Numerical analysis of a new Eulerian–Lagrangian finite element method applied to steady‐state hot rolling processes

Abstract

SUMMARYA finite element code for steady-state hot rolling processes of rigid-visco-plastic materials under planestrain conditions was developed in a mixed Eulerian-Lagrangian framework. This special set up allows for a direct calculation of the local deformations occurring at the free surfaces outside the contact region between the strip and the work roll. It further simplifies the implementation of displacement boundary conditions, such as the impenetrability condition. When applied to different practical hot rolling situations, ranging from thick slab to ultra-thin strip rolling, the velocity-displacement based model (briefly denoted as vu-model) in this mixed Eulerian-Lagrangian reference system proves to be a robust and efficient method.The vu-model is validated against a solely velocity-based model (vv-model) and against elementary methods based on the Kármán-Siebel and Orowan differential equations. The latter methods, when calibrated, are known to be in line with experimental results for homogeneous deformation cases. For a massive deformation it is further validated against the commercial finite-element software package Abaqus/Explicit. It is shown that the results obtained with the vu-model are in excellent agreement with the predictions of the vv-model and that the vu-model is even more robust than its vv-counterpart. Throughout the study we assumed a rigid cylindrical work roll; only for the homogeneous test case, we also investigated the effect of an elastically deformable work roll within the frame of the Jortner Green's function method.The new modelling approach combines the advantages of conventional Eulerian and Lagrangian modelling concepts and can be extended to three dimensions in a straightforward manner.

A novel mixed Eulerian–Lagrangian finite-element method for steady-state hot rolling processes

“…Further aspects of the numerical methodology of our vu-model and its application to various test cases of practical interest will be subject of a subsequent publication (cf. [15]). …”

“…In addition, this ÿnite-element type when applied to di erent hot rolling situations, ranging from strip to slab rolling, appears to be even more robust than its vv-model counterparts (see Ref. [15]). …”

“…Then, we can write the transformation rules for surface and volume elements between (E-L) and (L) as dã =Ĵ · dA ·F −1 =Ĵ ·F −T · dA and dṽ =Ĵ · dV (15) and, respectively, between (E-L) and (E) as da =J · dã ·F −1 =J ·F −T · dã and dv =J · dṽ:…”

“…Only then the system of linear equations, solved via special sparse-matrix techniques, including scaling of the di erent quantities, will yield a stable and su ciently accurate solution to our problem. For a detailed discussion on the various techniques applied in the numerical solution process, the reader is referred to [15,19,20], respectively.…”

Contact mechanics and work roll wear in cold rolling of thin strip

“…Jiang et al [6][7][8] calculated the elastic deformation of thin strip, its shape, profile, and flatness. Based on a mixed Eulerian-Lagrangian finite element method, Synka et al [9] developed a finite element code to simulate different practical rolling processes, ranging from thick slab to ultra-thin strip rolling. Hacquin et al [10] developed a refinement of the roll bending and flattening models for strip rolling.…”