Remarks on explicit strong ellipticity conditions for anisotropic or pre-stressed incompressible solids

We present a set of explicit conditions, involving the components of the elastic stiffness tensor, which are necessary and sufficient to ensure the strong ellipticity of an orthorhombic incompressible medium. The derivation is based on the procedure developed by Zee & Sternberg (Arch. Rat. Mech. Anal., 83, 53-90 (1983)) and, consequently, is also applicable to the case of the homogeneously pre-stressed incompressible isotropic solids. This allows us to reformulate the results by Zee & Sternberg in terms of components of the incremental stiffness tensor. In addition, the resulting conditions are specialized to higher symmetry classes and compared with strong ellipticity conditions for plane strain, commonly used in the literature.The first author’s work and the second author’s visit to Brunel University were partly supported by Brunel University’s ‘BRIEF’ award scheme

Explicit formulation for the Rayleigh wave field induced by surface stresses in an orthorhombic half-plane

We develop an explicit asymptotic model for the Rayleigh wave field arising in case of stresses prescribed on the surface of an orthorhombic elastic half-plane. The model consists of an elliptic equation governing the behaviour within the half-plane, with boundary values given on the half-plane surface by a wave equation. Consequently, propagation along the surface is entirely accounted for by the hyperbolic equation, which, besides, may be immediately recast in terms of the associated surface displacement. The model readily solves otherwise involved dynamic problems for prescribed surface stresses, and its effectiveness is demonstrated for the classical Lamb's problem, as well as for the steady-state moving load problem. The latter example shows that the proposed model is really obtained by perturbation around the steady-state solution for a load moving at the Rayleigh speed

Asymptotic strategy for matching homogenized structures. Conductivity problem

The paper is concerned with application of the homogenization theory to bodies containing macroinhomogeneities or bodies, parts of which cannot be homogenized (partial homogenization). This situation arises, in particular, for problems of joining homogeneous and periodically inhomogeneous bodies, or combining inhomogeneous bodies of different periodic structure. The peculiarity of the problem is related to a boundary layer, possibly arising on the interface of the matched components. Moreover, this boundary layer may be either real or fictitious, with the latter occurring due to inaccurate formulation of boundary conditions along the interface, ignoring the effect of the micro-stresses. The consideration is carried out within the framework of the steadystate heat equation. The focus of current investigation is on formulation of the problem for the periodicity cell in case of discontinuous homogenized deformations, when these cannot be treated as independent of the “fast” variables. The first order correctors are constructed. The issue of consistent matching procedure, avoiding emergence of fictitious boundary layers, is discussed. It is shown that the temperature of an inhomogeneous fragment on the boundary may be determined from the solution of the homogenized problem, whereas the derivatives (temperature gradients) require fast correctors of the homogenization theory to be taken into account. The analytical consideration is confirmed by results of numerical simulations

Buckling of a spherical shell under external pressure and inward concentrated load: asymptotic solution

An asymptotic solution is suggested for a thin isotropic spherical shell subject to uniform external pressure and concentrated load. The pressure is the main load and a concentrated lateral load is considered as a perturbation that decreases buckling pressure. First, the post-buckling solution of the shell under uniform pressure is constructed. A known asymptotic result for large deflections is used for this purpose. In addition, an asymptotic approximation for small post-buckling deflections is obtained and matched with the solution for large deflections. The proposed solution is in good agreement with numerical results. An asymptotic formula is then derived, with the load-deflection diagrams analyzed for the case of combined load. Buckling load combinations are calculated as limiting points in the load-deflection diagrams. The sensitivity of the spherical shell to local perturbations under external pressure is analyzed. The suggested asymptotic result is validated by a finite element method using the ANSYS simulation software package

On a Hyperbolic Equation for the Rayleigh Wave

A 1D hyperbolic equation is derived for the Rayleigh wave induced by prescribed surface loading. The wave operator turns out to be independent of the vertical coordinate, which appears only in the right hand side of the equation as a parameter within the pseudo-differential operator acting on the given load. It is shown that in case of the classical 2D Lamb problem this operator causes smoothening of the surface concentrated impulse as the depth increases. The suggested formulation enables revealing of the peculiarities of surface elastic wave

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