# Characterization of the Bulk Flow Properties of Industrial Powders from Shear Tests

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}is the one occurring during consolidation. It is usually considered that the Mohr circle representing the state of stress during the material consolidation in the critical state shear closes the yield locus on the consolidation side. Therefore, σ

_{1}is estimated from the largest intercept on the σ axis of the Mohr circle tangent to the yield locus and passing through the consolidation point. The unconfined yield strength, f

_{c}, is the material strength under unconfined uniaxial compression and, therefore, corresponds to the finite intercept on the σ axis of the Mohr circles, which is tangent to the yield locus line and passes through the origin of σ–τ plane. The unconfined yield strength represented as a function of the major principal stress is the so-called “flow function”.

_{t}. It represents the resistance stress necessary to separate two layers of materials by means of an isostatic tensile strain. Despite its direct experimental measurement, which is not standardized, the tensile strength is represented by the intersection between the negative side of the σ-axis and the yield locus. In powders, it is the evident macroscopic manifestation of the attractive forces between the constituent particles. Like cohesion and unconfined yield strength, a finite value of the tensile strength is possible only if attractive interparticle interactions are present. Cohesion, unconfined yield strength and tensile strength are macroscopic evidence of attractive of interparticle forces, such as van der Waals, electrostatic and capillary forces, which depend on the state of powder consolidation that, in turn, is a function on the packing state of the powder and the stress history [10].

_{i}σ + C = (C/σ

_{t}) σ + C

_{t}, are the line intercepts on the τ and the σ axis, respectively. The angle of internal friction, ϕ

_{i}, is the slope angle of the yield line.

_{t})

^{1/n}

## 2. Methodology

_{X}, of specific degree, N, that best fits the input data in a least-squares sense. In the case when N = 1, the output coefficients P

_{1}and P

_{2}are considered. All the linear yield locus parameters expressed by Equation (1) can be calculated as a function of the polynomial coefficients:

_{2}

_{t}= P

_{2}/P

_{1}

_{i}= arctan(P

_{1})

_{1}, is calculated by the intersection of the σ-axis and the Mohr circle tangent to the yield locus and passing through the point (σ

_{pre}, τ

_{pre}), which is representative of the pre-shear stresses. The tangent point between the Mohr circle and the yield locus, as well as the Mohr circle radius and center, is calculated according to tangency condition between a line and a circle. In particular, the first-order Taylor series approximation of the Warren–Spring curve about the tangent point is used to obtain a linear function.

_{c}, representing the state of stress in the unconfined material at yield, is estimated by the intersection of the σ-axis and the Mohr circle tangent to the yield locus and passing through the origin of the axis.

- Linear YL, which allows for the generation of a linear yield locus, the related consolidation Mohr circle and the related unconfined yield Mohr circle, starting from experimental data;
- Warren–Spring, which allows for the generation of the curved yield locus, the related consolidation Mohr circle and the related unconfined yield Mohr circle, starting from experimental data;
- Compare, which allows for the comparing of the results obtained by the two previous approaches.

_{pre}, τ

_{pre}) pre-shear data points. These values are used as input data for the subprocess that returns the main bulk flow properties and the statistics data as output. In particular, the coefficient of determination (R-squared), the root-mean-square error (RMSE) and Pearson’s coefficient are reported as statistical indexes of the fitting process.

_{pre}, τ

_{pre}).

_{pre}, τ

_{pre}) is considered to belong to the yield locus only in the case in which the resulting model value for the yield locus, τ, at σ

_{pre}is larger than the experimental value, τ

_{pre}. Such an approach is well described elsewhere [28], and it is also highlighted in Figure 5 and Figure 6, where hyphenated lines report the repeated procedure applied in case the condition mentioned above on τ

_{pre}is met.

## 3. Data Analysis Using cYield

_{sv}) and particle size distributions by weight (PSD). Bulk densities were calculated from data supplied from the Schulze shear tester. PSDs and d

_{sv}were measured by a laser scattering particle size analyzer (Mastersizer 2000, Malvern Panalytical Ltd., Malvern, UK), and the 10th, the 50th and the 90th percentile sizes (d

_{10}, d

_{50}and d

_{90}, respectively) are reported.

