# Joint behaviour during shear process using an innovative equivalent geometrical and 3D printing technology

Emilio Abi Aad, Marianne Conin, Olivier Deck, Marwan Al Heib

► To cite this version:

Emilio Abi Aad, Marianne Conin, Olivier Deck, Marwan Al Heib. Joint behaviour during shear process using an innovative equivalent geometrical and 3D printing technology. Rock and Fracture Mechanics in Rock Engineering and Mining (EUROCK 2022), Sep 2022, Espoo, Finland. pp.012051, 10.1088/1755-1315/1124/1/012051. [https://doi.org/10.1088/1755-1315/1124/1/012051](https://doi.org/10.1088/1755-1315/1124/1/012051). [hal-04016771](https://HAL.inistre.fr/hal-04016771)

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# Joint behaviour during shear process using an innovative equivalent geometrical and 3D printing technology

E Abi Aad¹, M Conin¹, O Deck¹ and M Al Heib²,

¹Université de Lorraine, CNRS, GeoRessources Laboratory, Ecole des Mines de Nancy, Campus Artem CS14234, 54042 Nancy Cedex, France,

²Institut National de l'Environnement Industriel et des Risques, Campus Artem 54042 Nancy Cedex, France

emilio.abi-aad@univ-lorraine.fr

**Abstract.** The morphology of rock joints has been recognized as the key factor controlling its mechanical behavior, including its pre-peak and post peak phases. It is playing a significant role in joint aperture in association with asperities crushing, and rock matrix yielding deformation. The purpose of the paper is to examine how the joint aperture or closure occur in the joint, discussing the relationship with normal stress, joint morphology and intact matrix mechanical properties. To do this, an innovative methodology based on 3DP technology using a sand and phenolic binder was applied to construct the matrix and joint. The surface roughness was considered to have fractal properties with a self-affine replication, in accordance with natural observations. The constructed joint underwent eighteen shear tests with constant normal stress. Preliminary results revealed that, for each of pre-peak and post-peak phases, the joint behavior was controlled by a specific range of asperities sizes. Three behaviors were observed depending on the applied normal stress: (i) at low normal stress the waviness causes dilation; (ii) at normal stress over 40% of the UCS value, tensile and/or slip cracks are observed around the waviness, led to a crushing and beheading of these asperities; (iii) at intermediate normal stress, the two mechanisms were conjointly observed.

## 1. Introduction

Shear behavior of rock joints has long been recognized as a critical element in understanding the mechanical behavior of fractured rock masses and their flow transport processes [1-2]. The rock fracture morphology is often referred to as roughness, of which couplet with the rock mechanical parameters, and the in-situ conditions, play a significant role in the shear behavior. However, groundwater flow and transport solutes are driven by the voids developed between joint contacts during the shearing mechanism and referred to as dilatation. Patton, [3] acknowledged that roughness occurs at many scales. He classified asperities into two categories: (i) unevenness i.e., small-scale asperity that predominantly controls the shear behavior during small displacement, (ii) waviness i.e., large-scale asperity that governs the shearing behavior for large displacement. Barton, [4] also reported as a result of a set of shear tests performed on a natural rough joint that the unevenness asperity controls the shear behavior at low applied normal stress, while by increasing the applied normal stress the unevenness asperity is sheared off and the waviness asperity becomes the controlling component. Yang et al., [5] further confirm these investigations by performing shear tests on rough surfaces assimilated to a 1D fractal object and molded by polystyrene. Many studies have been conducted to examine the effect of asperity shape and degradation on shear behaviour during the shearing process [6-8]. Asadi, [9] showed, on the basis of shear test conducted on mortar triangular and wavy shaped joints, that by increasing the asperity dip angle, the failure mode changes from asperity sliding to cut-off and considerable tensile cracking; therefore, the failure occurs within the matrix and the post-peak behaviour depends on the material mechanical properties. Later on, Song et al., [10] concluded that for a self-similar micrometer-scale asperity, that peak shearing stress increases when the asperity magnitude decreases. When the

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wavelength or the amplitude of an asperity are large, the asperity starts to fracture; otherwise, the plasticity is noticed within the matrix under the asperity.

