# A Short Review on the Finite Element Method for Asymmetric Rolling Processes

^{*}

## Abstract

**:**

## 1. Introduction

_{33}reduced its magnitude vs. rolling asymmetry, A, meaning, Σ

_{33}is lower for A = 1.05 and A = 1.3 (asymmetric rolling) than for A = 1.0 (symmetric rolling). Consequently, the applied rolling force decreased, which is an advantage to the rolling mill’s durability. In 2017, they introduced a self-consistent elastoplastic code into the finite element model to analyze texture variation across the sample thickness with considerable detail. The results were in line with the previous ones. The asymmetric rolling process modifies the material plastic anisotropy resulting in a higher average Lankford coefficient. Moreover, they identified an asymmetry ratio of 1.1 to obtain a nearly homogeneous plastic anisotropy across the sample thickness.

_{xz}and compression strain ε

_{zz}, where x, y, and z axes corresponded to rolling, transverse, and normal directions, respectively.

## 2. Finite Element Analysis Applied to Asymmetric Rolling Processes

- build the geometrical model;
- assign the material properties;
- define time steps;
- enforce boundary conditions;
- discretize the problem domain.

#### 2.1. Build the Geometrical Model

#### 2.2. Assign the Material Properties

_{0}), the hardening coefficient (C), and the initial accumulated shear strain (${\gamma}_{0}$). Studies without crystal plasticity models show a not applicable note (n/a).

#### 2.3. Define Time Steps

#### 2.4. Enforce Boundary Conditions

#### 2.5. Discretize the Problem Domain

#### 2.6. Crystallographic Models

#### 2.6.1. Crystallographic Homogenized Model

#### 2.6.2. Leffers-Wierzbanowski (LW) Model

#### 2.6.3. ALAMEL Model

#### 2.6.4. Visco-Plastic Self-Consistent (VPSC) Model

#### 2.7. Simulation

#### 2.8. Postprocessing

## 3. Summary and Final Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Geometrical models representation. (

**a**) A single-pass rolling process; (

**b**) A single-pass ASR process due to rolls’ diameters mismatch; (

**c**) A two-pass rolling process; (

**d**) Workpiece measurements.

**Figure 2.**Boundary conditions imposed to accomplish ASR results. (

**a**) A schematic of velocities and friction BC; (

**b**) An illustration of the additional BC to avoid bending; (

**c**) Symmetry plane RD–ND; (

**d**) Geometrical model without considering symmetry in the RD–ND plane.

**Figure 3.**Double scale structures, coordinates, RVE, and FCC crystal lattice structure. (

**a**) The macro-continuum; (

**b**) The micro-polycrystal structure; (

**c**) RVE; (

**d**) Crystal lattice. (Reproduced from [28], with permission from Elsevier, 2009).

Geometrical Parameters | Boundary Conditions | Ref. No. | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Workpiece Dimensions (mm) ^{1} | Rolls Diameters (mm) ^{1} | Passes (No.) | Total Red. (%) ^{2} | Angular Velocities (rpm) ^{1} | Friction Coefficients | — | |||||

Length | Width | Thickness | ${\varnothing}_{{\mathit{R}}_{\mathit{t}\mathit{o}\mathit{p}}}$ | ${\varnothing}_{{\mathit{R}}_{\mathit{b}\mathit{t}\mathit{m}}}$ | ${\mathit{\omega}}_{{\mathit{R}}_{\mathit{t}\mathit{o}\mathit{p}}}$ | ${\mathit{\omega}}_{{\mathit{R}}_{\mathit{b}\mathit{t}\mathit{m}}}$ | ${\mathit{\mu}}_{{\mathit{R}}_{\mathit{t}\mathit{o}\mathit{p}}}$ | ${\mathit{\mu}}_{{\mathit{R}}_{\mathit{b}\mathit{t}\mathit{m}}}$ | |||

254 | 127/ 177.8 | 63.5 | 193.8 | 193.8 | single | 25 | var | var | 1 | 1 | [1] |

