# Assessment of Deformation Flow in 1050 Aluminum Alloy by the Implementation of Constitutive Model Parameters

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## Abstract

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## Featured Application

**The manuscript describes the determination of the constitutive model parameters by a tensile test, while the validation of the material constants is performed by a simulation and experimental verification of the strain distribution during the symmetric and asymmetric rolling processes. A further validation was performed by modeling a depth-sensing indentation process.**

## Abstract

## 1. Introduction

_{pl}is the plastic strain, $\dot{\epsilon}$ is a strain rate, T is the temperature, σ

_{00}is the yield stress, B is the hardening variable, n is the hardening exponent, ${\dot{\epsilon}}_{ref}$ is the reference strain rate, C is the strain-rate sensitivity factor, T

_{ref}is the reference temperature, T

_{m}is the melting point of the material, and m* is the thermal softening factor.

_{t}is the total strain, ε

_{el}is the elastic strain, ε

_{pl}is the plastic strain, σ is the stress, E is the linear elastic modulus, and K is the hardening coefficient in the Hollomon equation [10] (second term of the equation), which takes into account the effect of the strain rate $\dot{\epsilon}$ on σ:

_{n}is the strain at the necking point.

_{B}is the maximum value of the engineering stress (it should be noted here that the engineering stress does not account for the reduction of the cross-section during tension, while this phenomenon is taken into consideration by the true stress).

_{00}is the yield stress, σ

_{sat}is the saturation stress, h

_{p}is the hardening parameter, and a is a function of the strain-hardening exponent. The model parameters σ

_{sat}, σ

_{00}, h, and a are fitted according to the procedure described in [15].

_{1}–D

_{5}are damage-related-type model parameters, σ

_{m}is the mean stress, ${\dot{\epsilon}}_{p}^{\ast}$ is the plastic strain rate, T

_{ref}is the reference temperature, and T

_{m}is the melting point of the material.

## 2. Materials, Experiments, and Numerical Methods

#### 2.1. Material and Experimental Procedures

_{0}) was 50 mm while the initial cross-section (A

_{0}) was 18.44 mm

^{2}. The load cell and the longitudinal extensometer provided information on the change of force and elongation during the test.

#### 2.2. Finite Element Modeling of Tensile Test

^{©}finite element software was employed for modeling the rotationally symmetric components. The simulation was performed for a simplified 2D geometry of the DIN 50125-A 10 × 50 tensile specimen. The gripping of the lower surface of the tensile specimen was replaced by the prescribed y-displacement defined for the lower edge. The calculation of stress and strain values and data post-processing were performed according to procedures described elsewhere [18,19].

#### 2.3. Finite Element Modeling of Hardness Measurement

_{m}can be estimated as [20]:

^{©}software, which is suitable for modeling the elastoplastic processes [26,27]. The study was performed on the fine-meshed workpiece geometry, shown in Figure 2b. To simplify the model, 2D rotationally symmetric geometries were assumed while both the indenter (sphere) and the lower support disc were assumed to be infinitely rigid bodies. The load function was divided into 200 equal steps. The initial mesh was generated by considering the geometry of the indent (sphere with a diameter of 3 mm) and updated every 20 steps to account for accurate strain and curvature caused by the indentation. The friction coefficient between the indenter and the workpiece was set to 0.12 [27].

#### 2.4. Finite Element Modeling of Symmetric and Asymmetric Rolling

_{i}is the initial thickness of the sheet, h

_{f}is the final thickness of the sheet, ε is the thickness reduction, z represents the normal direction of the rolled sheet, x corresponds to the rolling direction, γ is the derivative of displacement of the sheet in the rolling direction, ε

_{s}is the shear strain, and ε

_{vM}is the equivalent strain.

_{1}is the coefficient of proportionality between the theoretical minimum value of μ necessary for rolling [35] and the actual (real) one (k

_{1}~1.1–1.5 [40,41]).

## 3. Results and Discussion

#### 3.1. Tensile Test

_{sat}= 650.8 MPa, σ

_{00}= yield stress = 15.3 MPa, h

_{p}= 1255 MPa, and a = 15.5 (correlation coefficient = 0.9984). The hardening rate derived from this approximation predicts the occurrence of necking at a strain of ~0.34 (see Figure 5). The predicted strain value for the neck initiation (~0.34) is slightly underestimated compared to the experimentally observed one (~0.37); however, this model can accurately predict the behavior of a material under different strain modes at the advanced stages of deformation [15], which cannot be captured by tension.

