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Reverse modelling of natural rock joints using 3D scanning and 3D printing
Computers and Geotechnics 73 (2016) 210–220
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Technical Communication
Reverse modelling of natural rock joints using 3D scanning and 3D printing Quan Jiang ⇑, Xiating Feng, Yanhua Gong, Leibo Song, Shuguang Ran, Jie Cui State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 17 March 2015 Received in revised form 18 November 2015 Accepted 20 November 2015 Available online 8 January 2016 Keywords: Natural joint Joint replication 3D scanning 3D printing Shearing experiment
a b s t r a c t In order to overcome the deficiency of natural joint specimens with the same surface morphology for experimental studies, we present a technical method for replicating natural joint specimens that incorporates two advanced techniques – three-dimensional (3D) scanning and 3D printing – using computer-aided design (CAD) as the ‘bridge’. This method uses an optical scanning apparatus and CAD techniques to reconstruct a virtual joint specimen with the natural rock’s joint morphology, and a 3D printer then manufactures a physical mould based on the virtual joint specimen for casting concrete or plastic specimens quickly and accurately. Quality verification clearly indicated that this method reduces the experimental errors originating from the discrepancies between replicating specimens containing natural joint’s morphology. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The strength and stability of rock masses are significantly influenced by the presence of inherent joints and fractures. As has been thoroughly demonstrated, shearing failure is one of the common features of rock masses, and hence the shear strength that affects the deformation along natural joints or discontinuities is crucial to the stability of geotechnical engineering projects, such as underground tunnels, rock slopes and open-cut mines [1–7]. In the first ISRM International Congress on Rock Mechanics in 1966, Muller stated, ‘The deformation resistance of the material bridges takes effect at much smaller deformations than the joint friction: this joint friction makes up partly for lost strength’ [8,9]. In the recent years, the frictional behaviour of rock joints has been of interest to many researchers and engineers [10–15]. Shear and compression experiments are the most important ways to understand the mechanical behaviours of natural rock joints, such as shearing stress–displacement curves with peak strength, non-uniform shearing damage of natural surfaces and non-linear normal deformation during compression [16–20]. On the basis of these laboratory and field tests, several shearing strength criteria have been proposed to identify the strength of a natural rock joint, such as Patton’s model [21], Ladanyi’s empirical model [1], Barton’s empirical model [10], Amadei–Saeb’s analytical model [22,23] and Grasselli’s threedimensional (3D) model [24]. All of these distinguished works have
⇑ Corresponding author. Tel.: +86 27 87198805. E-mail address: [email protected] (Q. Jiang). http://dx.doi.org/10.1016/j.compgeo.2015.11.020 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.
indicated that the surface morphology of a rock joint, quantified as joint roughness coefficient (JRC), plays a key role in its shearing strength. During the experimental investigation of natural rock joints, a good experimental procedure requires that the variables in the experiment can be controlled such that only one variable can be isolated and selectively changed. Because no two natural rock joint samples, even those from the same deposit, are truly identical, classical experiments cannot be conducted if the testing scheme requires multiple specimens [25–28]. Therefore, the lack of sufficient joint specimens with the same natural surface morphology has always limited experimental studies. At present, joint samples are mainly produced using one of the following three general techniques [6,14,17,29–35]: (i) tensile fractures, usually made in a manner similar to the Brazilian test, (ii) sawn flat joints with undulated or irregular surfaces or (iii) casts of natural or stylized joints with silicon moulds, mated silicon rubber and aluminium moulds. Despite the widespread acceptance of the joint specimens produced using these techniques, there is an empirical concern that these methods lead to objective errors and increased time consumption, because most of these methods are unable to represent natural joints with complicated and irregular surface shapes or digitize their surface. This does not mean that the aforementioned traditional methods are inappropriate for rock joint behaviour studies, but rather they are more appropriate for local application with a lesser quantity than as the basis of a universal system [32,36–38]. Moreover, these traditional methods are inconvenient for data collection and statistical analysis for joint shapes. In fact,
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the current dilemma in the ways of producing duplicate specimens has reduced the interest towards intensive studies on the mechanical properties of natural rock joints. The recently developed 3D scanning (3DS) and 3D printing (3DP) techniques, combined with computer-aided design (CAD), acting as a bridge connecting 3DS and 3DP, provide a new way to manufacture experimental specimens with the same irregular joint morphology. The 3DS apparatus can help digitize and characterize the joint surfaces in three dimensions quickly, with high precision and no damage. On the basis of these digitization data for the natural joint, some 3D roughness parameters can be developed to characterize the joint surfaces and overcome the drawbacks of two-dimensional (2D) profiles [39–44]. Furthermore, 3DP, that is, additive manufacturing or additive layer manufacturing, allows the automated generation of free-form solids directly from a computer file to a real object [45–47]. This 3DP provides a new way to make a physical mould with the morphology of a natural joint surface, because this technology can be used to print a joint surface with the same shape as the original joint. In addition, CAD can help reconstruct a virtual model body with any shape for experimental use from the scanned data of the natural joint surface. The clonal joint mould can be produced by 3DP using CAD technology. Therefore, on the basis of the printed joint mould, massive experimental specimens with the same natural joint morphology can be replicated for shearing or compression tests. In this study, a new method for producing natural joint specimens that combines 3DS and 3DP techniques is presented. First, an optical scanning apparatus is used to digitize the 3D surface of a natural rock joint. With the resulting point cloud data, a virtual joint specimen with the natural joint’s morphology and given boundaries is reconstructed using CAD methods. The 3D printer then manufactures a physical mould quickly and accurately based on the virtual specimen. In this way, massive joint specimens can be replicated using concrete or plastic materials according to the printed mould. The error analysis between the original joint data, the printed joint surface and the actual surface of the physical specimen indicates that this reverse method can copy the natural morphology from the original rock joint to the artificial joint specimen with high efficiency and precision. Furthermore, direct shear tests show that the shearing displacement–force curves of the replicated joint specimens were similar to each other with little dispersion. These results clearly indicate that the presented method can reduce the experimental errors caused by the specimens themselves, and open a new door to further experimental study of natural joints.
tages of high precision and good repeatability as well as being fast and easy to use. Its measuring system consists of a central projector unit and two charge-coupled device (CCD) cameras, which are linked to a personal computer to drive the system and save the scanned data, respectively (Fig. 1). The working theory of the Holon3D system is based on the white-light fringe patterns with multiple-frequency phase shift grating for the joint surface. The images of these patterns, which become distorted because of the disturbance of the surface, are captured by the two CCD cameras. With the given relative offset and angle between the projector and the cameras, the corresponding software computes the 3D coordinates for each pixel in these images using the principle of triangulation [24,44,56]. 2.1. Digitization and reconstruction procedure of 3D joint surface The digitization and reconfiguration procedure for a virtual joint surface using the 3DS equipment and the computer-aided software includes four main steps, which are described as follows. Step 1: Pretreatment of the joint surface. Once the joint surface has been cleaned, a thin white toner powder needs to be sprayed on it to improve the reflectivity of the white light emitted from the central projector and the absorptivities of the two CCD cameras. The scanning system can calculate 3D coordinates for the visible pixels in only a limited area, within 10 cm in length. Therefore, the digitization of a joint surface typically requires several individual measurements from different positions. In order to combine the individually measured data clouds in a common point cloud using a global coordinate system, the reference points (adhesive circular black–white markers) need to be adhered to the surface of the natural joint (Fig. 2a). The distance between the markers should be onetenth to one-fifth of the length of the scanning window for the successful integration of the point clouds. Step 2: 3DS and data integration. When the joint specimen is placed under the measuring window of the scanning apparatus, a portion of the point cloud can be obtained by activating the scanner controlled by the corresponding software. If the region of interest of the joint surface is larger than the size of the measuring window, it is necessary to acquire multiple measurements by moving the specimen in a linear or curved manner (Fig. 2b). In principle, it is very important for each measuring window to include at least three reference points that overlap with the previous window. The scanning software can then automatically establish a global coordinate system and identify
2. 3D virtual reconstruction of natural joint Projecon unit
The general methods for determining the joint shape include contact and noncontact approaches. The contact approach requires the operator or a needle to physically touch the surface and record the height of the joint along the selected profiles or over a predefined area [48–51]. These early-contact methods have some drawbacks, because they are time-consuming in terms of obtaining the data and are insufficiently precise when recording at locally steep locations. The non-contact approaches, including the analytical photogrammetric method, He–Ne laser beam technique, laser scanning method and white-light fringe technology, use a technique to make the measurements without physically touching the fracture surfaces [52–56]. These methods have greatly increased the speed and accuracy of roughness measurements. In this study, the Holon3D system was used to digitize the entire rock joint surfaces using non-contact area scanning. This system, with an accuracy of ±0.01 mm over a region of 100 100 50 mm in a measuring time of <3 s, offers the advan-
Camera_1
Camera_2
White-light fringe
Rock joint
Controlling computer
Fig. 1. Working set-up of the scanner under the white-light fringe pattern.
