# ACTAS: A New Framework for Mechanical and Frictional Characterization in Axisymmetric Compression Test

## Abstract

**:**

## 1. Introduction

## 2. Problem Overview and the Current Common Solutions

#### 2.1. Cylindrical Profile Model

#### 2.2. Avitzur Model

#### 2.3. The Connection between Cylindrical Profile and Avitzur Models

## 3. Solving the Nonuniformity Problem

- dividing the sample into any desired number of zones,
- conditionally moving the zones’ boundaries,
- simulating the nonlinear material behavior, and
- simulating the discontinuities and singularities occurring after the onset of the foldover by handling the creation, distortion, and destruction of nodes.

## 4. Mathematical Implementation of ACTAS

#### 4.1. Total Power Dissipation and Deformation Load

#### 4.2. ACT Simulator

- (i)
- an admissible velocity field and the conditions for dynamically defining its internal boundaries (if exist),
- (ii)
- the geometrical domain of the initial sample in its pre-deformed shape, represented by field nodes,
- (iii)
- the ram speed U, which can be a constant or a function of other available variables,
- (iv)
- the initial distributions of the state variables of the defined constitutive law, such as stress and strain,
- (v)
- the constitutive law,
- (vi)
- the constant friction factor m,
- (vii)
- either the time step size $\mathsf{\Delta}t$ or the number of increments N, and
- (viii)
- the conditions for stopping the test, e.g., assigning a final height.

#### 4.3. ACT Analyzer

- (v)
- the load-displacement curve, and
- (vi)
- some geometry measurements of the sample at different time steps.

#### 4.3.1. Step 1: Estimating the Average Stress ($\tilde{\sigma}$)

#### 4.3.2. Step 2: Estimating the Friction Factor (${m}_{\mathrm{E}}$)

#### 4.3.3. Step 3: Updating the Average Stress ($\overline{\sigma}$)

## 5. Materials and Methods

^{−1}to its final height of ${H}_{\mathrm{f}}=10$ mm. Moreover, due to the axisymmetrical description of the selected velocity fields, the test sample is modeled as only a cross-section of the cylindrical sample bounded in $r=[0,{R}_{0}]$ and $z=[0,{H}_{0}/2]$.

