# A Finite Element Study of Thermo-Mechanical Fields and Their Relation to Friction Conditions in Al1050 Ring Compression Tests

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## Abstract

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## 1. Introduction

- Flow stress curves for Al1050 at different temperatures and strain rates are reported.
- Friction coefficients for lubricated and unlubricated contact between the specimen and tools at different temperatures and strain rates are determined.
- The heat loss during system RCT preparation is modeled and quantified demonstrating a non-negligible effect on friction and flow stress determination.
- A novel method for validating computed friction values by comparing the neutral radius from the FE model to metallurgical characterization is presented.

## 2. Experimental Setup for Al1050 Upsetting Tests

_{0}= 30 mm and its height was h

_{0}= 40 mm for all experiments. Since the FEM was later applied in order to characterize the materials mechanical behavior (see Section 3), two different test velocities were sufficient for identifying the strain rate sensitivity of the material as described in [11].

_{0}= 48 mm and D

_{i}= 24 mm, and their height was h

_{0}= 16 mm, so the relation between D

_{0}:D

_{i}:h

_{0}matches the 6:3:2 relation accepted as a standard for RCTs (see [7]). The experimental parameters for the RCTs included two different lubrication conditions (a liquid graphite-based lubricant, T-50, and no lubrication) in addition to different temperatures and ram velocities. Lubrication was applied to both the top and bottom faces of the pressing plates and the specimen. For the cylinders, all upsetting experiments were performed without application of a lubricant. The RCT experiment parameters are detailed in Table 2.

## 3. Computational Modeling of the Upsetting Tests

#### 3.1 Model Definitions

_{2}(Von Mises) based yield surface with isotropic strain hardening was used for the specimen while the plates were assumed to deform only in the elastic range due to their relative high yield stress even at the high testing temperatures, and the ram and matrix were defined as rigid. The rigid plastic constitutive law used for the specimen was assumed to be both temperature and strain rate dependent and encoded in tabular form into the FE program. A coupled transient analysis was conducted with a two-way coupling between the thermal and mechanical fields. Following initial computations and examination of the plastic strain rate (not shown herein), it was concluded that the heat generated by plastic deformation (using η = 0.9 in the examination) has negligible influence on the computed results and was therefore not considered in further analysis.

_{c}= 10

^{5}(W)/(m

^{2}·K)) and thermal convection coefficient (h = 10 (W)/(m

^{2}·K)), which is the limiting value for natural convection).

## 4. Determination of Flow-Stress Manifolds and Friction Conditions

#### 4.1. Iterative Process for Determining Flow Stress Manifolds

_{f}(F, h) = (4·F·(h

_{0}− ∆h))/(π·D

_{0}

^{2}·h

_{0}); ε

_{f}= ln(1−∆h/h

_{0})

_{0}is the initial diameter of the billet, h

_{0}is its initial height; F and ∆h are the experimentally measured load and ram displacements, respectively. The computed load-displacement curve and deformed shape (measurements of the diameters at the top, center, and bottom of the deformed specimens obtained for the experimentally obtained height) were compared to the experimental results. This iterative process was terminated once a relative error of less than 4% between all experimental and computed values was obtained. When comparing the specimen dimensions (outer and, for the ring, also inner diameters), the target function was simply computed by:

^{n}(known as Hollomon’s equation), was corrected (by changing either C, n, or both) and the set of computations was performed again. The iterative process is schematically presented in Figure 5.

#### 4.2. Flow Stress Manifolds for Al1050

^{n}) calibrated for Al1050 under different temperature and strain rate conditions, using the iterative process described in Section 4.1. The average experimental strain rate was calculated from the ram velocity by using (see [13]):

_{z}= 350 mm/min and an average strain rate of 〈$\stackrel{\u0307}{{\epsilon}_{zz}}$〉 = 0.38 for a ram velocity of v

_{z}= 1200 mm/min was obtained. The flow stress curves obtained from the iterative analysis, for the different strain rates examined, were not significantly different for all temperatures considered. It should be noted that, for different strain rates, the flow stress was sometimes expressed with the following power law, σ = C·${\epsilon}^{\stackrel{\u0307}{s}}$, where s is the strain rate sensitivity, or by the Sinh-Arrhenius type relation (the strain rate sensitivity is commonly indicated by m, however, this notation was modified to avoid confusion with the friction factor, also marked by m). Experiments on metals at different strain rates (at a constant temperature) were conducted in [14]. It was shown that s increases with temperature, moderately at low temperatures and then more rapidly above about half of the melting temperature (T > T

_{M}/2). However, this behavior depends not only on temperature, but also on the order of magnitude of the strain rates. The experimental results conducted in this study demonstrate that the strain rates within the examined range are not high enough for the flow stress (or s) to change significantly, even at high temperatures. As a consequence, the stress-strain relations at all temperatures were approximated by Hollomon’s equation. Table 3 provides average values (in terms of strain rate) of Hollomon’s coefficients for each temperature.

