A Refined Closed-Form Solution for Laterally Loaded Circular Membranes in Frictionless Contact with Rigid Flat Plates: Simultaneous Improvement of Out-of-Plane Equilibrium Equation and Geometric Equation

Abstract

Essential to the design and development of circular contact mode capacitive pressure sensors is the ability to accurately predict the contact radius, maximum stress, and shape of a laterally loaded circular membrane in frictionless contact with a concentric circular rigid flat plate. In this paper, this plate/membrane contact problem is solved analytically again by simultaneously improving both out-of-plane equilibrium equation and geometric equation, and a new and more refined closed-form solution is given to meet the need of accurate prediction. The new closed-form solution is numerically discussed in convergence and effectiveness and compared with the previous one, showing that it can greatly improve the prediction accuracy of the contact radius, maximum stress, and shape of the circular membrane in frictionless contact with the rigid flat plate.

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“…However, the plate/membrane contact problem involves both the plane-stretched membrane in the plate/membrane contact area of 0 ≤ r ≤ b and the large deflection membrane in the plate/membrane non-contact area of b ≤ r ≤ a (see Figure 1 d). But the existing analytical solutions of the plate/membrane contact problem [ 26 , 27 , 28 , 29 ] are all obtained using the classic in-plane equilibrium equation that is only applicable to the plane-stretched membrane problems, which inevitably introduces calculation errors. In this paper, this plate/membrane contact problem is further analytically solved by giving up the equi-biaxial constant stress state assumption and using more accurate out-of-plane and in-plane equilibrium equations and geometric equations, and a new and more refined closed-form solution is presented, which is detailed as follows.…”

“…Lian et al also presented a closed-form solution of this plate/membrane contact problem [ 28 ], where the equi-biaxial constant stress state assumption was given up, and the out-of-plane equilibrium equation used was established by giving up the small rotation angle assumption of the membrane. Li et al presented a more refined closed-form solution of this plate/membrane contact problem [ 29 ], where the out-of-plane equilibrium equation and geometric equations used were established by giving up the small rotation angle assumption of the membrane, except for giving up the equi-biaxial constant stress state assumption. However, the in-plane equilibrium equation used in [ 26 , 27 , 28 , 29 ] is the classic one, which does not take into account the contribution of deflection to in-plane equilibrium at all and is only applicable to plane-stretched or compressed membranes and not to large deflection membranes.…”

“…Li et al presented a more refined closed-form solution of this plate/membrane contact problem [ 29 ], where the out-of-plane equilibrium equation and geometric equations used were established by giving up the small rotation angle assumption of the membrane, except for giving up the equi-biaxial constant stress state assumption. However, the in-plane equilibrium equation used in [ 26 , 27 , 28 , 29 ] is the classic one, which does not take into account the contribution of deflection to in-plane equilibrium at all and is only applicable to plane-stretched or compressed membranes and not to large deflection membranes. In other words, the classic in-plane equilibrium equation used in [ 26 , 27 , 28 , 29 ] is only applicable to the plane-stretched membrane in the plate/membrane contact area of 0 ≤ r ≤ b and not to the large deflection membrane in the plate/membrane non-contact area of b ≤ r ≤ a (see Figure 1 d).…”

“…However, the in-plane equilibrium equation used in [ 26 , 27 , 28 , 29 ] is the classic one, which does not take into account the contribution of deflection to in-plane equilibrium at all and is only applicable to plane-stretched or compressed membranes and not to large deflection membranes. In other words, the classic in-plane equilibrium equation used in [ 26 , 27 , 28 , 29 ] is only applicable to the plane-stretched membrane in the plate/membrane contact area of 0 ≤ r ≤ b and not to the large deflection membrane in the plate/membrane non-contact area of b ≤ r ≤ a (see Figure 1 d). So, in [ 26 , 27 , 28 , 29 ], the use of the classic in-plane equilibrium equation inevitably introduces calculation errors.…”

“…In other words, the classic in-plane equilibrium equation used in [ 26 , 27 , 28 , 29 ] is only applicable to the plane-stretched membrane in the plate/membrane contact area of 0 ≤ r ≤ b and not to the large deflection membrane in the plate/membrane non-contact area of b ≤ r ≤ a (see Figure 1 d). 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.…”