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Considered is an elastic, incompressible, isotropic material whose constitutive law is specified by a strainenergy function W which is a function of two strain invariants I sub 1 and I sub 2. In the mathematical theory of large deformations of axially symmetrical elastic mem branes, the governing equations are a set of nonlinear ordinary differential equations. Three types of deformation of thin circular cylindrical rubber tubes are discussed. In the first type a rubber tube is deformed into another circular cylindrical tube of different length and diameter by simultaneous inflation and extension of the tube. The second type of deformation considered is a stretching of the tube without internal pressure. The third type is a tube inflated by internal pressure, with or without a change in total length or end diameter. In these two types the deformed tube is a curved surface of revolution; the analysis is more complicated, and the calculations are restricted to Mooney-Rivlin materials.
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Stochastic elasticity is a fast developing field that combines nonlinear elasticity and stochastic theories in order to significantly improve model predictions by accounting for uncertainties in the mechanical responses of materials. However, in contrast to the tremendous development of computational methods for large-scale problems, which have been proposed and implemented extensively in recent years, at the fundamental level, there is very little understanding of the uncertainties in the behaviour of elastic materials under large strains. Based on the idea that every large-scale problem starts as a small-scale data problem, this book combines fundamental aspects of finite (large-strain) elasticity and probability theories, which are prerequisites for the quantification of uncertainties in the elastic responses of soft materials. The problems treated in this book are drawn from the analytical continuum mechanics literature and incorporate random variables as basic concepts along with mechanical stresses and strains. Such problems are interesting in their own right but they are also meant to inspire further thinking about how stochastic extensions can be formulated before they can be applied to more complex physical systems.