On the limit velocity and buckling phenomena of axially moving orthotropic membranes and plates

In this paper, we consider the static stability problems of axially moving orthotropic membranes and plates. The study is motivated by paper production processes, as paper has a fiber structure which can be described as orthotropic on the macroscopic level. The moving web is modeled as an axially moving orthotropic plate. The original dynamic plate problem is reduced to a two-dimensional spectral problem for static stability analysis, and solved using analytical techniques. As a result, the minimal eigenvalue and the corresponding buckling mode are found. It is observed that the buckling mode has a shape localized in the regions close to the free boundaries. The localization effect is demon…

Optimization and analysis of processes with moving materials subjected to fatigue fracture and instability

We study systems of traveling continuum modeling the web as a thin elastic plate of brittle material, traveling between a system of supports at a constant velocity, and subjected to bending, in-plane tension and small initial cracks. We study crack growth under cyclic in-plane tension and transverse buckling of the web analytically. We seek optimal in-plane tension that maximizes a performance vector function consisting of the number of cycles before fracture, the critical velocity and process effectiveness. The present way of applying optimization in the studies of fracture and stability is new and affords an analytical tool for process analysis. peerReviewed

Dynamic analysis for axially moving viscoelastic panels

In this study, stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis. We use the flat panel theory combined with the Kelvin–Voigt viscoelastic constitutive model, and we include the material derivative in the viscoelastic relations. Complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity, and the corresponding eigenfunctions are found using central finite differences. The governing equation for the transverse displacement of the panel is of fifth order in space, and thus five boundary conditions are set for the problem. The fifth condition is derived and set at the in-…

The stress-strain state and stabilization of viscoelastoplastic, imperfect moving web continuum

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-dime…

A review of the analytical and numerical modeling of composites

This review article is dedicated to the materials that are made from two or more constituent materials with different physical and/or chemical properties. The focus is on the materials, where the individual components remain separate and distinct within the final structure. The new combined material usually have some additional characteristic properties compared to the individual components, or, in other case, some critical properties of combined material may follow almost equally one of the components. Typically, preferred properties of the new material can be such as strength, porosity, conductivity or cheapness. The ultimate goal of this study is to find methods and tools for achieve adequ…

The origin of in-plane stresses in axially moving orthotropic 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 the 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 into a solid continuum flow problem. Mass conservation in the flow problem, and the behaviour of free edges in th…