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3 mm
(a)
1 mm
(b)
Integrate
A piece of point cloud
(c)
Z (mm)
Adjust
Y (m X (mm)
m)
(d)
CAD modeling
Fig. 2. Digitization and reconstruction procedure for a 3D joint surface. (a) Joint sample and attached reference points, (b) multiple pieces of point cloud and the merged common point cloud representing the joint, (c) adjusting the initial intermediate plane to the horizontal unique plane, and (d) 3D reconstruction of the natural joint.
all the reference points included in the window. For all the following point cloud measurements, the system can automatically recognize the new and predefined reference points and use them to automatically adjust the new measurement data into the global coordinate system. Then, a global integration of point clouds is performed with the help of the supporting software by cleaning a few noise points induced by the
scanning errors, and then merging the overlaps of multiplepoint cloud collections. The major contributors of the noise points are the unstable light reflectivity and environmental disturbance. In general, noise points are small in number (several hundred) and the number of point clouds is approximately several million. Thus, the effect of noise points on the global precision of the measurement is very little.
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Step 3: Adjusting the common plane. The common/intermediate best-fit plane (O–X–Y–Z) of the obtained point cloud is initially inclined because the irregular joint surface is not horizontal and the specimen is moved frequently during the procedure [43,57]. Thus, adjusting the common plane for the merged point cloud from an inclined to a horizontal position is necessary for the subsequent reconstruction of the 3D joint specimen with CAD, whose expression is provided in Eq. (1) (Fig. 2c). Considering this changed unique plane as the reference plane, the incision of the joint surface is more convenient, and the statistical results for the surface data are more stable and universal [43]. Here, a matrix with 4 4 formation is needed when operations with transformation and rotation for a graph by the matrix multiplication (Eq. (1)).
Pu ¼ Pðx; y; z; 1Þ M R 2
1 0
60 1 6 M¼6 40 0 0
0 0 1
0
ð1Þ
3
07 7 7 05
ð2Þ
0 a 1 3
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 þ1 b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 b2 þc2 þ1 6 ðb2 þc2 þ1Þ 6 bc 1 ffi c 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 ðc2 þ1Þðb2 þc2 þ1Þ c2 þ1 ðb2 þc2 þ1Þ R¼6 6 b c ffi 1 ffi pffiffiffiffiffiffiffi 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ðc2 þ1Þðb2 þc2 þ1Þ c2 þ1 ðb2 þc2 þ1Þ 0 0 0
0
7 7 07 7 7; 7 07 5 1
ð3Þ
where a, b and c are the parameters of the initially inclined plane equation (z ¼ a þ bx þ cy), Pðx; y; z; 1Þ is the initial position of a 3D point with added switching component ‘1’, P u is the 3D point’s adjusted position relative to the horizontal unique plane and M and R are the transformational and rotational matrices [58], respectively. Step 4: Reconstruction of the 3D virtual specimen. With the current CAD technology, the joint surface can be reconstructed from the point clouds using a specifically developed triangulated irregular network (TIN) mesh [24,29]. This approach discretizes the point cloud data of the joint surface into contiguous triangles, which are defined by their vertices and the orientation of the vector normal to the triangular plane surface. Then, a 3D virtual body containing the natural joint surface is obtained by adding other sides (Fig. 2d). This method of discretizing the joint surfaces is particularly advantageous for building a virtual 3D model.
2.2. Characterization of the point cloud data of a natural joint Although the above-described method accomplishes the transition from the natural joint specimen to the virtual joint model using digitization and the CAD technique, a second treatment of the data obtained for the joint surface is still required to further estimate other morphological characteristics, such as the magnitude of the joint, angularity, undularity and anisotropy. The spacing of the joint’s point cloud obtained with the 3D scanner is a variable, because of irregular optical reflection of small surfaces with irregular inclination and size. Therefore, the output data for quantitative analysis should be normalized from the original point cloud into constant data point intervals using a mathematical interpolation program (e.g. Griddata function of Matlab [59] or Inverse Distance Weighted Interpolation, seeing the Appendix A). Non-linear interpolation is generally adopted for this process. Consideration of an appropriate interpolation interval is important to avoid information loss during the data collection and analysis of
the joint’s digital morphology, because a too large point interval would damage the data’s usefulness and a too small point interval would create large file sizes and long processing times. Two methods for estimating the appropriate interpolation interval are presented in this study for the ‘Z’ direction and the ‘X–Y plane’. Considering the sample depicted in Fig. 2a, a tentative comparison was first performed among different interpolation intervals ranging from 0.03 to 1.0 mm along the axes ‘x’ and ‘y’. The statistical average intermediate height of the unique plane is used as a valuating index (Eq. (4)):
D¼
N 1X jzi j: N i¼1
ð4Þ
The deviations of average intermediate heights under different interpolation intervals are listed in Table 1. The gradient of average intermediate heights tended to decrease with decreasing interval value. When the interpolation interval was set to 0.1 mm, the corresponding gradient of average intermediate height was approximately 0.02%. Then, the distribution density in the ‘X–Y plane’ was also analyzed. In general, the effective height information (‘Z’) can be considered during the interpolation if more than one original cloud point existed on the interpolating box with a given length, and this interpolation can reduce the redundancy degree of the cloud point data, such as in ‘Case 3’ in Fig. 3. However, the interpolation can partly lose the characteristic of the point cloud data if the interval value is too large, such as in ‘Case 1’ in Fig. 3. Moreover, the interpolation interval is not valid if no cloud point is in a box whose edge length is too small, such as in ‘Case 2’ in Fig. 3. The range of statistical number of cloud points in a given box is approximately ½X s; X þ s, where (X) is the means and s is the standard deviation under the given interpolation interval. The interpolation interval is valid in principle if X s P 1, because this interval can achieve the effect of interpolation and reduce the redundancy degree of the cloud point data. However, the interpolation interval is not valid if X s is too large, such as X s > 10, because this interval would damage the characteristic of the point cloud data for a rock joint. The statistical value for the 0.1-mm interpolation interval shows that the corresponding X s > 1 (Table 2). Therefore, considering the calculating power of the computer, the 0.1-mm interval was accepted for the subsequent analysis of the shape of the joint. Precisely, the above-described method still cannot yield the optimal resolution of the interpolation interval, although this elementary analysis can provide meaningful information for estimating an appropriate interval. A clear contour cloud map can be obtained with this interval (Fig. 4), and 2D profile lines in different positions can also be abstracted. With the normalized digital data at equal intervals, the statistical analysis of the joint shape can be easily implemented based on the characteristic indices, such as Z2 [60], Rs [61] and Zs2 [39] (Eqs. (5–7)): Table 1 Average intermediate heights under different interpolation intervals and corresponding deviations. Interval (mm)
Num. X/Y points
Ave. height (mm)
Deviation (%)
1.0 0.50 0.30 0.15 0.10 0.075 0.05 0.037 0.03
150 300 500 1000 1500 2000 3000 4000 5000
1.513423 1.511014 1.510026 1.509294 1.509048 1.508924 1.508833 1.508769 1.508750
– 0.160 0.070 0.048 0.020 0.008 0.006 0.004 0.001
Note: the deviation is defined as [(Ave. height)previous (Ave. height)current]/(Ave. height)current.