— | — | 8–15 | 1000 | 1000 | multi (2) | — | var | var | 0.2 | 0.3 | [2] |

25.1 | 18.82 | 6.27 | 158.76 | 158.76 | single | 14.17 | 19.20 | 19.20 | — | — | [5] |

200 | var | 6/10 | 210/ 186 | 210/ 200 | single | 5:5:20/30 | 0.92/ 1.42 | 1/ 1.53 | 0.2/0.4/ 0.8/1.0 | 0.6 | [6] |

— | — | 5/10 | var | var | single | 20:10:40 | var | var | 0.3 | 0.3 | [8] |

— | — | 3.2 | 189/ 126 | 126 | multi (3) | 50 | var | var | 0.3 | 0.3 | [9] |

— | — | 2 | 189 | 126 | multi | 80 | — | — | 0.4 | 0.4 | [14] |

— | — | 206/ 54.2 | 1200 | 1200 | single | 10:10:40 | var | var | var | var | [7] |

— | — | 4 | 128 | 128 | single | 37.5 | 18 | 12 | 0.2 | 0.2 | [13] |

— | — | 5/10 | var | var | single | 12/25/40 | var | var | var | var | [11] |

— | 80 | 2 | var | 210 | single | var | var | 28.65 | 0.15 | 0.15 | [12] |

36 | 0.4 | 6 | — | — | multi (3) | 83 | var | $2{\omega}_{{R}_{top}}$ | 0.2 | 0.2 | [15] |

— | — | 16 | var | 500 | single | 50 | var | 9.55 | var | 0.7 | [16] |

2.5 | 30 | 60 | 66 | 66 | single | 10–30 | 26.45 | 25.50 | 0.25 | 0.25 | [25] |

40 | — | 6 | 450 | 450 | multi (2) | 50 | var | var | — | — | [28] |

40 | — | 6 | 450 | 450 | multi (2) | 75 | 1080 | 2160 | 0.5 | 0.5 | [29] |

— | — | 3 | 400 | 400 | single | 10:20:70/80 | var | 47.75 | 0.5 | 0.5 | [17] |

— | — | 2 | var | 480 | single | 10/25/40 | 38.2 | 38.2 | 0.3 | 0.3 | [18] |

200 | var | 1.5/1.8/ 2.0 | 125 | 125 | single | 10/20 30 | var | var | 0.035 ^{4}0.025 ^{5} | 0.035 ^{4}0.025 ^{5} | [19] |

60 | — | 1.2 | 180 | 180 | multi | 50 | var | 2 | 0.05 | 0.15 | [30] |

— | — | — | 180 | 180 | single | 36 | var | var | 0.1/0.4 | 0.1/0.4 | [55] |

25 | 20 | 2 | 200 | 200 | two | 75 | var | 0.95 | 0.3 | 0.3 | [21] |

— | — | 2.9 | 180 | 180 | single | 36 | var | 10 | 0.1–0.4 | 0.1–0.4 | [27] |

50–100 | — | 1–8 | var | var | single | 10:10:60 | var | var | 0.3 | 0.3 | [22] |

— | — | 1–8 | 50–500 | 50–500 | single | 10:10:60 | var | var | var | var | [23] |

40 | 40 | 1 | 500 | 500 | single | 60 | 3.82 | var | 0.1–0.4 | 0.1–0.4 | [24] |

— | — | — | — | — | single | 20:10:40 | var | var | 0.4 | 0.4 | [34] |

— | 100 | 2 | 100 | 120 | single | 10 | var | 50.6 | 0.359 | 0.359 | [35] |

50 | — | 0.2 | 80 | 270 | single | var | — | — | 0.2 | 0.2 | [38] |

— | — | 7 | var | $1.5{\varnothing}_{{R}_{top}}\phantom{\rule{0ex}{0ex}}2.0{\varnothing}_{{R}_{top}}$ | multi | 70 | 23 | 23 | — | — | [39] |

35 | 20 | var | var | $1.5{\varnothing}_{{R}_{top}}$ | single | 50 | 24 | 24 | 0.9 | 0.9 | [40] |

510 | 230 | 6.7 | 295 | 295 | single | 33/44 | var | var | 0.45 | 0.45 | [41] |

— | — | 250 | 1000 | 1000 | single | 12/20 | 1 ^{3} | 1.3 ^{3} | 0.4 | 0.4 | [42] |

^{1}var—various values were used;

^{2}v1:v2:v3—various values were used, starting on v1 and ending on v3 with steps by increments of v2;

^{3}values in m/s according to [42];

^{4}static friction coefficient;

^{5}kinetic friction coefficient.

