Analysis of a Compressed Thin Film Bonded to a Compliant Substrate: The Energy Scaling Law

Kohn, Robert V.; Nguyên, Hoài-Minh

doi:10.1007/s00332-012-9154-1

Cited by 26 publications

(40 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although Eq. 1 may be relevant also for more complex types of wrinkle patterns [e.g., under biaxial compression (20) or depressurizing a shell with a stiff core (21)], confinement of sheets in the absence of an imposed tension often leads to patterns with deep folds or stressfocusing zones (22)(23)(24), rather than to the oscillatory wrinkles described by Eqs. 1 and 2 and manifested in the following experimental examples.…”

Section: Theorymentioning

confidence: 99%

Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

Paulsen

Hohlfeld

King

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

105

180

View full text Add to dashboard Cite

Wrinkle patterns in compressed thin sheets are ubiquitous in nature and technology, from the furrows on our foreheads to crinkly plant leaves, from ripples on plastic-wrapped objects to the protein film on milk. The current understanding of an elementary descriptor of wrinkles-their wavelength-is restricted to deformations that are parallel, spatially uniform, and nearly planar. However, most naturally occurring wrinkles do not satisfy these stipulations. Here we present a scheme that quantitatively explains the wrinkle wavelength beyond such idealized situations. We propose a local law that incorporates both mechanical and geometrical effects on the spatial variation of wrinkle wavelength. Our experiments on thin polymer films provide strong evidence for its validity. Understanding how wavelength depends on the properties of the sheet and the underlying liquid or elastic subphase is crucial for applications where wrinkles are used to sculpt surface topography, to measure properties of the sheet, or to infer forces applied to a film.elastic sheets | wrinkles | curved topography W rinkles emerge in response to confinement, allowing a thin sheet to avoid the high energy cost associated with compressing a fraction e Δ of its length ( Fig. 1) (1-7). The wavelength, λ, of wrinkles reflects a balance between two competing effects: the bending resistance, which favors large wavelengths, and a restoring force that favors small amplitudes of deviation from the flat, unwrinkled state. Two such restoring forces are those due to the stiffness of a solid foundation or the hydrostatic pressure of a liquid subphase (Fig. 1A). Cerda and Mahadevan (1) realized that a tension in the sheet can give rise to a qualitatively similar effect ( Fig. 1B) and thereby proposed a universal law that applies in situations where the wrinkled sheet is nearly planar and subjected to uniaxial loading:Here the bending modulus B = Et 3 =½12ð1 − Λ 2 Þ (with E the Young's modulus, t the sheet's thickness, and Λ the Poisson ratio), whereas out-of-plane deformation is resisted by an effective stiffness, K eff , which can originate from a fluid or elastic substrate, an applied tension, or both. Eq. 1 is appealing in its simplicity, but it applies only for patterns that are effectively one-dimensional. In particular, it does not apply when the stress varies spatially or when there is significant curvature along the wrinkles. Here, we study two experimental settings in which these limitations are crucial: (i) indentation of a thin polymer sheet floating on a liquid, which leads to a horn-shaped surface with negative Gaussian curvature, and (ii) a circular sheet attached to a curved liquid meniscus with positive Gaussian curvature. In both cases, wrinkle patterns live on a curved surface, show spatially varying wavelengths, and are limited in spatial extent. The extent of finite wrinkle patterns in a variety of such 2D situations has recently been addressed (6,(8)(9)(10)(11) and was found to depend largely on external forces and boundary conditions. Howeve...

show abstract

Section: Theorymentioning

confidence: 99%

Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

Paulsen

Hohlfeld

King

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

105

180

View full text Add to dashboard Cite

show abstract

“…The energy (6) is still useful, even when it is not a -limit, i.e., when it is interpreted as a functional that depends on h, the small parameter. Indeed, the energy (6) is used extensively in the physics literature (e.g., in [16,17,19,22,23]) as well as for rigorous analysis (e.g., in [20,21,24,25]) even in situations where the energy does not scale like h 5 . For blistering in thin films attached to a substrate with elastic mismatch [26,27], there is post-facto justification for using the elastic energy (6); the results from this reduced energy [24] agree with the results obtained from a fully 3D elastic energy [28], although the elastic energy scales like h 2 h 5 .…”

