# Numerical simulation of flatness defects: application to rolling process

Kékéli Kpogan, Hamid Zahrouni, Michel Potier-Ferry, Hachmi Ben Dhia

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Kékéli Kpogan, Hamid Zahrouni, Michel Potier-Ferry, Hachmi Ben Dhia. Numerical simulation of flatness defects: application to rolling process. 12e Congrès de Mécanique, Apr 2015, Agadir, Morocco. hal-01868101

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Distributed under a Creative Commons Attribution 4.0 International License
# NUMERICAL SIMULATION OF FLATNESS DEFECTS: APPLICATION TO ROLLING PROCESS

KPOGAN K.¹*, ZAHROUNI H.¹, POTIER-FERRY M.¹, BEN DHIA H.²

¹DAMAS, Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures, Université de Lorraine, France

²Ecole Centrale Paris, Laboratoire de Mécanique des Sols, Structures et Matériaux (MSSMat), Grande Voie des Vignes, 92295 Châtenay-Malabry Cedex, France

* Corresponding author

**Abstract:** In this paper we present a new numerical technique to model the buckling phenomena under residual stresses induced by rolling process. This technique consists in coupling two finite element models: the first one consists in a three dimensional model based on 8-node tri-linear hexahedron which is used to model the three dimensional behaviour of the sheet in the roll bite; the second model is based on a shell formulation well adapted to large displacements and rotations, it will be used to compute buckling of the strip out of the roll bite. We propose to couple these two models by using Arlequin method.

**Keywords:** Rolling, Residual stresses, Buckling, Arlequin Method, Asymptotic Numerical Method

## 1. Introduction

Flatness defects are among the major problems encountered in strip rolling. Their direct origin is out-of-bite stress gradients resulting in buckling in the compressive stress areas. The most important flatness defects are “edge-waves” and “center-waves” buckles. These waves are the result of buckling due to self-equilibrating longitudinal residual stresses with a compressive longitudinal membrane force state in the middle of the strip (“center buckles”) or in the edge zones (“edge buckles”), respectively. During the rolling process, the buckling waves are usually suppressed by global traction. Thus, in some cases, the sheet may appear more or less flat, or even perfectly flat on the rolling line. Nevertheless we can still talk about flatness defects, insofar as there may be residual stresses in the sheet. This is why the post-bite stress profile is called “latent flatness defects”. If defects are only latent, the stress field computed beyond the bite by a 3D finite element model (FEM) should be correct. One can find in the literature three ways to model rolling, depending on the manner to account for flatness defects. The first way is based on 3D FEM that generally do not permit to capture buckling effects; they can only predict latent defects. Other models permit to account for buckling in an uncoupled approach: the 3D model yields residual stresses that are further included as the starting part of a buckling analysis. Last, few FEM are able to model rolling and buckling in a coupled manner. Several publications have been focused on uncoupled approaches. Counhaye [5] was the first to propose a coupling model of

rolling process and buckling phenomena. The model is based on a stress-relaxation algorithm applied only in the out-of-bite areas. They added an additional term to the elastic/plastic strain rate decomposition which represents the local shortening of a material segment when it becomes wavy due to buckling. Abdelkhalek et al. [6] used a different approach for coupling 3D FEM and Asymptotic Numerical Method (ANM) using a shell formulation. In this approach, a full 3D model of rolling is computed; the post-bite stress field obtained is introduced in the shell formulation where a buckling analysis under residual stresses is performed. The new distribution of the stress field obtained by buckling is considered as a new boundary condition for the 3D FEM. These computations are repeated until convergence.

In the present work, we propose a new approach to simulate rolling process taking into account buckling phenomena. A simplified model is used which consists in coupling a 3D continuum model and a shell model using Arlequin method. The buckling is due to residual stresses which are introduced in the 3D model and propagated in the shell model during the simulation.