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Y
Case 1
Note: Inial cloud point Interpolated point Interval value is too large, the interpolaon partly lost the characteriscs of cloud point data
X O
Reduce the redundancy degree of cloud point data
Case 2
Case 3
Z
Interpolated interval
Interpolated line
Reasonable
X Interval value is too small, no inial cloud point for Interpolaon
O
More than one cloud point for Interpolaon
Fig. 3. Sketch map for estimating the reasonability of interval value for interpolation.
Table 2 Statistical values of different interpolation interval. Interval
Mean
Standard deviation
X±s
0.1 0.3 0.5 1.0
1.7984 16.1856 44.9599 179.8395
0.6006 2.5535 6.0861 21.7658
[2.3990, 1.1978] [18.7391, 13.6321] [51.046, 38.8738] [201.6053, 158.0737]
N1 X ðziþ1 zi Þ2 2 ðN 1ÞðDsÞ i¼1
1
Z2 ¼
!1=2 ð5Þ
Rs ¼ At =An
ð6Þ
" Z s2
¼
" 1 ðN x 1ÞðNy 1Þ
þ D1y2
1 D x2
N y 1N x 1 X X
y 1 N x 1N X X
JRCs ¼ 61:79ðZ 2 Þ 3:47:
ðziþ1;jþ1 zi;jþ1 Þ2 þðziþ1;j zi;j Þ2 2
i¼1 j¼1 ðziþ1;jþ1 ziþ1;j Þ2 þðzi;jþ1 zi;j Þ2 2
ð7Þ
##1=2
the number of points along the x-axis, Ny is the number of points along the y-axis, Dx and Dy are the sampling steps along the x and y axes, respectively, zi is the height of point i (i = 1, N) for the 2D joint surface and zi;j is the height at position (xi, yi) for the 3D joint surface. Comparing the abstracted 2D profile lines with Barton and Choubey’s standard map of JRC [10], it can be preliminarily determined that the JRC of the type of natural joint (Fig. 3) ranges from 10 to 14. Furthermore, statistical analysis for this joint showed that the corresponding Z2 was approximately 0.243, according to Eq. (5), and the corresponding statistical JRCs was approximately 11.9, according to the experimental expression of Yu and Vayssade (Eq. (8)) [62]:
;
j¼1 i¼1
where N is the number of points along the 2D profile line, Ds is the interval of points along the 2D profile line, At is the true surface area of the 3D joint, An is the nominal surface area of the 3D joint, N x is
ð8Þ
3. Replication of joint specimens Once the 3D virtual joint specimen has been built, a plastic joint mould can be produced using 3DP technology. Considering the printed joint mould as the parent template, many joint specimens with almost identical surface shapes can be replicated by casting concrete or other similar material into the joint mould.
Fig. 4. Natural joint’s contour cloud map and the corresponding 2D profile lines at different positions.
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3.1. 3DP for joint mould Currently, some general 3DP equipment and the corresponding printing materials have been developed for producing physical items using different forming methods [63]. Of them, 3D printers using fused deposition modelling (FDM) are widely used. The FDM printer contains a print head that acts like an X–Y–Z plotter. During printing, the raw material, mostly acrylonitrile butadiene styrene (ABS) or biodegradable polylactic acid (PLA), is melted in the print head and deposited layer-by-layer on the building table to produce the 3D object [64].
Before printing, the digital file of the virtual joint model constructed using CAD needs to be converted to a standard format that is readable by the printing software. In general, the standard template library or stereo lithography (STL) format is the current industry standard for printing models. The STL file represents the model using information about the coordinates and outward surface normal of the triangles. The basic printing parameters, such as the filled ratio, printing precision and layer thickness, should be set before the printer’s operating software reads the virtual model. The printing operation can start only when sufficient raw materials are available. In fact, this printing method realizes the
Prinng PLA mould
Fig. 5. Producing a PLA joint mould using a 3D printer.
(a)
(b)
=
+
Printed PLA mould
(c)
(d)
Fig. 6. Process of creating concrete specimens with natural joints based on the PLA mould. (a) Install the PLA mould, (b) cast the lower concrete block, (c) cast the upper concrete block, and (d) a pair of joint specimens with occluded upper and lower blocks.
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1:2:0.5 by weight is prepared as the raw material for all the replicas [6,15,31,39]. Then, this concrete is grouted into the iron box containing the joint mould (Fig. 6b). Once the concrete is set, the iron box is opened and the joint mould and concrete block are detached. The natural joint is thus cloned on the concrete block. Step 3: Casting the upper concrete joint block. The cast lower concrete block is replaced in the iron box with its joint surface oriented upwards. Then, the mixed high-strength concrete is grouted into the box after spraying a shield powder onto the joint surface of the concrete block (Fig. 6c and d). Similarly to the previous step, the iron box is opened, and the upper and lower concrete blocks are detached when the concrete is set. In this way, a pair of joint specimens with upper and lower blocks that fully couple with each other is obtained. The concrete specimens are carefully maintained in accordance with the general guidelines [64,65].
Preparaon • Clean joint surface • Spray white powder • Aach reference points
Digizaon joint surface • 3DS for rock joint • Merge point cloud • Adjust unique plane
Advantages: Save me and easy operaon Digitally stasc & save joint data
Physical replicaon • 3DP for joint mold • Cast concrete specimen
Advantages: Easily produce parent mould but save material Replicated specimens have the same shape
Fig. 7. Basic flowchart of reverse joint specimen modelling and the corresponding characteristics.
direct transformation of the digital joint model to a physical joint mould made of PLA material (Fig. 5).
Overall, the reverse modelling process of rock joints includes three basic technical sections: preparation of the natural joint sample, 3D digitization of the joint surface and physical replication of the joint sample (Fig. 7). In particular, the application of 3DS technology for the 3D digitization of the joint surface is beneficial in terms of time savings and digital data preservation.
3.2. Replication of the physical joint specimen In order to obtain mated replicas of natural rock joints, the upper or lower surface of the natural joint can be constructed using the aforementioned printed PLA mould with plaster or concrete. Then, the other surface of the joint replica is made by pouring the concrete onto the first surface. Replication of these joint specimens using the PLA mould includes the following main steps: Step 1: Installation of the PLA mould. The printed parent joint mould should be placed inside an iron box for casting (Fig. 6a). The height of the joint surface of the PLA mould should be half that of the iron box. In this way, the casted natural joint face will be on the middle of the specimen, which is convenient for the direct shear test. Step 2: Casting the lower concrete joint block. A high-strength concrete mixture of cement, sand and additives in a ratio of
4. Quality verification for reversed joint specimen Because the final reversed joint specimen is the result of 3DS, computer-aided reconstruction, 3DP and concrete casting, some differences in surface shape between the replicated and original joints exist, which become the magnitude of the error between the joints that in turn affects the experimental results. Therefore, a geometrical error analysis and a mechanical direct shear test of the replicated joint specimen are required to be performed. 4.1. Error analysis of the replicated joint surface Three models – the 3D original/virtual joint model, the printed PLA joint mould and the concrete joint specimen – are involved in the reverse modelling of a natural rock joint. First, a direct comparison of the 2D profile lines of the three models revealed that the
5
Surface height (mm)
Surface height (mm)
5 Original model
3
Printed PLA mold Concrete specimen
1 -1 -3 -5
Original model
3
Printed PLA mold Concrete specimen
1 -1 -3 -5
-80
-60
-40
-20
0
20
40
60
80
-80
-60
-40
Profile posion (mm)
Original model
3
Printed PLA mold Concrete specimen
1 -1 -3
-60
-40
-20
0
20
Profile posion (mm)
40
60
80
Surface height (mm)
Surface height (mm)
0
20
40
60
80
40
60
80
5
5
-5 -80
-20
Profile posion (mm)
Original model
3
Printed PLA mold Concrete specimen
1 -1 -3 -5 -80
-60
-40
-20
0
20
Profile posion (mm)
Fig. 8. Visual comparison of 2D profile lines between the three different models.
