Strain-based formulas for stresses in profiled center-wound rolls

A new solution to the problem of modeling elastic stresses in center-wound rolls is proposed. Its premise is that residual strains from the web relax to create stresses within the roll. Stress terms-obtained with linear and non-linear elastic solutions-are weighted averages of web stresses in centerwound rolls. However, there are major differences between these calculated stresses and previous solutions. The solution here rigorously satisfies two boundary conditions and is in closed form. The proposed approach contains four stress terms. Two are general stresses from the homogeneous part of the solution, while two new solutions come from the residual strains in the roll generated by web tension stresses. The radial dependence of the residual strains (i.e., the roll profile) is very important, since the residual roll stress depend on the winding conditions. All winding profiles lead to complete, although algebraically complicated, elastic solutions in analytic form. Nonlinear elastic solutions are also investigated using the effective residual web stress and a stress-dependent radial elastic modulus. Numerical nonlinear equations developed from expressions derived here have straightforward solutions. A selection of material constitutive laws and roll profiles are eraluated for comparison. Finally, the modeled roll stresses are compared with pull-tab-measured stresses from instrumented wound rolls. The radial plateau stresses in the midroll are further approximated with several simple closed-form algebraic expressions. Application : a new approach for modeling the elastic stresses in center-wound rolls.