On displacement-velocity coupling and the origin of in-plane stress in orthotropic moving continua
Published inReports of the Department of Mathematical Information Technology. Series B, Scientific computing
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. ...