Acoustic waves at the interface of a pre-stressed incompressible elastic solid and a viscous fluid

TL;DR

This paper investigates how pre-stress affects interfacial wave propagation between an incompressible hyperelastic solid and a viscous fluid, deriving boundary conditions, modeling ultrasonic waves for aneurysm assessment, and analyzing stability in compression.

Contribution

It provides a rigorous derivation of boundary conditions at the interface and applies the model to ultrasonic wave analysis and stability assessment under pre-stress.

Findings

Viscous fluid slightly stabilizes the compressed solid half-space.

High-frequency ultrasonic waves justify the half-space model for aneurysm assessment.

Traditional standing wave analysis is inadequate for fluid-contacted solids.

Abstract

We analyse the influence of pre-stress on the propagation of interfacial waves along the boundary of an incompressible hyperelastic half-space that is in contact with a viscous fluid extending to infinity in the adjoining half-space. One aim is to derive rigorously the incremental boundary conditions at the interface; this derivation is delicate because of the interplay between the Lagrangian and the Eulerian descriptions but is crucial for numerous problems concerned with the interaction between a compliant wall and a viscous fluid. A second aim of this work is to model the ultrasonic waves used in the assessment of aortic aneurysms, and here we find that for this purpose the half-space idealization is justified at high frequencies. A third goal is to shed some light on the stability behaviour in compression of the solid half-space, as compared with the situation in the absence of fluid; we find that the usual technique of seeking standing waves solutions is not appropriate when the half-space is in contact with a fluid; in fact, a correct analysis reveals that the presence of a viscous fluid makes a compressed neo-Hookean half-space slightly more stable. For a wave travelling in a direction of principal strain, we obtain results for the case of a general (incompressible isotropic) strain-energy function. For a wave travelling parallel to the interface and in an arbitrary direction in a plane of principal strain, we specialize the analysis to the neo-Hookean strain-energy function.