Elastic Building Blocks in a Wrinkle Cascade Robert Schroll and - - PowerPoint PPT Presentation

Elastic Building Blocks in a Wrinkle Cascade

Robert Schroll and Benny Davidovitch Eleni Katifori

UMass Amherst Rockefeller University

2011 IMA Workshop – p. 1

Curtain Problem

Consider elastic sheet subject to stretching (Ustretch ∼ Et ≡ Y) + bending (Ubend ∼ Yt2) energies Sheet subject to uniform uniaxial confinement in ˆ y direction Compressive stress relieved by buckles with wavelength λb At x = 0, impose shape with λe < λb

y x

Photo by v1ctory_1s_m1ne

Transitioning between wavelengths requires ˆ x curvature

⇒Gaussian curvature ⇒In-plane strain

System gets to choose how much strain to accommodate

2011 IMA Workshop – p. 2

Wrinkle Cascade

Thin polystyrene sheet floating on water ⇒Surface tension favors small amplitude at edge

Huang et al, PRL 105, 038302 (2010)

Transition to optimal wavelength happens through cascade What is basic shape of the unit cell? Is it smooth or sharp?

Belgacem et al, J. Nonlinear Sci. 10, 661 (2000); Jin and Sternberg, J. Math. Phys. 42, 192 (2001); Das et al, PRL 98, 014301 (2007); Pomeau,

• Phil. Mag. B 78, 235 (1998)

2011 IMA Workshop – p. 3

Outline Curtain Problem Types of Building Blocks Smooth & Smooth Together Phase Space of Wrinkling

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Sharp & Smooth Building Blocks

We wish to characterize the building blocks that make up the shapes of thin elastic sheets. Sharp — Stress focused to corners and ridges.

Witten, RMP 79, 643 (2007)

Smooth — Stress does not focus.

Cerda and Mahadevan, PRL 91, 074302 (2003)

2011 IMA Workshop – p. 5

Sharp Building Blocks

Features where curvatures diverge as t → 0

⇒Stress is focused into vanishingly small areas

Shape reflects geometric principle: “Mostly developable” configuration to avoid Gaussian curvature Working Definition: Let AS be the area of the sheet with significant elastic energy density. A feature is sharp if

AS ATot → 0 as t → 0

Examples: d-cones: AS ∼ t2/3 minimal ridge: AS ∼ t1/3

2011 IMA Workshop – p. 6

Smooth Building Blocks

Both the curvature and stress are diffuse throughout the sheet A mechanical property reigns: Compressive stresses vanish with t (relaxed energy / membrane lmit) Working definition: A feature is smooth if:

AS ATot

0 as t → 0 Examples: Mahadevan-Cerda wrinkles from tension Lamé geometry — annulus under tension

2011 IMA Workshop – p. 7

Simplest Curtain Problem

Model a single generation in the cascade

• 1. Confine an elastic sheet

⇒Euler buckle

• 2. Force one edge into

3-buckle shape

• 3. Make sheet long enough

to achieves single buckle

W L ΔW x y z

Simulate with the Surface Evolver

• Program by Brakke to minimize energies over a mesh
• Built-in elastic and bending energies
• Use conjugate gradient and Hessian searches for local minima

http://www.susqu.edu/brakke/evolver/evolver.html

2011 IMA Workshop – p. 8

Simplest Curtain Problem

Model a single generation in the cascade

• 1. Confine an elastic sheet

⇒Euler buckle

• 2. Force one edge into

3-buckle shape

• 3. Make sheet long enough

to achieves single buckle

W L ΔW x y z

Simulate with the Surface Evolver

• Program by Brakke to minimize energies over a mesh
• Built-in elastic and bending energies
• Use conjugate gradient and Hessian searches for local minima

http://www.susqu.edu/brakke/evolver/evolver.html

2011 IMA Workshop – p. 8

Simplest Curtain Problem

Model a single generation in the cascade

• 1. Confine an elastic sheet

⇒Euler buckle

• 2. Force one edge into

3-buckle shape

• 3. Make sheet long enough

to achieves single buckle

W L x y z

Simulate with the Surface Evolver

• Program by Brakke to minimize energies over a mesh
• Built-in elastic and bending energies
• Use conjugate gradient and Hessian searches for local minima

http://www.susqu.edu/brakke/evolver/evolver.html

2011 IMA Workshop – p. 8

Two Prominent Features

W Lt

Plotting Gaussian curvature

Long, apparently smooth transition region Grows as thickness decreases, confinement increases Terminates in crescent with large Gaussian curvature

2011 IMA Workshop – p. 9

Two Prominent Features

W Lt

Plotting Gaussian curvature

Long, apparently smooth transition region Grows as thickness decreases, confinement increases Terminates in crescent with large Gaussian curvature

2011 IMA Workshop – p. 9

Two Prominent Features

W Lt

Plotting Gaussian curvature

Long, apparently smooth transition region Grows as thickness decreases, confinement increases Terminates in crescent with large Gaussian curvature

2011 IMA Workshop – p. 9

Two Prominent Features

W

Plotting Gaussian curvature

Long, apparently smooth transition region Grows as thickness decreases, confinement increases Terminates in crescent with large Gaussian curvature