## 6. Creating a Virtual Laboratory Using ACTAS

#### 6.1. Case 1: A One-Zone Sample

#### 6.1.1. Velocity Field

#### 6.1.2. ACT Simulator

#### 6.1.3. ACT Analyzer

#### 6.2. Case 2: Capturing the Foldover of the Side Surface

#### 6.2.1. Velocity Field

#### 6.2.2. ACT Simulator

#### 6.2.3. ACT Analyzer

## 7. Benchmarking ACTAS against FEM Virtual Experiments

#### 7.1. ACT Simulator

#### 7.2. ACT Analyzer

## 8. Results and Discussion

#### 8.1. Average Stress

#### 8.2. Stress Distribution throughout the Sample

#### 8.3. Strain Distributions

#### 8.4. Flow Behavior at the Center of the Sample

#### 8.5. Friction Factor

## 9. Summary and Conclusions

- To employ the ACTA module, closed-form solutions for the unknowns of the model must be available, e.g., the barreling parameters of the velocity fields. Due to the formulation of VFs, the derivation of the closed-form solutions may not be trivial or possible. Instead, one can use kinematics of the VF to estimate the solution of its unknowns; see, e.g., Equation (19).
- As the first step in building a virtual laboratory based on ACTAS, a one-zone VF (LAKT) [33,34] is used. The setup is selected such that the percentage volume change error becomes a negligible value of $\delta \left(V\right)\approx 0.02\%$. In this setup, ACTA correctly estimates all samples’ constitutive behavior and friction factors for $m\lesssim 0.6$, which is found to be the upper limit of LAKT to meaningfully model ACT.
- Due to the incapability of LAKT in modeling the foldover phenomenon, an extension of it is proposed; see Section 6.2 for the details. By implementing the newly proposed two-zone velocity field, the ACTA module obtains accurate results for the full range of $0\le m\le 1$.
- ACTAS is benchmarked against ten reference solutions obtained from FEM-based experiments (Table 1). The ACTS module shows low percentage errors for the deformation load ($\delta \left(L\right)\lesssim 6\%$) and geometrical measurements (Figure 13). The ACTA module accurately estimated the average stress-strain curves for all samples.
- Investigating the pointwise stress-strain curves at the center of the samples between ACTAS and FE models, ACTAS provides improved results compared to those of the conventional methods. Moreover, ACTAS results in no false identification of peak stress, misleading to interpretations about the onset of dynamic recrystallization.
- This paper also addresses the shortcoming of employing the CPM without a priori knowledge of friction. As a solution, the Avitzur model is coupled with the CPM (called A-CPM) and is represented as:$$\frac{{p}_{\mathrm{ave}}}{\overline{\sigma}}\approx 1+{\left(\right)}^{\frac{\overline{R}}{H}}2,$$
- Comparing the solutions of A-CPM and FE models, it is found that A-CPM can be used to accurately identify the friction-free average stress-strain curves regardless of the severity of friction.
- Because the solutions of A-CPM are almost identical to those of the Avitzur model (Figure 2), it infers the high accuracy of the Avitzur model in estimating the average stress-strain curves.
- The unreliability of the Avitzur model for estimating friction factor is confirmed once more, aligned with previous investigations.
- For microstructural analyses, the study is usually focused on the center of the sample, where the flow curve may differ from the average one. For such studies, the analyses of A-CPM should be considered with extra care, as it underestimates both the stress and the strain values. It may even provide misleading results regarding the onset of dynamic recrystallization.

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ACT | Axisymmetric Compression Test |

ACTAS | ACT Analyzer and Simulator (the framework proposed in this work) |

ACTS | Simulator module of ACTAS |

ACTA | Analyzer module of ACTAS |

CPM | Cylindrical Profile Model |

A-CPM | The coupled version of the Avitzor and the Cylindrical Profile models |

VF | Velocity Field |

LAKT | Lee and Altan version of the VF proposed by Kobayashi and Thomsen [33,34] |

LHS | Latin Hypercube Sampling |

## Appendix A. Effects of b_{A} on A-CPM

**Figure A1.**The assigned stress-strain curve for sample #6 and those obtained from A-CPM using the estimates of ${b}_{\mathrm{A}}$ that are developed with a static method for the (

**a**) forward and (

**b**) backward Euler kinematics. (The formulation of ${b}_{\mathrm{EN}}$ and ${b}_{\mathrm{S}}$ is the same for the forward Euler kinematics in the static method).

**Figure A2.**The assigned and A-CPM estimates of the flow stress curve of sample #6. Various estimtes of ${b}_{\mathrm{A}}$ used in A-CPM are developed with a dynamic method for the (

**a**) forward and (

**b**) backward Euler kinematics.

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**Figure 1.**The schematic of the sample geometry in axisymmetric compression test (left) before and (right) after the deformation. ${R}_{0}$ and ${H}_{0}$ are the sample’s initial radius and height, respectively. By deforming the sample with a constant velocity of $-U/2$ parallel to the z axis, three characteristic radii can be identified: the mid-plane (${R}_{\mathrm{M}}$), top-plane (${R}_{\mathrm{T}}$) and slip (${R}_{\mathrm{S}}$) radii with foldover being $F={R}_{\mathrm{T}}-{R}_{\mathrm{S}}$. (${R}_{\mathrm{S}}$ is the result of the expansion of ${R}_{0}$, and F is the contribution of the side surface foldover, resulting in a larger top-plane radius ${R}_{\mathrm{T}}$). In a flow-based model, any point $p(r,z)$ moves based on the radial and axial components of the velocity field (${U}_{r},{U}_{z}$).

**Figure 3.**The percentage volume change error as a function of numbers of simulation steps and grids. With an arbitrarily small threshold of $\delta {\left(V\right)}_{\mathrm{max}}=0.02\%$, the features for the forthcoming tests of the current study are selected.

**Figure 4.**Discretized sample of Case 1 with a grid of 50 at the (

**a**) initial and (

**b**) final stage of the ACT.

**Figure 5.**The results of Case 1. (

**a**) The variation of deformation load (left y-axis), mid-plane, and top-plane radii (right y-axis) as functions of displacement d. (

**b**) The values of ${b}_{\mathrm{L}}$ obtained from the minimization process (${b}_{\mathrm{L}\left(\mathrm{min}\right)}$) and the kinematics estimation (${b}_{\mathrm{L}\left(\mathrm{K}\right)}$) from Equation (20).

