# A Study on the Failure Behavior of Sand Grain Contacts with Hertz Modeling, Image Processing, and Statistical Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Materials, Experimental Setup, and Testing Program

#### 2.2. Analytical Method

_{d}is the vertical force at failure (note that a discussion on the failure force through representative force–displacement curves from the experiments is presented in Section 3.5).

_{yz}, RB

_{xz}, RT

_{yz}, and RT

_{xz}were obtained from the images of bottom and top grains from two different directions.

_{d}is the peak load or failure load during the crushing test, and σ is the maximum tensile stress.

_{B}, E

_{T}and ν

_{B}, ν

_{T}are the Young’s moduli and Poisson’s ratio values of the bottom and top particles.

## 3. Results and Discussion

#### 3.1. Analysis of Local Radius and Contact Radius

#### 3.2. Contact Strength Based on Weibull Analysis

_{m}is the mean value of particle tensile stress, and σ

_{stdev}is the standard deviation.

_{m}is the position of the central tendency of the normal distribution, and the value is the abscissa of the peak of the curve. The expected value was 44.8 MPa at 0.01 mm/min and 42.6 MPa at the speed of 1 mm/min. At different loading rates, the probability of breaking peaks close to σ

_{m}is large, and the value farther from σ

_{m}is less likely to occur, which is consistent with the breaking frequency displayed by the bar chart. At the two different loading rates, the values of σ

_{stdev}(standard deviation) were 12.8 MPa and 11.8 MPa, respectively, resulting in a very small coefficient of variation of the two data. The σ

_{stdev}of 0.01 mm/min was slightly larger than 1 mm/min, and the curve of 0.01 mm/min mildly flat.

_{c}), which describes the minimum strength that 37% of the population possessed. Based on the data in Figure 7 and the implementation of Weibull analysis, Equation (7a) was used. By applying natural logarithm function on both sides twice, as shown in Equation (7b), as well as sorting the stress data obtained by different methods (from smaller to larger values) to obtain P

_{s}, the log plot of P

_{s}against the tensile strength (Figure 8) can provide the Weibull modulus (m).

_{s}is the survival probability, m is the Weibull modulus and σ

_{c}is the characteristic strength.

#### 3.3. Implementation of Hertz Analysis of the Force–Displacement Curves to Derive Contact Young’s Modulus and Failure Stress

#### 3.4. Clustering Analysis

#### 3.5. Failure Modes

#### 3.5.1. General Observations and Classification of Breakage Patterns

#### 3.5.2. Influence of Breakage Patterns on the Load–Displacement Curves

## 4. Conclusions

- (1)
- The analysis of the data showed that even though the grain-to-grain systems in the present study had a similar range of strength (peak stress) with that of single- and multiple-contact crushing tests as presented in the literature, m-modulus values were, in general, higher in magnitude in the present work. This leads to the conclusion that the restriction of the grains to rotate during their compression leads to a smaller variability of peak stress values, even though it did not influence greatly the absolute values of strength. This may have important implications in integrating laboratory grain-scale tests with discrete-based numerical simulations, as currently the majority of numerical works implement in their calibration results from single-grain crushing tests. However, under in situ conditions, it is not necessary for the particles to be represented by the single-grain crushing test model.
- (2)
- The method used to estimate the contact radius and the equivalent radius, thus the contact area (by applying Hertz expressions), heavily influenced the calculated m-modulus. In general, the image processing using Matlab to compute the local radius in the vicinity of grain’s contacts led to much lower values of m-modulus compared with a manual estimation based on the obtained digital images. Despite that, the computed equivalent radius had a significant influence on the resultant peak stress, though this influence was more pronounced for the data where Matlab analysis was used compared with the manual method. It is noted that the manual method is not conceptually incorrect, in that the user defines, based on a digital image, a circle in the vicinity of the particle peak (where the contact with the neighboring grain is expected to occur), through which the local morphology is assessed. However, the Matlab analysis can capture more precisely the small differences in the morphologies of different grains, and this is very important to be considered in natural materials that display irregular shapes. The manual method homogenizes to some extend the assessed morphological characteristics of natural grains, giving rise to increased m-modulus values that inherently represent the variation of the characteristics within a population. Previous studies examining the breakage of grains have considered more “global” shapes rather than the locality of the particles, which, based on the present study, may lead to inaccurate estimation of the failure stress (as the failure stress is influenced by the estimated contact area).
- (3)
- The application of the Hertz model was implemented in two different steps. First, it was used to estimate the contact area to derive stresses from the measured forces; second, it was used to compute the contact Young’s modulus (E *) of each individual sample. Implementing the Hertz model to compute E * for each individual test had an influence on the resultant peak stress compared with the manual method (eyeball fitting), in which E * was assumed to be the same (30 GPa) for all the pairs of grains. This means that in the analysis of the breakage of particles, assessment of the elastic (and morphological) characteristics of the given sample is desirable.
- (4)
- Despite the significant differences in the estimated peak stress based on the method of analysis used, experiments at two different speeds (0.01 and 1 mm/min) provided similar results in terms of peak stress and m-modulus values. However, the speed of the tests had some influence on the mode of failure of the samples and also on the shape of the force–displacement curves. As the speed in the present study was examined within two folders, it is recommended that in future research works a wider range of velocities could be applied to examine the problem of grain breakage.
- (5)
- In general, the majority of the tests had a conservative-destructive or conservative mode of failure, where conservative implies micro-scale damage and formation of debris in the vicinity of grain contacts, while destructive implies macroscopic grain breakage. In some of the tests, a fragmentary mode of failure was observed, in which case meso-scale breakage and formation of small fragments occurred. A greater portion of fragmentary-type failure was observed at a higher speed. Thus, in the calibration of discrete-based numerical samples, it is important to consider how grains interact with respect to their breakage behavior rather than implementing the modes of failure as single-grain crushing tests would suggest (i.e., at the breakage point, a particle should not be seen as an isolated body, but its interaction with neighboring grains must be taken into account).
- (6)
- Four classes of force–displacement curves were distinguished, in which case most of the samples had a brittle failure with clear peak stress (with or without the occurrence of hardening prior to reaching peak stress). In some of the tests, more prominently at the higher speed, multiple peaks were observed with a significant shift of the displacements (corresponding to peak stress) to higher values. This behavior was ascribed to possible creep influences; however, it is recommended that future studies could further explore this behavior to draw firm conclusions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic representation of crushing test setup and representative force—displacement curve.

