On displacement-velocity coupling and the origin of in-plane stress in orthotropic moving continua

In this paper, we address the problem of the origin of in-plane stresses in continuous, two-dimensional high-speed webs. In the case of thin, slender webs, a typical modeling approach is the application of a stationary in-plane model, without considering the effects of in-plane velocity field. However, for high-speed webs this approach is insufficient, because it neglects the coupling between the total material velocity and the deformation experienced by the material. By using a mixed Lagrange–Euler approach in model derivation, the solid continuum problem can be transformed to solid a continuum flow problem. Mass conservation in the flow problem, and the behaviour of free edges in the two-dimensional case, are both seen to influence the velocity field. We concentrate on the steady-state solutions of the model, and study briefly the coupled nature of material viscoelasticity and transport velocity in one dimension. Analytical solutions of the one-dimensional equation are presented with both elastic and viscoelastic material assumptions. Numerical solution of the two-dimensional elastic problem is also presented. Due to the nature of the velocity-dependent contraction, a nonlinear FEM solution procedure is used. The results indicate that inertial effects produce an additional contribution to elastic contraction in unsupported, free webs.