Prediction of Peak Shear Strength of Natural, Unfilled Rock Joints Accounting for Matedness Based on Measured Aperture

The peak shear strength criterion developed by Johansson and Stille (2014) for fresh, unweathered rock joints is based on the understanding of the different failure modes for a single asperity, the adhesion theory of friction and an idealised description of a rock joint surface roughness based on fractal theory. Furthermore, the criterion accounts for the change in the dilation angle at grain scale and the variation of the number and size of the active asperities at contact due to scale and matedness.

To understand how a single asperity can fail, Johansson and Stille (2014) studied the transition between different failure modes of an idealised asperity at different dilation angles (\(i\)), where the side of the asperity facing the shear direction was assumed to be subjected to a normal stress (\({\sigma }_{\mathrm{n}}^{{^{\prime}}}\)) equal to the uniaxial compressive strength of the intact rock (\({\sigma }_{\mathrm{ci}}\)). The results of the analysis showed that for low values of \(i\), it is sliding failure that controls the shear strength. At higher inclinations, shearing through the asperity becomes the governing failure mode. Finally, when the asperity inclination increases further, tensile failure occurs. They concluded that the value of \(i\), where the transition between sliding and shearing failure occurs, depends on the strength of the individual asperity. However, asperity inclinations above this transition value will not contribute to reach a higher mobilised shear strength. For this reason, according to Johansson and Stille (2014), the shear strength of a single asperity with a mobilised friction angle below the transition between sliding and shear failure can be expressed as was originally suggested by Patton (1966):where \({\phi }_{\mathrm{p}}\) is the peak friction angle, \({\phi }_{\mathrm{b}}\) is the basic friction angle for a dry and sawn surface and \({i}_{\mathrm{n}}\) is the dilation angle at sample scale.

This peak shear strength criterion extends the analysis performed for a single asperity to sample scale based on the idea that surface roughness can be described with fractal theory (Mandelbrot 1985; Renard et al. 2006; Stigsson and Mas Ivars 2019). Based on this assumption, surface roughness may be idealised as a superposition of different asperities at multiple scales (i.e., with different heights and lengths). Furthermore, Brown (1987) and Malinverno (1990) both showed that a self-affine fractal profile keeps a power law relationship between the variation of the asperity height and the span of the measured profile. Taking up these findings, Johansson and Stille (2014) assumed that there also exists a scaling relationship between the asperity height (\({h}_{\mathrm{asp}}\)) and the asperity base length (\({L}_{\mathrm{asp}}\)) of different sized asperities on a rock joint profile. This scaling relationship is given bywhere \({a}^{*}\) is an amplitude constant and \(H\) is the Hurst exponent.

In addition, for an idealised asperity there exists a geometric relation between the parameters \({h}_{\mathrm{asp}}\) and \({L}_{\mathrm{asp}}\), given by

By combining Eqs. (2) and (3), Johansson and Stille (2014) established an expression that relates the length and inclination of the asperities at contact, given by

Based on the adhesion theory, Johansson and Stille’s peak shear strength criterion assumes that for a given \({\sigma }_{\mathrm{n}}^{{^{\prime}}}\), the contact area (\({A}_{\mathrm{c}}\)) for a fresh and unweathered rock joint increases proportionally to the area of the sample (\(A\)). This can be expressed as

Previous observations on sheared rock joint surfaces have shown that the total contact area during the shearing process is only a small portion of the total area of the rock joint sample (Grasselli et al. 2002). Furthermore, these contact areas occur where the steepest asperities facing the shear direction are located (Kimura and Esaki 1995; Grasselli et al. 2002). After analysing a large number of scanning data from rock joint surfaces, Grasselli (2001) proposed the following empirical relationship to express the potential contact area ratio (\({A}_{\mathrm{c},\mathrm{p}}\)) at different asperity inclinations for a perfectly mated rock joint:where \({A}_{0}\) is the maximum possible contact area ratio, \({\theta }_{\mathrm{max}}^{*}\) is the maximum apparent dip angle in the shearing direction, \({\theta }^{*}\) is the apparent dip angle and \(C\) is a roughness parameter that governs the concavity of the curve.

