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Influence of work-roll grinding error and high-fidelity corrective grinding in cold sheet rolling

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Abstract

High-fidelity flatness defects in cold-rolled strip and sheet, arising from highly localized thickness strain variations, present an ongoing challenge to the metal industry. A primary cause of such defects, based on rolling practice, but for which the effects have not been rigorously investigated, is the transfer of localized work-roll diameter deviations due to roll grinding error. This study addresses high-fidelity work-roll diameter deviation transfer in the cold rolling of stainless steel, aluminum, and copper. Parametric studies are performed on a 4-high mill to examine the influences of roll diameter, reduction, strip width, and material on the transfer of high-fidelity work roll diameter deviations. Studies are conducted using an efficient 3D roll-stack model that predicts strip thickness profile deviations via the simplified-mixed finite element method. Reduction deviations on the outgoing strip, which correlate to strip flatness/shape defects, are quantified and analyzed to understand the transfer characteristics of work-roll grinding deviations relative to perfectly ground (smooth) work rolls. The results reveal that high-fidelity transfer depends not only on roll grinding deviation amplitudes and mill loading, but also on the specific locations of deviations along the roll face length due to 3D bulk roll-stack deformations as well as effective stiffness ratio between the work roll and the strip. Concluding the study is a novel approach to identify customized work roll grinding profiles tailored specifically to eliminate pre-existing high-fidelity strip flatness defect patterns, wherein “corrective” high-fidelity roll diameter profiles account for the predicted 3D mill deflections, contact force distributions, and coupled micro-/macro-scale deformation mechanics.

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Availability of data and material

Available upon request.

Code availability

Available upon request.

Notes

  1. Ref. [23] showed the coupling effects between micro- and macro-scale deformation.

  2. Secant is preferred over tangent particularly in dynamic analysis with direct time integration.

  3. An effective and adequate debris removal system would be required.

Abbreviations

\([{{\varvec{K}}}_{{\varvec{G}}}]\) :

Global stiffness matrix

\({\varvec{u}}\) :

Global displacement vector

\({\varvec{f}}\) :

Global force/load vector

\({u}_{j}\) :

Translational displacement in x-direction (m)

\({v}_{j}\) :

Translational displacement in y-direction (m)

\({w}_{j}\) :

Translational displacement in z-direction (m)

\([{{\varvec{K}}}_{{\varvec{F}}}]\) :

Elastic foundation stiffness contribution in the global stiffness matrix

\(\left[{{\varvec{K}}}_{{\varvec{T}}}\right]\) :

Timoshenko beam stiffness contribution in the global stiffness matrix

\({k}_{{f}_{1}}\) :

Elastic foundation stiffness of body 1 at a contact interface (Pa)

\({k}_{{f}_{2}}\) :

Elastic foundation stiffness of body 2 at a contact interface (Pa)

\({k}_{{f}_{eq}}\) :

Equivalent elastic foundation stiffness at a contact interface (Pa)

\(\left[{\varvec{N}}\right]\) :

Shape function matrix

\({l}_{i}\) :

Length of the element \(i\) (m)

\({d}_{12}\) :

Distance between the roll center axes for bodies 1 and 2 (m)

\(x\) :

Location along the width-wise direction (x-direction)

\({y}_{c}\) :

Initial coordinate (m)

\(D\left(x\right)\) :

Diameter profile as a function of location \(x\) along the x-direction (m)

\(\delta\) :

Contact interference between two contacting bodies (m)

\({P}_{c}\) :

Total contact load (N)

\(E\) :

Young’s modulus (Pa)

\(w\) :

Strip width (m)

\(H\) :

Entry thickness (m)

\(h\) :

Exit thickness (m)

\(F\) :

Rolling force per unit width (N/m)

\(\overline{r }\) :

Average reduction (thickness strain) across the strip width

\(\Delta r\left(x\right)\) :

Deviation in thickness strain at location \(x\) in the x-direction from the average reduction across the strip width

\(\Delta {r}_{in}\left(x\right)\) :

Deviation in thickness strain at location \(x\) in the x-direction from the average reduction across the strip width for the entry strip

\(\Delta {r}_{e}\left(x\right)\) :

Deviation in thickness strain at location \(x\) in the x-direction from the average reduction across the strip width for the exit strip

\(n\) :

Total number of Gaussian points

\(\Delta {r}_{e,e}\) :

Total reduction deviation of the exit strip considering the errored roll profile

\(\Delta {r}_{e,s}\) :

Total reduction deviation of the exit strip considering the smooth roll profile

\({d}_{wr}\) :

Diameter of the work roll (m)

\({l}_{wr}\) :

Face length of the work roll (m)

\({F}_{norm}\) :

Normalized specific rolling force

\(RM{S}_{lit}\) :

RMS error calculated analogously to the calculations in the literature

\(RM{S}_{t}\) :

RMS error calculated per this work using thickness profile

\(h{\left(x\right)}_{e}\) :

Exit thickness profile as a function of axial location \(x\) considering the grinding error on the work roll (m)

\(h{\left(x\right)}_{u}\) :

Exit thickness profile as a function of axial location \(x\) considering the smooth profile of the work roll (m

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Funding

This research was funded by National Science Foundation, grant number CMMI-1555531.

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Authors and Affiliations

  1. The University of Texas at Dallas, 800 W. Campbell Rd, Richardson, TX, 75080, USA

    Akash Patel, Arif Malik, Feng Zhang & Ritin Mathews

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  1. Akash Patel
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  2. Arif Malik
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Contributions

Patel, A.: Conceptualization, methodology, formal analysis, investigation, data curation, writing, and visualization. Malik, A.: Conceptualization, methodology, software, resources, data curation, writing, supervision, project administration, and funding acquisition. Zhang, F.: Software. Mathews, R.: Formal analysis, writing, and visualization.

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Correspondence to Arif Malik.

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Patel, A., Malik, A., Zhang, F. et al. Influence of work-roll grinding error and high-fidelity corrective grinding in cold sheet rolling. Int J Adv Manuf Technol 120, 7389–7413 (2022). https://doi.org/10.1007/s00170-022-09228-7

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