2011 IMA Workshop – p. 9

Transition Region Scales with t

Compliance given by

ǫ ≡ t2/W 2∆

Centerlines collapse under scaling x → xǫ1/4 Recalls scaling argument: Bending ∼ Stretching Bκ2 ∼ Yt2 √

∆

W

2

∼ Y u2

xx ∼ Y

√

∆W

Lt

4

⇒ Lt ∼ W/ǫ1/4

Mahadevan, Vaziri, and Das, EPL 77, 40003 (2007) Recent experiments: Vandeparre et al, arXiv:1012.4325

2011 IMA Workshop – p. 10

Transition Region Scales with t

Compliance given by

ǫ ≡ t2/W 2∆

Centerlines collapse under scaling x → xǫ1/4 Recalls scaling argument: Bending ∼ Stretching Bκ2 ∼ Yt2 √

∆

W

2

∼ Y u2

xx ∼ Y

√

∆W

Lt

4

⇒ Lt ∼ W/ǫ1/4

Mahadevan, Vaziri, and Das, EPL 77, 40003 (2007) Recent experiments: Vandeparre et al, arXiv:1012.4325

2011 IMA Workshop – p. 10

Diffuse Stress in Transition

Argument suggests stretching energy is not focused This diffuse stress region does not shrink with t Suggests scaling solution in ¯ x

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Collapse of Compressive Stress

Airy (stress) potential has same scaling solution:

χ = ǫ1/2∆YW 2g(¯

x, y/W)

σxx ∼ Y∆ǫ1/2 σyy ∼ Y∆ǫ ⇒ σyy σxx ∼ ǫ1/2 − − →

t→0 0

Suggests mechanical principle for diffuse-stress areas: compressive stress vanishes relative to tensile stress

Stein and Hedgepeth, NASA TN D-813 (1961)

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Two Prominent Features

W

Plotting Gaussian curvature

Long, apparently smooth transition region Grows as thickness decreases, confinement increases Terminates in crescent with large Gaussian curvature

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Two Prominent Features

Plotting Gaussian curvature

Long, apparently smooth transition region Grows as thickness decreases, confinement increases Terminates in crescent with large Gaussian curvature

2011 IMA Workshop – p. 13

Focused Stress

There is a concentration of Gaussian curvature near x∗ Associated length scales vanish with ǫ Energy negligible in thin limit: Ufoc ∼ Yt5/3 ≪ Udif ∼ Yt3/2

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Focused Stress

There is a concentration of Gaussian curvature near x∗ Associated length scales vanish with ǫ Energy negligible in thin limit: Ufoc ∼ Yt5/3 ≪ Udif ∼ Yt3/2

2011 IMA Workshop – p. 14

Focused and Diffuse Stress

Focused and diffuse stress zones can coexist, despite differing constraints. Open questions: How to match geometric and mechanical constraints at junction of diffuse and focused stress zones? What is role of focused structure in minimizing elastic energy? How does focused structure compare to d-cone?

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Boundary Conditions Matter

Results up to now: Imposed shape must be planar Rotation of plane has little impact Releasing planarity constraint

⇒Removes diffuse-stress region

Only focused structure; Lt ∼ W

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Phase Diagram for Curtains?

Thickness (ε) Sharp features Buckling Transition Thickness provides cut-off length

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Phase Diagram for Curtains?

Thickness (ε) Tension (T/σ) Sharp features Buckling Transition Thickness provides cut-off length

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Tension Smooths Sheet

Apply tension T ˆ x (⊥ to confinement) For T < compressive stress σ, no effect For T > σ, focused structure “melts” Shape well-described by two Fourier modes

Davidovitch, PRE 80, 025202 (2009).

2011 IMA Workshop – p. 18

Phase Diagram for Curtains?

Thickness (ε) Tension (T/σ) Sharp features Buckling Transition Thickness provides cut-off length Tension irons

• ut shape

2011 IMA Workshop – p. 19

Phase Diagram for Curtains?

Thickness (ε) Tension (T/σ) Sharp features Buckling Transition Thickness provides cut-off length Tension irons

• ut shape

2011 IMA Workshop – p. 19

Conclusions

Smooth and sharp bulding blocks reflect different principles Smooth features reflect mechanical property: compressive stress vanishes Sharp features reflect geometric property: focus Gaussian curvature Model curtain shows coexistence of diffuse stress (smooth) and focused stress (sharp) regions Thickness and tension control degree of focusing

⇒Beginning of phase diagram for wrinkling

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Directions

Where else will we see coexistance of smooth and sharp features? What is influence of boundary conditions? How analogous is focused structure to d-cone? Can we get insight into core size from “melting” under tension? See PRL 106, 074301 (2011)

Funding: NSF-MRSEC on Polymers at UMass (DMR-0820506); Petroleum Research Fund of ACS

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Focused Scaling

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Boundary Conditions

W ∆W ~ x y z a) b) c)

2 4 6 0.2 0.1 0.0 0.1 z

x d)

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Elastic Energies

Surface Evolver models sheet as triangulated 2D plane Two edges of each triangle define a local “metric” Strain measured as difference between current and target metrics Stretching energy: integrated quadratics of strain Bending energy calculated per vertex

• Mean curvature from gradient of volume
• Gaussian curvature from excess angle

2011 IMA Workshop – p. 24