For a fresh and unweathered rock joint, the value of \({A}_{\mathrm{c}}\) can be obtained by the product of \({A}_{\mathrm{c},\mathrm{p}}\) and \(A\). By combining the adhesion theory and the potential contact area ratio in Eqs. (5) and (6), Johansson and Stille (2014) suggested the following expression to estimate the dilation angle at grain scale (\({i}_{\mathrm{g}}\)) for a perfectly mated rock joint:

Johansson and Stille (2014) further discussed how the size of the asperities at contact varies due to scale and matedness. According to the adhesion theory, the ratio between \({A}_{\mathrm{c}}\) and \(A\) for a rock joint surface is only dependent on the applied \({\sigma }_{\mathrm{n}}^{{^{\prime}}}\) and its \({\sigma }_{\mathrm{ci}}\) [see Eq. (5)]. Consequently, this ratio is independent of scale, and the following expression can be derived:

The subscripts \(\mathrm{n}\) and \(\mathrm{g}\) in Eq. (8) indicate sample and grain size, respectively (Johansson and Stille 2014).

Furthermore, the true contact area is the sum of all the individual contact points on the rock joint which, together with Eq. (8), expresses how the number and size of the contact asperities on a rock joint change with the sample scale under constant normal load (CNL) conditions. Based on these assumptions, Johansson and Stille (2014) suggested an expression to estimate the average length of the asperities in contact at a size associated with sample size (\({L}_{\mathrm{asp},\mathrm{n}}\)), given bywhere \({L}_{\mathrm{asp},\mathrm{g}}\) is the average length of the asperities in contact at grain size, \({L}_{\mathrm{n}}\) is the length of the sample, \({L}_{\mathrm{g}}\) is the scale of the asperities associated with grain size and \(k\) is the matedness constant.

Finally, by combining Eqs. (2), (3), (4), (7) and (9), it is then possible to derive the equation that expresses \({i}_{\mathrm{n}}\), given by

The peak shear strength criterion developed by Johansson and Stille (2014) has the ability of explaining how the compressive strength, roughness, scale and matedness of a fresh, unweathered rock joint surface interact to form the peak shear strength under CNL conditions. However, the criterion has only been tested against tensile induced rock joints (Johansson and Stille 2014; Johansson 2016).

The parameter \(k\) proposed by Johansson and Stille (2014) describes how the number and size of the active asperities taking part in the shearing process between two joint surfaces vary proportionally with shear displacement. The value of this parameter varies between 0 for a rock joint with perfect match and 1 for a totally mismatched rock joint. The equation for \(k\) is given bywhere \(u\) is the total shear displacement at the peak (Johansson 2016).

During laboratory experiments on perfectly mated rock joints, Johansson (2016) proposed a relationship between an initial relative shear displacement or dislocation between lower and upper parts prior the shear test (\({u}_{\mathrm{i}}\)) and the length of the analysed rock joint sample. This relationship was used to estimate \(k\) with Eq. (11). According to Johansson (2016), for a perfectly mated rock joint, the size of the active asperities at contact at peak shear strength is associated with grain scale, \({L}_{\mathrm{asp},\mathrm{n}}={L}_{\mathrm{g}}\). This means that when using Eq. (11), \(k=0\), and consequently, the peak shear strength in a shear test occurs at a \(u={L}_{\mathrm{g}}/2\). Conversely, for a rock joint sample where the upper and lower surfaces have undergone a certain displacement \({u}_{\mathrm{i}}\), the number of contact points decreases, and the size of the asperities at contact increases. This also means that an additional shear displacement (\(\Delta u\)) in the performed shear test is needed to reach the peak shear strength of the analysed unmated rock joint. As a result, \({u=u}_{\mathrm{i}}+\Delta u\), and the parameter \(k\) differs from 0 in these cases. Based on the principles of self-affine fractal theory and the idealisation of surface roughness as a superposition of different asperities at multiple scales, Johansson (2016) explained that it is reasonable to assume that a direct association exists between \({u}_{\mathrm{i}}\) of an unmated rock joint and the average length of the asperities in contact during the shearing process (\({L}_{\mathrm{asp},\mathrm{n}}\)). Johansson (2016) further explained that due to the association between \({u}_{\mathrm{i}}\) and \({L}_{\mathrm{asp},\mathrm{n}}\), the additional deformation \(\Delta u\) that is needed to reach the peak must also be directly proportional to \({u}_{\mathrm{i}}\) and \({L}_{\mathrm{asp},\mathrm{n}}\). The additional deformation of \(\Delta u\) required to reach the peak shear strength therefore becomes equal to \({L}_{\mathrm{asp},\mathrm{n}}/2\). This implies that \(\Delta u={u}_{\mathrm{i}}\), leading to \(u={2u}_{\mathrm{i}}\). The limitation of this assumption is connected to the possibility of describing surface roughness using self-affine fractal theory. For instance, this assumption should not be applied to predict the peak shear strength of blasted joint surfaces, saw-tooth rock joints or rock-concrete contact surfaces.

