Radially Oscillating Incompressible Hyperelastic Multi-Layer Tubes: Interface Effects and Energy Approach

Abstract

The finite amplitude radial motion of thick walled hyperelastic tubes has been extensively studied since the work of Knowles in the context of material incompressibility. This allows for explicit integration of the balance equation for linear momentum in the radial direction. We use this procedure to examine the effect of layering upon the oscillatory response of such tubes. For a suite of materials characterized by different hyperelastic constitutive laws and different material densities, we consider the parametric effect of property mismatch, relative layer thickness, and layer ordering on the qualitative shape of closed orbits in a phase space of radial displacement vs. radial velocity. Even the simple case of a two-layer system shows how changing a single parameter can give significant qualitative variation in orbital shape (e.g., eccentricity, crowding of maximum velocity locations toward minimum displacement locations, etc.) which in turn alters the orbital period (oscillation frequency). Changes in more than one parameter can either exacerbate or reduce such effects, in the latter case by for example a balancing of shear wave speeds. Equivalent results are obtained by a somewhat more direct energy treatment, either using Lagrangian dynamics or Hamilton’s principle, both of which bypass the notion of stress.