The Quarterly Journal of Mechanics and Applied Mathematics - Advance Access

SYMMETRY ANALYSIS AND EXACT SOLUTIONS OF MAGNETOGASDYNAMIC EQUATIONS

Pandey, M., Radha, R., Sharma, V. D. — 2008-05-09

Using the invariance group properties of the governing system of partial differential equations (PDEs), admitting Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing one-dimensional unsteady planar and cylindrically symmetric motions in magnetogasdynamics involving shock waves. Some appropriate canonical variables are characterised that transform the equations at hand to an equivalent autonomous form, the constant solutions of which correspond to non-constant solutions of the original system. The governing system of PDEs includes as a special case the Euler's equations of non-isentropic gasdynamics. It is interesting to remark that in the absence of magnetic field, one of the exact solutions obtained here is precisely the blast wave solution obtained earlier using a different method of approach. A particular solution to the governing system, which exhibits space–time dependence, is used to study the wave pattern that finally develops when a magnetoacoustic wave impacts with a shock. The influence of magnetic field strength on the evolutionary behaviour of incident and reflected waves and the jump in shock acceleration, after collision, are studied.

Homogenization of Magneto-Electro-Elastic Multilaminated Materials

2008-04-23

In this work, based on the periodic unfolding homogenization technique, the limiting equations modelling the behaviour of three-dimensional magneto-electro-elastic periodic structures are rigorously established. The local problems and the corresponding homogenized coefficients of the elastic, dielectric, magnetic permittivity, piezoelectric, piezomagnetic and magneto-electric (ME) tensors are explicitly described. The homogenization model is exemplified for laminated composites and a unified general formula for all effective properties of periodic multilaminated magneto-electro-elastic composites is obtained. This formula is applied to investigate the global behaviour for the important case of transversely isotropic constituents and any finite number of layers in each periodic cell. Examples that provide theoretical evidence of the presence of both a product property and the ME effect are given.

Asymptotic Results For Bifurcations In Pure Bending Of Rubber Blocks

Coman, C. D., Destrade, M. — 2008-04-16

The bifurcation of an incompressible neo-Hookean thick block with a ratio of thickness to length , subject to pure bending, is considered. The two incremental equilibrium equations corresponding to a nonlinear pre-buckling state of strain are reduced to a fourth-order linear eigenproblem that displays a multiple turning point. It is found that for 0 < < , the block experiences an Euler-type buckling instability which in the limit -> degenerates into a surface instability. Singular perturbation methods enable us to capture this transition, while direct numerical simulations corroborate the analytical results.

A PRIORI ESTIMATES ON THE PARTIAL REALIZED GAIN OF ULTRA-WIDEBAND (UWB) ANTENNAS

Sohl, C., Gustafsson, M. — 2008-04-02

A sum rule valid for a large class of linear and reciprocal antennas is presented in terms of the electric and magnetic polarizability dyadics. The identity is based on the holomorphic properties of the forward scattering dyadic and includes arbitrarily shaped antennas modelled by linear and time-translational invariant constitutive relations. In particular, a priori estimates on the partial realized gain are introduced, and lower bounds on the onset frequency are derived for two important archetypes of ultra-wideband antennas: those with a constant partial realized gain and those with a constant effective antenna aperture. The theoretical findings are illustrated by an equiangular spiral antenna, and comparison with numerical simulations show great potential for future applications in antenna design.

A NEW APPROXIMATION METHOD FOR SCATTERING BY LONG FINITE ARRAYS

Thompson, I., Linton, C. M., Porter, R. — 2008-03-27

The scattering of water waves by a long array of evenly spaced, rigid, vertical circular cylinders is analysed under the usual assumptions of linear theory. These assumptions permit the reduction of the problem to that of solving the Helmholtz equation in two dimensions, with appropriate circular boundaries. Our primary goal is to show how solutions obtained for semi-infinite arrays can be combined to provide accurate and numerically efficient solutions to problems involving long, but finite, arrays. The particular diffraction problem considered here has been chosen both for its theoretical interest and for its applicability. The design of offshore structures supported by cylindrical columns is commonplace and understanding how the multiple interactions between the waves and the supports affect the field is clearly important. The theoretical interest comes from the fact that, for wavelengths greater than twice the geometric periodicity, the associated infinite array can support Rayleigh–Bloch surface waves that propagate along the array without attenuation. For a long finite array, we expect to see these surface waves travelling back and forth along the array and interacting with the ends. For particular sets of parameters, near-trapping has previously been observed and we provide a quantitative explanation of this phenomenon based on the excitation and reflection of surface waves by the ends of the finite array.

OSCILLATIONS OF A LOAD SUPPORTED BY INCOMPRESSIBLE, ISOTROPIC LIMITED ELASTIC SHEAR MOUNTS

Beatty, M. F. — 2008-03-21

The small-amplitude, free vibrational motion of a load supported symmetrically by incompressible, isotropic and homogeneous rubber-like shear mounts is studied for a general class of materials having limiting molecular chain extensibility. The oscillational frequency of the small motion of the load superimposed on a finite static shear is determined for all materials in this class of limited elastic materials. It is shown that, for the same static shear, the normalized vibrational frequency of a load on neo-Hookean shear mounts is a lower bound for its normalized vibrational frequency on any limited elastic shear mounts in the class. Formal relations are derived for the finite-amplitude oscillations of the load. Specific exact results for both small- and large-amplitude horizontal motions are provided for the limited elastic Gent material model, including a specific upper bound on the period of the finite motion, and the effects of limited extensibility are described both analytically and graphically. The intermediate amplitude solution is given explicitly in terms of a Jacobian elliptic function, and the corresponding second-order solution for the inclined motion of the load in terms of elliptic functions is noted.