However, a drawback of the approach introduced by Johansson (2016) to account for the matedness is that it cannot be used with natural, unmated rock joints that exhibit an aperture. The main reason for this is that the estimation of \({u}_{\mathrm{i}}\) is in itself difficult when calculation of the peak shear strength of natural rock joints in the field is intended.

To account for this, we propose a revised version of Johansson and Stille’s peak shear strength criterion. This revised peak shear strength criterion incorporates the use of objective measurements of a natural rock joint average aperture (\(a\)) to account for its matedness as a step in the calculation of its \({\phi }_{\mathrm{p}}\). In this new approach, it is suggested that the natural rock joint is assumed to be equal to an initially perfectly mated rock joint where the upper and lower parts have been dislocated a certain \({u}_{\mathrm{i}}\). If sliding along active asperities is assumed, this virtual \({u}_{\mathrm{i}}\) in the perfectly mated rock joint will lead to changes in the aperture until a value of \(a\) equal to the one measured in the natural rock joint is reached. This principle is illustrated in Fig. 1. Additionally, this implies that aperture changes may theoretically be associated with the inclination of the active asperities in contact that contributes to the peak shear strength. By extension, this means that if \(a\) for a natural rock joint is known, a virtual \({u}_{\mathrm{i}}\) can be estimated by applying the relationship suggested by Ladanyi and Archambault (1969) and Oh and Kim (2010):

Since it can be assumed that \(u={2u}_{\mathrm{i}}\) as described above, the parameter \(k\) in Eq. (11) can be redefined as

Therefore, the revised peak shear strength criterion uses measurements of \(a\) to estimate \({i}_{\mathrm{n}}\) and \(k\) by performing an iterative process with Eqs. (7), (10) and (13) and calculating \({\phi }_{\mathrm{p}}\) with Eq. (1).

The analysed rock joint samples were obtained by over-drilling through an existing rock joint adjacent to the foundation of the Storfinnforsen buttress dam. The intact rock beneath the dam’s foundation consists of grey coarse-grained granite. Figure 3 shows two of the analysed samples from Storfinnforsen.

Additionally, two rock joint samples taken from an existing rock joint adjacent to the foundation of the Långbjörn concrete dam were included in this analysis. These rock joint samples consisted of grey coarse-grained granite and were slightly weathered. These samples were previously sheared in the laboratory by Johansson (2009).

The dimensions of the natural, unfilled rock joint samples from Storfinnforsen (S1 to S8) and Långbjörn (L1 and L2), together with the test conditions during the direct shear tests, are shown in Table 1.

High-resolution optical scanning of the rock joints from Storfinnforsen (S1 to S8) was performed with an ATOS Compact Scan 5M system. The performed measurements on these eight samples had an accuracy of ± 0.02 mm. The rock joint samples from Långbjörn (L1 and L2) were scanned with an ATOS III system by Johansson (2009). The measurements of these two samples had an accuracy of ± 0.05 mm. Before performing the direct shear tests, the upper and lower joint surfaces were scanned separately. The upper and lower parts of each rock joint sample were then put together and scanned again, making it possible to analyse the degree of contact between both surfaces prior to the direct shear tests. To maintain consistency with the global reference system, circular reference points were placed around the upper and lower parts of all analysed samples, see Fig. 3. The scanned rock joint surfaces were re-generated with a resolution of 0.3 by 0.3 mm. This resolution was assumed to be appropriate to capture \({L}_{\mathrm{g}}\) on the rock joints according to previous recommendations by Grasselli and Egger (2003) and Tatone and Grasselli (2009). The parameters \({\theta }_{\mathrm{max}}^{*}\), \(C\) and \({A}_{0}\) for the analysed joint surfaces were determined based on the digitised surfaces using the following methodology: first, normal vectors (\({{\varvec{n}}}_{\mathbf{i}}\)) were generated for each element on the 0.3 by 0.3 mm grid. By defining the shear direction (\({\varvec{t}}\)), the values of \({\theta }^{*}\) for each asperity facing the shear test direction were determined using

This principle is illustrated in Fig. 4a. As an example, the measured values of \({\theta }^{*}\) with respect to the defined \({\varvec{t}}\) for the lower and upper parts of sample S1 are illustrated in Fig. 4b, c, respectively. The calculated values of \({\theta }^{*}\) for the lower and upper parts of each rock joint sample were sorted separately in descending order. For each \({\theta }^{*}\), the parameter \({A}_{\mathrm{c},\mathrm{p}}\) defined as the sum of all the areas with a certain inclination facing the shear direction was determined using Eq. (6). Measured values of the relationship between \({\theta }^{*}\) and \({A}_{\mathrm{c},\mathrm{p}}\) for the lower and upper surfaces of sample S1 at a resolution of 0.3 by 0.3 mm are shown in Fig. 4d. Data from these two curves were used to calculate \({\theta }_{\mathrm{max}}^{*}\), \(C\) and \({A}_{0}\) by best-fit regression analysis [see Eq. (6)]. Table 2 presents the results of the regression analysis for the analysed rock joint samples.

