Faraday waves in soft elastic solids

Recent experiments have observed the emergence of standing waves at the free surface of elastic bodies attached to a rigid oscillating substrate and subjected to critical values of forcing frequency and amplitude. This phenomenon, known as Faraday instability, is now well understood for viscous fluids but surprisingly eluded any theoretical explanation for soft solids. Here, we characterize Faraday waves in soft incompressible slabs using the Floquet theory to study the onset of harmonic and subharmonic resonance eigenmodes. We consider a ground state corresponding to a finite homogeneous deformation of the elastic slab. We transform the incremental boundary value problem into an algebraic eigenvalue problem characterized by the three dimensionless parameters, that characterize the interplay of gravity, capillary and elastic waves. Remarkably, we found that Faraday instability in soft solids is characterized by a harmonic resonance in the physical range of the material parameters. This seminal result is in contrast to the subharmonic resonance that is known to characterize viscous fluids, and opens the path for using Faraday waves for a precise and robust experimental method that is able to distinguish solid-like from fluid-like responses of soft matter at different scales.

Effect of Rotation in an Orthotropic Elastic Slab

The fundamental equations of the two dimensional generalized thermoelasticity (L-S model) with one relaxation time parameter in orthotropic elastic slab has been considered under effect of rotation. The normal mode analysis is used to the basic equations of motion and heat conduction equation. Finally, the resulting equations are written in the form of a vector-matrix differential equation which is then solved by the eigenvalue approach. The field variables in the space time domain are obtained numerically. The results corresponding to the cases of conventional thermoelasticity CTE), extended thermoelasticity (ETE) and temperature rate dependent thermoelasticity (TRDTE) are compared by means of graphs.

Wave Propagation Through a Micropolar Slab Sandwiched by Two Elastic Half-Spaces

The problem of wave propagation through a micropolar elastic slab sandwiched by two classical elastic half-spaces is studied in this paper. Different from the classical elastic solids, the particle in micropolar solids can bear not only the displacements but also the rotations. The additional kinetic freedom results in four kinds of wave modes, namely, the longitudinal displacement (LD) wave, the longitudinal microrotational (LR) wave, and two coupled transverse (CT) waves. Apart from the LD wave, the other three waves are dispersive. The existence of couple stresses and the microrotations also makes the interface conditions between the micropolar slab and the classic elastic half-spaces different from that between two classic solids. The nontraditional interface conditions lead to a set of algebraic equations from which the amplitude ratios of reflection and transmission waves can be determined. Further, the energy fluxes carried by various waves are evaluated and the energy conservation is checked to validate the numerical results obtained. The influences of the micropolar elastic constants and the thickness of slab are discussed based on the numerical results. Two situations of incident P wave and incident SH wave are both considered.

Two independent methods for mapping the grounding line of an outlet glacier – example from the Astrolabe Glacier, Terre Adélie, Antarctica

The grounding line is a key element acting on the dynamics of coastal outlet glaciers. Knowing its position accurately is fundamental for both modelling the glacier dynamics and establishing a benchmark to which one can later refer in case of change. Here we map the grounding line of the Astrolabe Glacier in East Antarctica (66°41´ S; 140°05´ E), using hydrostatic and tidal methods. The first method is based on new surface and ice thickness data from which the line of buoyant flotation is found. We compare this hydrostatic map with kinematic GPS measurements of the tidal response of the ice surface. By detecting the transitions where the ice starts to move vertically in response to the tidal forcing we find control points for the grounding line position along GPS profiles. %If it can be shown that the long-term viscous mechanical behaviour of the ice slab validates the hydrostatic approach, mapping the grounding line from the ice supper surface displacements conversely requires correcting for the rigid elastic slab effect that dominates at tidal frequencies. With the help of a 2-dimensional elastic plate model, rigid elastic deviations are computed and applied to these control points. Once the extent of the grounding zone, the kinematic approach is consistent with the hydrostatic map. These two approaches lead us to propose a grounding line for the Astrolabe Glacier that significantly deviates from those obtained so far from satellite imagery.

On a new class of electro-elastic bodies. II. Boundary value problems

In part I of this two-part paper, a new theoretical framework was presented to describe the response of electro-elastic bodies. The constitutive theory that was developed consists of two implicit constitutive relations: one that relates the stress, stretch and the electric field, and the other that relates the stress, the electric field and the electric displacement field. In part II, several boundary value problems are studied within the context of such a construct. The governing equations allow for nonlinear coupling between the electric and stress fields. We consider boundary value problems wherein both homogeneous and inhomogeneous deformations are considered, with the body subject to an electric field. First, the extension and the shear of an electro-elastic slab subject to an electric field are studied. This is followed by a study of the problem of a thin circular plate and a long cylindrical tube, both subject to an inhomogeneous deformation and an electric field. In all the boundary value problems considered, the relationships between the stress and the linearized strain are nonlinear, in addition to the nonlinear relation to the electric field. It is emphasized that the theories that are currently available are incapable of modelling such nonlinear relations.

A Numerical Simulation of Interfacial Slip and its Role in Friction

The transition from static friction to kinetic friction results from the attainment of a point of instability, whereby interfacial slip becomes more energetically favorable than sticking. Such an instability is explored in this work via a plane-strain elastostatic analysis. A rigid pin of prescribed geometry is placed in contact with an elastic slab and translated horizontally under conditions of constant load. An intrinsic static coefficient of friction is prescribed, which limits the ratio of shear stress to contact pressure at each location within the interface. Additionally, the surface of the elastic slab is given a desired undulation to simulate the effects of surface roughness. As the pin is translated horizontally, a lateral reaction force (i.e., friction force) is developed and is observed to grow nearly linearly with increasing lateral displacement. At a critical point, a substantial portion of the interface experiences slip, leading to a large decrease in the friction force and thereby revealing a stick-slip behavior. It is found that the overall (macroscopic) static friction coefficient can be significantly less than the intrinsic friction coefficient and that the presence of even a small amount of roughness can have a large effect on the friction force.