# Percent Reduction in Transverse Rupture Strength of Metal Matrix Diamond Segments Analysed via Discrete-Element Simulations

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## Abstract

**:**

_{TRS}, which is the percent reduction of the transverse rupture strength of metal matrix diamond segments with or without diamonds, is a key metric for evaluating the bonding condition of diamonds in a matrix. In this work, we build, calibrate, and verify a discrete-element simulation of a metal matrix diamond segment to obtain D

_{TRS}for diamond segments with various diamond-grain sizes, concentrations, and distributions. The results indicate that D

_{TRS}increases with increasing diamond-grain concentration and decreases with increasing diamond-grain size. Both factors can be explained by the total diamond contact length, the increase of which causes the increase in D

_{TRS}. The distribution of diamond grains in segments also strongly influences the increase of D

_{TRS}. The use of D

_{TRS}as a metric to assess the bonding condition of diamonds in matrixes is not valid unless the diamond-grain size, concentration, and distribution and total diamond contact length are the same for all diamond segments under consideration.

## 1. Introduction

_{TRS}, which represents the percent reduction in TRS between a segment without diamonds and one with diamonds, is used to determine the bonding condition between diamond and metal matrix [2,3,4].

_{TRS}indicates that diamonds are weakly bonded to the matrix. Previous studies experimented with various ingredients for the metal matrix and various diamond coatings [2,3,4,5] and found that D

_{TRS}sometimes changes unexpectedly, although the metal matrix, diamond-grain size, and diamond concentration were the same in each experiment. In other words, D

_{TRS}becomes unexpectedly invalid during applications. The reason behind this unexpected change is tentatively attributed to the assumption used in deriving D

_{TRS}, that all diamonds in segments are exactly the same and uniformly distributed. Although it is true that the diamond distribution in segments is difficult to control with precision, it is not certain whether the diamond-grain distribution does affect D

_{TRS}.

## 2. Materials and Experimental Details

#### 2.1. Fabrication of Diamond Segments

#### 2.2. The Three-Point Bending Tests and Compression Tests

#### 2.3. Single Grit Shearing Test

## 3. Establishment, Calibration, and Verification of Discrete-Element Model

^{2D}(Particle Flow Codes in two dimensions). The DEM model was composed of particles and the bonds between them. In such a model, a contact bond is an interaction between two particles because the particles themselves are deformable, and a parallel bond is an approximation of the physical behaviour of a bond substance between particles [6,11].

#### 3.1. Discrete-Element Model of Metal Matrix

#### 3.2. Discrete-Element Model of Diamond

#### 3.3. Discrete-Element Model of the Diamond Segment

#### 3.4. Verification of Discrete-Element Model of the Diamond Segment

^{3}or 0.88g (4.4 Carat) of diamonds by cm

^{3}is defined as the 100% concentration of diamonds in the segment [17]. The TRS obtained by simulation and experiment are compared in Table 5, the error of which is within the satisfactory value of less 10%.

## 4. Simulation of Percent Reduction of TRS

_{TRS}, is generally used to describe the retention ability of diamonds in diamond segments. Thus, the D

_{TRS}was calculated by: D

_{TRS}= (σ

_{0}− σ)/σ

_{0}× 100%, where σ

_{0}is the TRS of the metal matrix and σ is the TRS of the diamond segment with diamonds.

_{TRS}equated to a generally stronger diamond retention [18]. In order to study the feasibility of this approach, the DEM model with the microcosmic parameters obtained from Section 4 was applied to study the factors that influence the D

_{TRS}. To do this, D

_{TRS}was calculated by the DEM simulation in which diamond concentration, diamond size, and diamond distribution were varied. In this section, the dimension of the segment’s DEM model was 30 mm × 6 mm.

#### 4.1. Effect of Diamond Concentration

_{TRS}, diamonds were added to the Co-based matrix with concentrations of 25%, 50%, 75%, and 100%. The simulated D

_{TRS}are plotted in Figure 7. It can be found that the D

_{TRS}increases with increasing diamond concentration.

^{3}, and the number of diamond grains were 43, 88, 129, and 172. As a typical of composite material, diamonds can be considered as a heterogeneous material to the matrix. Due to the DEM simulation in this work being in 2D mode, the total diamond contact length was used to quantitate the contacting boundary of the diamonds in the matrix. As the diamond-grain size was 550 µm, the total diamond contact length L

_{total}for the four concentrations were 77, 154, 231, and 308 mm, respectively. The relationship between D

_{TRS}and L

_{total}is presented in Figure 7. It can be easily concluded that D

_{TRS}increases nearly linearly with L

_{total}.

#### 4.2. Effect of Diamond-Grain Size

_{TRS}and L

_{total}versus the increase of diamond-grain size. Figure 8 also shows that, with the same grain concentration, D

_{TRS}decreases with increasing diamond size.