The relationship between \({h}_{\mathrm{asp}}\) and \({L}_{\mathrm{asp}}\) established in Eq. (3) was estimated by calculating the root mean square of the first derivative (\({Z}_{2}\)) for different sampling intervals along the shear direction (\(\Delta x\)) in the digitised joint surfaces. Each digitised joint surface was divided into profiles separated by a constant distance perpendicular to the shear direction (\(\Delta y\)). The parameter \({Z}_{2}\) describes therefore the average inclination of the asperities over a certain \(\Delta x\) and for each profile separated a distance \(\Delta y\). This can be expressed aswhere \({N}_{x}\) and \({N}_{y}\) are the number of coordinate points over a digitised rock joint surface parallel and perpendicular to the shear direction, respectively. The pairs \(\left({x}_{i,j}, {z}_{i,j}\right)\) and \(\left({x}_{i+1,j}, {z}_{i+1,j}\right)\) are adjacent coordinates in the same profile along the shear direction separated by a constant distance \(\Delta x\) (Myers 1962).

The values of \(\Delta x\) and \(\Delta y\) were equal and varied between 0.3 mm and 9.6 mm for the digitised rock joint surfaces. As an example, Table 3 shows the results of the calculation of \({h}_{\mathrm{asp}}\) for different \({L}_{\mathrm{asp}}\) for sample S1. Based on these results, the values of \(H\) for the analysed rock joint samples were determined through regression analysis using Eq. (2).

Additionally, the aperture of all the tested rock joint samples was measured after superposing the lower and upper digitised surfaces and calculating the difference in elevation between points with same \(x\)- and \(y\)-coordinates. This was possible since both upper and lower joint surfaces of the tested rock samples were scanned in the same global reference system. Figure 5 shows the rock joint aperture for sample S1. After calculating the difference in elevation for each point in the upper and lower digitised surfaces the parameter \(a\) was obtained bywhere \({z}_{i,j}^{\mathrm{upper}}\) and \({z}_{i,j}^{\mathrm{lower}}\) are the elevation of two points with same \(x\)- and \(y\)-coordinates situated in the upper and lower digitised surfaces with a resolution of 0.3 by 0.3 mm, respectively.

The values of \(H,\) together with the measurements of \(a\) and the estimation of \(k\) and \({i}_{\mathrm{n}}\) for the analysed rock joints, are shown in Table 4.

Conventionally, the results of direct shear tests are presented by plotting shear stress and normal displacement versus shear displacement (Muralha et al. 2014). However, the revised peak shear strength criterion expresses the \({\phi }_{\mathrm{p}}\) in degrees. To maintain consistency in the comparison between the conducted shear tests and the predicted \({\phi }_{\mathrm{p}}\) of the analysed rock joint samples with the revised criterion, plots containing mobilised friction angle (\(\phi\)) versus \({\delta }_{\mathrm{s}}\) have been presented in this study.

The results of \(\phi\) and \({\delta }_{\mathrm{n}}\) versus \({\delta }_{\mathrm{s}}\) measured during the laboratory direct shear tests conducted on the analysed rock joint samples are presented in Fig. 7. Additionally, values of measured \({\phi }_{\mathrm{p}}\), dilation angle at the peak (\({i}_{\mathrm{peak}}\)) and shear displacement at the peak (\({\delta }_{\mathrm{peak}}\)) are presented in Table 5. Rock joint samples S1 to S8 showed a measured \({\phi }_{\mathrm{p}}\) varying between 50.5° and 71.8°. On average, the measured \({\phi }_{\mathrm{p}}\) was 59.3°. These rock joint samples showed a measured \({i}_{\mathrm{peak}}\) between 9.0° and 35.1°. The measured values of \({\phi }_{\mathrm{p}}\) and \({i}_{\mathrm{peak}}\) were 44.6° and 7.3° for sample L1, and 42.4° and 7.6° for L2.