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Q. Jiang et al. / Computers and Geotechnics 73 (2016) 210–220 Table 3 Average heights of the unique plane and errors between different joint models. Avg. intermediate height (mm)
A (JRC = 11.9) B (JRC = 8.5)
Shearing force (kN)
(a)
Errors
Original 3D model
Printed PLA mould
Concrete joint specimen
Printed PLA mould
Concrete joint specimen
1.509 0.530
1.493 0.517
1.472 0.530
0.010 0.00080
0.024 0.022
20
(b)
16
Shearing force (kN)
Samples
12 8
Speciment A-1 Speciment A-2 Speciment A-3
4 0
0
1
2
3
4
5
6
7
Shear displacement (mm)
16 12 8 Speciment B-1 Speciment B-2 Speciment B-3
4 0
0
1
2
3
4
5
6
7
Shear displacement (mm)
Fig. 9. Typical shearing displacement–shearing force curves of two kinds of specimens containing joints. (a) Type A joint specimen, JRC = 11.9, and (b) type B joint specimen, JRC = 8.5.
1#
2#
3#
Fig. 10. Damaged surfaces of natural joint specimens under the same shearing experimental condition.
features of the concrete joint profile are closely associated with those of the original joint model and the PLA mould (Fig. 8). In order to quantitatively investigate the errors in joint morphology between these models, two joint specimens with JRCs 11.9 and 8.5 were selected. Following the reverse modelling method described earlier, a corresponding virtual joint model, a PLA joint mould and a concrete joint specimen were made for each natural specimen. The digital surface data of the printed PLA mould
and the concrete joint specimen were also obtained using 3DS. Assuming the average height of the joint’s unique plane as the index (Eq. (4)), the corresponding average heights of the unique plane and errors between different models can be calculated (Table 3). Table 3 shows that for joints A and B, the errors in average height between the original 3D model and the printed PLA mould were 1 and 0.8%, respectively, and that the errors between the orig-
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inal 3D model and the concrete joint specimen were 2.4% and 2.2%, respectively. This investigation indicated that the reverse modelling method for natural rock joints based on 3DS and 3DP technologies was fairly precise, making it successful in overcoming the problem of creating replicates of joint samples with the same surface shape. 4.2. Direct shear test In order to check the mechanical stability and uniformity of the joint specimens produced by this method, a direct shear test was performed on the rock shear machine, which is a servo-control experiment system. Two types of joint specimens – type A joint (JRC = 11.9) and type B joint (JRC = 8.5) – were used for this test. During the testing procedure, the normal stress on the joint and the shearing speed were set to 0.5 MPa and 0.01 mm/s, respectively. Three direct shear tests were performed for each type of natural joint, whose shearing displacement–force curves are shown in Fig. 9. The experimental results showed that the shearing displacement–force curves of the type A and type B joint specimens were identical. The maximum deviation of shearing strength (individual peak force/average peak force) for the type A joint specimen was approximately 6.7%, and that for the type B joint specimen was approximately 4.7%. In general, other direct shear tests of replicated rock joints have indicated that the common deviation in shearing strength due to objective factors is approximately 8– 20% [66–69]; our experimental results have smaller deviations. This above comparison indicates that this method can increase the accuracy of the shearing test results, because all the objective deviations induced by the random shearing failure of joint surfaces cannot be avoided. Moreover, the actual damage positions of the three pairs of shearing joint blocks were also the same (Fig. 10). These mechanical testing results indicate that our modelling method for natural joints is highly uniform and precise, favouring the summarization of experimental rule. This verification indicated that the reverse joint specimens created using the 3DS and 3DP technologies have small modelling errors and uniform material strength, and they produce the same mechanical results, which can obviously improve the quality of replications of natural joints and facilitate the further study of the mechanism of the formation of natural joints. 5. Discussion and conclusion In this study, a new method for replicating natural joint specimens was described, which uses two advanced techniques – 3DS and 3DP – with CAD as the ‘bridge’. The mechanisms underlying the modelling process for natural joints are the preparation of the natural joint sample, 3D digitization of the joint surface by white-light scanning and physical replication of the joint sample by casting the concrete into a 3DP parent mould. The 3D morphology of a natural joint can be easily and accurately digitized in the laboratory or field using the 3DS technology, particularly when the joint sample cannot be get back to the laboratory. The obtained 3D digital data provide a convenient data set, which can be used to quantitatively analyze and calculate the characteristic indices for the natural joint, such as the roughness, magnitude, angularity, undularity and anisotropy. With the help of CAD and 3DP technology, the virtual 3D joint model can be reconstructed and printed based on the scanned digital surface of the natural joint. Moreover, 3DP can directly transform the digital joint model to a physical mould. This means
that any imaged joint specimen can be created with a given surface shape and material strength, in accordance with the experimental specifications and regardless of the difficulties in processing the testing samples. Because the printed parent mould and the original joint are identical in their shapes and the concrete material for the joint specimen is homogeneous, the replicated joint specimens are almost identical. Consequently, the experimental results from these joint specimens are stable and uniform, enhancing the experimental knowledge and avoiding the introduction of random error caused by differences among specimens, simultaneously. Quality verification of the replicated joint specimens indicated that this method causes little shape loss from the original natural joint, and the corresponding experimental results for the produced joint specimens confirmed that not only the shearing force–displacement curves but also the failure position remained uniform under similar testing conditions. All these verifying tests clearly indicate that this method can reduce the experimental errors caused by variations in the specimens themselves and directly facilitate the further study of the mechanism of the formation of natural joints by providing abundant specimens. Acknowledgement The authors gratefully acknowledge the financial support from National Natural Science Foundation of China (Grant Nos. 51379202 and 41172284). Appendix. A %MATLAB program of ‘‘Inverse Distance Weighted Interpolation Method (IDW)”for interpolation calculation according to Data file ‘data.txt’. clear all load data.txt% Input the spatial point cloud data [n,c,p]=size(data); fori=1:n X(i,1)=data(i,1); X(i,2)=data(i,2); X(i,3)=data(i,3); xw(i)=X(i,1); yw(i)=X(i,2); end dx=input(‘Input interpolating interval in X direction’) dy=input(‘Input interpolating interval in Y direction’) R=input(‘Deciding the searching radius’) %’Searching radius’ depends on the interpolating interval, 3D scanningprecision, joint’s morphology, etc. P=input(‘Input index for weighting function’) % P=1, 2 or 3 xwmax=max(xw); xwmin=min(xw); ywmax=max(yw); ywmin=min(yw); nx=round((xwmax-xwmin)./dx+1); ny=round((ywmax-ywmin)./dy+1); fori=1:nx for j=1:ny w((i-1).⁄ny+j,1)=xwmin+(i-1).⁄dx; w((i-1).⁄ny+j,2)=ywmin+(j-1).⁄dy; end end % Calculating the distances for interpolating position fori=1:(nx.⁄ny) for j=1:n
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dx=abs(X(j,1)-w(i,1)); dy=abs(X(j,2)-w(i,2)); dxy(j)=sqrt(dx.⁄dx+dy.⁄dy); end [dij,bj]=sort(dxy); [s,t]=size(dij); di=[]; for k=1:t ifdij(k)==0 di(k,1)=1; di(k,2)=bj(k+1); elseifdij(k)<=R di(k,1)=1./(dij(k+1))^P; di(k,2)=X(bj(k+1),3)./(dij(k+1))^P; end end sum1=sum(di(:,1)); [p,q]=size(di); for g=1:p temp(g)=di(g,2)./sum1; end w(i,3)=sum(temp); end dlmwrite(‘outdata.txt’, w, ‘delimiter’,’ ‘,’precision’, ‘%10.5f’) % Output the result %--- interpolation function provided by MATLAB software ---%[XI,YI,ZI1] = griddata(X(:,1),X(:,2),X(:,3),Pmx(:,1),Pmy(:,2),’ method’); %--- method: linear, cubic, nearest, v4---
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Technical Communication
Reverse modelling of natural rock joints using 3D scanning and 3D printing Quan Jiang ⇑, Xiating Feng, Yanhua Gong, Leibo Song, Shuguang Ran, Jie Cui State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 17 March 2015 Received in revised form 18 November 2015 Accepted 20 November 2015 Available online 8 January 2016 Keywords: Natural joint Joint replication 3D scanning 3D printing Shearing experiment