#### 4.3. Determining the Friction Conditions

_{y}(k = 0.577σ

_{y}according to the Huber-Von Mises yield criterion or 0.5σ

_{y}according to Tresca), where σ

_{y}is the material yield stress. μ is therefore theoretically limited to 0.577 in metal forming.

#### 4.4. Validation

## 5. Relation Between Friction Induced Texture and Thermo-Mechanical Fields

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Verification of the Computational Models

**Figure A1.**Verification of cylinder upsetting simulations: convergence in equivalent plastic strains (

**a**), von-mises stress (

**b**), and temperature (

**c**). Convergence in strain energy values (

**d**). Examples of the specimen mesh used in the convergence study (

**e**).

**Figure A2.**Verification of ring upsetting simulations: convergence in equivalent plastic strains (

**a**), von-mises stress (

**b**), and temperature (

**c**). Convergence in strain energy values (

**d**). Examples of the specimen mesh used in the convergence study (

**e**).

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**Figure 1.**Experimental setup of: (

**a**) a ring upsetting test, (

**b**) a cylindrical specimen upsetting test. The upper pressing plate is not shown in this setup.

**Figure 2.**An finite element (FE) geometric model of the upsetting process of the (

**a**) cylindrical and (

**b**) ring upsetting specimens with T.C. indicating the thermo-couple locations.

**Figure 3.**An example of the computational stages for simulating the system preparation of specimen R13.

**Figure 4.**Thermo-mechanical fields (equivalent stress, equivalent strain, and temperature) for the upsetting conditions of specimen A7 as obtained from: (

**a**) a thermo-mechanical model, including calculation of the pre-upsetting system preparation; (

**b**) a thermo-mechanical model without preparation (constant initial temperatures).

**Figure 5.**Iterative process for determining the flow stress and friction conditions at different temperatures and strain rates: (

**A**) initial guess of the flow stress manifold based on analytical analysis, (

**B**) FE analysis simulations of all experiments, (

**C**) comparison between computed and experimental results, (

**D**) modification of the flow stress data and friction coefficients, (

**E**) input of new flow stress manifolds and friction conditions into the analysis.

**Figure 6.**Experimental vs. numerical force-displacement curve for specimen A1, flow stress determined from cylinder upsetting alone (

**a**). Experimental vs. numerical force-displacement curve for specimen R1, which has the same experimental conditions (

**b**).

**Figure 9.**Dependence of the minimum (middle height) inner diameter on μ for specimens with a ram velocity of 350 mm/min at various temperatures.

**Figure 10.**Comparison between experimental and computed force-displacement curves for the Al1050 upsetting process, at different temperatures for 〈$\stackrel{\u0307}{{\epsilon}_{zz}}$〉 = 0.11 (

**a**) and 〈$\stackrel{\u0307}{{\epsilon}_{zz}}$〉 = 0.38 (

**b**).

**Figure 11.**Comparison between computed and measured inner ring diameter for experiments R1-R16 (

**a**) and an example of a computed and measured force-displacement curve for specimen R15 (

**b**).

**Figure 12.**Comparison between predicted and experimental ring neutral radius and flow patterns for specimens R2 (

**top**) and R8 (

**bottom**), which represent the effect of temperature for unlubricated rings loaded at V = 1200 mm/min.

**Figure 13.**Comparison between predicted and experimental ring neutral radius and flow patterns for specimens R5 (

**top**) and R13 (

**bottom**), which represent the effect of lubrication at 450 °C, V = 350 mm/min.

**Figure 14.**Comparison between predicted and experimental ring neutral radius and flow patterns for specimens R3 (

**top**) and R4 (

**bottom**), which represent the effect of strain rate, on unlubricated rings at 350 °C.

**Figure 15.**Metallographic specimen of the pre-deformed ring (

**a**), and of the deformed unlubricated specimen R5 (

**top**) and lubricated specimen R13 (

**bottom**) (

**b**) both tested at 450 °C.

**Figure 16.**Segmentation into parts with different texture: Pre-deformed microstructure of (

**a**) specimen R5 (

**b**) and R13 (

**c**). Metallography (

**top**) is presented against numerically obtained flow lines (

**bottom**). Locations of nodes from which data was extracted are marked in red. Metallography includes also the notation of the zones.