Two different mechanical behaviours were observed when comparing the measured \(\phi\) versus \({\delta }_{\mathrm{s}}\) of the analysed rock joint samples (see Fig. 7a). The results of the direct shear tests for samples S1, S3, S4 and S5 showed a clear peak and post-peak behaviour. For these rock joint samples, the registered \({\delta }_{\mathrm{peak}}\) varied between 0.26 and 1.2 mm (see Table 5). On the contrary, samples S2, S6, S7 and S8 had a smoother transition into post-peak behaviour and a clear peak could not be observed. The registered \({\delta }_{\mathrm{peak}}\) for these samples varied between 0.69 and 4.1 mm. The rock joint samples L1 and L2 showed also a smooth transition into post-peak behaviour with a registered \({\delta }_{\mathrm{peak}}\) of 3.22 and 2.20 mm, respectively. This observed mechanical behaviour with a smooth transition into post-peak shear strength is consistent with other results reported by Johansson (2009, 2016) on unmated rock joints.

The comparison between measured \({\delta }_{\mathrm{n}}\) versus \({\delta }_{\mathrm{s}}\) in Fig. 7b showed a similar mechanical behaviour for all the analysed rock joint samples. The results showed an initial closure followed by a relative opening between the lower and upper parts of the analysed samples. The measured \({\delta }_{\mathrm{n}}\) at 5 mm shear displacement varied between 0.9 and 2.1 mm for samples S1 to S8. Samples L1 and L2 had a measured \({\delta }_{\mathrm{n}}\) of 0.24 and 0.31 mm, respectively. Sample L1 was sheared 4 mm and sample L2 was sheared 5 mm. In comparison with samples S1 to S8, samples L1 and L2 had a lower measured \({\delta }_{\mathrm{n}}\). The observed mechanical behaviour of L1 and L2 was as expected since they presented a smoother surface roughness and a higher measured \(a\) than samples S1 to S8 (see Tables 2, 4).

A comparison between calculated \({\phi }_{\mathrm{p}}\) with the revised criterion and measured \({\phi }_{\mathrm{p}}\) in the laboratory for the analysed rock joint samples is presented in Fig. 8a. The value of \({\phi }_{\mathrm{b}}\) obtained by means of tilt tests was estimated as 31° for the analysed rock joint samples (S1 to S8, L1 and L2). The \({\sigma }_{\mathrm{ci}}\) of these ten rock joint samples was estimated using the Schmidt Hammer Index as suggested by Barton and Choubey (1977). The \({\sigma }_{\mathrm{ci}}\) obtained for samples S1 to S8 was 110 MPa. Rock joint samples L1 and L2 had a \({\sigma }_{\mathrm{ci}}\) of 140 MPa (Johansson 2009). The applied \({\sigma }_{\mathrm{n}}^{{^{\prime}}}\) on the analysed rock joints is presented in Table 1. Parameters \({\theta }_{\mathrm{max}}^{*}\), \(C\) and \({A}_{0}\) are provided in Table 2. Values of \(H\), \(a\), \(k\) and \({i}_{\mathrm{n}}\) are provided in Table 4.

The calculated values of \({\phi }_{\mathrm{p}}\) with the revised criterion were in good agreement with the results from the direct shear tests conducted in the laboratory. The results show that for samples S1 to S8, the calculated \({\phi }_{\mathrm{p}}\) varied between 52.4° and 64.7°. The average value of the calculated \({\phi }_{\mathrm{p}}\) on these samples with the revised criterion was 58.0°. The difference between measured and calculated \({\phi }_{\mathrm{p}}\), as expressed in absolute values, varied between 0.4° and 4.8°. The exception was sample S3, which showed a less good agreement. In this rock joint sample, the difference between measured and calculated \({\phi }_{\mathrm{p}}\) was 9.3°. Samples L1 and L2 had a calculated value of \({\phi }_{\mathrm{p}}\) of 41.1° and 47.6°, respectively. For these two samples, the difference between measured and calculated \({\phi }_{\mathrm{p}}\), as expressed in absolute values, was 3.5° and 5.2°, respectively.

To increase the completeness of this study, Fig. 8a also includes the results from six direct shear tests previously performed by Johansson (2016) on fresh tensile-induced rock joints with dimensions of 60 by 60 mm and 200 by 200 mm. These rock joint samples came from the Flivik quarry in Sweden and consisted of grey coarse-grained granite. The values of \({\phi }_{\mathrm{p}}\) were predicted using the peak shear strength criterion developed by Johansson and Stille (2014) with \(k=0\). This assumption is considered reasonable, since tensile-induced rock joints exhibit a perfect match between the upper and lower parts (Johansson 2016). The main reason for including them in this analysis is to provide a comparison between three different groups of rock joints with different joint matedness. In total, the comparison between measured and calculated \({\phi }_{\mathrm{p}}\) comprises 16 rock joint samples.