Material | Temp. (°C) | $\mathit{\rho}$ (kg/m ^{3})
| E (GPa) | $\mathsf{\nu}$ | Constitutive Law | Crystal Plasticity Parameters ^{1} | Ref. No. | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{\tau}}_{0}$ $\left(\mathrm{MPa}\right)$ | n | ${\mathit{h}}_{0}$ $\left(\mathrm{MPa}\right)$ | C | ${\mathit{\gamma}}_{0}$ | |||||||

Mild steel | 1200 | 6920 | — | — | $\overline{\mathsf{\sigma}}=9{\left(\overline{\mathsf{\epsilon}}\right)}^{0.132}$ | n/a | n/a | n/a | n/a | n/a | [1] |

Low carbon steel | 1027 | — | — | — | $\overline{\mathsf{\sigma}}=\frac{2}{3}\frac{{\mathsf{\sigma}}_{\mathrm{i}}}{{\mathsf{\epsilon}}_{\mathrm{i}}}\mathrm{I}\overline{\mathsf{\epsilon}}$ | n/a | n/a | n/a | n/a | n/a | [2] |

Aluminum | RT 420 | 2600 | 68.5 | 0.33 | — | n/a | n/a | n/a | n/a | n/a | [5] |

Steel | RT | — | — | — | — | n/a | n/a | n/a | n/a | n/a | [6] |

C15 | — | — | — | — | — | n/a | n/a | n/a | n/a | n/a | [8] |

Steel | 700 | — | 110 | — | — | n/a | n/a | n/a | n/a | n/a | [9] |

Aluminum | RT | — | — | — | $\overline{\mathsf{\sigma}}=179{\left(\overline{\mathsf{\epsilon}}\right)}^{0.22}$ | — | — | — | — | — | [14] |

Low carbon steel | var | — | — | — | — | n/a | n/a | n/a | n/a | n/a | [7] |

AA 1100 | RT | — | — | — | — | — | — | — | — | — | [13] |

Aluminum | RT | — | 68 | 0.3 | $\overline{\mathsf{\sigma}}=50.3{\left(1+\frac{\overline{\mathsf{\epsilon}}}{0.05}\right)}^{0.26}$ | n/a | n/a | n/a | n/a | n/a | [11] |

AA 1050P | RT | — | 69 | — | $\overline{\mathsf{\sigma}}=162.3{\left(\overline{\mathsf{\epsilon}}\right)}^{0.0353}$ | n/a | n/a | n/a | n/a | n/a | [12] |

AA 6022 | 250 | — | 70.2 | 0.3 | — | 16.4 | 0.26 | 24 | 6 | 0.004 | [15] |

AISI-1015 | 1200 | — | — | — | $\overline{\mathsf{\sigma}}=84.5{\left(\overline{\epsilon}\right)}^{0.31}{\left(\dot{\overline{\epsilon}}\right)}^{0.25}$ | n/a | n/a | n/a | n/a | n/a | [16] |

Low carbon steel | RT | — | 210 | 0.3 | — | — | — | — | — | — | [25] |

AA 6022 | 250 | — | — | — | $\overline{\mathsf{\sigma}}=131{\left(\overline{\epsilon}\right)}^{0.13}{\left(\dot{\overline{\epsilon}}\right)}^{0.058}$ | 16.5 | 0.26 | 22.5 | 5.6 | 0.005 | [28] |

AA 6022 | RT | — | — | — | $\overline{\sigma}=373{\left(\overline{\epsilon}\right)}^{0.13}{\left(\dot{\overline{\epsilon}}\right)}^{0.058}$ | 67.3 | 0.26 | 28.7 | 6 | 0.044 | [29] |