Three different methods have focused on the quantitative description of fracture surface roughness. (i) subjective empirical method JRC (joint roughness coefficient) proposed by Barton and Choubey, [11-12], and adopted by the ISRM. However, in order to minimize the subjectivity of the JRC determination, alternate approaches have been proposed: (ii) statistical analysis [13-15] which has established many parameters to quantify the roughness shape, classified into three groups: amplitude, spacing, and hybrid parameters; (iii) fractal method first proposed by Mandelbort [16]. Further on, many studies have been devoted to identifying that a fractal surface with a self-affine replication having great potential to describe surface roughness of a natural rock joint e.g. granite, sandstone, or marble, over several orders of magnitude [7, 17-21]. Such surfaces satisfy a scale invariance, and are statistically invariant by transformation.

$$x \to \lambda x; y \to \lambda y; z(x, y) \to \lambda^H z(x, y) \qquad (1)$$

Where $H$ is called the Hurst exponent.

Therefore, the objective of this present work is to experimentally analyse the relation between fracture roughness and shear behaviour and dilatation at various normal load for a rock joint constituted with cohesive granular material. For this purpose, an innovative approach based on 3D printing (3DP) technology [22-25] using sand and liquid binder is adopted to form the artificial rough rock joint because of its high accuracy and repeatability [26-27]. Different rough surfaces replicating natural rock joints (smooth to rougher) are generated numerically as self-affine fractal 3D object by precisely controlling their geometrical parameters.

## 2. Artificial 3DP Material

### 2.1. 3DP process

In this paper, the rock-like material are constructed from artificial materials based on liquid binder and sand using a binder jetting 3DP process. This process applies a selective binder through the inkjet head to bind sand and shape the required forms. In this study, a 3D sand printer VX1000 from Volxejet was chosen to manufacture specimens. The type of used sand is silica in reference GS140, with a phenolic resin as binder. The printed layer thickness is between 250 µm and 300 µm. The properties of the material depend on the level of binder saturation, the printing direction, the storage time and conditions, and the curing time and temperature [23, 26].

### 2.2. Specimens preparation for mechanical testing

A series of uniaxial compressive and three-point bending tests were performed to characterize the macro mechanical behaviour of the 3D printed rock. Two types of specimens: cylinder (D=38 mm, H=76 mm), and beam (l=e=34 mm, L=150 mm) were printed. Since the 3D printing process involves possible mechanical anisotropy, the specimens are printed in three perpendicular X, Y and Z directions, with three different levels of binder saturation 6.5% (stander percentage), 5.7% (intermediary percentage) and 5% (minimum percentage to assure the cohesion of the material).

### 2.3. Uniaxial compression and three bending tests

Uniaxial compressive tests are performed on printed cylindrical samples, while three bending tests are conducted on printed beam specimens. During the uniaxial compressive tests, all specimens are loaded at the controlled rate of 0.2 mm/s, until failure, while a loading rate of 0.1 mm/s or 0.05 mm/s is applied during the three points bending tests. The axial strain ($\varepsilon$) is provided by the displacement jack. A minimum of 3 tests are performed in each configuration (Printing-direction; saturation binder rate $W$). All specimens are tested at room temperature and after the same storage time (two months). The uniaxial compressive strength ($UCS$) is taken as the peak stress that the specimens can sustain during the test, and the Young's modulus ($E$) is taken as the slope of the linear component of the stress-strain curve. Whereas, the material's tensile strength ($\sigma_t$) is determined with three points bending test as the peak
stress that the specimens can withstand. The results show the repeatability of the compression tests. In 90% of the cases, brittle behavior is observed for all specimens printed in X and Y direction, while ductile behavior is observed for Z. Results indicate that UCS, σt, and E of the 3DP specimens are highly dependent on the saturation binder rate (W) and printing direction (Figure 1 and Figure 2). In detail, UCS, σt, and E increase with the increasing of the saturation binder rate, and the highest properties are measured for specimens printed in the Y-direction. Therefore, the mechanical behavior of this material will be considered as anisotropic, and might be a useful replica of an anisotropic weathered intact rock material low to fairly stiff, very low to low strength, and very low to low brittle material (UCS/σt<10).