The analysis performed on the rock joint samples from Storfinnforsen (S1 to S8) showed that the calculated values of \({\phi }_{\mathrm{p}}\) with the revised criterion were in good agreement with the measured values of \({\phi }_{\mathrm{p}}\) in the laboratory. However, the comparison between calculated and measured \({\phi }_{\mathrm{p}}\) for sample S3 deviated from the expected behaviour. The reason for this discrepancy (9.3°) is not clear. During the laboratory shear tests, the observed \({\delta }_{\mathrm{peak}}\) for this sample was 0.36 mm. The measured \({\delta }_{\mathrm{peak}}\) gives an indication of the size of the effective asperities contributing to the shear resistance at the peak. The results from the laboratory direct shear tests on the six perfectly mated rock joints performed by Johansson (2016) showed an average \({\delta }_{\mathrm{peak}}\) of approximately 0.3 mm. This may indicate that the joint surfaces of sample S3 had similar conditions to a perfectly mated rock joint. Consequently, the application of the methodology proposed in this paper might underestimate the matedness of sample S3, leading to a larger discrepancy between measured and calculated \({\phi }_{\mathrm{p}}\).

The comparison between measured and calculated values of \({\phi }_{\mathrm{p}}\) for the rock joint samples from Långbjörn (L1 and L2) was also in good agreement. Note that the revised criterion was able to predict well the \({\phi }_{\mathrm{p}}\) of samples L1 and L2, which, compared with the other group of analysed rock joints (S1 to S8), had a higher degree of weathering, a lower \({\phi }_{\mathrm{p}}\) and a higher registered \({\delta }_{\mathrm{peak}}\) measured in the laboratory (see Fig. 7a). The reason for this is that L1 and L2 had, on average, higher values of \(C\) in combination with lower values of \({\theta }_{\mathrm{max}}^{*}\) than samples S1 to S8 (see Table 2). Higher values of \(C\) are associated with low surface roughness where a large portion of the asperities facing the shear direction is less steep than \({\theta }_{\mathrm{max}}^{*}\) (Grasselli 2001). In addition, L1 and L2 had a high measured \(a\), which led to high values of \(k\) and low values of \({i}_{\mathrm{n}}\) (see Table 4). This combination between \(C\), \({\theta }_{\mathrm{max}}^{*}\) and \(a\) shows how surface roughness interacts with the matedness in the revised criterion to form \({\phi }_{\mathrm{p}}\). Furthermore, these results also show how the description of surface roughness using self-affine fractal theory and the direct association between asperities at multiple scales is accounted for in the revised criterion. As it is observed in Fig. 7a, b, the analysed rock joint samples with higher measured \(a\) had a higher registered \({\delta }_{\mathrm{peak}}\) and a lower \({\delta }_{\mathrm{n}}\) at maximum shear displacement.

The calculated values of \({\phi }_{\mathrm{p}}\) for the rock joint samples from Flivik were also in good agreement with the results from the direct shear tests conducted in the laboratory by Johansson (2016).

A discrepancy was observed when comparing the measured \({i}_{\mathrm{peak}}\) during the conducted direct shear tests and the calculated \({i}_{\mathrm{n}}\) with the revised criterion of the analysed rock joint samples. The average values of measured \({i}_{\mathrm{peak}}\) and calculated \({i}_{\mathrm{n}}\) for all ten samples were 18.4° and 24.3°, respectively (see Tables 4, 5). The average calculated \({i}_{\mathrm{n}}\) was 5.9° higher than the average measured \({i}_{\mathrm{peak}}\). A possible reason for this discrepancy between measured \({i}_{\mathrm{peak}}\) and calculated \({i}_{\mathrm{n}}\) may be due to the existence of an asperity failure component in the performed shear tests. The peak shear strength criterion by Johansson and Stille (2014) and the revised criterion both assume that sliding along the active asperities is the predominant failure mode governing the shearing process. However, the contribution from an asperity failure component is close to the contribution from sliding—at least under the assumption that the contact pressure on the asperities is equal to \({\sigma }_{\mathrm{ci}}\) (Johansson and Stille 2014; Johansson 2016). Therefore, even though a discrepancy was observed between average \({i}_{\mathrm{peak}}\) and \({i}_{\mathrm{n}}\), the comparison between measured and calculated \({\phi }_{\mathrm{p}}\) will show a good agreement. Furthermore, this discrepancy will not influence the calculation of \({\phi }_{\mathrm{p}}\) with the revised criterion, since the calculation of \(k\) and \({i}_{\mathrm{n}}\) originates from an initial measured \(a\) of the natural rock joint.

