# Wrinkling Problems for Non-Linear Elastic Membranes

McKay, Barry (1996) Wrinkling Problems for Non-Linear Elastic Membranes. PhD thesis, University of Glasgow.

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## Abstract

In this thesis we study several examples of finite deformations of non-linear, elastic, isotropic membranes consisting of both incompressible and compressible materials which result in the membrane becoming wrinkled. To investigate the nature and occurrence of these wrinkled regions we adapt ordinary membrane theory by using a systematic approach developed by Pipkin (1986) and Steigmann (1990) which accounts for wrinkling automatically. In each problem considered, we employ the relaxed strain-energy function proposed by Pipkin (1986) and assume that the in-plane principal Cauchy stresses are non-negative. A discussion of the basic equations for a membrane from the three-dimensional theory and the derivation of the relaxed strain-energy function from tension field theory is given. The resulting equations of equilibrium are then used to formulate various problems considered and solutions are obtained by analytical or numerical means for both the tense and wrinkled regions. In particular we consider the deformation of a membrane annulus of uniform thickness which is subjected to either a displacement or a stress on the inner and outer radii. We present the first analytical solution for such a problem, for incompressible and compressible materials, for both the tense and wrinkled regions. This first problem therefore provides a simple example to illustrate the theory of Pipkin (1986). The second problem studies an elastic, circular, cylindrical membrane which is inflated by an internal pressure and subjected to a flexural deformation. The equations of equilibrium are solved numerically, two different solution methods being described, and results are presented graphically showing the deformed cross-section of the cylinder for incompressible and compressible materials. Particular attention is given to the value of curvature at which wrinkling begins. An incremental deformation is also considered to investigate possible bifurcation solutions which could occur at some finite value of curvature. The final problem considers two butt jointed, incompressible, elastic, circular, cylindrical membranes of different material and geometric properties. In particular we fix the cylinders to have different initial radii which ensures that wrinkling will occur. The composite cylinder is inflated and subjected to axial loading on either end. This deformation may have useful applications in surgery as it could be considered as a first approximation model for arterial grafts, the wrinkled surface having important implications in the formation of blood clots as blood flows through such a region. Again the equations of equilibrium are solved numerically and graphical results of the deformed, axial length against the deformed radius for a range of values of the parameters are given showing the tense and wrinkled regions.

Item Type: Thesis (PhD) Doctoral Adviser: D M Haughton Mechanical engineering, Mechanics 1996 Enlighten Team glathesis:1996-76442 Copyright of this thesis is held by the author. 19 Nov 2019 14:20 19 Nov 2019 14:20 http://theses.gla.ac.uk/id/eprint/76442