a b s t r a c t In order to overcome the deficiency of natural joint specimens with the same surface morphology for experimental studies, we present a technical method for replicating natural joint specimens that incorporates two advanced techniques – three-dimensional (3D) scanning and 3D printing – using computer-aided design (CAD) as the ‘bridge’. This method uses an optical scanning apparatus and CAD techniques to reconstruct a virtual joint specimen with the natural rock’s joint morphology, and a 3D printer then manufactures a physical mould based on the virtual joint specimen for casting concrete or plastic specimens quickly and accurately. Quality verification clearly indicated that this method reduces the experimental errors originating from the discrepancies between replicating specimens containing natural joint’s morphology. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The strength and stability of rock masses are significantly influenced by the presence of inherent joints and fractures. As has been thoroughly demonstrated, shearing failure is one of the common features of rock masses, and hence the shear strength that affects the deformation along natural joints or discontinuities is crucial to the stability of geotechnical engineering projects, such as underground tunnels, rock slopes and open-cut mines [1–7]. In the first ISRM International Congress on Rock Mechanics in 1966, Muller stated, ‘The deformation resistance of the material bridges takes effect at much smaller deformations than the joint friction: this joint friction makes up partly for lost strength’ [8,9]. In the recent years, the frictional behaviour of rock joints has been of interest to many researchers and engineers [10–15]. Shear and compression experiments are the most important ways to understand the mechanical behaviours of natural rock joints, such as shearing stress–displacement curves with peak strength, non-uniform shearing damage of natural surfaces and non-linear normal deformation during compression [16–20]. On the basis of these laboratory and field tests, several shearing strength criteria have been proposed to identify the strength of a natural rock joint, such as Patton’s model [21], Ladanyi’s empirical model [1], Barton’s empirical model [10], Amadei–Saeb’s analytical model [22,23] and Grasselli’s threedimensional (3D) model [24]. All of these distinguished works have
⇑ Corresponding author. Tel.: +86 27 87198805. E-mail address: [email protected] (Q. Jiang). http://dx.doi.org/10.1016/j.compgeo.2015.11.020 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved.
indicated that the surface morphology of a rock joint, quantified as joint roughness coefficient (JRC), plays a key role in its shearing strength. During the experimental investigation of natural rock joints, a good experimental procedure requires that the variables in the experiment can be controlled such that only one variable can be isolated and selectively changed. Because no two natural rock joint samples, even those from the same deposit, are truly identical, classical experiments cannot be conducted if the testing scheme requires multiple specimens [25–28]. Therefore, the lack of sufficient joint specimens with the same natural surface morphology has always limited experimental studies. At present, joint samples are mainly produced using one of the following three general techniques [6,14,17,29–35]: (i) tensile fractures, usually made in a manner similar to the Brazilian test, (ii) sawn flat joints with undulated or irregular surfaces or (iii) casts of natural or stylized joints with silicon moulds, mated silicon rubber and aluminium moulds. Despite the widespread acceptance of the joint specimens produced using these techniques, there is an empirical concern that these methods lead to objective errors and increased time consumption, because most of these methods are unable to represent natural joints with complicated and irregular surface shapes or digitize their surface. This does not mean that the aforementioned traditional methods are inappropriate for rock joint behaviour studies, but rather they are more appropriate for local application with a lesser quantity than as the basis of a universal system [32,36–38]. Moreover, these traditional methods are inconvenient for data collection and statistical analysis for joint shapes. In fact,
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the current dilemma in the ways of producing duplicate specimens has reduced the interest towards intensive studies on the mechanical properties of natural rock joints. The recently developed 3D scanning (3DS) and 3D printing (3DP) techniques, combined with computer-aided design (CAD), acting as a bridge connecting 3DS and 3DP, provide a new way to manufacture experimental specimens with the same irregular joint morphology. The 3DS apparatus can help digitize and characterize the joint surfaces in three dimensions quickly, with high precision and no damage. On the basis of these digitization data for the natural joint, some 3D roughness parameters can be developed to characterize the joint surfaces and overcome the drawbacks of two-dimensional (2D) profiles [39–44]. Furthermore, 3DP, that is, additive manufacturing or additive layer manufacturing, allows the automated generation of free-form solids directly from a computer file to a real object [45–47]. This 3DP provides a new way to make a physical mould with the morphology of a natural joint surface, because this technology can be used to print a joint surface with the same shape as the original joint. In addition, CAD can help reconstruct a virtual model body with any shape for experimental use from the scanned data of the natural joint surface. The clonal joint mould can be produced by 3DP using CAD technology. Therefore, on the basis of the printed joint mould, massive experimental specimens with the same natural joint morphology can be replicated for shearing or compression tests. In this study, a new method for producing natural joint specimens that combines 3DS and 3DP techniques is presented. First, an optical scanning apparatus is used to digitize the 3D surface of a natural rock joint. With the resulting point cloud data, a virtual joint specimen with the natural joint’s morphology and given boundaries is reconstructed using CAD methods. The 3D printer then manufactures a physical mould quickly and accurately based on the virtual specimen. In this way, massive joint specimens can be replicated using concrete or plastic materials according to the printed mould. The error analysis between the original joint data, the printed joint surface and the actual surface of the physical specimen indicates that this reverse method can copy the natural morphology from the original rock joint to the artificial joint specimen with high efficiency and precision. Furthermore, direct shear tests show that the shearing displacement–force curves of the replicated joint specimens were similar to each other with little dispersion. These results clearly indicate that the presented method can reduce the experimental errors caused by the specimens themselves, and open a new door to further experimental study of natural joints.
tages of high precision and good repeatability as well as being fast and easy to use. Its measuring system consists of a central projector unit and two charge-coupled device (CCD) cameras, which are linked to a personal computer to drive the system and save the scanned data, respectively (Fig. 1). The working theory of the Holon3D system is based on the white-light fringe patterns with multiple-frequency phase shift grating for the joint surface. The images of these patterns, which become distorted because of the disturbance of the surface, are captured by the two CCD cameras. With the given relative offset and angle between the projector and the cameras, the corresponding software computes the 3D coordinates for each pixel in these images using the principle of triangulation [24,44,56]. 2.1. Digitization and reconstruction procedure of 3D joint surface The digitization and reconfiguration procedure for a virtual joint surface using the 3DS equipment and the computer-aided software includes four main steps, which are described as follows. Step 1: Pretreatment of the joint surface. Once the joint surface has been cleaned, a thin white toner powder needs to be sprayed on it to improve the reflectivity of the white light emitted from the central projector and the absorptivities of the two CCD cameras. The scanning system can calculate 3D coordinates for the visible pixels in only a limited area, within 10 cm in length. Therefore, the digitization of a joint surface typically requires several individual measurements from different positions. In order to combine the individually measured data clouds in a common point cloud using a global coordinate system, the reference points (adhesive circular black–white markers) need to be adhered to the surface of the natural joint (Fig. 2a). The distance between the markers should be onetenth to one-fifth of the length of the scanning window for the successful integration of the point clouds. Step 2: 3DS and data integration. When the joint specimen is placed under the measuring window of the scanning apparatus, a portion of the point cloud can be obtained by activating the scanner controlled by the corresponding software. If the region of interest of the joint surface is larger than the size of the measuring window, it is necessary to acquire multiple measurements by moving the specimen in a linear or curved manner (Fig. 2b). In principle, it is very important for each measuring window to include at least three reference points that overlap with the previous window. The scanning software can then automatically establish a global coordinate system and identify
2. 3D virtual reconstruction of natural joint Projecon unit
The general methods for determining the joint shape include contact and noncontact approaches. The contact approach requires the operator or a needle to physically touch the surface and record the height of the joint along the selected profiles or over a predefined area [48–51]. These early-contact methods have some drawbacks, because they are time-consuming in terms of obtaining the data and are insufficiently precise when recording at locally steep locations. The non-contact approaches, including the analytical photogrammetric method, He–Ne laser beam technique, laser scanning method and white-light fringe technology, use a technique to make the measurements without physically touching the fracture surfaces [52–56]. These methods have greatly increased the speed and accuracy of roughness measurements. In this study, the Holon3D system was used to digitize the entire rock joint surfaces using non-contact area scanning. This system, with an accuracy of ±0.01 mm over a region of 100 100 50 mm in a measuring time of <3 s, offers the advan-
Camera_1
Camera_2
White-light fringe
Rock joint
Controlling computer
Fig. 1. Working set-up of the scanner under the white-light fringe pattern.