<figure><img src="image_3.png" /><figcaption>**Figure 1.** UCS and Young's Modulus (E) of 3DP specimens function of the binder rate (W)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Tensile Strength (σt) of 3DP specimens function of the binder rate (W)</figcaption></figure>

### 3. Artificial 3DP rock joint

#### 3.1. Morphological characterization of rough rock joint

The Power Spectrum Density (PSD) is the Fourier transform of the autocorrelation function of waves composing any signal (1D) or surface (2D). It has been successfully used as a quantitative description to generate a 3D fractal surface by summing different periodic waves of different scales [28-31]. For a self-affine fractal surface, the log-log (PSD) plot reveals a linear trend between the logarithm of the power and the wave-vector (as function of wavelength) range equal to -2(H+1), where H is called the Hurst exponent related to the fractal dimensions Df via H=3-Df [32-33]. The PSD of a self-affine fractal depends on six parameters characterizing the surface topography (Table 1). The qmin, qmax, qroll-off are three parameters that define the maximum and minimum limits in space for the application of fractal correlation.

<table><caption>**Table 1.** Self-affine fractal surface parameters used for PSD method</caption><thead><tr><th>Parameter</th><th></th><th>Description</th><th>Rough correlation</th></tr></thead><tbody><tr><td><i>H</i></td><td>Hurst exponent</td><td>It varies between 0.5 (rougher surface) and 0.99 (smoother surfaces);</td><td>-</td></tr><tr><td><i>σ</i></td><td>Root mean square</td><td>Control the amplitude of the asperity;</td><td>+</td></tr><tr><td><i>q</i><sub>min</sub></td><td>Smaller wave-vector</td><td>The smallest possible wave-vector, depending on the size of the surface;</td><td>+</td></tr><tr><td><i>q</i><sub>max</sub></td><td>Cut-off wave-vector</td><td>The largest possible wave-vector, depending on the system under consideration;</td><td>+</td></tr><tr><td><i>q</i><sub>roll-off</sub></td><td>Roll-off wave-vector</td><td>The self-affine correlation length i.e. for a wavelength greater or equal to the long-distance roll-off wave, no more fractal correlation between the surface topography is present.</td><td>+</td></tr></tbody></table>

A set of ideal self-affine rough surfaces similar to a natural rock joint of dimension 150 mm × 150 mm are generated by the PDS method. It includes six rough surfaces with the same roughness but with different values of the couple (Ht= 0.5-0.75-0.99; σj=0.75-1). The smallest wavelength is 1mm and the
radial roll-off wavelength is fixed approximately at 42mm. Figure 4 presents two examples of numerical surfaces.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Numerical 150×150cm self-affine surface, with $δx=δy=1mm$, a) (H=0.5; σ=1mm) rough surface b) (H=0.99; σ=0.75mm) smooth surface</figcaption></figure>

<table><caption>**Table 2** Fractal properties of the numerical rough surfaces</caption><thead><tr><th>Surface</th><th>[H<sub>i</sub>]<sub>initial</sub></th><th>[σ<sub>j</sub>]<sub>initial</sub></th><th>10<sup>-3</sup></th><th>Roughness</th></tr></thead><tbody><tr><td>S1</td><td>0.99</td><td>0.75</td><td></td><td rowspan="2">Smoother</td></tr><tr><td>S2</td><td>0.75</td><td>0.75</td><td></td></tr><tr><td>S3</td><td>0.99</td><td>1.0</td><td></td><td rowspan="3">Rougher</td></tr><tr><td>S4</td><td>0.5</td><td>0.75</td><td></td></tr><tr><td>S5</td><td>0.75</td><td>1.0</td><td></td></tr><tr><td>S6</td><td>0.5</td><td>1.0</td><td></td><td></td></tr></tbody></table>

### 3.2. 3DP and scanning artificial joint surface

These surfaces were printed with an estimated W between 5.7% and 6.5% in the Z-printing direction and then scanned using a 3D scanner. The scanning rate is 5mm/s with an accuracy of 50µm, and $δ_{x,y}=300µm$. The scanning results confirm that the printed surfaces are isotropic in the X-Y plan. Discrepancies between the numerical surface (black) and the printed surface (orange/red) highlight the resolution limits (wavelength between 42 and 2.5mm) of the 3D Printer (Figure 5).

