Adhesion of an elastic plate to a sphere

Majidi, Carmel; Fearing, Ronald S.

doi:10.1098/rspa.2007.0341

Cited by 41 publications

(58 citation statements)

References 14 publications

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“…The above picture changes if we consider crystals constrained to a spherical surface of radius R. In this case, an additional energy enters the free energy that takes into account the elastic cost of bending the crystal to accommodate the curved template [11,12]. First consider again the circular crystal.…”

Section: B On a Spherementioning

confidence: 99%

“…To make the analysis more straightforward, we scale the free energy to the spherical surface area, 4πR 2 , times Young's modulus Y . This leads to a reduced unit η := a/R and a dimensionless free energy [9][10][11][12],…”

Section: B On a Spherementioning

confidence: 99%

“…The dimensionless free energy for the ribbon on a spherical surface is given in terms of dimensionless width ω = w/R and length λ = l/R by [11,12] …”

Section: B On a Spherementioning

confidence: 99%

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Impact of interaction range and curvature on crystal growth of particles confined to spherical surfaces

2017

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When colloidal particles form a crystal phase on a spherical template, their packing is governed by the effective interaction between them and the elastic strain of bending the growing crystal. For example, if growth commences under appropriate conditions, and the circular crystal that forms reaches a critical size, growth continues by incorporation of defects to alleviate elastic strain. Recently it was found experimentally that, if defect formation is somehow not possible, the crystal instead continues growing in ribbons that protrude from the original crystal. Here we report on computer simulations in which we observe both the formation of ribbons at short interaction ranges and packings that incorporate defects if the interaction is longer-ranged. The ribbons only form above some critical crystal size, below which the nucleus is roughly spherically shaped.We find that the scaling of the critical crystal size differs slightly from the one proposed by the Manoharan group, and reason this is because the actual process is a two-step heterogeneous nucleation of ribbons on top of roughly circular crystals.

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Section: B On a Spherementioning

confidence: 99%

Section: B On a Spherementioning

confidence: 99%

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Impact of interaction range and curvature on crystal growth of particles confined to spherical surfaces

2017

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“…These latter structures are treated as a thin elastic plate in contact with a rigid, non-flat substrate (Persson & Gorb 2003;Majidi & Fearing 2008). In biology, cell adhesion is also an active area of study and has been addressed by various plate theories and adhesion models (Seifert 1991;Rosso et al 2000;Wan & Liu 2001).…”

Section: Introductionmentioning

confidence: 99%

A simplified formulation of adhesion problems with elastic plates

Majidi

Adams

2009

Proc. R. Soc. A.

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The solution of adhesion problems with elastic plates generally involves solving a boundary-value problem with an assumed contact area. The contact region is then found by minimizing the total potential energy with respect to the contact area (i.e. the contact radius for the axisymmetric case). Such a procedure can be extremely long and tedious. Here, we show that the inclusion of adhesion is equivalent to specifying a discontinuous internal bending moment at the contact region boundary. The magnitude of this moment discontinuity is related to the work of adhesion and flexural rigidity of the plate. Such a formulation can greatly reduce the algebraic complexity of solving these problems. It is noted that the related plate contact problems without adhesion can also be solved by minimizing the total potential energy. However, it has long been recognized that it is mathematically more efficient to find the contact area by specifying a continuous internal bending moment at the boundary of the contact region. Thus, our moment discontinuity method can be considered to be a generalization of that procedure which is applicable for problems with adhesion.

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“…The orthoradial compression responsible for the formation of wrinkles is a consequence of Gauss Theorema Egregium: wrapping a sphere with a planar sheet implies stretching or compression in addition to bending [22]. In terms of scaling, the perfect contact between the plate and the stamp involves a typical strain ∼ (R/ρ) 2 and a stretching energy EhR 6 /ρ 4 [21,23]. The ratio of the stretching energy to the typical bending energy Eh 3 R 2 /ρ 2 thus yields the dimensionless parameter R/ √ ρh.…”

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confidence: 99%

Stamping and Wrinkling of Elastic Plates

Hure¹,

Roman²,

Bico³

2012

Phys. Rev. Lett.

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We study the peculiar wrinkling pattern of an elastic plate stamped into a spherical mold. We show that the wavelength of the wrinkles decreases with their amplitude, but reaches a maximum when the amplitude is of the order of the thickness of the plate. The force required for compressing the wrinkled plate presents a maximum independent of the thickness. A model is derived and verified experimentally for a simple one-dimensional case. This model is extended to the initial situation through an effective Young modulus representing the mechanical behavior of wrinkled state. The theoretical predictions are shown to be in good agreement with the experiments. This approach provides a complement to the "tension field theory" developed for wrinkles with unconstrained amplitude.Wrinkling patterns are observed when thin plates are put under compression, spanning scales from geological patterns [1], skin wrinkles resulting from aging processes or scars [2,3] to cells locomotion generating strains on substrates [4]. They have important applications in micro-engineering such as the formation of controlled patterns [5], or the estimation of mechanical properties from the number and the extent of the wrinkles [6][7][8][9]. While most studies have focused on near threshold patterns, recent contributions have pushed further the description of finely wrinkled plates, i.e., well above the initial buckling threshold [10,11]. These studies consider plates submitted to strong in-plane tension on the boundaries, which prevents stress focusing commonly observed in crumpled paper [12]. In these descriptions, wrinkles are assumed to totally relax compressive stresses, as in traditional tension field theory [13,14]. In addition, both amplitude and wavelength of the wrinkles vanish with the thickness of the plate. We propose to study a conceptually different wrinkling regime where the boundaries are free from tension but the amplitude of the wrinkles is highly constrained. We focus on the simplest example, an elastic plate compressed in a spherical mold with a confinement defined by a gap δ (Fig. 1). This stamping configuration is common in industrial processes where metal plates are plastically embossed, the mismatch in Gaussian curvature generally leading to regular wrinkles [15][16][17]. In the elastic case, crumpling singularities first appear as the mold is progressively closed down (Fig. 1c) and evolve into a pattern of apparently smooth radial wrinkles for high confinement (Fig. 1d-e). We show that constraining the amplitude does not lead to the collapse of compressive stress. Instead, the wrinkling pattern derives from a nontrivial balance between compression and bending stresses, with surprising consequences : the wavelength of the wrinkles does not vanish and the constraining force reaches a maximum independent of the thickness of the plate.One-dimensional problem. We start by considering the simpler problem of a plate of length L, thickness h and unit width. In-plane displacements are imposed at both ends u x (±L/2) = ±∆/2 (Fig...

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Adhesion of an elastic plate to a sphere

Cited by 41 publications

References 14 publications

Impact of interaction range and curvature on crystal growth of particles confined to spherical surfaces

Impact of interaction range and curvature on crystal growth of particles confined to spherical surfaces

A simplified formulation of adhesion problems with elastic plates

Stamping and Wrinkling of Elastic Plates

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