A Variational Formulation for Thin Membranes

A Variational Formulation for Thin Membranes

De Rooij, R.

Abdalla, M.M. (mentor)

Aerospace Engineering

Aerospace Structures and Materials

Aerospace Structures and Computational Mechanics

2013-07-08

Membrane structures have a rich history of use across many disciplines and are widely used in aerospace and structural engineering applications. A few examples can be found in solar sails, atmospheric balloons and parachutes. These membranes have many advantages, including their ability to take complex shapes and their low mass to surface ratio, which is especially important in aerospace engineering. Although membranes can carry tensile loads very well, they tend to wrinkle under the slightest compressive load. As these wrinkles affect the load carrying capabilities of the membrane it is important to model the stress distribution in the membrane accurately to assess these capabilities. To model the wrinkles in a membrane, the mesh size in a numerical model needs to be at least as small as the wrinkles to detect them. Considering the small scale of the wrinkles with respect to the membrane as a whole, this requirement often results in unacceptably high computational costs. This issue can be solved by modeling the wrinkles as a continuous in-plane contraction of the membrane, which is known as the tension field theory method. This method is incorporated into the theory of finite deformations by a relaxed strain energy function defined in the literature. In this thesis it was investigated whether a variational formulation of the relaxed strain energy function can be used to obtain a robust and efficient method to model the stress distribution in membranes in the presence of wrinkles. Using the interior-point method, the relaxed strain energy function was formulated as an unconstrained optimization problem based on the original strain energy function. The convexity properties of this relaxation were proven to guarantee a stable energy minimization in the membrane. Using the variational principle three governing equations were obtained which uniquely described the deformation of the membrane under applied loading and the stresses and wrinkling strains in the membrane. The solution to these equations was obtained by discretization of the membrane structure and a linearization of the governing equations. Several numerical examples were considered which assessed, verified and validated the proposed method. It was shown that the method is efficient and robust based on the properties of the governing equations and the numerical results.

membranes
wrinkling
relaxed strain energy function
interior-point method
variational formulation
convexity
tension field theory
finite element method
isogeometric analysis

http://resolver.tudelft.nl/uuid:407b1564-04d7-4f24-badf-61bb960ec9d2

Student theses

master thesis

(c) 2013 De Rooij, R.