Summary
Some stability problems for finitely deformed elastic membranes are considered. The first part treats the stability of a stretched elastic membrane which is orthotropic in its reference state. The stability criteria for a uniformly deformed orthotropic plane sheet under conditions of dead loading are obtained. The potential energy for a uniformly extended and inflated circular cylindrical membrane (the preferred directions being axial and circumferential) with axially symmetric deformations and under constant pressure is found to have the same form as that for an isotropic material. Stability criteria for a long tube are obtained and some numerical results are given through stability diagrams using strain energy functions appropriate for biological tissues. The effect on stability of the flow of an incompressible fluid through the tube is considered next. To model the internal fluid motion, small perturbations superimposed on a state of steady flow with velocity profiles of a slug flow and a Poiseuille flow are assumed, and a one-dimensional analysis is used as a third approach. Equations for the fundamental frequencies of vibration are derived. The fluid motion is found to have a stabilizing or destabilizing influence on the membrane depending on the approach used. The results are compared to Rayleigh’s results for a circular liquid jet and they are found to agree for the slug flow and the Poiseuille flow models.
Übersicht
Es werden einige Stabilitätsprobleme für Membranen mit endlichen Deformationen betrachtet. Im ersten Teil wird die Stabilität einer gedehnten Membran betrachtet, die im Ausgangszustand orthotrop ist. Es werden die Stabilitätsbedingungen für ein gleichförmig deformiertes ebenes Blatt erhalten. Die potentielle Energie für eine gleichförmig gedehnte und aufgeblasene kreiszylindrische Membran mit achsialsymmetrischer Deformation unter konstantem Druck hat dann dieselbe Form wie bei isotropem Material. Es werden Stabilitätskriterien für lange Röhren abgeleitet und einige numerische Ergebnisse für biologische Gewebe mitgeteilt. Als nächstes wird der Einfluß auf die Stabilität der Strömung eines inkompressiblen Fluids durch die Röhre untersucht. Dabei werden kleine Störungen der stationären Strömung überlagert, wobei als Geschwindigkeitsprofil Schleichströmung und Poiseuille-Strömung angenommen werden. Die eindimensionale Analyse wird bis zur dritten Ordnung durchgeführt. Es werden Gleichungen für die Frequenzen der Grundschwingungen abgeleitet. Dabei zeigt sich, daß die Fluidströmung je nach der Art der Näherungen einen stabilisierenden oder auch destabilisierenden Einfluß haben kann. Die Ergebnisse werden mit denen von Rayleigh für einen Fluidstrahl mit kreisförmigem Querschnitt verglichen. Dabei wird Übereinstimmung sowohl für Schleich- wie auch für Poiseuille-Strömung festgestellt.
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Wong, J.S.K., Shield, R.T. The stability of a cylindrical elastic membrane of biological tissue and the effect of internal fluid flow. Ing. arch 49, 393–412 (1980). https://doi.org/10.1007/BF02426918
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DOI: https://doi.org/10.1007/BF02426918