<figure><img src="image_6.png" /><figcaption>**Figure 4.** 3DP specimens manufactured using the jetting process</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Comparison between the PSD of the numerical surface (black line) and the printed surfaces (orange for surface S4, red for surface S3)</figcaption></figure>

**4. Direct shear tests**

Direct shear test are performed in rock joint apparatus. The test system includes a shear box designed for specimens up to size 1500 mm × 1500 mm, composed of upper and lower halves. The upper part moves vertically in (−Z̄) direction and horizontally in (−X̄) direction, while the lower part could moves horizontally in (X̄). The device is equipped with one and two load cells to measure normal and shear loads, respectively, as well as four Linear Variable Differential Transducers (LVDTs) in the normal direction to record the normal displacement of the upper halve. The average value of the two shear load cells and the four LVDTs are used as the system shear loads and the normal displacement, respectively. Tests are conducted under a constant normal load condition, equal to 0.4MPa (σn/UCS=10%), 0.8MPa (σn/UCS=25%) and 1.2MPa (σn/UCS=35%). Eighteen tests are carried out on the six printed surfaces.

**5. Direct shear tests results**

The shear results exhibit the same tendency for the six surfaces, with a classically observed increase of peak and residual shear stresses (τp, τr) with an increase of normal stress. The peak shear stresses are approximately equal regardless of the variation In Hurst exponent (H° and standard deviation (σ) (Table 3), which the breaking of unevenness may explain. Figure 6 shows the results only for two surfaces S3 and S4. We observe that the shear stress reaches a peak value, followed by a smooth decline to a constant residual value of approximately 70%-90% of the peak value. This difference between the peak and the residual values is classically observed on shear experiments on natural rock joints [35] and is interpreted as the consequence of the damaging of the surface. This damaging is mainly observed on the unevenness asperities (figure 8). Figure 7 shows that the asperities degradation evolves with the increase of the applied normal stress. Under lower stress σn/UCS=10%, the waviness and unevenness asperities are significantly less damaged, and accompanied by a dilatation behaviour, while in the case of σn/UCS=35% all scale asperities are completely sheared off, and contraction is observed instead of dilatation (Figure 6b, e and Figure 7). For the intermediary stress, a dilatation phase followed by a contraction phase are identified (Figure 6c, f). Figure 8 shows pictures of the S4 shear tests for different shear displacements under normal stress equal to 0.4, 0.8 and 1.2MPa. The observations show that: (i) the joint dilatation is associated with a mechanical aperture (blue zone), (ii) areas of contraction and damaging (red zone) are detected even under low normal stress at the exact location, and are more present as the normal stress increase, (iii) cracks develop progressively during tests, and initiate at the edge of the waviness asperities. They progressively extend within the matrix. Three series of cracks of a dip between 20° and 30° (blue, yellow and green lines) are identified. They always initiate at the same location, whatever the normal stress, but they are clearer under intermediate and higher normal stress.

The different rough surfaces are decomposed into four wavelength (λ) ranges: [150mm, 42mm], [150mm, 8mm], [150mm, 2.5mm], and [150mm, 0.5mm] using FFT technique. The ranges are defined
according to the variation of the Hurst exponent detected on the printed surfaces (Figure 5). Table 4 presents the computations of the slopes at 95% quantile of each wavelength range of each surface. The results show that the dilatation angle is approximately equal to the slope at 95% quantile of the wavelength above 42mm, proving that the waviness asperities mostly control the dilatancy regardless of unevenness.

