformation Crocheting & knitting [Nath et-al 003] Uncontrolled tearing [E. Coen et-al 003] Controlled tearing Crocheted segment of the hyperbolic plane [Taimina 1997].

36 Computer simulations Defects in flexible membranes with crystalline order [Seung & Nelson 1988] Recent implementation to macroalgae blades [Koehl et al 008] Three layers of identical array of springs of spatially varying rest length. [Marder et-al 004]

37 Outline: Growth and geometric frustration. Incompatible 3D hyperelasticity. Non Euclidean plates: Examples. Reduced D elastic theory of non Euclidean plates. Non Euclidean plates: Analysis. An application: Almost minimal surfaces.

38 Differential geometry II : Surfaces The first quadratic form (metric) The second quadratic form (curvatures) a αβ + - α r β r b αβ r Nˆ αβ The mean curvature The Gaussian curvature ( ) ( β κ + κ Tr ) 1 H 1 1 b α K ( β ) κ κ det b 1 α K F ( aαβ, γaαβ, γ δaαβ) Gauss Theorema Egregium

39 Differential geometry II : Surfaces The first quadratic form (metric) The second quadratic form (curvatures) a αβ + - α r β r b αβ r Nˆ αβ The mean curvature The Gaussian curvature ( ) ( β κ + κ Tr ) 1 H 1 1 b α K ( β ) κ κ det b 1 α K F ( aαβ, γaαβ, γ δaαβ) Gauss Theorema Egregium 0 G ( a, a, b, b 1, αβ γ αβ αβ γ αβ Peterson Mainardi Codazzi eq. )

40 Differential geometry II : Surfaces The first quadratic form (metric) The second quadratic form (curvatures) a αβ + - α r β r b αβ r Nˆ αβ Given two quadratic forms which satisfy all three GPMC equations, they define a surface uniquely. K F ( aαβ, γaαβ, γ δaαβ) Gauss Theorema Egregium 0 G ( a, a, b, b 1, αβ γ αβ αβ γ αβ Peterson Mainardi Codazzi eq. )

41 A toy model Trapezoidal inclusion Similar lengths Similar lengths Large thickness: Flat configuration. The trapezoid is under compression. Too long Small thickness: Buckled configuration. The trapezoid is bent to accommodate the excess length.

42 A toy model No stress free configuration Rest lengths are continuous The body possess no internal structure across the thin dimension

43 Neither plates nor shells Long Short A plate may be considered as a stack of identical surfaces A shell may be considered as a continuous collection of non-identical surfaces Similarly to plates: no structural variation across the thin dimension Unlike plates: D metric of mid-surface non-euclidean

44 Neither plates nor shells Long Short A plate may be considered as a stack of identical surfaces A shell may be considered as a continuous collection of non-identical surfaces Existing plate/shell theories do no apply Similarly to plates: no structural variation across the thin dimension Unlike plates: D metric of mid-surface non-euclidean

45 Neither plates nor shells t 3 x t denotes the coordinate along the thin dimension Non-Euclidean plate metric g a a , a1 a 3aαβ 0. K( ad ) 0 Rijkl( g3d ) 0

46 Reduced D elastic energy Goal: Describe the elastic behavior, using only mid-surface properties; First and second fundamental forms. E [ + ] k i ν i k ε ε k 1 ν ε ε i k t t Y (1 + ν ) 1 i g dx dx dx 3 Integrate out the thin dimension. Assumption: Faces of sheet are free of forces.

47 Reduced D elastic energy Goal: Describe the elastic behavior, using only mid-surface properties; First and second fundamental forms. E t 1 8 A αβγδ ( a αβ a αβ )( a γδ a γδ ) a 1 dx dx + t A αβγδ b αβ b γδ a 1 dx dx + h. o. t A αβγδ ( αγ βδ αβ γδ ) ν a a a a Y 1+ ν + 1 ν a αβ r r b r Nˆ α The first quadratic form (metric) of the mid-surface β αβ αβ The second quadratic form (curvatures) of the mid-surface

48 Reduced D elastic energy Stretching content Bending content E te + t 3 S e B e S γδ 1 αβγδ 1 8 A ( aαβ aαβ)( aγδ a ) a dx dx Stretching content measures deviations from the given D metric. Vanishes only if the midsurface is an isometric embedding of a. e B γδ 1 αβγδ 1 4 A bαβb a dx dx Bending content measures the magnitude of curvatures. Vanishes only for flat configuration. Generalization of F.V.K and Koiter theories for large displacements and arbitrary intrinsic geometry.

