Small superimposed radial oscillations for a class of damaged limited elastic tubes

Abstract

Small superimposed radial oscillations for a class of damaged limited elastic, incompressible, isotropic and homogeneous thick-walled circular cylindrical tube are examined. Both the damage function and the considered limited elastic class depend only on the first invariant $$ I_{1} $$\n of the left Cauchy–Green deformation tensor. The present work puts some light in the context of the surgical procedure of angioplasty, used especially to treat cardiovascular diseases. Our simplified isotropic assumption allows for a much faster and analytical computation of comparably accurate results. The examples of three limited elastic material models are presented, namely the Arruda–Boyce, Gent, and Demiray. The radial oscillations are further investigated for a damaged limited elastic thin membrane, and the examples above are explored. Universal relations are found for the three material membranes. The obtained results are matched with the clinical data of the existing literature. It appears that the pressure ratio for the virgin and damaged arterial tissue is higher before angioplasty and vice versa. A similar trend for the frequency ratio also follows.