_{c}and material flowability ffc = σ

_{1}/fc) obtained for all the samples considered at the various consolidation levels, as a function of the model yield locus. Moreover, values of the extrapolated isostatic tensile strength, σ

_{t}, and the Warren–Spring coefficient, n, are reported, as well. The flow functions obtained for all the samples are displayed in Figure 7, where the flow functions obtained by using the linear yield locus model are compared to those obtained through the Warren–Spring model.

^{2}, and the root-mean-square error (RMSE) are reported in Figure 9 and Figure 10, respectively.

_{t}, shows significant differences in the two cases. Therefore, it must be recognized that the extrapolation in the traction plane of the yield locus is a rather strong assumption, as it is not possible to consider shear data in the traction half-plane. Indeed, such an extrapolation does not certainly reflect the reality, as it may estimate an incorrect value of the tensile strength.

_{t}, and in some cases resulting in unrealistic values [24]. However, as demonstrated by García-Triñanes et al. [10], the Warren–Spring model is the only model capable of extracting with good agreement to the experimental evidence the flow parameters that characterize the non-linearity of cohesive powder yield loci.

## 4. Conclusions

_{pre}, τ

_{pre}), contrary to other software previously developed and presented in the literature.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**CYield App general view reporting, as an example, the use of the experimental results for the sample CP2 at the pre-shear load of 0.614 kg.

Sample | ρ_{b} (kg/m^{3}) | d_{10} (μm) | d_{50} (μm) | d_{90} (μm) | d_{sv} (μm) |
---|---|---|---|---|---|

Ceramic Powder (CP1) | 700 | 3 | 12 | 28 | 7 |

Ceramic Powder (CP2) | 1000 | 18 | 35 | 61 | 22 |

Ceramic Powder (CP3) | 1200 | 38 | 61 | 95 | 41 |

Ceramic Powder (CP4) | 1350 | 55 | 87 | 130 | 51 |

Ceramic Powder (CP5) | 1400 | 90 | 184 | 423 | 104 |

Rutile (RU) | 2300 | 91 | 205 | 444 | 146 |

Calcium Carbonate (CaC) | 500 | 2 | 7 | 40 | 4 |

Dolomitic Lime (DL) | 1000 | 2 | 23 | 257 | 6 |

Sample | Pre-Shear Load (kg) | Number of Repetitions Per Point | Linear Yield Locus | Warren–Spring | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

C (Pa) | f_{c} (Pa) | σ_{1} (Pa) | σ_{t} (Pa) | C (Pa) | f_{c} (Pa) | σ_{1} (Pa) | σ_{t} (Pa) | n (-) | |||

