An exact solution for the stretching of elastic membranes containing a hole or inclusion (Q751880)

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.