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.
Highlights
A vast body of literature is devoted to modeling dynamics and deformations of axially moving strings, beams or plates, see a review paper by Chen [2]
We are dealing with an open system, and particles are continuously entering and leaving the active material volume, this condition is fulfilled at each time step of a quasistatic simulation owing to the kinematic nature of the boundary conditions at both ends of the domain, see the discussion in Sect. 6 below
We studied the mesh convergence and compared the solutions with the proposed mixed Eulerian-Lagrangian description and with the conventional Lagrangian finite elements by finding the maximal and the minimal values of the displacement uz, which are observed at the edges y = ±w/2 of the plate
Summary
A vast body of literature is devoted to modeling dynamics and deformations of axially moving strings, beams or plates, see a review paper by Chen [2]. Previous attempts to obtain a model for an axially moving deformable plate [7,8] did not account for particular kinematic conditions at the boundaries of the domain in the form of time and space variations of the prescribed velocity profiles. These effects are relevant in applications of hot rolling, strip coiling or belt drive simulations. Further extensions of the approach are discussed in the concluding remarks below
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