Section: Plate Theories Through -Convergencementioning

confidence: 99%

“…The zig-zag and Eckhaus instabilities are generically present in pattern forming systems where the preferred patterns are rolls and the homogeneous state is unstable to perturbations with a range of wave numbers [2]. We therefore expect that they may also occur in elastic sheet in situations that lead to nearly periodic patterns, e.g., the wrinkling of an elastic thin film bonded to a compliant substrate [22,25,35]. The variational analysis in [25] identifies the minimum energy states in this system with a herringbone pattern and determines the associated wavelength.…”

Section: Phase Surfacesmentioning

confidence: 99%

“…We therefore expect that they may also occur in elastic sheet in situations that lead to nearly periodic patterns, e.g., the wrinkling of an elastic thin film bonded to a compliant substrate [22,25,35]. The variational analysis in [25] identifies the minimum energy states in this system with a herringbone pattern and determines the associated wavelength. This analysis also shows that the homogeneous flat state is unstable to a range of wave numbers.…”

Section: Phase Surfacesmentioning

confidence: 99%

See 1 more Smart Citation

Elastic Sheets, Phase Surfaces, and Pattern Universes

Newell

Venkataramani

2017

Stud Appl Math

View full text Add to dashboard Cite

We connect the theories of the deformation of elastic surfaces and phase surfaces arising in the description of almost periodic patterns. In particular, we show parallels between asymptotic expansions for the energy of elastic surfaces in powers of the thickness h and the free energy for almost periodic patterns expanded in powers of ε, the inverse aspect ratio of the pattern field. For sheets as well as patterns, the resulting energy can be expressed in terms of natural geometric invariants, the first and second fundamental forms of the elastic surface, respectively, the phase surface. We discuss various results for these energies and also address some of the outstanding questions. We extend previous work on point (in two dimensional) and loop (in three dimensional) disclinations and connect their topological indices with the condensation of Gaussian curvature of the phase surface. Motivated by this connection with the charge and spin of pattern quarks and leptons, we lay out an ambitious program to build a multiscale universe inspired by patterns in which the short (spatial and temporal) scales are given by a nearly periodic microstructure and whose macroscopic/slowly varying/averaged behaviors lead to a hierarchy of structures and features on much longer scales including analogs to quarks and leptons, dark matter, dark energy, and inflationary cosmology. One of our new findings is an interpretation of dark matter as the energy density in a pattern field. The associated gravitational forces naturally result in galactic rotation curves that are consistent with observations, while simultaneously avoiding some of the small‐scale difficulties of the standard ΛCDM (cold dark matter) paradigm in cosmology.

show abstract

“…The cylinder-mandrel problem is similar in spirit to that of [19]: in some sense, the obstacle in our analysis plays the role of their elastic substrate. In this case, our best upper and lower bounds do not match (so that at least one of them is suboptimal).…”

Section: Introductionmentioning

confidence: 99%

Axial Compression of a Thin Elastic Cylinder: Bounds on the Minimum Energy Scaling Law

Tobasco

2017

Comm Pure Appl Math

View full text Add to dashboard Cite

We consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core. Treating the core as an obstacle, we prove upper and lower bounds on the minimum energy of the cylinder that depend on its relative thickness and the magnitude of axial compression. We focus exclusively on the setting where the radius of the core is greater than or equal to the natural radius of the cylinder. We consider two cases: the "large mandrel" case, where the radius of the core exceeds that of the cylinder, and the "neutral mandrel" case, where the radii of the core and cylinder are the same. In the large mandrel case, our upper and lower bounds match in their scaling with respect to thickness, compression, and the magnitude of pre-strain induced by the core. We construct three types of axisymmetric wrinkling patterns whose energy scales as the minimum in different parameter regimes, corresponding to the presence of many wrinkles, few wrinkles, or no wrinkles at all. In the neutral mandrel case, our upper and lower bounds match in a certain regime in which the compression is small as compared to the thickness; in this regime, the minimum energy scales as that of the unbuckled configuration. We achieve these results for both the von Kármán-Donnell model and a geometrically nonlinear model of elasticity.

show abstract

Analysis of a Compressed Thin Film Bonded to a Compliant Substrate: The Energy Scaling Law

Cited by 26 publications

References 39 publications

Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets

Elastic Sheets, Phase Surfaces, and Pattern Universes

Axial Compression of a Thin Elastic Cylinder: Bounds on the Minimum Energy Scaling Law

Contact Info

Product

Resources

About