AA 6022 | 250 | — | — | — | $\overline{\sigma}=119{\left(\overline{\epsilon}\right)}^{0.084}{\left(\dot{\overline{\epsilon}}\right)}^{0.058}$ | 26.9 | 0.26 | 6.5 | 6 | 0.044 | [29] |

AISI-1045 | RT | — | — | — | — | n/a | n/a | n/a | n/a | n/a | [17] |

Q235 | RT | — | 210 | — | n/a | n/a | n/a | n/a | n/a | [18] | |

Steel | RT | 7850 | 117 | 0.3 | — | n/a | n/a | n/a | n/a | n/a | [19] |

AA 182 | RT | 2700 | 68.1 | 0.3 | $\overline{\sigma}=136+315{\left(\overline{\epsilon}\right)}^{0.45}$ | 55 | — | — | — | — | [30] |

AA 6061 | RT | — | — | — | — | — | — | — | — | — | [55] |

AA 5083 | RT | — | — | — | — | n/a | n/a | n/a | n/a | n/a | [21] |

AA 6061 | RT | 2700 | 70 | 0.34 | — | 70 | — | 390 | — | — | [27] |

Al-6.2Mg-0.7Mn | 200 | 2700 | — | — | — | n/a | n/a | n/a | n/a | n/a | [22] |

Al-6.2Mg-0.7Mn | 200 | 2700 | — | — | — | n/a | n/a | n/a | n/a | n/a | [23] |

AA 5083 | RT | — | — | — | — | n/a | n/a | n/a | n/a | n/a | [24] |

AA 6016-HR AA 1050-CR | (Hot) | — | — | — | $\overline{\mathsf{\sigma}}=692{\left(0.01+\overline{\mathsf{\epsilon}}\right)}^{0.275}$ | — | — | — | — | — | [34] |

Aluminum | RT | 2710 | 70 | 0.3 | — | n/a | n/a | n/a | n/a | n/a | [35] |

9Cr2Mo | RT | — | 204 | 0.285 | — | n/a | n/a | n/a | n/a | n/a | [38] |

AA 1050 | RT | — | 71 | — | $\overline{\mathsf{\sigma}}=179{\left(\overline{\mathsf{\epsilon}}\right)}^{0.22}$ | n/a | n/a | n/a | n/a | n/a | [39] |

AA 1050 | 350 400 | — | — | — | — | n/a | n/a | n/a | n/a | n/a | [40] |

AA 5454 | RT | 2690 | 70.5 | 0.33 | — | n/a | n/a | n/a | n/a | n/a | [41] |

AA 7055 | 410 | — | — | — | (^{2}) | — | — | — | — | — | [42] |

^{1}n/a — not applicable;

^{2}$\dot{\overline{\epsilon}}=6.1192\times {10}^{9}{\left[\mathrm{sinh}\left(0.0147\overline{\sigma}\right)\right]}^{5.2212}exp\left(\frac{-1.36382\times {10}^{5}}{{R}_{0}T}\right)$, where ${R}_{0}=8.314J{K}^{-1}mo{l}^{-1}$, and T is the absolute temperature in K.

Initial Orientation | Rotation | Orientation before the 2nd Pass |
---|---|---|

Case 1 | 0° (no rotation) | |

Case 2 | 180° about the Transversal Direction | |

Case 3 | 180° about the Rolling Direction | |

Case 4 | 180° about the Normal Direction | |

**Table 4.**FEM models, mesh parameters, and commercial software used in ASR simulations by different authors.

Rolls Elements | Workpiece Elements | CP Grains ^{1} | Model | Software | Ref. No. | ||
---|---|---|---|---|---|---|---|

Type | No. | Type | No. | — | — | ||

triangular | 90 | triangular | 1120 | n/a | Elastic-plastic FEM | DEFEL | [1] |

— | — | — | — | n/a | Coupled FEM rigid-plastic with a general diffusion equation for the thermal phenomena in the deformation zone | — | [2] |

