Analytical Studies On The Small And Large Localizations In Pre-stressed Slender Cylinders Composed Of Compressible Hyperelastic Materials

In this thesis, we study two localization problems in pre-stressed slender cylin-ders composed of compressible hyperelastic materials subjected to axial forces. one is the small localization for materials with a general form, the other is the large localization for a specific material.Localization, represented as local strain concentration, is a manifestation of the degradation of material properties with localized large deformations. Due to its importance in structural safety assessment, much research has been, conducted to resolve experimental, theoretical and computational issues associated with localization problems. However, in a three-dimensional setting, the analytical studies on the small localization are quite few, and as far as we know, there is not any analytical result for the large localization available in literature.The main purpose of this thesis is to construct a three-dimensional.model and use it to present some analytical solutions to capture or predict some key experimental features.Our model is constructed on the basis of the theory of small elastic defor-mations superimposed on a finite elastic deformation,.which is represented as perturbations of a trivial (constant) solution of a dynamics system in mathe-matics. The coupled series-asymptotic expansion method is used to derive the normal form equation from the original complicated system of nonlinear PDEs. By writing the normal form equation into a first-order dynamical system and with a phase-plane analysis, we manage to solve the natural boundary value problems analytically. The asymptotic solutions in terms of integrals are obtained. And the influences of the geometry (radius-length ratio) of the cylinders on the properties of the non-trivial solutions are also discussed for the both two problems. The analytical results obtained for small strains can capture some key features in experiments by others, such as the snap-back phenomenon of the stress-strain response, the nonuniqueness of the post-peak behavior for different radius-length ratios, and so on.For the Blatz-Ko material, a specific isotropic hyperelastic material used to model large deformations, we develop the coupled series-asymptotic method, a localized method to deal with small strains, to a global one by introducing a novel methodology and use it to deal with large strains. This is the main contribution of this thesis. Our analytical results give some predictions for the behavior of Blatz-Ko material undergoing large deformations.