An Exact In-Plane Equilibrium Equation for Transversely Loaded Large Deflection Membranes and Its Application to the Föppl-Hencky Membrane Problem

Abstract

In the existing literature, there are only two in-plane equilibrium equations for membrane problems; one does not take into account the contribution of deflection to in-plane equilibrium at all, and the other only partly takes it into account. In this paper, a new and exact in-plane equilibrium equation is established by fully taking into account the contribution of deflection to in-plane equilibrium, and it is used for the analytical solution to the well-known Föppl-Hencky membrane problem. The power series solutions of the problem are given, but in the form of the Taylor series, so as to overcome the difficulty in convergence. The superiority of using Taylor series expansion over using Maclaurin series expansion is numerically demonstrated. Under the same conditions, the newly established in-plane equilibrium equation is compared numerically with the existing two in-plane equilibrium equations, showing that the new in-plane equilibrium equation has obvious superiority over the existing two. A new finding is obtained from this study, namely, that the power series method of using Taylor series expansion is essentially different from that of using Maclaurin series expansion; therefore, the recurrence formulas for power series coefficients of using Maclaurin series expansion cannot be derived directly from that of using Taylor series expansion.

An Improved Theory for Designing and Numerically Calibrating Circular Touch Mode Capacitive Pressure Sensors

“…How to use the analytical solution of the plate/membrane contact problem (derived in Section 2 ) and the analytical solution of the large deflection problem (derived in [ 25 ]), as well as the pressure–capacitance analytical relationship (i.e., Equation (57) derived in Section 3 ) to design and numerically calibrate a circular touch mode capacitive pressure sensor with a specified pressure detecting range is detailed as follows.…”

“…Therefore, the initially parallel gap g should be equal to the maximum deflection w m of the circular conductive membrane under the required or desired lower limit pressure, i.e., under the touch point pressure . To this end, the analytical solution for the large deflection problem of a circular membrane under transverse loading, which is presented in [ 25 ], is used to determine the maximum deflection w m of the circular conductive membrane under the touch point pressure , where the undetermined constants b 0 , c 0 and d 0 should be first determined, and then the maximum deflection w m and maximum stress σ m should be calculated. If the calculated maximum stress σ m is relatively small (about 0.2 times the yield strength σ y of the used membrane materials), then the calculated maximum deflection w m can be used as the initially parallel gap g ; otherwise, it is necessary to change the design parameters of the circular conductive membrane (such as radius a , thickness h , Poisson’s ratio v and Young’s modulus of elasticity E ) and repeat the above calculation until the requirement for σ m ≤ 0.2 σ y is met.…”

“…So, Li et al modified the classic in-plane equilibrium equation, but presented only an in-plane equilibrium equation partly taking into account the contribution of deflection to in-plane equilibrium [ 4 ]. The in-plane equilibrium equation that fully takes into account the contribution of deflection to in-plane equilibrium was established by Sun et al [ 25 ], and is given by where σ t denotes the circumferential stress.…”

“…In fact, the Föppl–Hencky membrane problem has been very well solved analytically [ 22 , 23 , 24 , 25 ], but the analytical solution of the plate/membrane contact problem still needs to be further improved. Xu and Liechti solved this plate/membrane contact problem based on the assumptions of an equi-biaxial constant stress state and small rotation angle of the membrane [ 26 ].…”

“…So, in [ 26 , 27 , 28 , 29 ], the use of the classic in-plane equilibrium equation inevitably introduces calculation errors. In this paper, the plate/membrane contact problem is reformulated using a more accurate in-plane equilibrium equation which fully takes into account the contribution of deflection to the in-plane equilibrium [ 25 ], resulting in a new and more accurate analytical solution of the problem. On this basis, the design and numerical calibration theory for circular touch mode capacitive pressure sensors have been greatly improved and perfected.…”