An exact solution for the stretching of elastic membranes containing a hole or inclusion
type: Item
instance of: scholarly article
title: An exact solution for the stretching of elastic membranes containing a hole or inclusion
zbMATH Open document ID: 0715.73007
DOI: 10.1016/0093-6413(91)90024-Q
author: D. Kharzeev
published in: Mechanics Research Communications
publication date: 1991-01-01T00:00:00Z
review text: In this note we consider the problem of an isotropic elastic membrane disc, of uniform thickness, containing a circular hole or inclusion at the centre and subjected to a radially symmetric expansion. This problem has been considered by many authors [e.g.: \textit{C. H. Wu}, J. Appl. Mech. 45, 553-558 (1978; Zbl 0386.73080); \textit{S. J. Lee} and \textit{R. T. Shield}, Z. Angew. Math. Phys. 31, 454-472 (1980; Zbl 0436.73009)] since the original work by \textit{R. S. Rivlin} and \textit{A. G. Thomas} [Philos. Trans. R. Soc. Lond., Ser. A 243, 289-298 (1951; Zbl 0042.425)]. In every case the governing equilibrium equations have been solved numerically after assuming that the membrane is composed of incompressible material with either the Mooney-Rivlin or neo-Hookean strain-energy. Here we demonstrate for the first time how exact analytic solutions can be obtained by an appropriate choice of a particular, simple, strain-energy function. Both the incompressible and compressible cases are considered. The principal Cauchy stresses are found to vary monotonically with the radius and, apart from one special case for compressible materials, are positive throughout the membrane, as would be expected. The resulting deformation has one unexpected feature in that the deformed thickness of the disc remains uniform, though, in general, it is a monotonically decreasing function of increasing deformation.
Mathematics Subject Classification ID: 74E05
Mathematics Subject Classification ID: 74K20
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74S30
Mathematics Subject Classification ID: 74P10
zbMATH DE Number: 4178857
zbMATH Keywords: isotropic elastic membrane disc
zbMATH Keywords: uniform thickness
zbMATH Keywords: radially symmetric expansion
zbMATH Keywords: composed of incompressible material
zbMATH Keywords: Mooney-Rivlin
zbMATH Keywords: neo- Hookean strain-energy
zbMATH Keywords: compressible cases
MaRDI profile type: Publication
full work available at URL: 0093-6413(91)90024-q
OpenAlex ID: W2015859974
description: scientific article; zbMATH DE number 4178857