Kinematics of deformable media

We investigate the kinematics of deformations in two and three dimensional media by explicitly solving (analytically) the evolution equations (Raychaudhuri equations) for the expansion, shear and rotation associated with the deformations. The analytical solutions allow us to study the dependence of the kinematical quantities on initial conditions. In particular, we are able to identify regions of the space of initial conditions that lead to a singularity in finite time. Some generic features of the deformations are also discussed in detail. We conclude by indicating the feasibility and utility of a similar exercise for fluid and geodesic flows in flat and curved spacetimes.Comment: 28 pages, 12 figure

On the Static and Dynamic Contact Problem of an Inflated Spherical Viscoelastic Membrane

Inflated membrane structures, useful in vibration/shock isolation devices, terrestrial and space structures, etc., rely on the internal dissipation in the membrane for vibration attenuation. In this work, using the Christensen viscoelastic material model, we study the contact mechanics, displacement-controlled relaxation response, force-controlled creep response, dynamic contact, and energy dissipation due to oscillatory contact in an inflated spherical nonlinear viscoelastic membrane. We consider an inflated spherical membrane squeezed between two large rigid, frictionless, parallel plates. The effective stiffness and damping in the membrane-plate assembly are determined, and a phenomenological model is developed. Under oscillatory contact condition, the energy dissipation per cycle is determined. Further, using the free-vibration test, the damped natural frequency of the membrane-plate system is calculated

Kinematics of flows on curved, deformable media

In this article, we first investigate the kinematics of specific geodesic flows on two dimensional media with constant curvature, by explicitly solving the evolution (Raychaudhuri) equations for the expansion, shear and rotation along the flows. We point out the existence of singular (within a finite value of the time parameter) and non-singular solutions and illustrate our results through a `phase' diagram. This diagram demonstrates under which initial conditions (or combinations thereof) we end up with a singularity in the congruence and when, if at all, we encounter non--singular solutions for the kinematic variables. Our analysis illustrates the differences which arise due to a positive or negative value of the curvature. Subsequently, we move on to geodesic flows on two dimensional spaces with varying curvature. As an example, we discuss flows on a torus, where interesting oscillatory features of the expansion, shear and rotation emerge, which are found to depend on the ratio of the radii of the torus. The singular (within a finite time)/non--singular nature of the solutions are also discussed. Finally, we arrive at some general statements and point out similarities or dissimilarities that arise in comparison to our earlier work on media in flat space.Comment: Corrections in some equations and in one figure

Vibrations and Waves in Continuous Mechanical Systems

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