Theory of gradient-elastic membranes and its application in the wrinkling analysis of stretched thin sheets

In this paper, a gradient theory for large deformation analysis of elastic membranes is developed. Thin membranes are modelled as material surfaces, and the formulation starts with an internal energy density function, which is assumed to be dependent upon the first as well as the second spatial derivatives of deformation field. Governing equilibrium equations and boundary conditions are derived in a variational framework. General forms of constitutive equations for the developed model are discussed. In particular, constitutive laws incorporating two material length-scale parameters and suitable for isotropic as well as anisotropic materials are introduced. Since the strong form of the governing differential equations is highly nonlinear, a finite element formulation for numerical solution of initially flat gradient-elastic membranes is developed. As an application of the introduced theory, wrinkling phenomenon in a thin elastic membrane under uniaxial stretching is studied. Numerical simulations show that by proper selection of the material length scale parameters in the proposed constitutive law, the theory is capable of capturing the detailed wrinkling patterns in stretched membranes. Notably, the maximum wrinkling amplitude obtained by the present formulation has good agreement to the experimental data reported in the literature. Applicability of the developed formulation to predict the deformation of anisotropic materials is also investigated. In particular, it is also shown that the formulation is able to successfully capture the buckling of pantographic lattice as well as wrinkling of engineering fabrics. (C) 2019 Elsevier Ltd. All rights reserved.