## 2. Formulation

We propose an algorithm which consists in coupling 3D FEM and shell model by Arlequin method. So the idea is to combine these two models in the same finite element simulation that we solve by ANM. In the three dimensional part, we consider small strain and residual stresses which come from a full model taking into account the elastoplastic law adapted to rolling.

$$ \left\{ \begin{array}{l} \displaystyle \int_{\Omega_{3D}}^{t} \sigma : \delta\varepsilon d\Omega - \lambda P_{ext} (\delta u^{3D}) = 0 \\ \\ \sigma = D : \varepsilon + \lambda_{res} \sigma^{res} \end{array} \right. \qquad (1) $$

where $\varepsilon$ and $\delta\varepsilon$ are respectively the strain and the virtual strain tensors. $P_{ext}$ corresponds to the energy density due to external load, which is calibrated by a load parameter $\lambda$; $\sigma$ and $\sigma^{res}$ are respectively the Cauchy stress tensor and the residual stress. We use small strain in this part of the model since we assume that the sheet in upstream of the roll mill is flat. But progressively, the 3D model will transmit the residual stresses to the shell model. Thus only the shell model will buckle during the simulation.

Finally in the shell part, we consider a geometrically nonlinear model. This shell model, proposed by Büchter et al. [7], avoids locking by incorporating an additional parameter describing the shell thickness variation. It is distinguished from classical shell models that are usually based on degenerated constitutive relations, since the present formulation uses the unmodified and complete three-dimensional constitutive law.
$$
\left\{
\begin{array}{l}
\displaystyle \int_{\Omega_{\text{shell}}} {}^t S : [ \gamma^l (\delta u^{\text{shell}}) + 2\gamma^{nl} (u^{\text{shell}}, \delta u^{\text{shell}}) ] d\Omega = \lambda P_{\text{ext}} (\delta u^{\text{shell}}) \\[1.5em]
\displaystyle \int_{\Omega_{\text{shell}}} {}^t S : \delta \tilde{\gamma} d\Omega = 0 \\[1.5em]
S = D : [ \gamma^l (u^{\text{shell}}) + \gamma^{nl} (u^{\text{shell}}, u^{\text{shell}}) + \tilde{\gamma} ] + \lambda_{\text{res}} S^{\text{res}}
\end{array}
\right.
\qquad (2)
$$

The total Green-Lagrange strain tensor is decomposed into
the compatible strain $\gamma^u = \gamma^l + \gamma^{nl}$ and an additional assumed
strain term $\tilde{\gamma}$. Stress tensor is the second Piola-Kirchhoff
stress $S$ and $S^{\text{res}}$ the residual stress.

3. Arlequin coupling

The Arlequin method allows to couple two different
mechanical states through reliable coupling operators as
well as consistent energy distribution between the two
coupling zones (see Fig. 1).

<figure><img src="image_2.png" /><figcaption>**Figure 1:** The Arlequin method in a general mechanical problem; A general mechanical problem (a) becomes a superposed domains $\Omega_1$ and $\Omega_2$ that intersect in the gluing zone $\Omega$.</figcaption></figure>

The central point of the model is the coupling operator which is selected by analogy with the deformation energy of the shell. Otherwise the coupling area varies during the rolling process. In this paper, we limit ourselves to the $H^1$ coupling, and hence

$$
c(\mu, u) = \int_{\Omega} \mu u + \gamma^l(\mu) : D : \gamma^l(u) d\Omega \quad (3)
$$

We denote respectively $u$ and $\mu$ the displacement and the
Lagrange multiplier fields.

The integration domain is three-dimensional and the
Lagrange multiplier is discretized using shell kinematic.
We note that the integration points and interpolation, for
Lagrange multiplier, are not necessarily the same as for
the shell domain.

In addition we introduce weighting functions in order to
share the energy between the two models in the coupling
area. These weighting functions are used respectively for
strain energy and for work induced by external forces.
They satisfy the following relations :

$$
\left\{
\begin{array}{l}
\alpha^{3D} + \alpha^{\text{shell}} = 1 \quad \mathbf{in} \ \Omega_c \\[1.5ex]
\alpha^{3D} = \mathbf{1} \ \mathbf{in} \ \Omega_{3D} \setminus \Omega_c \\[1.5ex]
\alpha^{\text{shell}} = \mathbf{1} \ \mathbf{in} \ \Omega_{\text{shell}} \setminus \Omega_c
\end{array}
\right.
\qquad (4)
$$

Find the global displacement field in the domain $\Omega$ is
therefore to weight the displacement $u^{3D}$ and $u^{\text{shell}}$
respectively in the domains $\Omega_1$ and $\Omega_2$.