**Figure 6.**The correlation between the assigned (${m}_{\mathrm{A}}$) and estimated (${m}_{\mathrm{E}}$) values of the friction factor in Case 1. The continuous black line represents a perfect correlation.

**Figure 7.**The correlation between the assigned and estimated values of the material parameters (

**a**) k and (

**b**) n in Case 1.

**Figure 8.**The schematic of the newly proposed model that divides the sample into two zones, with zone 1 for $0\le r\le {R}_{\mathrm{T}}$ and zone 2 for ${R}_{\mathrm{T}}<r\le R\left(z\right)$. The border of zones 1 and 2 is at $r={R}_{\mathrm{T}}$.

**Figure 10.**The results of Case 2. (

**a**) The deformation load (left y-axis) and the sample’s geometry (right y-axis) as functions of displacement d. (

**b**) The values of ${b}_{\mathsf{\lambda}}$ obtained from the minimization process (${b}_{\mathsf{\lambda}}\left(\mathrm{min}\right)$) and the kinematics estimation (${b}_{\mathsf{\lambda}}\left(\mathrm{K}\right)$) from Equation (28) for two values of ${R}_{\square}$: ${R}_{\mathrm{S}}$ and ${R}_{\mathrm{T}}$. The zoomed-in inset shows that the ${b}_{\mathsf{\lambda}}\left(\mathrm{K}\right)\left({R}_{\mathrm{T}}\right)$ underestimates the barreling parameter from the onset of the foldover phenomenon.

**Figure 12.**The correlation between the assigned and estimated values of the material parameters (

**a**) k and (

**b**) n for Case 2.

**Figure 13.**The percent error of (

**a**) deformation load, and profile’s (

**b**) ${R}_{\mathrm{M}}$, (

**c**) ${R}_{\mathrm{T}}$, and (

**d**) ${R}_{\mathrm{S}}$ obtained from comparing ACTS and FE models. The symbols are to be read from the legend in the subplot (

**d**) that refers to the corresponding sample number in Table 1.

**Figure 14.**The correlation between the assigned (${m}_{\mathrm{FEM}}$) and estimated (${m}_{\mathrm{ACTA}}$) friction factors. The symbols point to different samples to be read using the legend of Figure 13d.

**Figure 15.**The correlation between the assigned and estimated values of the material parameters (

**a**) k and (

**b**) n for the samples investigated in the benchmark.

**Figure 16.**The stress analysis of the ten reference samples (see Table 1). (

**a**) average flow curves obtained from A-CPM (Equation (4)) and ACTA (according to the algorithm developed for Case 2) in comparison with the assigned Hollomon models. (

**b**) the correlation between the estimated average stresses (A-CPM and ACTA) and the assigned values. (Sample #5 is randomly selected for further detailed discussions in this section. For that, its flow curve is identified in the subset (

**a**)).

**Figure 17.**Comparison of the von Mises stress distributions in a randomly selected sample (#5) for (

**a**) FEM and the proposed (

**b**) ACTS and (

**c**) ACTA models. (The presented solutions for ACTA are essentially the solutions of ACTS using the parameters identified from the ACTA analyses of the FE virtual experiments). The values are in the units of MPa.

**Figure 18.**Comparison of the effective strain distributions in a randomly selected sample (#5) for (

**a**) FEM, (

**b**) ACTS, and (

**c**) ACTA models.

**Figure 19.**The stress-strain curves at the center of the samples (

**a**) 1, (

**b**) 2, (

**c**) 3, (

**d**) 4, (

**e**) 5, (

**f**) 6, (

**g**) 7, (

**h**) 8, (

**i**) 9, and (

**j**) 10. The data are obtained from different models of FEM, ACTS, and ACTA. The results of A-CPM, which are uniform throughout the sample, are added for comparison. Note that the ranges for both stress and strain vary for different samples, and $max\left(\right)open="("\; close=")">{\overline{\epsilon}}_{\mathrm{A}-\mathrm{CPM}}$ for all samples.

