A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations

Abstract

In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption of the membrane was given up. In particular, a more effective geometric equation was, for the first time, established to replace the classic one, and finally the resulting new boundary value problem due to the improvement of geometric equation was successfully solved by the power series method. The conducted numerical example indicates that the closed-form solution presented in this study has higher computational accuracy in comparison with the existing solutions of the well-known Föppl–Hencky membrane problem. In addition, some important issues were discussed, such as the difference between membrane problems and thin plate problems, reasonable approximation or assumption during establishing geometric equations, and the contribution of reducing approximations or relaxing assumptions to the improvement of the computational accuracy and applicability of a solution. Finally, some opinions on the follow-up work for the well-known Föppl–Hencky membrane were presented.

A Refined Theory for Characterizing Adhesion of Elastic Coatings on Rigid Substrates Based on Pressurized Blister Test Methods: Closed-Form Solution and Energy Release Rate

“…In order to investigate the influence of the above-mentioned assumptions (i)–(iii) on the analytic solution of the well-known Hencky problem, the calculation results of the deflection of the polymer thin film obtained by using four existing analytical solutions of the well-known Hencky problem are presented, as shown in Figure 9 , where the dashed lines represent the results obtained by the well-known Hencky solution (i.e., the solution used in Sun et al [ 9 ], in which assumptions (i)–(iii) were all adopted), the dotted lines by the solution presented in Sun et al [ 23 ] (only assumption (i) was given up), the dash-dotted lines by the solution presented in Sun et al [ 25 ] (assumptions (i) and (ii) were simultaneously given up), and the solid lines by the refined closed-form solution presented here (assumptions (i)–(iii) were all given up). From Figure 9 , it can be seen that the dashed line, dotted line, and the dash-dotted line are very close to the solid line when q takes 0.0001 MPa, which also demonstrates the validity of the refined closed-form solution presented here.…”

“…Yang et al [15] obtained the closed-form solution of the axisymmetric deformation problem of prestressed membrane also by giving up the so-called small-rotation-angle assumption. Recently, Sun et al [25] presented a new closed-form solution of the well-known Hencky problem by simultaneously giving up the assumptions (i) and (ii), but the assumption (iii) is still where it is. So, it is necessary to take in account the effect of the deflection on the in-plane equilibrium equation, and give up the approximations used in the geometric equation.…”

“…The detailed derivation of Equation 15is shown in Appendix A. Moreover, if the approximations adopted in the geometric equations are further given up, the geometric equations may be written as [25] e r = [(1…”

“…The detailed derivation of Equation (15) is shown in Appendix A . Moreover, if the approximations adopted in the geometric equations are further given up, the geometric equations may be written as [ 25 ] and where is the radial strain, is the circumferential strain, and u is the radial displacement. Further, if the change in membrane thickness is ignored, then the relations of the stress and strain may be written as, and where E is the Young’s modulus of elasticity and is the Poisson’s ratio.…”

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.…”