<table><caption>**Table 3.** Resulting values of peak ($\tau_p$) and residual shear ($\tau_r$) stresses measured for the six surfaces</caption><thead><tr><th rowspan="2">Surface</th><th colspan="2">σ<sub>n</sub>/ UCS =10%</th><th colspan="2">σ<sub>n</sub>/ UCS =25%</th><th colspan="2">σ<sub>n</sub>/ UCS =35%</th></tr><tr><th>τ<sub>p</sub> [MPa]</th><th>τ<sub>r</sub> [MPa]</th><th>τ<sub>p</sub> [MPa]</th><th>τ<sub>r</sub> [MPa]</th><th>τ<sub>p</sub> [MPa]</th><th>τ<sub>r</sub> [MPa]</th></tr></thead><tbody><tr><td>S1</td><td>0.23</td><td>0.21</td><td>0.49</td><td>0.43</td><td>0.72</td><td>0.65</td></tr><tr><td>S2</td><td>0.27</td><td>0.24</td><td>0.50</td><td>0.44</td><td>0.70</td><td>0.63</td></tr><tr><td>S3</td><td>0.32</td><td>0.29</td><td>0.54</td><td>0.43</td><td>0.76</td><td>0.51</td></tr><tr><td>S4</td><td>0.19</td><td>0.16</td><td>0.45</td><td>0.38</td><td>0.76</td><td>0.66</td></tr><tr><td>S5</td><td>0.24</td><td>0.20</td><td>0.47</td><td>0.41</td><td>0.79</td><td>0.66</td></tr><tr><td>S6</td><td>0.23</td><td>0.17</td><td>0.49</td><td>0.40</td><td>0.76</td><td>0.64</td></tr></tbody></table>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Shear (a,d) and dilatation (b,c,e,f) behavior for S3 and S4 surfaces for $\sigma_n$/UCS=10% (blue curve), $\sigma_n$/UCS=25% (orange), and $\sigma_n$/UCS=35% (red)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 7.** S4 before and after shearing for $\sigma_n$/UCS=10%, $\sigma_n$/UCS=25%, and $\sigma_n$/UCS=35%. Green and yellow zones are the same in the 4 cases (before and afters shearing). The green zone corresponds to damaged area (for sheared surfaces) in the direction perpendicular to the shear, and yellow zone corresponds to damaged area (for sheared surfaces) in the shear direction</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 8.** Shearing tests under normal stress $\sigma_n$= 0.4,0.8, and 1.2MPa, mechanical aperture (blue zones), contraction and damaging (red zones), cracks (blue, yellow, and green lines)</figcaption></figure>

<table><caption>**Table 4.** Comparison between the experimental dilatancy angle and 95% quantile of the angle distribution of the numerical surface for different wavelengths.</caption><thead><tr><th rowspan="2"></th><th colspan="4">Θ [°]-95% Quantile</th><th rowspan="2">Experimental<br>Dilatancy<br>angle ψ [°]</th></tr><tr><th>λ > 42mm</th><th>λ > 8mm</th><th>λ > 2.5mm</th><th>λ > 0.5mm</th></tr></thead><tbody><tr><td>S1</td><td>3.8</td><td>12.5</td><td>14.7</td><td>19.7</td><td>4.1</td></tr><tr><td>S2</td><td>3.3</td><td>13.5</td><td>17.2</td><td>21.4</td><td>3.8</td></tr><tr><td>S3</td><td>5.0</td><td>16.4</td><td>18.9</td><td>22.3</td><td>4.2</td></tr><tr><td>S4</td><td>3.2</td><td>14.9</td><td>21.8</td><td>25.7</td><td>4.1</td></tr><tr><td>S5</td><td>5.0</td><td>18.2</td><td>23.2</td><td>26.6</td><td>C -0.9</td></tr><tr><td>S6</td><td>4.3</td><td>19.7</td><td>28.0</td><td>31.5</td><td>C -1.9</td></tr></tbody></table>

## 6. Conclusion

This study introduces a new method to replicate artificial rock joints with controllable geometric and mechanical parameters using 3DP binder jetting technology based on sand and liquid binder. This technology may be employed to replicate a weathered natural rock joint consisting of cohesive granular rock material of low strength. To mechanically characterize the artificial joint behaviour, compression and three-point bending tests are first conducted on the material, followed by shear tests on different joint configurations. The salient conclusions of the study are:

(i) By increasing the normal stress over 25% of the UCS, a closure of the apertures associated with zones of contracting and damaging is noticed. Thus, the matrix yields, and cracks propagate within the matrix, (ii) For a weathered rock joint, the dilatancy angle depends on the waviness asperity, and the shear behaviour depends on the smallest asperity, not broken.

Further on, a set of shear test will be performed on rough surfaces with other fractal geometries to support this findings, and to find out what range-scale of asperities that controls the pre-peak behaviour (friction angle), as well as the post peak behaviour.

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