**Figure 3.**Example of the application of two different methods (Matlab and manual) to obtain the local radius. (

**a**) Manual method from image of a grain with a circle fitted to grain boundary, (

**b**) Binary image of the grain, (

**c**) Pixelated boundary of the grain and the range of boundary used for local radius calculation in Matlab method.

**Figure 4.**Comparison of (

**a**,

**b**) local radii of bottom and top grains and (

**c**) equivalent radii based on Matlab and manual analysis of local morphology of the grains (experiments at 0.01 mm/min speed).

**Figure 5.**Effect of particle equivalent radius on peak stress at a loading rate of (

**a**) 0.01 mm/min; (

**b**) 1 mm/min.

**Figure 6.**Frequency distribution of peak stress for different loading rates: (

**a**) 0.01 mm/min; (

**b**) 1 mm/min.

**Figure 7.**Survival probability against normalized stress curves at different loading rates applying Matlab and manual methods for local morphology analysis.

**Figure 8.**Weibull distribution plots at different loading rates applying Matlab and manual methods for local morphology analysis.

**Figure 9.**(

**a**,

**b**) Two examples of Hertz fitting to the load–displacement curves to estimate contact Young’s modulus (corresponding to 0.01 mm/min speed).

**Figure 13.**Clustering analysis investigating (

**a**) the relationship between the equivalent modulus and the peak stress, (

**b**) the relationship between the equivalent radius and the peak stress, (

**c**) the relationship between the maximum displacement (corresponding to peak load) and the peak stress.

**Figure 14.**Representative examples of the different failure modes (

**a**) conservative, (

**b**) destructive, and (

**c**) fragmentary.

**Figure 15.**Failure pattern for loading rate 0.01 mm/min: (

**a**) system failure mode; (

**b**) system failure mode when considering particle position; (

**c**) failure mode for top and bottom grains.

**Figure 16.**Failure pattern for loading rate 1 mm/min: (

**a**) system failure mode; (

**b**) system failure mode when considering particle position; (

**c**) failure mode for top and bottom grains.

**Figure 17.**Comparison between different system failure modes on the survival probability for loading rate 0.01 mm/min.

**Figure 18.**Comparison between different system failure modes on the Weibull distribution for loading rate 0.01 mm/min.

**Figure 19.**Different classes (types) of load–displacement curves: (

**a**) class-A; (

**b**) class-B; (

**c**) class-C; (

**d**) class-D.

**Figure 20.**Distribution of the different classes of load–displacement curves for different loading rates: (

**a**) 0.01 mm/min; (

**b**) 1 mm/min.

**Figure 21.**Peak load, maximum displacement (corresponding to peak load) and peak stress for the different classes of curves at velocities of: (

**a**) 0.01 mm/min; (

**b**) 1 mm/min.

Materials: Leighton Buzzard Sand (LBS); Size: 1.18–2.36 mm | ||||
---|---|---|---|---|

Loading Rate | Number of Tests | Test for Radius | Hertz Fitting | |

Manual | Matlab | |||

0.01 mm/min | 122 | • | • | • |

1 mm/min | 122 | • | • |

**Table 2.**Summary of test results in terms of local radius and equivalent radius computation based on Matlab and manual analysis of local morphology.

Local Radius (R): mm | Equivalent Radius (R*): mm | ||||||||
---|---|---|---|---|---|---|---|---|---|

Position | Mean | Stdev | Max | Min | Mean | Stdev | Max | Min | |

Matlab | Bottom | 0.46 | 0.22 | 1.34 | 0.11 | 0.2 | 0.09 | 0.54 | 0.07 |

Top | 0.44 | 0.25 | 1.49 | 0.09 | |||||

Manual | Bottom | 0.42 | 0.12 | 0.94 | 0.16 | 0.2 | 0.04 | 0.33 | 0.11 |

Top | 0.41 | 0.12 | 0.83 | 0.17 |

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**MDPI and ACS Style**

Li, S.; Kasyap, S.S.; Senetakis, K.
A Study on the Failure Behavior of Sand Grain Contacts with Hertz Modeling, Image Processing, and Statistical Analysis. *Sensors* **2021**, *21*, 4611.
https://doi.org/10.3390/s21134611

**AMA Style**

Li S, Kasyap SS, Senetakis K.
A Study on the Failure Behavior of Sand Grain Contacts with Hertz Modeling, Image Processing, and Statistical Analysis. *Sensors*. 2021; 21(13):4611.
https://doi.org/10.3390/s21134611

**Chicago/Turabian Style**

Li, Siyue, Sathwik S. Kasyap, and Kostas Senetakis.
2021. "A Study on the Failure Behavior of Sand Grain Contacts with Hertz Modeling, Image Processing, and Statistical Analysis" *Sensors* 21, no. 13: 4611.
https://doi.org/10.3390/s21134611