Sample | m | k (MPa) | n |
---|---|---|---|

1 | 0.05 | 388 | 0.05 |

2 | 0.15 | 433 | 0.75 |

3 | 0.25 | 343 | 0.95 |

4 | 0.35 | 163 | 0.65 |

5 | 0.45 | 298 | 0.55 |

6 | 0.55 | 253 | 0.15 |

7 | 0.65 | 73 | 0.25 |

8 | 0.75 | 478 | 0.35 |

9 | 0.85 | 208 | 0.45 |

10 | 0.95 | 118 | 0.85 |

**Table 2.**The von Mises stress at the center of the sample in FEM, ACTS, ACTA, and A-CPM models, in the units of MPa. The numbers in the parentheses are the calculated percentage errors.

Sample | FEM | ACTS | ACTA | A-CPM | |||
---|---|---|---|---|---|---|---|

1 | 377 | 374 | (0.8) | 370 | (1.9) | 374 | (0.8) |

2 | 265 | 260 | (1.9) | 250 | (5.7) | 246 | (7.2) |

3 | 188 | 184 | (2.1) | 178 | (5.3) | 167 | (11.2) |

4 | 116 | 110 | (5.2) | 108 | (6.9) | 100 | (13.8) |

5 | 232 | 220 | (5.2) | 218 | (6.0) | 197 | (15.1) |

6 | 248 | 237 | (4.4) | 239 | (3.6) | 226 | (8.9) |

7 | 69 | 66 | (4.9) | 66 | (4.9) | 60 | (13.5) |

8 | 438 | 411 | (6.2) | 419 | (4.3) | 367 | (16.2) |

9 | 183 | 172 | (6.0) | 177 | (3.3) | 148 | (19.1) |

10 | 84 | 78 | (7.4) | 83 | (1.4) | 62 | (26.4) |

**Table 3.**The effective strain at the center of the FEM, ACTS, and ACTA models; the effective strain for the A-CPM model is uniform throughout each sample and the same for all samples, that is ${\overline{\epsilon}}_{\mathrm{A}-\mathrm{CPM}}=0.47$. The numbers in the parentheses are the calculated percentage errors.

Sample | FEM | ACTS | ACTA | A-CPM | ||
---|---|---|---|---|---|---|

1 | 0.57 | 0.49 | (14.0) | 0.50 | (12.3) | (17.4) |

2 | 0.52 | 0.51 | (2.6) | 0.50 | (3.6) | (9.5) |

3 | 0.53 | 0.52 | (2.4) | 0.52 | (3.0) | (11.7) |

4 | 0.59 | 0.55 | (7.2) | 0.54 | (8.1) | (20.3) |

5 | 0.64 | 0.58 | (9.7) | 0.57 | (11.3) | (26.4) |

6 | 0.82 | 0.65 | (20.5) | 0.64 | (21.5) | (42.7) |

7 | 0.82 | 0.65 | (19.8) | 0.64 | (22.1) | (42.5) |

8 | 0.78 | 0.66 | (15.8) | 0.64 | (17.9) | (39.7) |

9 | 0.75 | 0.65 | (12.9) | 0.65 | (13.2) | (37.5) |

10 | 0.67 | 0.61 | (8.5) | 0.62 | (7.7) | (30.1) |

Sample | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

${m}_{\mathrm{FEM}}$ | 0.05 | 0.15 | 0.25 | 0.35 | 0.45 | 0.55 | 0.65 | 0.75 | 0.85 | 0.95 |

${m}_{\mathrm{ACTA}}$ | 0.07 | 0.13 | 0.22 | 0.31 | 0.41 | 0.54 | 0.58 | 0.68 | 0.85 | 1.00 |

${m}_{\mathrm{Avitzur}}$ | 0.04 | 0.04 | 0.05 | 0.08 | 0.11 | 0.21 | 0.19 | 0.19 | 0.19 | 0.17 |

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**MDPI and ACS Style**

Solhjoo, S.
ACTAS: A New Framework for Mechanical and Frictional Characterization in Axisymmetric Compression Test. *Materials* **2023**, *16*, 441.
https://doi.org/10.3390/ma16010441

**AMA Style**

Solhjoo S.
ACTAS: A New Framework for Mechanical and Frictional Characterization in Axisymmetric Compression Test. *Materials*. 2023; 16(1):441.
https://doi.org/10.3390/ma16010441

**Chicago/Turabian Style**

Solhjoo, Soheil.
2023. "ACTAS: A New Framework for Mechanical and Frictional Characterization in Axisymmetric Compression Test" *Materials* 16, no. 1: 441.
https://doi.org/10.3390/ma16010441