Nonlinear large deformation problem of rectangular thin plates and its perturbation solution under cylindrical bending: Transform from plate/membrane to beam/cable

Abstract

The Föppl‐von Kármán equations describing deformation of flexible thin plates are established on the basis of moderately large‐deflection and small rotation angle. For many years, the equations have been regarded as a classical and effective model used for the analysis of flexible thin plate problems, and at the same time, this model has also been constantly evolving and improving. The improvements for the model, however, come mainly from properties of materials perspective, but seldom from the deformation of plate perspective. In this study, we revisit Föppl‐von Kármán equations from the viewpoint of deformation concerning rotation angle. A new form for the classical equations without a small‐rotation‐angle assumption is derived, for the first time, by giving up the basic assumption that the sine function of the rotation angle equals to the first‐order derivative of the corresponding displacement, thus improving the governing equations while enhancing the nonlinearity. The abandonment of the small‐rotation‐angle assumption reveals such a fact that the second‐order derivative of displacement in classical equations originates from the curvature and twist of plates. Due to the complexity of the equations derived, its perturbation solution is obtained under cylindrical bending with two opposites fully fixed. Results indicate that via cylindrical bending, a two‐dimensional plate or membrane problem is easily associated with a one‐dimensional beam or cable problem. Results also show that the abandonment of the small‐rotation‐angle assumption will contribute to the free development of the deflection curve rotation of the plate, while at the same time, the deflection value will tend to decrease to agree with the deformation of the plate.

Finite element analysis of the interaction between high-compliant balloon catheters and non-cylindrical vessel structures: towards tactile sensing balloon catheters

“…Associated with this, He et al (He et al 2022 ) recently revisited the classical treatment of the Föppl-von Kármán equations used to describe thin plate/membrane deflections and addressed the related basics of moderately large deflections and small rotations. To overcome the issues associated with the small-rotation-angle assumption they reformulate the classical equations without this assumption and show that for cylindrical bending the problem can be associated with an one-dimensional beam (thin plate) or cable (membrane) problem leading to analytical solutions with the help of the perturbation method (He et al 2022 ). As the ICS in the current study is relatable to a cylindrical bending problem with large deflections and large rotations, the obtained relations from He et al (He et al 2022 ) may serve as suitable starting point for lumped models and reverse identification procedures.…”

“…Adapting the solutions from He et al (He et al 2022 ), the formulation for the maximum thin plate deflection is given by or treated as membrane deflection problem where is the current inflation pressure, and are the current membrane length and thickness, and is the effective Young’s modulus of the balloon including the sensors. Whether the thin plate (19) or the membrane (20) formulation is applicable, mainly depends on the balloon material and the thickness.…”

“…Adapting the solutions from He et al (He et al 2022), the formulation for the maximum thin plate deflection 0 (i) is given by or treated as membrane deflection problem ( 14)…”