_{total}were 77, 110, and 150 mm, respectively, for the three types of diamond segments. It can be found that with the same diamond-grain concentration, the number of diamond grains and L

_{total}in the DEM model depend on the diamond-grain size. From Figure 8, it also can be found that D

_{TRS}increases linearly with total contact length, which agrees with the result shown in Figure 7.

_{TRS}was calculated for the four segments (Figure 9 and Table 6). Ordering the segments from smallest D

_{TRS}to largest gives S1 < S2 = S3 < S4. Based on Table 6, ordering the segments from smallest diamond-grain concentration to largest gives the same result: S1 < S2 = S3 < S4. By comparing these two ordered lists, it can be concluded that: (i) D

_{TRS}generally increases with increasing grain concentration, which is consistent with the result discussed in the Section 3.1; and (ii) for the same L

_{total}and diamond-grain concentration, D

_{TRS}is independent of diamond-grain size. The diamond-grain concentration and total contact length of diamonds strongly impact the D

_{TRS}. The total contact length of the diamonds corresponds with the contact surface area between the diamond grains and the matrix in actual three-dimensional diamond segments. It is reasonable that with the increase of diamonds in the matrix, the TRS reduction increases.

#### 4.3. Effect of Diamond Distribution

_{TRS}, in the Figure 10, the diamond particle size was 550 μm and the concentration was 50% (with 90 diamonds in each segment) for all four samples. The four distributions are based on the following considerations:

_{TRS.}Thus, distribution D2 is a random distribution, which is the same as the distribution (a) in Figure A1 in Appendix.

_{TRS}of the segments with the different diamond-grain distributions are listed and compared in Table 7. The 5.1% difference in D

_{TRS}of distributions D1 and D2 is caused by the spatial difference in distributions D1 and D2. For the ordered distributions, the difference of 18% between D

_{TRS}for D3 and D4 is caused by changing the number of rows from three to six. It can be concluded that the diamond-grain distribution is also a key factor determining the D

_{TRS}.

_{TRS}are presented in Figure 12. It can be seen that the D

_{TRS}decreased with the increase of either row’s number or the horizontal distance.

**a**–

**d**) in Figure 11. L1 is the greatest and L4 is the least in L1–L4. The bond of the matrix/diamond boundary is weaker than the bond of the matrix/matrix. According to the features of the composite material, the closer the weak bond is to the edge, the easier it is for the material to be destroyed. However, from Figure 12, we can see the D

_{TRS}of the segment with three rows is larger than the segment with six rows, which means that the D

_{TRS}of the segment with L1 (1 mm) is larger than the segment with L4 (0.5 mm), and the segment with L1 is easier to be destroyed. Thus, we can find that the distance from the specimen edge to the nearest diamond interface has less affect than the distance of the changed horizontal direction.

#### 4.4. Discussion

_{TRS}in Section 4.1, Section 4.2 and Section 4.3 were induced only by the different diamond distributions, rather than by the diamond bonding conditions with the matrix. This observation can reasonably explain the unexpected change in D

_{TRS}while evaluating the diamond bonding condition by TRS. Therefore, D

_{TRS}is not a reasonable metric to evaluate the bonding condition of diamonds in segments unless the grain size, concentration, and distribution of diamonds in the segments are the exactly same.

## 5. Conclusions

_{TRS}, of diamond segments are the diamond-grain size, concentration, and distribution within the diamond segment. These factors were analysed by DEM simulation, which also considered the diamond total contact length L

_{total}to explain the results.

_{TRS}for diamond segments increases with diamond-grain concentration but decreases with diamond-grain size. The percent of TRS reduction increases with the increase of the L

_{total}in the segment. For a given L

_{total}, diamond-grain concentration, and diamond-grain distribution, the diamond grains of differing sizes have no effect on D

_{TRS}. However, the spatial distribution of the diamond grains with the segment strongly affects the D

_{TRS}. Finally, the D

_{TRS}is not a reasonable metric for evaluating the bonding conditions of diamonds in segments unless the diamond-grain size, concentration, and distribution are the same in all segments under consideration.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Illustration of shearing test: (

**a**) shearing test device; (

**b**) schematic of shearing; (

**c**) real shearing; (

**d**) shearing force signal.

**Figure 2.**Illustration of establishing the DEM of a diamond segment: (

**a**) the morgraphy of diamond segment (observed by Scanning Electron Microscope); (

**b**) DEM of a diamond segment.

**Figure 3.**Illustration of Co–Cu–Sn-based metal matrix: (

**a**) before sintering, (

**b**) after sintering, and (

**c**) DEM Co–Cu–Sn metal matrix.

**Figure 4.**Failure of Co-based metal matrix after being tested: (

**a**,

**c**) by DEM simulation; (

**b**,

**d**) by experiment.

**Figure 8.**Percent TRS reduction D

_{TRS}and total diamond boundary length L

_{total}versus diamond size.

