A nonlinear model of center-wound rolls incorporating refined boundary conditions

This article presents a novel approach for determining the displacement, strains and stresses developed in a center-wound roll. The approach consists of a nonlinear model and some refined boundary conditions. The investigation starts with a linear, displacement-based formulation and refined boundary conditions previously developed by the author. The choice of displacement-based formulation, unlike the stress-, or strain-based counterpart, allows for easy determination of relative and absolute displacements, and stress and strain components caused by the relative displacement and by the wound-in stress. The refined boundary conditions, on the other hand, take into account radial displacement compatibility at the core-roll interface, as well as between the current outer surface of the roll and the next wound-on lap, resulting in relaxation of the wound-in stress, or the loss of it. Conventional boundary conditions, on the contrast, consider only radial displacement compatibility at the core-roll interface. The nonlinear model incorporates nonlinear radial modulus of elasticity of the wound roll. It is linearized lap-by-lap, translating the nonlinear problem into a series of linear equations of banded structure. A number of examples are chosen to demonstrate that the approach is accurate, rather simple, able to account for wound-in stress loss and capable of updating web thickness. Furthermore, it can accommodate any form of nonlinear radial modulus of elasticity and any profile of wound-in stress.