Mini-Workshop: Mathematical Problems in the Nonlinear Elastodynamics of Rubber-Like Materials

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Mathematical modelling of Stoneley wave in a transversely isotropic thermoelastic media

This paper is concerned with the study of propagation of Stoneley waves at the interface of two dissimilar transversely isotropic thermoelastic solids without energy dissipation and with two temperatures. The secular equation of Stoneley waves is derived in the form of the determinant by using appropriate boundary conditions i.e. the stresses components, the displacement components, and temperature at the boundary surface between the two media are considered to be continuous at all times and positions . The dispersion curves giving the Stoneley wave velocity and Attenuation coefficients with wave number are computed numerically. Numerical simulated results are depicted graphically to show the effect of two temperature and anisotropy on resulting quantities. Copper material has been chosen for the medium and magnesium for the medium Some special cases are also deduced from the present investigation

Thermal convection with a Cattaneo heat flux model

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one