On the Description of Large Deformation in Curvilinear Coordinate Systems: Application to Thick-Walled Cylinders

Abstract

This chapter presents a description of the large deformation of solids in a curvilinear coordinate system with the application to the elastic–plastic deformation of thick-walled cylinders under simultaneous radial and axial loading. The tensor-based presentation is founded on the continuum theory. Both tensor and component notations are adopted. The analysis assumes the material to be a homogeneous and isotropic continuous medium. Material-independent fundamentals are first discussed in detail, including strain, deformation, and velocity gradients. Different forms of stress tensor in different space- or material-based coordinate systems are presented and discussed. The stress and strain definitions are used to derive the equilibrium equations and constitutive laws for elastic–plastic material behavior. The presented definitions have been used to develop a solution of radially and axially loaded thick-walled elastic–plastic cylinders with nonlinear hardening, adopting an associative constitutive law. The solution is capable of accurately providing continuous distribution of stress and strain gradients throughout the cylinder. It can be used therefore to calculate the current state in the bifurcation analysis of radially and axially loaded thick-walled cylinders and to establish the basis for further research on the safety of pressurized thick-walled cylindrical structures.