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

Abstract

Summary The paper is concerned with the modeling of the planar motion of a horizontal sheet of metal in a rolling mill. Inhomogeneous velocity profiles, with which the material is generated at one roll stand and enters the next one, lead to the time evolution of the deformation of the metal strip. We propose a novel kinematic description in which the axial coordinate is an Eulerian one, while the transverse motion of the sheet is modeled in a Lagrangian framework. The material volume travels across a finite element mesh, whose boundaries are in contact with the roll stands. The concise mathematical formulation of the method is different from the classical form of the arbitrary Lagrangian–Eulerian approach with a rate form of constitutive relations. The undeformed state of the strip is incompatible owing to the varying time rate of the generation of material. We treat this phenomenon by introducing the notion of intrinsic strains, which are mathematically described using the multiplicative decomposition of the deformation gradient. We currently present the approach for quasistatic simulations of in‐plane elastoplastic deformations of the strip. A practically relevant problem with two strip segments and three roll stands is studied in a numerical example. Copyright © 2016 John Wiley & Sons, Ltd.

Transient modelling of flexible belt drive dynamics using the equations of a deformable string with discontinuities

“…more power is lost at higher transmitted moments. This simple expression for the coefficient of efficiency is a consequence of the relation (25), which is based upon the idealized contact model. The analysis with the finite slip zones results in more complicated expressions, see, e.g., [13].…”

“…This helps avoiding the mentioned polygon effect and allows for finer meshes near the contact regions. See also Vu-Quoc and Li [23], Vetyukov et al [24,25] for non-material finite element formulations in application to axially moving structures.…”

Mechanics of Axially Moving Structures at Mixed Eulerian-Lagrangian Description

“…As it is typical for axially moving structures, Lagrangian (material) kinematic description is not an optimal choice for this sort of problems, in particular for numerical methods with spatial discretization. Purely Eulerian (spatial) (Eliseev and Vetyukov, 2012;Vetyukov et al, 2017b) or mixed Eulerian-Lagrangian (Vetyukov et al, 2016(Vetyukov et al, , 2017a formulations are advantageous as we discretize the problem in a domain-specific manner. Exact solutions of the problem of an extensible belt moving between the pulleys with dry friction law of contact were provided by Rubin (2000); Bechtel et al (2000); Morimoto and Iizuka (2012) for stationary regimes of motion.…”

“…12.21. Imperfect geometry of the belt is accounted for using the multiplicative decomposition from the undeformed state to the reference one (Vetyukov et al, 2017a), and then we apply the mixed Eulerian-Lagrangian kinematic description to prevent the mesh from motion in the circumferential direction. We observe the effects, known from the practical experience: skew hanging of the belt, partial loss of contact with the drums (indicated by missing integration points in the figure) and lateral run-off during the motion.…”

Flexible belt hanging on two pulleys: Contact problem at non-material kinematic description

“…Modelling the deflection of the belt in the field of gravity becomes challenging when the system starts moving. Referring the interested reader to a detailed discussion of the voluminous literature on axially moving continua and Eulerian (spatial) kinematic description in structural mechanics to Chen (2005); Vetyukov et al (2016Vetyukov et al ( , 2017a; Vetyukov (2018), here we point out the intrinsic drawbacks of conventional finite element schemes with Lagrangian (material) kinematics when applied to the considered sort of problems as for example done by Dufva et al (2007). Letting the finite element mesh move between the qualitatively different domains, namely the free spans and the zones of sticking or sliding contact between the belt and the pulleys, one inevitably experiences numerically induced oscillations in the solution, see also the benchmark example in (Oborin et al, 2018).…”