49 Reduced D elastic energy Stretching content Bending content E te + t 3 S e B e S γδ 1 αβγδ 1 8 A ( aαβ aαβ)( aγδ a ) a dx dx e B γδ 1 αβγδ 1 4 A bαβb a dx dx Large thickness: Bending term dominates Flat solutions (unbuckled) Small thickness: Stretching term dominates D isometric solution Note: a αβ and b αβ are not independent variables. [Efrati Shron & Kupferman JMPS 009]

50 Reduced D elastic energy Current literature: Elastic substrate, gravitational force, capillary forces, tension, confinement or some other mechanism necessitates buckling. Competition between the two energy terms yields the buckling length scales. [Huang et-al 007] [Cerda & Mahadevan 003-4] [Vella et-al 009]

51 Reduced D elastic energy Current literature: Elastic substrate, gravitational force, capillary forces, tension, confinement or some other mechanism necessitates buckling. Competition between the two energy terms yields the buckling length scales. [Huang et-al 007] [Cerda & Mahadevan 003-4] [Vella et-al 009] No external force; Confinement is replaced by an embedding problem. [Efrati Klein & Sharon. Physica D 007]

52 Outline: Growth and geometric frustration. Incompatible 3D hyperelasticity. Non Euclidean plates: Examples. Reduced D elastic theory of non Euclidean plates. Non Euclidean plates: Analysis. - Buckling - The vanishing thickness limit - Equipartition An application: Almost minimal surfaces.

53 Hemispherical plate We numerically minimize the elastic energy E te + t 3 S e B a 1 0 sin 0 ( x 1 ) We use a reference metric of a spherical cap x 1. 1 The solution is assumed to be axially symmetric

54 Hemispherical plate E te + t 3 S e B

55 Hemispherical plate te S E te + t 3 S e B t 3 0 e B

56 Hemispherical plate Similar flat configurations te S E te + t 3 S e B t 3 0 e B Almost isometric embedding

57 Hemispherical plate Similar flat configurations te S E te + t 3 S e B t 3 0 e B Slightly buckled Almost isometric embedding

58 Hemispherical plate Analytic results: Similar flat configurations The unbuckled state must contain both compression and tension. K( a) 0 If, there exists a finite buckling thickness. Classification in terms of the plane stress in the unbuckled configuration to super-critical and sub-critical bifurcations. te S [Efrati et-al PRE 009]

59 Hemispherical plate 0 We have proved that In the limit i.e. an isometric embedding of mid-surface. t a a αβ αβ

60 Hemispherical plate 0 We have proved that In the limit i.e. an isometric embedding of mid-surface. t a αβ a αβ The isometry is (usually) not unique. The limit configuration will minimize the bending content amongst all isometries. Recent Gamma limit proof for non Euclidean plates directly from 3D [Lewicka & Pakzad 009].

61 Hemispherical plate No mutual zero to the two terms; No equipartition. E te + t 3 S e B e S e B t 0 0 t 0 e 0 B t 3 0 e B 7 t

62 Hemispherical plate e & t & S e B E te + t 3 S e B e S e B t 0 0 t 0 e 0 B The variation of the bending energy about its limit t 3 ( ) 7 0 e e t B B

63 Hemispherical plate The power 7/ comes from a boundary layer whose thickness scales as l B. L t k The variation of the bending energy about its limit t 3 ( ) 7 0 e e t B B [Efrati et-al PRE 009]

64 Outline: Growth and geometric frustration. Incompatible 3D hyperelasticity. Non Euclidean plates: Examples. Reduced D elastic theory of non Euclidean plates. Non Euclidean plates: Analysis. An application: Almost minimal surfaces.

