Preface to the Third Edition

Lai, W. Michael; Rubin, David T.; Krempl, E.

doi:10.1016/b978-0-08-041700-4.50004-4

Cited by 1 publication

(1 citation statement)

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“…Another possibility is to use an Eulerian flow description, and by this the actual deformation of the moving continuum can be handled using a mixed Lagrange-Euler description [34]. The Eulerian description is a standard approach in fluid dynamics where the observer is watching a control volume, where possible deformations will appear [64,38].…”

Section: The Governing Equationsmentioning

confidence: 99%

The origin of in-plane stresses in axially moving orthotropic continua

Kurki

Jeronen

Saksa

et al. 2016

International Journal of Solids and Structures

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Highlights• In-plane motion of an axially travelling orthotropic sheet was modelled• Velocity difference between supports generates strain due to mass conservation• In a travelling viscoelastic material the axial strain varies along the length• Small deformations of free edges due to the Poisson effect affect the velocity• Inertial effects lead to additional cross-directional contraction AbstractIn 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 the two-dimensional case, are both seen to influence the velocity field. We concentrate on solutions of a steady-state type, 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 models. The two-dimensional elastic problem is solved numerically using a nonlinear finite element procedure. An important new fundamental feature of the model is the coupling of the driving velocity field to the deformation of the material, while accounting for small deformations of the free edges. The results indicate that inertial effects produce an additional contribution to elastic contraction in unsupported, free webs.

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Section: The Governing Equationsmentioning

confidence: 99%