Quasistatic inflation processes within rigid tubes

Abstract

In this paper the authors consider mechanical devices of rotational symmetry that can be seen as segments of an artificial worm or as a balloon for angioplasty. Continuing former work [7] the segment is now placed within a cylindrical or constricted rigid tube that will be touched or pressed during inflation of the segment. Both the segment's shape and the forces of contact are investigated. The main mathematical tool is the Principle of Minimal Potential Energy -handled as an optimal control problem with state constraint. The necessary optimality conditions are carefully analyzed and simulation results for characteristic examples are presented. The treatment of the problem is primarily mathematical but aiming at application.

Quasistatic inflation processes within compliant tubes

“…In this paper we continue investigations published in . There, the authors firstly considered quasistatic ballooning processes of originally cylindrical compliant “segments” ‐ by means of internal pressure expanding either freely or within a rigid tube.…”

“…The background system can be seen as part of a worm crawling in a compliant tube or as a system in medical endoscopy. The investigations continue former work, that concerned freely inflating segments and rigid surrounding tubes, respectively, . The mathematical basis is taken from , where the governing boundary value problem was derived out of the Principle of Minimal Potential Energy formulated as an optimal control problem with state constraint.…”

“…Regarding membrane material: Drop the Hooke law and switch to some other material constitutions, maybe of Mooney‐Rivlin type and different for segment and vein; easily done within the source code by exchanging the ψ‐functions with any others (see ). Take into consideration finite strength of the membranes: introduce bounds for the stress resultants n 11 and n 22 (see ) and therefrom find a maximal feasible pressure . Relax the constraint of meridional inextensibility; this might give a more realistic rheology ‐ but at the expense of losing the (theoretically and computationally valuable) invariance of arc‐lengths and the existence of the maximum volume configuration of the inflated () segment. …”

“…Take into consideration finite strength of the membranes: introduce bounds for the stress resultants n 11 and n 22 (see ) and therefrom find a maximal feasible pressure .…”

“…(5) Geometrically, the supposed meridional inextensibility guarantees the existence of a shape of maximal volume of the segment (formally approached by ). This shape can be described by means of elliptic integrals (see ) and may be used as a starting object in iteration. For this reason it seems useful to have some corresponding data at hand:…”