A New Solution to Well-Known Hencky Problem: Improvement of In-Plane Equilibrium Equation

Abstract

In this paper, the well-known Hencky problem—that is, the problem of axisymmetric deformation of a peripherally fixed and initially flat circular membrane subjected to transverse uniformly distributed loads—is re-solved by simultaneously considering the improvement of the out-of-plane and in-plane equilibrium equations. In which, the so-called small rotation angle assumption of the membrane is given up when establishing the out-of-plane equilibrium equation, and the in-plane equilibrium equation is, for the first time, improved by considering the effect of the deflection on the equilibrium between the radial and circumferential stress. Furthermore, the resulting nonlinear differential equation is successfully solved by using the power series method, and a new closed-form solution of the problem is finally presented. The conducted numerical example indicates that the closed-form solution presented here has a higher computational accuracy in comparison with the existing solutions of the well-known Hencky problem, especially when the deflection of the membrane is relatively large.

Polymer Conductive Membrane-Based Non-Touch Mode Circular Capacitive Pressure Sensors: An Analytical Solution-Based Method for Design and Numerical Calibration

“… In the vertical direction perpendicular to the initially flat circular membrane, there are two vertical forces acting the free body, that is, the πr 2 q produced by the loads q within r , and the 2πrσ r h sin θ produced by the membrane force σ r h , where σ r is radial stress. So, the out-of-plane equilibrium condition is where Substituting Equation (A2) into Equation (A1) yields While in the direction parallel to the initially flat circular membrane, the equilibrium condition may be written as [ 36 ] where σ t denotes circumferential stress. The derivation of Equation (A4) is detailed in [ 36 ].…”

“…So, the out-of-plane equilibrium condition is where Substituting Equation (A2) into Equation (A1) yields While in the direction parallel to the initially flat circular membrane, the equilibrium condition may be written as [ 36 ] where σ t denotes circumferential stress. The derivation of Equation (A4) is detailed in [ 36 ]. If the radial and circumferential strain and the radial displacement are denoted by e r , e t and u , respectively, then the relationships between strain and displacement for large deflection problems may be written as [ 37 ] and Moreover, the relationships between stress and strain are still assumed to satisfy linear elasticity and expressed in terms of generalized Hooke’s law [ 38 ] and Substituting Equations (A5) and (A6) into Equations (A7) and (A8) yields and By means of Equations (A4), (A9) and (A10), one has After substituting the u in Equation (A11) into Equation (A9), we obtain an equation containing only the radial stress σ r and deflection w ( r ) Equations (A3) and (A12) are two equations for solving the radial stress σ r and deflection w ( r ).…”

“…While in the direction parallel to the initially flat circular membrane, the equilibrium condition may be written as [ 36 ] where σ t denotes circumferential stress. The derivation of Equation (A4) is detailed in [ 36 ].…”

A Theoretical Study on an Elastic Polymer Thin Film-Based Capacitive Wind-Pressure Sensor

“…The problem before the peripherally fixed wind-driven circular polymer elastic thin film touches the spring-driven movable electrode plate is simplified into the well-known Föppl–Hencky membrane problem, i.e., the problem of axisymmetric deformation of a peripherally fixed circular membrane under the action of uniformly distributed transverse loads q [ 34 , 35 , 36 , 37 , 38 , 39 , 40 ], as shown in Figure 2 a. The effectiveness of the well-known Hencky solution is recognized.…”

A Closed-Form Solution without Small-Rotation-Angle Assumption for Circular Membranes under Gas Pressure Loading

“…A computational error in the power series solution which was presented originally by Hencky in 1915 [4] was corrected subsequently by Chien in 1948 [5] and Alekseev in 1953 [6], respectively. This solution is usually called the well-known Hencky solution, and is often cited in some studies of related issues [7][8][9][10][11][12].…”