A modeling and resolution framework for wrinkling in hyperelastic sheets at finite membrane strain

Abstract

Wrinkles commonly occur in uniaxially stretched rectangular hyperelastic membranes with clamped-clamped boundaries, and can vanish upon excess stretching. Here we develop a modeling and resolution framework to solve this complex instability problem with highly geometric and material nonlinearities. We extend the nonlinear Föppl-von Kármán thin plate model to finite membrane strain regime for various compressible and incompressible hyperelastic materials. Under plane stress condition, 2D hyperelastic constitutive models can be systematically deduced based on general 3D strain energy potentials, e.g., Saint-Venant Kirchhoff, neo-Hookean, Mooney-Rivlin, Gent model and Gent-Gent model. Moreover, we establish a novel and efficient numerical resolution framework combining a path-following continuation technique by Asymptotic Numerical Method (ANM) and a discretization by a spectral method. The main advantages of this framework include the generality for both compressible and incompressible materials, ease of programming, high precision and efficient continuation predictor. Based on the proposed approach, effect of different incompressible constitutive models on the post-buckling response is investigated, which shows that restabilization points and wrinkling amplitudes are quantitatively influenced. However, for compressible materials, Poisson’s ratio plays a critical role in the wrinkling and restabilization behavior. We find that smaller Poisson’s ratio makes later onset of wrinkling, lower amplitude and earlier disappearance of wrinkles. Besides, severe strain-stiffening phenomena are explored by accounting for phenomenological models such as Gent model and Gent-Gent model. Efficiency and accuracy of the proposed modeling and resolution framework were examined by comparing with some benchmarks.