By introducing the weighting functions in equations 1 and
2, the Arlequin method allows us to write the following
variational form :

$$
\left\{
\begin{array}{l}
\displaystyle \int_{\Omega_{3D}} \alpha^{3D} {}^t \sigma : \delta \varepsilon d\Omega + c(\mu, \delta u^{3D}) = \lambda P_{\text{ext}} (\alpha^{3D} \delta u^{3D}) \\[2ex]
\left\{
\begin{array}{l}
\displaystyle \int_{\Omega_{\text{shell}}} \alpha^{\text{shell}} {}^t S : [ \gamma^l (\delta u^{\text{shell}}) + 2\gamma^{nl} (u^{\text{shell}}, \delta u^{\text{shell}}) ] d\Omega - \\
\qquad c(\mu, \delta u^{\text{shell}}) = \lambda P_{\text{ext}} (\alpha^{\text{shell}} \delta u^{\text{shell}}) \\[1.5ex]
\displaystyle \int_{\Omega_{\text{shell}}} \alpha^{\text{shell}} {}^t S : \delta \tilde{\gamma} d\Omega = 0
\end{array}
\right. \\[2ex]
\sigma = D : \varepsilon + \lambda_{\text{res}} \sigma^{\text{res}} \\[2ex]
S = D : [ \gamma^l (u^{\text{shell}}) + \gamma^{nl} (u^{\text{shell}}, u^{\text{shell}}) + \tilde{\gamma} ] + \lambda_{\text{res}} S^{\text{res}}
\end{array}
\right.
\qquad (5)
$$

4. Asymptotic numerical method

ANM is a technique for solving nonlinear equations based
on the Taylor expansion to high order [3]. The technique
consists in transforming a given nonlinear problem into a
sequence of linear ones to be solved successively, leading
to a numerical representation of the solution in the form of
power series truncated at relatively high orders. Compared
to iterative methods, ANM allows significant reduction of
computation time since only one decomposition of the
stiffness matrix is used to describe a large part of the
solution branch. Thus, the procedure allows developing the
unknown variables of the problem in the form of power
series with respect to a path parameter "a" (see [1, 3] for
more details).

5. Results

The presented test case concerns flatness defects generated
by a rolling process. The procedure we proposed is non-
iterative coupling: (i) first a rolling calculation (LAM3)
allows us to provide residual stresses, (ii) to compute
flatness defects, a second calculation using the coupling
code 3D/shell/Arlequin is then performed considering
residual stresses resulting from the previous calculation.

The results are compared with Counhaye's [5] and
Abdelkhalek's [6] models and with experimental
measurements.

Note that Counhaye's model is a modification of rolling
code LAM3 which only relax membrane stresses in
compression zones but the model is limited for the
representation of flatness defects in the sheet.

Moreover, the procedure proposed by Abdelkhalek is based
on an iterative coupling between the rolling code LAM3
and a thin shell code. It is also able to relax the excessive
stresses and predicts flatness defects of various shapes. But
compared to our new procedure, it has some limitations.
Firstly, it's too complicated; the first iteration is a complete
rolling calculation, the following describes only the
downstream domain of the structure. Secondly, the transfer
of the data between subdomains is not really standard. In
particular, clamped boundary condition which is used on
the shell model side of the bite is not very convincing.

We apply the proposed model to a rolling case described in
Table 1, which refers to the rolling pass, the last stand of a
tinplate sheet mill with very low thickness.