**Figure 17.**Average values of thermo-mechanical fields within specimen R5 at each zone: Out-of-plane (circumferential) principal strain (

**a**), maximum in-plane (

**b**), and minimum in-plane (

**c**) principal strains, and the shear stress (

**d**) evolution with time.

**Figure 18.**Average values of thermo-mechanical fields within specimen R13 at each zone: out-of-plane (circumferential) principal strain (

**a**), maximum in-plane (

**b**), and minimum in-plane (

**c**) principal strains, and the shear stress (

**d**) evolution with time.

**Table 1.**Upsetting experiment parameters for cylindrical specimens: Initial temperatures and ram velocity.

Experiment Notation | A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 |
---|---|---|---|---|---|---|---|---|

Specimen temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Pressing plates temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Ram and matrix temp [°C] | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 |

Ram velocity [mm/min] | 350 | 1200 | 350 | 1200 | 350 | 1200 | 350 | 1200 |

**Table 2.**Ring compression tests (RCTs) parameters: Initial temperatures, ram velocity, and lubrication conditions.

Experiment Notation | R1 | R2 | R3 | R4 | R5 | R6 | R7 | R8 |

Specimen temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Pressing plates temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Ram and matrix temp [°C] | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 |

Ram velocity [mm/min] | 350 | 1200 | 350 | 1200 | 350 | 1200 | 350 | 1200 |

Experiment Notation | R9 | R10 | R11 | R12 | R13 | R14 | R15 | R16 |

Specimen temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Pressing plates temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Ram and matrix temp [°C] | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 |

Ram velocity [mm/min] | 350 | 1200 | 350 | 1200 | 350 | 1200 | 350 | 1200 |

**Table 3.**Flow-stress relations for Al1050 at different temperatures as determined from the iterative process.

Specimen Temp [°C] | 25 | 350 | 450 | 540 |

Flow stress [MPa] | $\sigma =123\xb7{\epsilon}^{0.3}$ | $\sigma =46\xb7{\epsilon}^{0.315}$ | $\sigma =26\xb7{\epsilon}^{0.215}$ | $\sigma =12.5\xb7{\epsilon}^{0.135}$ |

Experiment Notation | R1 | R2 | R3 | R4 | R5 | R6 | R7 | R8 |

Specimen temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Ram velocity [mm/min] | 350 | 1200 | 350 | 1200 | 350 | 1200 | 350 | 1200 |

Lubriction | none | none | none | none | none | none | none | none |

$\mu $ | 0.4 | 0.4 | >0.45 | >0.45 | >0.45 | >0.45 | >0.45 | >0.45 |

Experiment Notation | R9 | R10 | R11 | R12 | R13 | R14 | R15 | R16 |

Specimen temp [°C] | 25 | 25 | 350 | 350 | 450 | 450 | 540 | 540 |

Ram velocity [mm/min] | 350 | 1200 | 350 | 1200 | 350 | 1200 | 350 | 1200 |

Lubriction | T-50 | T-50 | T-50 | T-50 | T-50 | T-50 | T-50 | T-50 |

$\mu $ | 0.15 | 0.15 | 0.17 | 0.16 | 0.17 | 0.16 | 0.28 | >0.45 |

**Table 5.**A comparison between the predicted and observed neutral radius location for different temperatures, loading rates, and friction conditions.

Experiment | Computed [mm] | Measured [mm] | Relative Error [%] |
---|---|---|---|

R2 | 15.52 | 15.05 | 3.12 |

R8 | 17.10 | 16.92 | 1.06 |

R5 | 16.24 | 16.53 | 1.75 |

R13 | 15.68 | 15.10 | 3.84 |

R3 | 16.60 | 16.49 | 0.67 |

R4 | 17.14 | 16.78 | 2.14 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Mittelman, B.; Priel, E.; Navi, N.U. A Finite Element Study of Thermo-Mechanical Fields and Their Relation to Friction Conditions in Al1050 Ring Compression Tests. *J. Manuf. Mater. Process.* **2018**, *2*, 83.
https://doi.org/10.3390/jmmp2040083

**AMA Style**

Mittelman B, Priel E, Navi NU. A Finite Element Study of Thermo-Mechanical Fields and Their Relation to Friction Conditions in Al1050 Ring Compression Tests. *Journal of Manufacturing and Materials Processing*. 2018; 2(4):83.
https://doi.org/10.3390/jmmp2040083

**Chicago/Turabian Style**

Mittelman, Brigit, Elad Priel, and Nissim U. Navi. 2018. "A Finite Element Study of Thermo-Mechanical Fields and Their Relation to Friction Conditions in Al1050 Ring Compression Tests" *Journal of Manufacturing and Materials Processing* 2, no. 4: 83.
https://doi.org/10.3390/jmmp2040083