A similar discrepancy could also be observed when comparing the average values of measured \({\phi }_{\mathrm{p}}-{i}_{\mathrm{peak}}\) (37.8°) with \({\phi }_{\mathrm{b}}\) (31°) measured in the tilt tests (see Table 5). The average measured \({\phi }_{\mathrm{p}}-{i}_{\mathrm{peak}}\) was 6.8° higher than the measured \({\phi }_{\mathrm{b}}\). This discrepancy may be due to the existence of an asperity failure component as previously discussed. Another possible reason is an uncertainty in the measurements of \(i\) in the performed shear tests since it is influenced by the selected \({\mathrm{d}\delta }_{\mathrm{s}}\). The values of \(i\) were calculated with a constant \({\mathrm{d}\delta }_{\mathrm{s}}\) of 0.1 mm. Selecting a value of \({\mathrm{d}\delta }_{\mathrm{s}}\) higher than 0.1 mm will result in lower values of measured \(i\) in the performed shear tests. On the other hand, values of \({\mathrm{d}\delta }_{\mathrm{s}}\) lower than 0.1 mm will result in a large scattering when measuring \(i\) since the measured \({\mathrm{d}\delta }_{\mathrm{n}}\) for each \({\mathrm{d}\delta }_{\mathrm{s}}\) is becoming close to the accuracy of the installed LVDTs. Additionally, Johansson (2016) suggested that it is necessary that the selected \({\mathrm{d}\delta }_{\mathrm{s}}\) to calculate \(i\) from the shear tests is of the same magnitude as the micro-scale roughness of the sawn samples used to determine \({\phi }_{\mathrm{b}}\) with tilt tests in order for them to be compatible. The ISRM Suggested Method for determining \({\phi }_{\mathrm{b}}\) recommends using saw blades with teeth or rim grits on the range of 60–100 US Mesh during the specimen preparation (Alejano et al. 2018). Bruce et al. (1989) also observed that reasonable values of \({\phi }_{\mathrm{b}}\) for quartzite and dolostone were obtained when their surfaces were roughened with a No. 80 grit. They observed that the roughness of the prepared plates had a Centre Line Average (CLA) of 200 µm for the quartzite and 150 µm for the dolostone. The selected \({\mathrm{d}\delta }_{\mathrm{s}}\) (100 µm) for the calculation of \(i\) in the performed shear tests in this study is in the same range as the observed values of CLA by Bruce et al. (1989) and therefore it is considered suitable for the measurements of \(i\). The uncertainties in the measured \({i}_{\mathrm{peak}}\) and the compatibility between \({\phi }_{\mathrm{b}}\) and \(i\) may also be a possible reason for the observed discrepancy between measured \({i}_{\mathrm{peak}}\) during the conducted direct shear tests and the calculated \({i}_{\mathrm{n}}\) previously discussed.

To study the influence of \(a\) on the estimation of \(k\) and the calculation of \({\phi }_{\mathrm{p}}\) with the revised criterion, a sensitivity analysis was performed. This sensitivity analysis included samples S1 to S8, L1 and L2. Figure 9 shows the variation of \(k\) and calculated \({\phi }_{\mathrm{p}}\) with different values of measured \(a\). The results showed that both the parameter \(k\) and the calculated \({\phi }_{\mathrm{p}}\) showed steep variation when the measured \(a\) was within a range of between 0 and approximately 1 mm. For instance, for the values of \(a\) within the range between 0 and 1 mm, the parameter \(k\) increased between 0.42 and 0.67 in the analysed rock joint samples (see Fig. 9a). For the same range of \(a\), the calculated \({\phi }_{\mathrm{p}}\) decreased between 7.9° and 13.9° for the analysed samples (see Fig. 9b). This variation in the estimated value of \(k\) and calculated \({\phi }_{\mathrm{p}}\) was less abrupt at higher ranges of the measured \(a\).