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3 mm
(a)
1 mm
(b)
Integrate
A piece of point cloud
(c)
Z (mm)
Adjust
Y (m X (mm)
m)
(d)
CAD modeling
Fig. 2. Digitization and reconstruction procedure for a 3D joint surface. (a) Joint sample and attached reference points, (b) multiple pieces of point cloud and the merged common point cloud representing the joint, (c) adjusting the initial intermediate plane to the horizontal unique plane, and (d) 3D reconstruction of the natural joint.
all the reference points included in the window. For all the following point cloud measurements, the system can automatically recognize the new and predefined reference points and use them to automatically adjust the new measurement data into the global coordinate system. Then, a global integration of point clouds is performed with the help of the supporting software by cleaning a few noise points induced by the
scanning errors, and then merging the overlaps of multiplepoint cloud collections. The major contributors of the noise points are the unstable light reflectivity and environmental disturbance. In general, noise points are small in number (several hundred) and the number of point clouds is approximately several million. Thus, the effect of noise points on the global precision of the measurement is very little.
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Step 3: Adjusting the common plane. The common/intermediate best-fit plane (O–X–Y–Z) of the obtained point cloud is initially inclined because the irregular joint surface is not horizontal and the specimen is moved frequently during the procedure [43,57]. Thus, adjusting the common plane for the merged point cloud from an inclined to a horizontal position is necessary for the subsequent reconstruction of the 3D joint specimen with CAD, whose expression is provided in Eq. (1) (Fig. 2c). Considering this changed unique plane as the reference plane, the incision of the joint surface is more convenient, and the statistical results for the surface data are more stable and universal [43]. Here, a matrix with 4 4 formation is needed when operations with transformation and rotation for a graph by the matrix multiplication (Eq. (1)).
Pu ¼ Pðx; y; z; 1Þ M R 2
1 0
60 1 6 M¼6 40 0 0
0 0 1
0
ð1Þ
3
07 7 7 05
ð2Þ
0 a 1 3
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 þ1 b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 b2 þc2 þ1 6 ðb2 þc2 þ1Þ 6 bc 1 ffi c 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 ðc2 þ1Þðb2 þc2 þ1Þ c2 þ1 ðb2 þc2 þ1Þ R¼6 6 b c ffi 1 ffi pffiffiffiffiffiffiffi 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ðc2 þ1Þðb2 þc2 þ1Þ c2 þ1 ðb2 þc2 þ1Þ 0 0 0
0
7 7 07 7 7; 7 07 5 1
ð3Þ
where a, b and c are the parameters of the initially inclined plane equation (z ¼ a þ bx þ cy), Pðx; y; z; 1Þ is the initial position of a 3D point with added switching component ‘1’, P u is the 3D point’s adjusted position relative to the horizontal unique plane and M and R are the transformational and rotational matrices [58], respectively. Step 4: Reconstruction of the 3D virtual specimen. With the current CAD technology, the joint surface can be reconstructed from the point clouds using a specifically developed triangulated irregular network (TIN) mesh [24,29]. This approach discretizes the point cloud data of the joint surface into contiguous triangles, which are defined by their vertices and the orientation of the vector normal to the triangular plane surface. Then, a 3D virtual body containing the natural joint surface is obtained by adding other sides (Fig. 2d). This method of discretizing the joint surfaces is particularly advantageous for building a virtual 3D model.
2.2. Characterization of the point cloud data of a natural joint Although the above-described method accomplishes the transition from the natural joint specimen to the virtual joint model using digitization and the CAD technique, a second treatment of the data obtained for the joint surface is still required to further estimate other morphological characteristics, such as the magnitude of the joint, angularity, undularity and anisotropy. The spacing of the joint’s point cloud obtained with the 3D scanner is a variable, because of irregular optical reflection of small surfaces with irregular inclination and size. Therefore, the output data for quantitative analysis should be normalized from the original point cloud into constant data point intervals using a mathematical interpolation program (e.g. Griddata function of Matlab [59] or Inverse Distance Weighted Interpolation, seeing the Appendix A). Non-linear interpolation is generally adopted for this process. Consideration of an appropriate interpolation interval is important to avoid information loss during the data collection and analysis of
the joint’s digital morphology, because a too large point interval would damage the data’s usefulness and a too small point interval would create large file sizes and long processing times. Two methods for estimating the appropriate interpolation interval are presented in this study for the ‘Z’ direction and the ‘X–Y plane’. Considering the sample depicted in Fig. 2a, a tentative comparison was first performed among different interpolation intervals ranging from 0.03 to 1.0 mm along the axes ‘x’ and ‘y’. The statistical average intermediate height of the unique plane is used as a valuating index (Eq. (4)):
D¼
N 1X jzi j: N i¼1
ð4Þ
The deviations of average intermediate heights under different interpolation intervals are listed in Table 1. The gradient of average intermediate heights tended to decrease with decreasing interval value. When the interpolation interval was set to 0.1 mm, the corresponding gradient of average intermediate height was approximately 0.02%. Then, the distribution density in the ‘X–Y plane’ was also analyzed. In general, the effective height information (‘Z’) can be considered during the interpolation if more than one original cloud point existed on the interpolating box with a given length, and this interpolation can reduce the redundancy degree of the cloud point data, such as in ‘Case 3’ in Fig. 3. However, the interpolation can partly lose the characteristic of the point cloud data if the interval value is too large, such as in ‘Case 1’ in Fig. 3. Moreover, the interpolation interval is not valid if no cloud point is in a box whose edge length is too small, such as in ‘Case 2’ in Fig. 3. The range of statistical number of cloud points in a given box is approximately ½X s; X þ s, where (X) is the means and s is the standard deviation under the given interpolation interval. The interpolation interval is valid in principle if X s P 1, because this interval can achieve the effect of interpolation and reduce the redundancy degree of the cloud point data. However, the interpolation interval is not valid if X s is too large, such as X s > 10, because this interval would damage the characteristic of the point cloud data for a rock joint. The statistical value for the 0.1-mm interpolation interval shows that the corresponding X s > 1 (Table 2). Therefore, considering the calculating power of the computer, the 0.1-mm interval was accepted for the subsequent analysis of the shape of the joint. Precisely, the above-described method still cannot yield the optimal resolution of the interpolation interval, although this elementary analysis can provide meaningful information for estimating an appropriate interval. A clear contour cloud map can be obtained with this interval (Fig. 4), and 2D profile lines in different positions can also be abstracted. With the normalized digital data at equal intervals, the statistical analysis of the joint shape can be easily implemented based on the characteristic indices, such as Z2 [60], Rs [61] and Zs2 [39] (Eqs. (5–7)): Table 1 Average intermediate heights under different interpolation intervals and corresponding deviations. Interval (mm)
Num. X/Y points
Ave. height (mm)
Deviation (%)
1.0 0.50 0.30 0.15 0.10 0.075 0.05 0.037 0.03
150 300 500 1000 1500 2000 3000 4000 5000
1.513423 1.511014 1.510026 1.509294 1.509048 1.508924 1.508833 1.508769 1.508750
– 0.160 0.070 0.048 0.020 0.008 0.006 0.004 0.001
Note: the deviation is defined as [(Ave. height)previous (Ave. height)current]/(Ave. height)current.
