On localised vibrations in incompressible pre-stressed transversely isotropic elastic solids

This paper is concerned with 2D localised vibration in incompressible pre-stressed fibre-reinforced elastic solids and the closely related problem of surface wave propagation in such materials. An appropriate constitutive model is derived and its stability discussed within the context of the strong ellipticity condition. Surface wave propagation in an associated half-space is considered, with the particular cases of propagation along a principal direction of primary deformation and that of almost inextensible fibres also investigated. The problems of free and forced edge vibration of a semi-infinite strip are analysed, revealing a link between the natural edge frequencies and the associated Rayleigh surface wave speed

Atmospheric oxygenation caused by a change in volcanic degassing pressure

International audienceThe Precambrian history of our planet is marked by two major events: a pulse of continental crust formation at the end of the Archaean eon and a weak oxygenation of the atmosphere (the Great Oxidation Event) that followed, at 2.45 billion years ago. This oxygenation has been linked to the emergence of oxygenic cyanobacteria1,2 and to changes in the compositions of volcanic gases3,4, but not to the composition of erupting lavas--geochemical constraints indicate that the oxidation state of basalts and their mantle sources has remained constant since 3.5 billion years ago5,6. Here we propose that a decrease in the average pressure of volcanic degassing changed the oxidation state of sulphur in volcanic gases, initiating themodern biogeochemical sulphur cycle and triggering atmospheric oxygenation. Using thermodynamic calculations simulating gas-melt equilibria in erupting magmas, we suggest that mostly submarine Archaean volcanoes produced gases with SO2/H2S,1 and low sulphur content. Emergence of the continents due to a global decrease in sea level and growth of the continental crust in the late Archaean then led to widespread subaerial volcanism, which in turn yielded gases much richer in sulphur and dominated bySO2. Dissolution of sulphur in sea water and the onset of sulphate reduction processes could then oxidize the atmosphere

Some comments on the dispersion relation for periodically layered pre-stressed elastic media

In this paper the dispersion relation associated with harmonic waves propagating in a periodically layered structure is derived and analysed. Specifically, the structure is made up of repeating unit cells, with each layer composed of an incompressible, pre-stressed elastic material, each interface perfectly bonded and the upper and lower surfaces of the structure free of incremental traction. The complexity of the problem is reduced using an approach involving the Cayley-Hamilton theorem. A numerical method is also used which eliminates positive exponential functions, thereby considerably reducing the complexity of solving the dispersion relation numerically. Numerical solutions are presented in respect of both a two-ply and symmetric four-ply unit cell. An interesting feature of these solutions is the grouping together of harmonics as the number of unit cells increases. In the case of n unit cells, n-1 harmonics group together in the moderate wave number region, with an additional harmonic joining the group at a higher wave number

Two-dimensional motion in a Bell-constrained plate

The so-called Bell constraint has been used for several years in plasticity theory and has additionally been the subject to several investigations within an elastic context. In this paper the effects of the Bell constraint on the propagation of harmonic waves in a finitely deformed elastic plate are considered. Strong ellipticity conditions are first derived for the unbounded case, and are shown to be dependent on the scalar multiplier associated with the Bell constraint. The dispersion relation, associated with harmonic wave propagation in a plate composed of such a material with zero incremental surface traction, is derived in respect of an arbitrary strain energy function. Asymptotic expansions are then obtained for high and low wave number. These expansions, which give phase speed as a function of wave number, harmonic number and pre-stress, are shown to give excellent agreement with numerical solutions

Dispersion phenomena in symmetric pre-stressed layered elastic structures

The dispersion relation associated with a symmetric three layer structure, composed of compressible, pre-stressed elastic layers, is derived. This mathematically elaborate transcendental equation gives phase speed as an implicit function of wave number. Numerical solutions are established to show a wide range of dispersion behaviour which is delicately dependent on the material parameters and pre-stress in each layer. Particularly interesting behaviour is observed within the short wave (high wave number) regime, with six possible cases of short wave liming behaviour shown possible. Within each of these, a short wave asymptotic analysis is carried out, resulting in a set of approximations which provide explicit relationships between phase speed and wave number. It is envisaged that these approximations may prove helpful to approximate numerical truncation errors associated with impact response, as well as providing excellent first approximations for particularly (numerically) challenging sets of material parameters