Localized Wrinkle Behavior near Fixed Boundaries in Flat and Cylindrical Membranes

Abstract

This study analyzes wrinkle behavior in both flat and cylindrical membranes undergoing shear displacement. The goal is to establish a relationship between the wrinkle behavior in the two types of membranes and understand how different geometric properties affect the behavior. The analysis is an equilibrium path-tracking analysis using the finite-element method. By applying cyclic boundary conditions to a flat membrane, a degree of similarity in the wrinkle behavior can be established between flat and cylindrical membranes. In both membranes, the initial wrinkles cover the entire membrane. These wrinkles are followed by smaller, localized wrinkles, referred to as collapsed sections. These collapsed sections form on existing wrinkles. The generation of new wrinkles, by splitting the existing wrinkles, is controlled by collapsed sections and is the same in both types of membranes. For the cylindrical membrane, the curved shape results in a more controlled wrinkle behavior. This is represented by fewer bifurcation points that require larger values of shear to occur. Second, the collapsed section frequency and position are fixed with respect to the wrinkle frequency and position. The cause of simplified and controlled wrinkle behavior is analyzed, and the control mechanism is explained. Nomenclature a = length, m b = width, m D = flexural rigidity, N · m 2 d = shell thickness, m E = Young's modulus, GPa f = force, N M = shell section moment, N N = shell section force, N∕m q = displacement, m R = radius, m t = time, s u = displacement, m δf = imposed force, N δu = imposed displacement, m ϵ = membrane strain κ = curvature λ = eigenvalue ν = Poisson's ratio σ = membrane stress, N∕m 2 ϕ = eigenvector

Buckling and wrinkling of thin membranes by using a numerical solver based on multivariate Taylor series

“…One remarks that the wrinkles are not uniformly distributed, their amplitude being larger near the edges. This amplifying effect of the boundary conditions has been observed in all the previous studies, except, of course, in the case of periodic boundary conditions [26].…”

“…). The latter relations define an affine subspace of polynomials that are easily computed because the right hand sides of (26) correspond to a first index lower or equal to m+2, while this index is equal to m+4 in the left hand side. To generate this subspace of polynomials, let us assume that the following coefficients {w 0,n , f 0,n , w 26) for m = 1, etc.…”

“…In this respect, the very thin plate under shear loading studied by Wong and Pellegrino [23] is a representative example of nearly slack membrane. Numerical studies of this problem based on plate bending models can be found in [23][7], the solver being the dynamic relaxation, and in [24][25] [26], the solver being the Newton-Raphson algorithm. In all these papers, the pure shear loading problem is modified, generally by the introduction of a prescribed transverse pretensioning displacement, this transverse displacement changing significantly the wrinkling problem.…”

“…In another paper [7], very large shears were applied, which is an alternative manner to stabilize the membrane. More recently, other authors [24][25] [26] were able to solve this shear problem with commercial computer codes by using classical Newton-like procedures, but always with the help of large transverse tension. In the present paper, this iconic benchmark is revisited with the help of an efficient path-following algorithm and by trying to disregard the additional tensile load.…”

“…) by collecting all the pairs {w m,n , f m,n } for 0 ≤ m ≤ 3, m + n ≤ N . To get a particular solution, one chooses c = 0 and one applies the recurrence formula(26): so one builts a pair of polynomials ( P w (x, y), P f (x, y)) that solves the FvK equations in the sense of Taylor series. To get the first solution of the homogeneous equations, let us choose g m,n = 0, f m,n = 0, c 1 = 1, c k = 0 for k = 1: then the application of the recurrence (26) yields a first pair of polynomials (N w 1 (x, y), N f 1 (x, y)).…”