— | — | — | — | n/a | Coupled thermo-elastic-plastic FEM | — | [5] |

— | — | isoparametric | 500 | n/a | — | DEFORM | [6] |

2-node linear rigid | 4-node bilinear with reduced integration and hourglass control | — | n/a | Elastic-plastic FEM | — | [8] | |

— | — | — | — | n/a | Elastic-plastic | — | [9] |

— | — | — | — | — | Elastic-plastic FEM coupled with Taylor and the Renouward–Wintenberger theories | — | [14] |

— | — | — | — | n/a | Elastic-plastic FEM | ABAQUS | [7] |

— | — | — | — | — | Elastic-plastic FEM | — | [13] |

— | — | 4-node quadrilateral | 800 | n/a | 2D elastic–plastic Arbitrary Lagrangian–Eulerian (ALE) | — | [11] |

— | — | — | — | n/a | Elastic-plastic FEM | ABAQUS | [12] |

— | — | — | 720 (1 × 144 × 5) | 19,440 (27 × 144 × 5) | Crystallographic homogenized FEM | — | [15] |

— | — | — | — | n/a | Rigid-viscoplastic FEM method | DEFORM | [16] |

— | — | — | — | 5000 | Elasto-plastic FEM coupled with LW model | ABAQUS | [25] |

— | — | 8-node isoparametric solid | 800 (160 × 5) | 100,000 (125 × 160 × 5) | Crystallographic homogenized elasto-viscoplastic FEM | [28] | |

— | — | — | 800 | 21,600 (27 × 800) | Crystallographic homogenized elasto-viscoplastic FEM and a discrete optimization method | [29] | |

— | — | — | — | n/a | — | DEFORM | [17] |

2-node linear discrete rigid | — | 4-node bilinear reduced integration and hourglass control | — | n/a | Elastic-plastic | ABAQUS | [18] |

8-node solid (SOLID164) | — | 8-node solid (SOLID164) | 121,378 | n/a | Elastic-plastic | LS-DYNA | [19] |

— | — | — | 9 elements in thickness | — | Elastic-plastic FEM coupled with VPSC | ABAQUS | [30] |

— | — | 8-node brick element with reduced integration (C3D8R) | 128 (16 × 1 × 8) | — | FEM coupled LW model | ABAQUS | [55] |

— | — | brick | 14,220 | n/a | Rigid-plastic | DEFORM 3D | [21] |

— | — | 8-node brick element with reduced integration (C3D8R) | 128 | 19,200 (150 × 128) | FEM coupled LW model | ABAQUS | [27] |

brick | 10,000 .. 12,000 | brick | 1000 .. 1200 | n/a | Rigid-plastic | DEFORM 2D | [22] |

brick | 10,000 .. 12,000 | brick | 1000 .. 1200 | n/a | Rigid-plastic | DEFORM 2D | [23] |

— | — | brick | 40,000 | n/a | Rigid-plastic | DEFORM 3D | [24] |

— | — | — | 1000 .. 2500 | — | FACET/ALAMEL | ABAQUS + PYTHON | [34] |

— | 11,368 | 4-node plane strain element | 28,000 | n/a | — | MSC.Marc | [38] |

— | — | — | — | n/a | Elastic-plastic | DEFORM | [39] |

— | — | — | — | n/a | Elastic-plastic | DEFORM 3D | [40] |

— | — | — | — | n/a | — | ABAQUS | [41] |

— | — | — | — | — | FEM coupled microstructure evolution model | — | [42] |

^{1}n/a—not applicable.

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**MDPI and ACS Style**

Graça, A.; Vincze, G.
A Short Review on the Finite Element Method for Asymmetric Rolling Processes. *Metals* **2021**, *11*, 762.
https://doi.org/10.3390/met11050762

**AMA Style**

Graça A, Vincze G.
A Short Review on the Finite Element Method for Asymmetric Rolling Processes. *Metals*. 2021; 11(5):762.
https://doi.org/10.3390/met11050762

**Chicago/Turabian Style**

Graça, Ana, and Gabriela Vincze.
2021. "A Short Review on the Finite Element Method for Asymmetric Rolling Processes" *Metals* 11, no. 5: 762.
https://doi.org/10.3390/met11050762