**Figure 9.**Percent TRS reduction for the various diamond segments (see Table 6).

**Figure 11.**Different order distributions of diamonds in the segment: three rows (

**a**), four rows (

**b**), five rows (

**c**), and six rows (

**d**).

**Figure 12.**Order distribution versus reduction rate of TRS. (

**a**) Effect of row number; (

**b**) effect of horizontal distance.

Mechanical Properties | Co-Based Metal Matrix (Segment without Diamonds) | ||
---|---|---|---|

Experimental Results | Simulation Results | Error | |

UCS/MPa | 1681 | 1842 | 9.6% |

Ec/GPa | 13.8 | 12.8 | 7.2% |

TRS/MPa | 1120 | 1046 | 4.2% |

**Table 2.**Main mechanical properties of the diamond obtained by the DEM simulation and ref. [15].

Mechanical Properties | Values from [16] | Values Simulated by DEM Model |
---|---|---|

Young’s modulus Ec (GPa) | 900–1000 | 940 |

Poisson’s ratio γ | 0.069–0.12 | 0.095 |

UCS (MPa) | 4500–5800 | 4853 |

Exposure Height (µm) | Critical Force of Shearing Test, F (N) | Error (%) | |
---|---|---|---|

By Experiment (N) | By Simulation (N) | ||

230 | 70.5 | 71.1 | 0.9 |

280 | 67.3 | 68 | 1.0 |

Values of Microcosmic Parameters | |||
---|---|---|---|

Matrix | Diamond | Diamond/Matrix Boundary | |

Particle density (kg/m^{3}) | 8900 | 3500 | / |

Particle contactmodulus, Ec (GPa) | 1.3 × e^{10} | 3 × e^{11} | / |

Particle stiffnessratio, kn/ks | 1 | 1 | / |

Particle friction coefficient | 0.8 | 0.1 | / |

Isotropics Stress, (Pa) | −2.0 × e^{7} | −5.0 × e^{7} | / |

Radius multiplier of parallel bond | 1 | 1 | 1 |

Elasticity modulus of parallel bond (Pa) | 1.3 × e^{9} | 11 × e^{11} | 4 × e^{7} |

Normal strength of parallel bond (Pa) | 3 × e^{8} | 1 × e^{10} | 4 × e^{3} |

Shear strength of parallel bond (Pa) | 3 × e^{8} | 1 × e^{10} | 4 × e^{3} |

Normal strength of contact bond (Pa) | 6 × e^{7} | / | 5 × e^{3} |

Shear strength of contact bond (Pa) | 6 × e^{7} | / | 5 × e^{3} |

Diamond Size (US Mesh) | Concentration (%) | TRS | Ec | ||||
---|---|---|---|---|---|---|---|

Simulation (MPa) | Experiment (MPa) | Error (%) | Simulation (N) | Experiment (N) | Error (%) | ||

30/40 | 50 | 743 | 786 | 5.5 | 17.4 | 15.9 | 9.4 |

Diamond Segments | Number of Diamonds with Given Grain Size | Total Contact Length (mm) | Diamond Concentration (%) | D_{TRS} (%) | ||
---|---|---|---|---|---|---|

550 µm | 380 µm | 270 µm | ||||

S1 | 0 | 92 | 0 | 110 | 25 | 21.6 |

S2 | 29 | 31 | 32 | 110 | 29.7 | 22.0 |

S3 | 23 | 32 | 37 | 110 | 29.7 | 22.0 |

S4 | 43 | 0 | 49 | 110 | 32 | 22.8 |

Diamond Distribution | TRS (MPa) | Percent TRS Reduction (%) |
---|---|---|

D1 | 788 | 24.7 |

D2 | 734 | 29.8 |

D3 | 729 | 30.3 |

D4 | 916 | 12.3 |

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**MDPI and ACS Style**

Chen, X.; Huang, G.; Tan, Y.; Yu, Y.; Guo, H.; Xu, X.
Percent Reduction in Transverse Rupture Strength of Metal Matrix Diamond Segments Analysed via Discrete-Element Simulations. *Materials* **2018**, *11*, 1048.
https://doi.org/10.3390/ma11061048

**AMA Style**

Chen X, Huang G, Tan Y, Yu Y, Guo H, Xu X.
Percent Reduction in Transverse Rupture Strength of Metal Matrix Diamond Segments Analysed via Discrete-Element Simulations. *Materials*. 2018; 11(6):1048.
https://doi.org/10.3390/ma11061048

**Chicago/Turabian Style**

Chen, Xiuyu, Guoqin Huang, Yuanqiang Tan, Yiqing Yu, Hua Guo, and Xipeng Xu.
2018. "Percent Reduction in Transverse Rupture Strength of Metal Matrix Diamond Segments Analysed via Discrete-Element Simulations" *Materials* 11, no. 6: 1048.
https://doi.org/10.3390/ma11061048