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Y
Case 1
Note: Inial cloud point Interpolated point Interval value is too large, the interpolaon partly lost the characteriscs of cloud point data
X O
Reduce the redundancy degree of cloud point data
Case 2
Case 3
Z
Interpolated interval
Interpolated line
Reasonable
X Interval value is too small, no inial cloud point for Interpolaon
O
More than one cloud point for Interpolaon
Fig. 3. Sketch map for estimating the reasonability of interval value for interpolation.
Table 2 Statistical values of different interpolation interval. Interval
Mean
Standard deviation
X±s
0.1 0.3 0.5 1.0
1.7984 16.1856 44.9599 179.8395
0.6006 2.5535 6.0861 21.7658
[2.3990, 1.1978] [18.7391, 13.6321] [51.046, 38.8738] [201.6053, 158.0737]
N1 X ðziþ1 zi Þ2 2 ðN 1ÞðDsÞ i¼1
1
Z2 ¼
!1=2 ð5Þ
Rs ¼ At =An
ð6Þ
" Z s2
¼
" 1 ðN x 1ÞðNy 1Þ
þ D1y2
1 D x2
N y 1N x 1 X X
y 1 N x 1N X X
JRCs ¼ 61:79ðZ 2 Þ 3:47:
ðziþ1;jþ1 zi;jþ1 Þ2 þðziþ1;j zi;j Þ2 2
i¼1 j¼1 ðziþ1;jþ1 ziþ1;j Þ2 þðzi;jþ1 zi;j Þ2 2
ð7Þ
##1=2
the number of points along the x-axis, Ny is the number of points along the y-axis, Dx and Dy are the sampling steps along the x and y axes, respectively, zi is the height of point i (i = 1, N) for the 2D joint surface and zi;j is the height at position (xi, yi) for the 3D joint surface. Comparing the abstracted 2D profile lines with Barton and Choubey’s standard map of JRC [10], it can be preliminarily determined that the JRC of the type of natural joint (Fig. 3) ranges from 10 to 14. Furthermore, statistical analysis for this joint showed that the corresponding Z2 was approximately 0.243, according to Eq. (5), and the corresponding statistical JRCs was approximately 11.9, according to the experimental expression of Yu and Vayssade (Eq. (8)) [62]:
;
j¼1 i¼1
where N is the number of points along the 2D profile line, Ds is the interval of points along the 2D profile line, At is the true surface area of the 3D joint, An is the nominal surface area of the 3D joint, N x is
ð8Þ
3. Replication of joint specimens Once the 3D virtual joint specimen has been built, a plastic joint mould can be produced using 3DP technology. Considering the printed joint mould as the parent template, many joint specimens with almost identical surface shapes can be replicated by casting concrete or other similar material into the joint mould.
Fig. 4. Natural joint’s contour cloud map and the corresponding 2D profile lines at different positions.
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3.1. 3DP for joint mould Currently, some general 3DP equipment and the corresponding printing materials have been developed for producing physical items using different forming methods [63]. Of them, 3D printers using fused deposition modelling (FDM) are widely used. The FDM printer contains a print head that acts like an X–Y–Z plotter. During printing, the raw material, mostly acrylonitrile butadiene styrene (ABS) or biodegradable polylactic acid (PLA), is melted in the print head and deposited layer-by-layer on the building table to produce the 3D object [64].
Before printing, the digital file of the virtual joint model constructed using CAD needs to be converted to a standard format that is readable by the printing software. In general, the standard template library or stereo lithography (STL) format is the current industry standard for printing models. The STL file represents the model using information about the coordinates and outward surface normal of the triangles. The basic printing parameters, such as the filled ratio, printing precision and layer thickness, should be set before the printer’s operating software reads the virtual model. The printing operation can start only when sufficient raw materials are available. In fact, this printing method realizes the
Prinng PLA mould
Fig. 5. Producing a PLA joint mould using a 3D printer.
(a)
(b)
=
+
Printed PLA mould
(c)
(d)
Fig. 6. Process of creating concrete specimens with natural joints based on the PLA mould. (a) Install the PLA mould, (b) cast the lower concrete block, (c) cast the upper concrete block, and (d) a pair of joint specimens with occluded upper and lower blocks.
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1:2:0.5 by weight is prepared as the raw material for all the replicas [6,15,31,39]. Then, this concrete is grouted into the iron box containing the joint mould (Fig. 6b). Once the concrete is set, the iron box is opened and the joint mould and concrete block are detached. The natural joint is thus cloned on the concrete block. Step 3: Casting the upper concrete joint block. The cast lower concrete block is replaced in the iron box with its joint surface oriented upwards. Then, the mixed high-strength concrete is grouted into the box after spraying a shield powder onto the joint surface of the concrete block (Fig. 6c and d). Similarly to the previous step, the iron box is opened, and the upper and lower concrete blocks are detached when the concrete is set. In this way, a pair of joint specimens with upper and lower blocks that fully couple with each other is obtained. The concrete specimens are carefully maintained in accordance with the general guidelines [64,65].
Preparaon • Clean joint surface • Spray white powder • Aach reference points
Digizaon joint surface • 3DS for rock joint • Merge point cloud • Adjust unique plane
Advantages: Save me and easy operaon Digitally stasc & save joint data
Physical replicaon • 3DP for joint mold • Cast concrete specimen
Advantages: Easily produce parent mould but save material Replicated specimens have the same shape
Fig. 7. Basic flowchart of reverse joint specimen modelling and the corresponding characteristics.
direct transformation of the digital joint model to a physical joint mould made of PLA material (Fig. 5).
Overall, the reverse modelling process of rock joints includes three basic technical sections: preparation of the natural joint sample, 3D digitization of the joint surface and physical replication of the joint sample (Fig. 7). In particular, the application of 3DS technology for the 3D digitization of the joint surface is beneficial in terms of time savings and digital data preservation.
3.2. Replication of the physical joint specimen In order to obtain mated replicas of natural rock joints, the upper or lower surface of the natural joint can be constructed using the aforementioned printed PLA mould with plaster or concrete. Then, the other surface of the joint replica is made by pouring the concrete onto the first surface. Replication of these joint specimens using the PLA mould includes the following main steps: Step 1: Installation of the PLA mould. The printed parent joint mould should be placed inside an iron box for casting (Fig. 6a). The height of the joint surface of the PLA mould should be half that of the iron box. In this way, the casted natural joint face will be on the middle of the specimen, which is convenient for the direct shear test. Step 2: Casting the lower concrete joint block. A high-strength concrete mixture of cement, sand and additives in a ratio of
4. Quality verification for reversed joint specimen Because the final reversed joint specimen is the result of 3DS, computer-aided reconstruction, 3DP and concrete casting, some differences in surface shape between the replicated and original joints exist, which become the magnitude of the error between the joints that in turn affects the experimental results. Therefore, a geometrical error analysis and a mechanical direct shear test of the replicated joint specimen are required to be performed. 4.1. Error analysis of the replicated joint surface Three models – the 3D original/virtual joint model, the printed PLA joint mould and the concrete joint specimen – are involved in the reverse modelling of a natural rock joint. First, a direct comparison of the 2D profile lines of the three models revealed that the
5
Surface height (mm)
Surface height (mm)
5 Original model
3
Printed PLA mold Concrete specimen
1 -1 -3 -5
Original model
3
Printed PLA mold Concrete specimen
1 -1 -3 -5
-80
-60
-40
-20
0
20
40
60
80
-80
-60
-40
Profile posion (mm)
Original model
3
Printed PLA mold Concrete specimen
1 -1 -3
-60
-40
-20
0
20
Profile posion (mm)
40
60
80
Surface height (mm)
Surface height (mm)
0
20
40
60
80
40
60
80
5
5
-5 -80
-20
Profile posion (mm)
Original model
3
Printed PLA mold Concrete specimen
1 -1 -3 -5 -80
-60
-40
-20
0
20
Profile posion (mm)
Fig. 8. Visual comparison of 2D profile lines between the three different models.