CP1 | 0.414 | 4 | 191 | 719 | 1132 | 283 | 122 | 731 | 1126 | 31 | 2.00 |

0.514 | 4 | 230 | 857 | 1397 | 346 | 161 | 891 | 1392 | 45 | 2.00 | |

0.614 | 4 | 250 | 963 | 1619 | 355 | 181 | 1006 | 1646 | 52 | 1.98 | |

0.714 | 4 | 284 | 1075 | 1866 | 415 | 180 | 1081 | 1877 | 49 | 1.92 | |

CP2 | 0.414 | 4 | 57 | 214 | 1036 | 84 | 21 | 121 | 1046 | 11 | 1.31 |

0.514 | 4 | 58 | 220 | 1265 | 86 | 27 | 140 | 1274 | 18 | 1.25 | |

0.614 | 4 | 59 | 228 | 1487 | 84 | 8 | 65 | 1538 | 3 | 1.32 | |

0.714 | 4 | 73 | 280 | 1741 | 106 | 34 | 173 | 1765 | 24 | 1.22 | |

CP3 | 0.414 | 4 | 31 | 116 | 1031 | 47 | 15 | 69 | 1038 | 15 | 1.13 |

0.514 | 4 | 38 | 145 | 1289 | 57 | 25 | 106 | 1298 | 25 | 1.11 | |

0.614 | 4 | 41 | 157 | 1498 | 60 | 25 | 112 | 1521 | 24 | 1.13 | |

0.714 | 4 | 53 | 203 | 1756 | 78 | 34 | 146 | 1772 | 34 | 1.12 | |

CP4 | 0.414 | 4 | 16 | 60 | 1006 | 23 | >1 | <1 | 1010 | >1 | 1.10 |

0.514 | 4 | 38 | 143 | 1279 | 56 | 38 | 143 | 1279 | 56 | 1.00 | |

0.614 | 4 | 31 | 120 | 1462 | 45 | 23 | 96 | 1470 | 27 | 1.06 | |

0.714 | 4 | 31 | 116 | 1670 | 45 | 15 | 65 | 1679 | 16 | 1.08 | |

CP5 | 0.414 | 4 | 17 | 67 | 1007 | 24 | <1 | 1 | 1008 | <1 | 1.10 |

0.514 | 4 | 18 | 71 | 1241 | 26 | 7 | 33 | 1242 | 8 | 1.07 | |

0.614 | 4 | 20 | 77 | 1461 | 27 | 9 | 38 | 1467 | 9 | 1.07 | |

0.714 | 4 | 25 | 96 | 1689 | 36 | <1 | <1 | 1701 | <1 | 1.11 | |

RU | 0.514 | 4 | 23 | 88 | 1271 | 32 | 0.03 | <1 | 1273 | <1 | 1.12 |

0.614 | 4 | 15 | 61 | 1498 | 20 | 0.07 | <1 | 1503 | <1 | 1.08 | |

0.714 | 4 | 15 | 59 | 1718 | 20 | 0.05 | <1 | 1720 | <1 | 1.05 | |

CaC | 1.6 | 2 | 330 | 1259 | 1614 | 477 | 260 | 1285 | 1597 | 96 | 1.90 |

3.2 | 2 | 654 | 2588 | 3321 | 889 | 470 | 2653 | 3277 | 129 | 2.00 | |

4.8 | 2 | 976 | 3887 | 5021 | 1310 | 698 | 3989 | 4955 | 188 | 2.00 | |

6.4 | 2 | 1178 | 4686 | 6237 | 1586 | 758 | 4772 | 6146 | 177 | 2.00 | |

7.9 | 2 | 1558 | 6314 | 8694 | 2032 | 1122 | 6460 | 8611 | 323 | 1.92 | |

9.5 | 2 | 1807 | 7364 | 10196 | 2335 | 1232 | 7517 | 10075 | 321 | 1.94 | |

11.1 | 2 | 2114 | 8753 | 12335 | 2664 | 1303 | 8936 | 12200 | 268 | 2.00 | |

DL | 1.6 | 2 | 283 | 1043 | 1503 | 437 | 177 | 1036 | 1488 | 46 | 2.00 |

3.2 | 2 | 432 | 1606 | 2794 | 652 | 95 | 1381 | 2768 | 8 | 2.00 | |

4.8 | 2 | 575 | 2109 | 4051 | 894 | 15 | 1473 | 4030 | <1 | 1.96 | |

6.4 | 2 | 675 | 2472 | 5281 | 1051 | 46 | 1508 | 5264 | 2 | 1.86 | |

7.9 | 2 | 872 | 3275 | 6809 | 1295 | 175 | 2412 | 6778 | 19 | 1.83 | |

9.5 | 2 | 932 | 3544 | 8167 | 1354 | 297 | 2635 | 8138 | 62 | 1.69 | |

11.1 | 2 | 1042 | 3869 | 9355 | 1580 | 15 | 1290 | 9362 | <1 | 1.75 |

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**MDPI and ACS Style**

Macri, D.; Chirone, R.; Salehi, H.; Sofia, D.; Materazzi, M.; Barletta, D.; Lettieri, P.; Poletto, M.
Characterization of the Bulk Flow Properties of Industrial Powders from Shear Tests. *Processes* **2020**, *8*, 540.
https://doi.org/10.3390/pr8050540

**AMA Style**

Macri D, Chirone R, Salehi H, Sofia D, Materazzi M, Barletta D, Lettieri P, Poletto M.
Characterization of the Bulk Flow Properties of Industrial Powders from Shear Tests. *Processes*. 2020; 8(5):540.
https://doi.org/10.3390/pr8050540

**Chicago/Turabian Style**

Macri, Domenico, Roberto Chirone, Hamid Salehi, Daniele Sofia, Massimiliano Materazzi, Diego Barletta, Paola Lettieri, and Massimo Poletto.
2020. "Characterization of the Bulk Flow Properties of Industrial Powders from Shear Tests" *Processes* 8, no. 5: 540.
https://doi.org/10.3390/pr8050540