Finite deformations of a toroidal shell

The stress-strain state of a nonlinear elastic shell exposed to the internal pressure is considered. A surface of the shell is toroidal in shape in the initial state. The Lagrangian coordinates of the shell are assigned to a cylindrical system. The kinematic characteristics of the process are shown: a law of the motion of points, vectors of a material basis, a strain affinor and its polar decomposition, the Cauchy-Green strain measure and tensor, the Finger measure, and the “left” and the“right” Hencky strain tensors. Neglecting the shear components of the stress tensor, a constitutive relation is obtained as a quasilinear relation between true stress tensor and the Hencky corotation tensor. A system of equilibrium equations is presented in terms of physical components of the true stress tensor in the Lagrangian coordinates. Using the equilibrium equations and the incompressibility condition, a closed system of nonlinear ordinary differential equations is obtained to determine six unknown functions, depending on the angle indicating a position of the points along the cross-section in the initial state. The method of successive approximations is applied to estimate stress tensor components and to derive logarithms of the elongations of material fibers.