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Q. Jiang et al. / Computers and Geotechnics 73 (2016) 210–220 Table 3 Average heights of the unique plane and errors between different joint models. Avg. intermediate height (mm)
A (JRC = 11.9) B (JRC = 8.5)
Shearing force (kN)
(a)
Errors
Original 3D model
Printed PLA mould
Concrete joint specimen
Printed PLA mould
Concrete joint specimen
1.509 0.530
1.493 0.517
1.472 0.530
0.010 0.00080
0.024 0.022
20
(b)
16
Shearing force (kN)
Samples
12 8
Speciment A-1 Speciment A-2 Speciment A-3
4 0
0
1
2
3
4
5
6
7
Shear displacement (mm)
16 12 8 Speciment B-1 Speciment B-2 Speciment B-3
4 0
0
1
2
3
4
5
6
7
Shear displacement (mm)
Fig. 9. Typical shearing displacement–shearing force curves of two kinds of specimens containing joints. (a) Type A joint specimen, JRC = 11.9, and (b) type B joint specimen, JRC = 8.5.
1#
2#
3#
Fig. 10. Damaged surfaces of natural joint specimens under the same shearing experimental condition.
features of the concrete joint profile are closely associated with those of the original joint model and the PLA mould (Fig. 8). In order to quantitatively investigate the errors in joint morphology between these models, two joint specimens with JRCs 11.9 and 8.5 were selected. Following the reverse modelling method described earlier, a corresponding virtual joint model, a PLA joint mould and a concrete joint specimen were made for each natural specimen. The digital surface data of the printed PLA mould
and the concrete joint specimen were also obtained using 3DS. Assuming the average height of the joint’s unique plane as the index (Eq. (4)), the corresponding average heights of the unique plane and errors between different models can be calculated (Table 3). Table 3 shows that for joints A and B, the errors in average height between the original 3D model and the printed PLA mould were 1 and 0.8%, respectively, and that the errors between the orig-
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inal 3D model and the concrete joint specimen were 2.4% and 2.2%, respectively. This investigation indicated that the reverse modelling method for natural rock joints based on 3DS and 3DP technologies was fairly precise, making it successful in overcoming the problem of creating replicates of joint samples with the same surface shape. 4.2. Direct shear test In order to check the mechanical stability and uniformity of the joint specimens produced by this method, a direct shear test was performed on the rock shear machine, which is a servo-control experiment system. Two types of joint specimens – type A joint (JRC = 11.9) and type B joint (JRC = 8.5) – were used for this test. During the testing procedure, the normal stress on the joint and the shearing speed were set to 0.5 MPa and 0.01 mm/s, respectively. Three direct shear tests were performed for each type of natural joint, whose shearing displacement–force curves are shown in Fig. 9. The experimental results showed that the shearing displacement–force curves of the type A and type B joint specimens were identical. The maximum deviation of shearing strength (individual peak force/average peak force) for the type A joint specimen was approximately 6.7%, and that for the type B joint specimen was approximately 4.7%. In general, other direct shear tests of replicated rock joints have indicated that the common deviation in shearing strength due to objective factors is approximately 8– 20% [66–69]; our experimental results have smaller deviations. This above comparison indicates that this method can increase the accuracy of the shearing test results, because all the objective deviations induced by the random shearing failure of joint surfaces cannot be avoided. Moreover, the actual damage positions of the three pairs of shearing joint blocks were also the same (Fig. 10). These mechanical testing results indicate that our modelling method for natural joints is highly uniform and precise, favouring the summarization of experimental rule. This verification indicated that the reverse joint specimens created using the 3DS and 3DP technologies have small modelling errors and uniform material strength, and they produce the same mechanical results, which can obviously improve the quality of replications of natural joints and facilitate the further study of the mechanism of the formation of natural joints. 5. Discussion and conclusion In this study, a new method for replicating natural joint specimens was described, which uses two advanced techniques – 3DS and 3DP – with CAD as the ‘bridge’. The mechanisms underlying the modelling process for natural joints are the preparation of the natural joint sample, 3D digitization of the joint surface by white-light scanning and physical replication of the joint sample by casting the concrete into a 3DP parent mould. The 3D morphology of a natural joint can be easily and accurately digitized in the laboratory or field using the 3DS technology, particularly when the joint sample cannot be get back to the laboratory. The obtained 3D digital data provide a convenient data set, which can be used to quantitatively analyze and calculate the characteristic indices for the natural joint, such as the roughness, magnitude, angularity, undularity and anisotropy. With the help of CAD and 3DP technology, the virtual 3D joint model can be reconstructed and printed based on the scanned digital surface of the natural joint. Moreover, 3DP can directly transform the digital joint model to a physical mould. This means
that any imaged joint specimen can be created with a given surface shape and material strength, in accordance with the experimental specifications and regardless of the difficulties in processing the testing samples. Because the printed parent mould and the original joint are identical in their shapes and the concrete material for the joint specimen is homogeneous, the replicated joint specimens are almost identical. Consequently, the experimental results from these joint specimens are stable and uniform, enhancing the experimental knowledge and avoiding the introduction of random error caused by differences among specimens, simultaneously. Quality verification of the replicated joint specimens indicated that this method causes little shape loss from the original natural joint, and the corresponding experimental results for the produced joint specimens confirmed that not only the shearing force–displacement curves but also the failure position remained uniform under similar testing conditions. All these verifying tests clearly indicate that this method can reduce the experimental errors caused by variations in the specimens themselves and directly facilitate the further study of the mechanism of the formation of natural joints by providing abundant specimens. Acknowledgement The authors gratefully acknowledge the financial support from National Natural Science Foundation of China (Grant Nos. 51379202 and 41172284). Appendix. A %MATLAB program of ‘‘Inverse Distance Weighted Interpolation Method (IDW)”for interpolation calculation according to Data file ‘data.txt’. clear all load data.txt% Input the spatial point cloud data [n,c,p]=size(data); fori=1:n X(i,1)=data(i,1); X(i,2)=data(i,2); X(i,3)=data(i,3); xw(i)=X(i,1); yw(i)=X(i,2); end dx=input(‘Input interpolating interval in X direction’) dy=input(‘Input interpolating interval in Y direction’) R=input(‘Deciding the searching radius’) %’Searching radius’ depends on the interpolating interval, 3D scanningprecision, joint’s morphology, etc. P=input(‘Input index for weighting function’) % P=1, 2 or 3 xwmax=max(xw); xwmin=min(xw); ywmax=max(yw); ywmin=min(yw); nx=round((xwmax-xwmin)./dx+1); ny=round((ywmax-ywmin)./dy+1); fori=1:nx for j=1:ny w((i-1).⁄ny+j,1)=xwmin+(i-1).⁄dx; w((i-1).⁄ny+j,2)=ywmin+(j-1).⁄dy; end end % Calculating the distances for interpolating position fori=1:(nx.⁄ny) for j=1:n
Q. Jiang et al. / Computers and Geotechnics 73 (2016) 210–220
dx=abs(X(j,1)-w(i,1)); dy=abs(X(j,2)-w(i,2)); dxy(j)=sqrt(dx.⁄dx+dy.⁄dy); end [dij,bj]=sort(dxy); [s,t]=size(dij); di=[]; for k=1:t ifdij(k)==0 di(k,1)=1; di(k,2)=bj(k+1); elseifdij(k)<=R di(k,1)=1./(dij(k+1))^P; di(k,2)=X(bj(k+1),3)./(dij(k+1))^P; end end sum1=sum(di(:,1)); [p,q]=size(di); for g=1:p temp(g)=di(g,2)./sum1; end w(i,3)=sum(temp); end dlmwrite(‘outdata.txt’, w, ‘delimiter’,’ ‘,’precision’, ‘%10.5f’) % Output the result %--- interpolation function provided by MATLAB software ---%[XI,YI,ZI1] = griddata(X(:,1),X(:,2),X(:,3),Pmx(:,1),Pmy(:,2),’ method’); %--- method: linear, cubic, nearest, v4---
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