65 Solving the vanishing thickness limit Provided there exists an isometric embedding with bounded energy content, the vanishing limit configuration minimizes the Willmore energy among all isometric embeddings. Gaussian curvature K 1 Mean curvature H ( k + k ) k 1 k 1 ( ) ( 1 ν k k + νk k da H K)dA

66 Solving the vanishing thickness limit Provided there exists an isometric embedding with bounded energy content, the vanishing limit configuration minimizes the Willmore energy among all isometric embeddings. Gaussian curvature K 1 Mean curvature H ( k + k ) k 1 k 1 H da [Willmore 1986] Constrained minimization problem: The GMPC equations.

67 Differential geometry III: Isometric embedding of surfaces in 3D The Gauss Peterson Mainardi Codazzi compatability conditions may be considered as evolution equations for the curvature form,. b αβ 0 < K For elliptic set of P.D.Es. K < 0 For hyperbolic set of P.D.Es. 1 K cm 9 t cm t 0. 06cm t 0. 05cm t cm 1 K cm 9

68 Almost minimal surfaces Minimal surfaces are surfaces of vanishing mean curvature H 0 Constitute trivial minima of the Willmore energy. Not every metric can be embedded as a minimal surface. H da

69 Almost minimal surfaces Minimal surfaces are surfaces of vanishing mean curvature H 0 Constitute trivial minima of the Willmore energy. Not every metric can be embedded as a minimal surface. H da The mean curvature of any hyperbolic metric can vanish along a curve

70 Almost minimal surfaces We considering a thin narrow hyperbolic strip t << w < < L E W w w H da w L

71 Almost minimal surfaces We considering a thin narrow hyperbolic strip t << w < < L ( 3 E W e0 + e1w + ew + e3w +...)dx w L

72 Almost minimal surfaces We considering a thin narrow hyperbolic strip t << w < < L E W O( w 5 ) On the mid-line of the strip the mean curvature vanishes. The resulting ribbons resemble minimal surfaces. Both chiralities possess the same energy, thus are equally probable.

73 Almost minimal surfaces Inspired by the result that every (narrow enough) hyperbolic metric can be embedded as an almost minimal surface, we found new family of pseudospherical embeddings (surfaces of constant Gaussian curvature: K-1).

74 Almost minimal surfaces Inspired by the result that every (narrow enough) hyperbolic metric can be embedded as an almost minimal surface, we found new family of pseudospherical embeddings (surfaces of constant Gaussian curvature: K-1). The new family of pseudospherical embeddings also admits embeddings of very large mean curvature.

75 Almost minimal surfaces Prescribing a hyperbolic geometry on a strip using responsive gels Do not Shrink Shrinks

76 Almost minimal surfaces Prescribing a hyperbolic geometry on a strip using responsive gels Do not Shrink Shrinks

77 Almost minimal surfaces The mean curvature quadratic in the width coordinate. Curvature Squared (mm - ) Edge Preliminary results K H -H /K Center H /K Distance from the Edge (mm)

78 Almost minimal surfaces The mean curvature quadratic in the width coordinate. Curvature Squared (mm - ) Edge Preliminary results K H -H /K Center H /K Distance from the Edge (mm) Results become irrelevant for wide strips.

79 Conclusion Residual stress inevitable in local shaping mechanisms. Incompatible elasticity is the theoretical tool to deal with residual stress. Non-Euclidean plates: - Derivation from 3D elasticity. - Buckling, Boundary layer. - The isometric embedding problem. Application: Almost minimal surfaces.

80 Conclusion Residual stress inevitable in local shaping mechanisms. Incompatible elasticity is the theoretical tool to deal with residual stress. Non-Euclidean plates: - Derivation from 3D elasticity. - Buckling, Boundary layer. - The isometric embedding problem. Application: Almost minimal surfaces. Natural extension: Non-Euclidean shells. Elastic theory of non-euclidean plates and shells [submmited to Nonlinearity 009] Geometric applications: Folding and controllability of non Euclidean plates.

81 The End