To increase the completeness of this sensitivity analysis, the \({\phi }_{\mathrm{p}}\) of the rock joint samples presented in this study was also calculated with the peak shear strength criteria previously developed by Grasselli and Egger (2003) and Johansson and Stille (2014). The input data required to apply these criteria to samples S1 to S8, L1, L2 and the perfectly mated rock joints from Flivik are provided in Tables 1, 2 and 4 and in the study performed by Johansson (2016). The peak shear strength criterion by Johansson and Stille (2014) was implemented in the analysed rock joint samples by setting the parameter \(k=0\). Since the upper and lower parts of the rock joint samples from Storfinnforsen and Långbjörn were not dislocated prior the laboratory shear tests, the values of \({u}_{\mathrm{i}}\) and \(k\) were equal to 0 according to Johansson (2016). A comparison was made between values of measured and calculated \({\phi }_{\mathrm{p}}\) with the peak shear strength criteria by Grasselli and Egger (2003) and Johansson and Stille (2014), which is illustrated in Fig. 8b. The results for the perfectly mated rock joints from Johansson (2016) showed that the calculated values of \({\phi }_{\mathrm{p}}\) obtained with Grasselli and Egger (2003) and Johansson and Stille (2014) peak shear strength criteria were in good agreement with the measured \({\phi }_{\mathrm{p}}\) in the laboratory. However, the comparison was less good for rock joint samples S1 to S8, L1 and L2. The calculated values of \({\phi }_{\mathrm{p}}\) with Grasselli and Egger (2003) and Johansson and Stille (2014) generally overestimated the \({\phi }_{\mathrm{p}}\) measured in the laboratory. Furthermore, the slope of the obtained linear fit through regression analysis for the peak shear strength criteria from Grasselli and Egger (2003) and Johansson and Stille (2014) was 0.3 and 0.4, respectively. This is not surprising, since both criteria assume a perfect match between the lower and upper parts joints, which is not the case of the natural, unfilled rock joint samples presented in this study.

The novelty of this paper lies in the use of high-resolution optical scanning to objectively derive \(a\) and to calculate the matedness of natural, unfilled rock joints as a step in the prediction of their \({\phi }_{\mathrm{p}}\). However, there are still some issues concerning the suggested methodology which require further study before it can be applied in the field.

Using the equipment and measurement techniques currently available makes it possible to obtain the necessary information from visible rock exposures, for example in tunnels or from core-drilling through existing rock joints under a dam foundation. For instance, aperture measurements can be determined through Borehole Image Processing System (BIPS) of the boreholes. The resolution with this technique, however, is limited to approximately 0.5 mm. Therefore, BIPS might be more suitable for rock joints with larger apertures where the upper and lower surfaces of the joints are not in contact with each other, while high-resolution optical scanning could be used directly on core samples for rock joints with smaller aperture, where contact between the upper and lower surfaces exists.

Taken together with additional information from the rock core samples, such as roughness parameters (\({\theta }_{\mathrm{max}}^{*}\), \(C\) and \({A}_{0}\)), \(H\) and \({\sigma }_{\mathrm{ci}}\) of the rock joint surfaces, these techniques might be sufficient to apply the methodology presented in this paper and to calculate \({\phi }_{\mathrm{p}}\) using the revised criterion. Furthermore, the information from the rock core samples would also be sufficient to account for surface anisotropy when applying the revised criterion. For instance, the parameters \({\theta }_{\mathrm{max}}^{*}\), \(C\) and \({A}_{0}\) presented in this study were calculated for a defined shear direction in the conducted direct shear tests. They would have been slightly different if another shear direction had been applied. In the field, it is therefore important to account for all potential shear directions when the rock joint surface roughness is characterised before the application of the revised criterion, unless the shear direction is known. The evaluation of roughness anisotropy using three-dimensional surface measurements can be done by applying the methodology described by Tatone and Grasselli (2009). The extent of the pre-investigations that need to be performed to realistically account for the aperture when predicting the \({\phi }_{\mathrm{p}}\) of large-scale natural rock joints is subject of further investigation. The size of a large-scale natural rock joint can be considered in the revised criterion by increasing the parameter that represents the sample size, \({L}_{\mathrm{n}}\).

A limitation with the current study is that the revised criterion has only been tested against coarse-grained granite samples. However, Johansson and Stille (2014) tested the criterion against perfectly mated rock joints of different rock types tested in the laboratory by Grasselli (2001). The results by Johansson and Stille (2014) showed a good agreement between measured and calculated contribution from roughness. The exception was the samples of gneiss, which deviated from the expected behaviour. According to Johansson and Stille (2014), this deviation was most likely due to the anisotropic \({\sigma }_{\mathrm{ci}}\) of the gneiss. They further explained that it was uncertain how the shear load during the direct shear tests performed by Grasselli (2001) had been applied with respect to the schistosity planes of the gneiss samples. The ability of the revised criterion to predict the \({\phi }_{\mathrm{p}}\) of natural, unfilled rock joints of other rock types is recommended to be further studied.

Finally, the original criterion by Johansson and Stille (2014) was developed to assess the safety of concrete dams against sliding in the rock foundation under low normal stress to joint compressive strength ratios, which are typical in civil engineering applications. In this paper, all samples were tested with low normal stress to joint wall compressive strength ratios of approximately 0.01. The ability of the proposed methodology together with the revised criterion to predict the \({\phi }_{\mathrm{p}}\) of natural, unfilled rock joints under higher normal loads needs to be further studied to verify its applicability under such conditions.