Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations

Abstract

In this paper, the problem of fluid–structure interaction of a circular membrane under liquid weight loading is formulated and is solved analytically. The circular membrane is initially flat and works as the bottom of a cylindrical cup or bucket. The initially flat circular membrane will undergo axisymmetric deformation and deflection after a certain amount of liquid is poured into the cylindrical cup. The amount of the liquid poured determines the deformation and deflection of the circular membrane, while in turn, the deformation and deflection of the circular membrane changes the shape and distribution of the liquid poured on the deformed and deflected circular membrane, resulting in the so-called fluid-structure interaction between liquid and membrane. For a given amount of liquid, the fluid-structure interaction will eventually reach a static equilibrium and the fluid-structure coupling interface is steady, resulting in a static problem of axisymmetric deformation and deflection of the circular membrane under the weight of given liquid. The established governing equations for the static problem contain both differential operation and integral operation and the power series method plays an irreplaceable role in solving the differential-integral equations. Finally, the closed-form solutions for stress and deflection are presented and are confirmed to be convergent by the numerical examples conducted.

Large Deflection Analysis of Peripherally Fixed Circular Membranes Subjected to Liquid Weight Loading: A Refined Design Theory of Membrane Deflection-Based Rain Gauges

“…Membranes are increasingly being used in a wide variety of applications [1][2][3][4]. In our earlier work [5], the statics problem of fluid-structure interaction of a peripherally fixed circular membrane subjected to liquid weight loading is investigated analytically. The mathematical formulation of this problem results in a boundary value problem including both differential operation and integral operation, and the resulting integro-differential equations are successfully solved by using the power series method.…”

“…Our primary motivation for investigating this fluid-structure interaction problem is to provide the closed-form solution needed for the development of a new type of membrane deflection-based rain gauge, a device for collecting and measuring the amount of rain which falls. In this study, the statics problem of fluid-structure interaction addressed in [5] is reformulated and solved, with an aim of giving a more refined closed-form solution than that given in [5], which is essential for the development of this membrane deflection-based rain gauge.…”

“…and solved, with an aim of giving a more refined closed-form solution than that given in [5], which is essential for the development of this membrane deflection-based rain gauge.…”

“…This statics problem of fluid-structure interaction is analytically addressed in [5] for the first time, where the out-of-plane equilibrium equation, in-plane equilibrium equation, geometric equations and physical equations are established, and a closed-form solution of the problem is presented. The out-of-plane equilibrium equation is obtained by the equilibrium condition in the direction perpendicular to the initially flat circular membrane, while the in-plane equilibrium equation is obtained by the equilibrium condition in the direction parallel to the initially flat circular membrane.…”

“…The geometric equations refer to the relationship between the strain and displacement of the deflected circular membrane, while the physical equations refer to the physical relationship between stress and strain following the generalized Hooke's law. In [5], the out-of-plane equilibrium equation established is an integro-differential equation involving both integral and differential operations, which makes it difficult to analytically solve the simultaneous equations of out-of-plane and inplane equilibrium equations, geometric equations and physical equations. The power series method has played an irreplaceable role in overcoming the difficulty of analytically solving these simultaneous equations, and due to its successful use, the closed-form solution for these simultaneous equations is finally given.…”