Nonlinear model of an axially moving plate in a mixed Eulerian–Lagrangian framework

Abstract

We consider finite deformations and bending of an elastic plate moving across a given domain. Velocities of the plate are kinematically prescribed at two parallel lines, which bound the region in the direction of motion. Inhomogeneity of the velocity profile at the exit from the domain results in planar deformations and out-of-plane buckling of the plate. The presented quasistatic analysis features a novel kinematic description, in which the coordinate in the direction of motion is a Eulerian one, while the displacements in transverse and the out-of-plane directions are modeled in a Lagrangian framework. The material volume is traveling across a finite element mesh, which is aligned to the boundaries of the domain. A concise mathematical formulation results in a robust numerical scheme without the need to solve the advection (transport) equation at each time step. The model is validated against solutions of a benchmark problem with a conventional Lagrangian finite element scheme. The approach is further demonstrated by modeling the time evolution of deformation of a moving plate.

Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells

“…We achieve this goal using a four-node finite element with the approximation scheme, suggested for linear plates in [46]: 16 The element has thus 48 mechanical degrees of freedom and belongs to the class of the so-called absolute nodal coordinate formulation (ANCF, [47,48]). A constant mass matrix and smooth approximation make it as well attractive for the transient analysis and non-material formulations for axially moving structures [49]. Despite inherent restrictions concerning the topology of the mesh and connections between shell segments, the present finite element has a relatively broad spectrum of potential applications with respect to both research and development.…”

Non‐material finite element rod model for out‐of‐plane bending of an elastic strip with natural curvature

“…Therefore, we enhance the mixed Eulerian-Lagrangian finite element model (MEL), presented for the two-dimensional case in [1]; see also [2] for the application to axially moving plates. Due to small imperfections the motion of the belt gains a lateral contribution, that is the belt runs off the pulleys.…”

Mixed Eulerian–Lagrangian description in materials processing: deformation of a metal sheet in a rolling mill

“…The initial transverse deflection appears owing to the unsymmetry of the speed of material generation v entry . The strip reaches the right roll stand inclined with @ x u y < 0 (Figure 9), and the contact line with the right roll stand starts moving according to (32). As both velocity profiles are time-independent, after a certain transient stage, the system arrives in a stationary regime without inclination at x D L ( Figure 10), and the axial strain at the right roll stand shall correspond to the entry velocity profile.…”

“…At the right roll stand, we specify a dependence of v exit on the spatial coordinate y (and not on Q y), such that the velocity profile varies when the contact line is traveling in the transverse direction, and the mean value of v exit may slightly differ from 1.05. As a technical detail, we mention that the local derivative @ x u in the boundary condition (32) is computed using the finite element approximation for the rightmost finite elements. The nominal strip velocity v D 1:00417 was chosen as the mean value of v entry for 0:5 6 y 6 0:5.…”