Let us call the method proposed in this work, Kpogan’s
model.
<table><caption>**Table 1.** Characteristics of investigated rolling pass</caption><tr><td>Strip width</td><td>855 mm</td></tr><tr><td>Strip thickness (strip crown)</td><td>0.355 mm (4.81%)</td></tr><tr><td>Exit thickness</td><td>0.252 mm</td></tr><tr><td>Rolling speed</td><td>22 m/s</td></tr><tr><td>Front tension</td><td>100 MPa</td></tr><tr><td>Back tension</td><td>170 MPa</td></tr><tr><td>Type of mill</td><td>4-high</td></tr><tr><td>diameter WR</td><td>555 mm</td></tr><tr><td>diameter BUR</td><td>1300 mm</td></tr><tr><td>Length WR</td><td>1400 mm</td></tr><tr><td>Length BUR</td><td>1295 mm</td></tr><tr><td>WR crown</td><td>0.0322%</td></tr><tr><td>BUR crown</td><td>No crown</td></tr><tr><td>Fraction law</td><td>Coulomb: τ = 0.03xσ<sub>s</sub></td></tr><tr><td>Young's modulus</td><td>E = 210 GPa</td></tr><tr><td>Poisson's ratio</td><td>ν = 0.3</td></tr></table>

Figure (2) shows the post-buckled state of the sheet. As Counhaye's model is limited to represent flatness defects, we compare the post-buckled state of the sheet with Abdelkhalek's model. Figures (2.a) and (2.b) show respectively the results of Kpogan and Abdelkhalek's model. We obtain, as Abdelkhalek's model, longitudinal folds near the roll mill.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Vertical displacement of the sheet buckled under residual stresses; Figure (a) : Kpogan's model; Figure (b) : Abdelkhalek's model</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Comparisons of results with reference models; Figure (a) : Kpogan's model; Figure (b) : Abdelkhalek's model; Figure (c) : Counhaye's model; Figure (d) : LAM3 code without buckling</figcaption></figure>

Figure (3) illustrates $\sigma_{xx}$ profiles for Kpogan, Abdelkhalek and Counhaye's models. Figure 15a shows the results obtained with Kpogan's model. We notice that with Kpogan's model, the stresses are relaxed and they are close

to the experimental measurements. Compared to the results of Abdelkhalek (Fig. 15b) and Counhaye's (Fig. 15c) models, Kpogan's model gives results much closer to the experiment.

## 6. Conclusion

We have proposed in this study a numerical model which consists in coupling Arlequin and asymptotic numerical methods to simulate flatness defects observed in rolling process. Our attention was focused on very thin sheets where these phenomena are often observed. Note that in thin sheet rolling, we need tri-dimensional model to compute correctly the deformations in the bite and a shell model far from the rolls to compute buckling modes. It is not obvious for a unique model to capture all these phenomena. Both models are coupled by using Arlequin method. This method allows us to model the entire structure while distinguishing upstream and downstream domain of the sheet.

We performed an industrial rolling test case taking into account complex residual stresses obtained by the rolling code LAM3. The results were compared with experimental data. The model we proposed gives satisfactory results. It predicts the relaxed stresses after buckling and shows the corresponding flatness defects.

## Références

[1] H. Zahrouni, B. Cochelin, M. Potier-Ferry, Computing finite rotations of shells by an asymptotic-numerical method, Computer methods in applied mechanics and engineering. 175 (1999) 71–85.

[2] H. Ben Dhia, Multiscale mechanical problems : the Arlequin method, Comptes Rendus de l'Académie des Sciences, Serie IIb, Paris, (1998) 899-904

[3] B. Cochelin, N. Damil, and M. Potier-Ferry, Méthode Asymptotique Numérique: une technique de résolution des équations non linéaires. Hermes Science Publishing, Paris, Londres (2007).

[4] A. Hacquin, P. Montmitonnet, P. Guillerault, A steady state thermo-elastoviscoplastic finite element model of rolling with coupled thermo-elastic roll deformation, Journal of materials processing technology. 60 (1996) 109–116.

[5] C. Counhaye, Modélisation et contrôle industriel de la géométrie des aciers laminés à froid (modelling and industrial control of the geometry of cold rolled steels). PhD thesis, University of Liege, (2000).

[6] S. Abdelkhalek, H. Zahrouni, M. Potier-Ferry, N. Legrand, P. Monmitonnet, P. Buessler, Coupled and uncoupled approaches for thin cold strip buckling prediction, International journal of material forming. 2 (2009) 833-836.

[7] N. Büchter, E. Ramm, D. Roehl, Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept, International journal for numerical methods in engineering 37 (15) (1994) 2551–2568.