#### 3.2. Defining the Model Parameters from Hardness Simulations

_{m}caused by the indentation-over-d/D ratio (indentation size per indenter size). The FEM simulation reveals a quasi-parabolic profile, which can be described by the following relation [20,23]:

#### 3.3. Rolling Processes: Simulation and Experiment

_{min}.

_{min}(for symmetric rolling, the best fit is observed for μ ≈ 1.1μ

_{min}, while for the asymmetric trial, μ ≈ 1.13μ

_{min}). This result is consistent with the recently reported simulations [36].

_{t}is the angular velocity of the upper roll, and ω

_{b}is the angular velocity of the lower roll.

_{t}/ω

_{b}= 1.474 and h

_{i}/h

_{f}= 1.476, and therefore the homogeneous strain distribution can be ensured, as shown in Figure 13. Comparable values are predicted by the model of Pustovoytov et al. [46] (ω

_{t}/ω

_{b}= 1.52 and h

_{i}/h

_{f}= 1.52). The discrepancy observed does not affect the result significantly in the examined case.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Geometric set-up used for the simulation of Brinell hardness; and (

**b**) fine mesh elements in the vicinity of an indent.

**Figure 7.**Effect of elastic and plastic terms in Equation (2) on the accuracy of FEM simulation: the continuous line takes into account both elastic and plastic parts, while the dotted line represents the simulation by considering only the plastic term.

**Figure 8.**FEM-computed and analytically fitted curves (Equations (20) and (21)) of surface pressure over indentation per indenter size ratio d/D.

**Figure 11.**Experimental and simulated displacement patterns across the thickness of asymmetrically rolled sheet.

**Figure 12.**Experimental and simulated displacement patterns across the half-thickness of symmetrically rolled sheet.

**Figure 15.**Strain rates in asymmetrically rolled sheet, calculated by FEM for different thickness layers: top and bot. surf.—are top and bottom surface layers; subsurf. layer is equidistant from both surface and middle; and mid. is the midthickness layer.

**Figure 16.**Strain rates in symmetrically rolled sheet, calculated by FEM for different thickness layers: surf.—is a surface layer; subsurf. layer is equidistant from both surface and middle; and mid. is the midthickness layer.

Description | Parameter | Value | |
---|---|---|---|

Asym. | Sym. | ||

Roll radii | R (mm) | 75 | 75 |

Initial thickness of the sheet | h_{i} (mm) | 1.92 | 5.02 |

Final thickness of the sheet | h_{f} (mm) | 1.3 | 3.61 |

Initial sheet length | L (mm) | 20 | 20 |

Material | - | Al-1050 | Al-1050 |

Minimal friction coefficient necessary for rolling | μ_{min} | 0.0672 | 0.118 |

Angular velocity of the top roll | ω_{t} (rad/s) | 1.1023 | 1.1023 |

Angular velocity of the bottom roll | ω_{b} (rad/s) | 0.748 | 1.1023 |

Velocity rate | ω_{t}/ω_{b} | 1.474 | 1.000 |

Model Parameter | Ramberg–Osgood Model | Hollomon Model |
---|---|---|

E (MPa) | 69,854 | 69,519 |

K (MPa) | 144.56 | 144.58 |

n | 0.3691 | 0.3703 |

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**MDPI and ACS Style**

Bátorfi, J.G.; Pál, G.; Chakravarty, P.; Sidor, J.J. Assessment of Deformation Flow in 1050 Aluminum Alloy by the Implementation of Constitutive Model Parameters. *Appl. Sci.* **2023**, *13*, 4359.
https://doi.org/10.3390/app13074359

**AMA Style**

Bátorfi JG, Pál G, Chakravarty P, Sidor JJ. Assessment of Deformation Flow in 1050 Aluminum Alloy by the Implementation of Constitutive Model Parameters. *Applied Sciences*. 2023; 13(7):4359.
https://doi.org/10.3390/app13074359

**Chicago/Turabian Style**

Bátorfi, János György, Gyula Pál, Purnima Chakravarty, and Jurij J. Sidor. 2023. "Assessment of Deformation Flow in 1050 Aluminum Alloy by the Implementation of Constitutive Model Parameters" *Applied Sciences* 13, no. 7: 4359.
https://doi.org/10.3390/app13074359