Wrinkle Generation Without Bifurcation in a Shear-Enforced Rectangular Membrane with Free Boundaries

Abstract

The object of this study is to clarify wrinkling behavior of a shear-enforced flat rectangular membrane with free boundaries. For this purpose, an equilibrium path tracking method using a finite element method is developed. This method includes bifurcation path tracking analysis that searches for bifurcation solutions. This method establishes an image of membrane behavior by calculating a series of successive equilibrium states before and after bifurcation buckling. Through detailed analysis of stress, displacement fields, and wrinkle interaction over a load parameter range, the analysis shows how existing wrinkles affect each other and the generation of new wrinkles. As a result, there is wrinkle generation with bifurcation and without bifurcation. The wrinkle generation mechanism without bifurcation is analyzed in detail. Wrinkle generation caused by bifurcations could potentially result in a large number of equilibrium paths. Each equilibrium path represents a specific wrinkle pattern. However, the analyzed results show that significantly fewer equilibrium paths are obtained than expected. These are due to wrinkle generation without bifurcations and to deformation symmetry. Nomenclature a = length, m b = width, m D = flexural rigidity, N m 2 E = Young's modulus, GPa f = force, N h = shell thickness, m M = shell section moment, N · m∕m N = shell section force, N∕m q; u = displacement, m t = time, s 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

Localized Wrinkle Behavior near Fixed Boundaries in Flat and Cylindrical Membranes

“…In the case of snap-through buckling and symmetric bifurcation buckling, the direction of the incremental displacement from the buckling point is equal to the eigenvector corresponding to a zero eigenvalue. For asymmetric bifurcation buckling, displacement and load should be changed to satisfy the relations of asymptotic theory discussed in Senda et al [13].…”

“…The perturbed state is in the eigenvector direction from the bifurcation point, where ϵ is an infinitesimal parameter. By assuming that the perturbed state is near a postbifurcation path, a convergence calculation, the Newton-Raphson iteration, is performed as described in Senda et al [13]. This static analysis searches for a new equilibrium point after the bifurcation.…”

“…e i2π∕N x mΔn x −1 e i2π∕N y nΔn y −1 (13) wheref i is the Fourier transformation of a finite set f i representing a solution u i x; y; z,f 2 indicates the complex conjugate off 2 , and Δn x and Δn y are discrete translations in their respective directions. The phase correlation product Rm; n represents the inner product between individual frequencies that represent the two solutions.…”

“…According to Senda et al [13], bifurcation points are bifurcation branching points and limit points [14]. When the solution passes through a limit point, snap-through bifurcation occurs, and the equilibrium path jumps in a discontinuous manner.…”

Review of Large Spacecraft Deployable Membrane Antenna Structures

“…(2) Multi-parameter and multi-objective optimizations of the high precision membrane antenna structures should be performed considering the membrane material creep property and structural nonlinearity. (3) Membrane wrinkle numerical simulation [74][75][76] aimed at large membrane antenna structures should be studied so that the specific locations and intensity of membrane wrinkles caused by external perturbations can be obtained, which is necessary for large membrane antenna structural design. (4) Dynamics of rigid-flexible coupling systems, such as spacecraft structures with large membrane antenna structures, need to be further studied, along with the dynamics of antennas in the deployment process.…”