Adhesive contact problems for a thin elastic layer : Asymptotic analysis and the JKR theory

Contact problems for a thin compressible elastic layer attached to a rigid support are studied. Assuming that the thickness of the layer is much less than the characteristic dimension of the contact area, a direct derivation of asymptotic relations for displacements and stress is presented. The proposed approach is compared with other published approaches. The cases are established when the leading-order approximation to the non-adhesive contact problems is equivalent to contact problem for a Winkler–Fuss elastic foundation. For this elastic foundation, the axisymmetric adhesive contact is studied in the framework of the Johnson–Kendall–Roberts (JKR) theory. The JKR approach has been generalized to the case of the punch shape being described by an arbitrary blunt axisymmetric indenter. Connections of the results obtained to problems of nanoindentation in the case that the indenter shape near the tip has some deviation from its nominal shape are discussed. For indenters whose shape is described by power-law functions, the explicit expressions are derived for the values of the pull-off force and for the corresponding critical contact radius

Surface and edge phenomena in pre-stressed incompressible elastic solids

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Asymptotic Formulation for the Rayleigh Wave on a Nonlocally Elastic Half-Space

This paper deals with the Rayleigh wave, propagating on a nonlocally elastic, linearly isotropic half-space, excited by a prescribed surface loading. The consideration develops the methodology of hyperbolic–elliptic models for Rayleigh and Rayleigh-type waves, and relies on the effective boundary conditions formulated recently, accounting for the crucial contributions of the nonlocal boundary layer. A slow-time perturbation scheme is established, leading to the reduced model for the Rayleigh wave field, comprised of a singularly perturbed hyperbolic equation for the longitudinal wave potential on the surface, acting as a boundary condition for the elliptic equation governing the decay over the interior. An equivalent alternative formulation involving a pseudo-differential operator acting on the loading terms, with parametric dependence on the depth coordinate, is also presented

Surface instability of sheared soft tissues

International audienceWhen a block made of an elastomer is subjected to large shear, its surface remains flat. When a block of biological soft tissue is subjected to large shear, it is likely that its surface in the plane of shear will buckle (apparition of wrinkles). One factor that distinguishes soft tissues from rubber-like solids is the presence -- sometimes visible to the naked eye -- of oriented collagen fibre bundles, which are stiffer than the elastin matrix into which they are embedded but are nonetheless flexible and extensible. Here we show that the simplest model of isotropic nonlinear elasticity, namely the incompressible neo-Hookean model, suffers surface instability in shear only at tremendous amounts of shear, i.e., above 3.09, which corresponds to a $72^\circ$ angle of shear. Next we incorporate a family of parallel fibres in the model and show that the resulting solid can be either reinforced or strongly weakened with respect to surface instability, depending on the angle between the fibres and the direction of shear, and depending on the ratio $E/\mu$ between the stiffness of the fibres and that of the matrix. For this ratio we use values compatible with experimental data on soft tissues. Broadly speaking, we find that the surface becomes rapidly unstable when the shear takes place ``against'' the fibres, and that as $E/\mu$ increases, so does the sector of angles where early instability is expected to occur

Degenerated Boundary Layers and Long-Wave Low-Frequency Motion in High-Contrast Elastic Laminates

The effect of high contrast on the multiscale behaviour of elastic laminates is studied. Mathematical modelling in this area is of significant interest for a variety of modern applications, including but not limited to advanced sandwich structures and photovoltaic panels. As an example, the antiplane shear of a symmetric, three-layered plate is considered. The problem parameters expressing relative thickness, stiffness and density are assumed to be independent. The high contrast may generally support extra length and time scales corresponding to degenerated boundary layers and propagating long-wave low-frequency vibration modes. The main focus is on the relation between these two phenomena. The developed multiparametric approach demonstrates that those do not always appear simultaneously. The associated explicit estimates on contrast parameters are established. In addition, the recent asymptotic extension of the classical Saint-Venant’s principle is adapted for calculating the contribution of the degenerate boundary layer or long-wave low-frequency propagation mode. The peculiarity of the limiting absorption principle in application to layered media is also addressed

The edge wave on an elastically supported Kirchhoff plate

This Letter deals with an analysis of bending edge waves propagating along the free edge of a Kirchhoff plate supported by a Winkler foundation. The presence of a foundation leads to a non-zero cut-off frequency for this wave, along with a local minimum of the associated phase velocity. This minimum phase velocity corresponds to a critical speed of an edge moving load and is analogous to that in the classical 1D moving load problem for an elastically supported beam