Determining the Cohesive Length of Rock Materials by Roughness Analysis

: In this research, the cohesive length of various rock types is measured using quantitative fractography alongside a recently developed multifractal analysis. This length is then utilized to gauge material cohesive stress through the theory of critical distances. Furthermore, the fracture process zone length of diﬀerent rings sourced from identical rocks is assessed as a function of ring dimensions and experimental measurements of fracture toughness, in accordance with the energy criterion of the ﬁnite fracture mechanics theory. Subsequently, employing the stress criterion within coupled ﬁnite fracture mechanics, the failure stress corresponding to the fracture process zone is determined for various rings. Ultimately, through interpolation, the critical stress corresponding to the cohesive length, quantiﬁed via quantitative fractography, is approximated. Remarkably, the cohesive stress values derived from both methodologies exhibit perfect alignment, indicating the successful determination of cohesive length for the analyzed rock materials. The study also delves into the signiﬁcant implications of these ﬁndings, including the quantiﬁcation of intrinsic tensile strength in quasi-brittle materials and the understanding of tensile strength variati ons under diverse stress concentrations and loading conditions.


Introduction
made pioneering efforts to bridge fracture physics and mechanics by quantifying both the fractal dimension of fractured surfaces and the fracture toughness of metals [1].They proposed a correlation between fracture toughness and the fractal dimension of metal surfaces.However, subsequent research by various scholars indicated that the fractal dimension of fractured surfaces is independent of the material type, suggesting the existence of a universal roughness exponent [2,3].
These findings spurred the development of models that conceptualize fracturing as a mixed-order phase transition at small length scales and first-order transition at length scales exceeding a critical threshold ξ [4][5][6][7].This critical length scale ξ can also be interpreted as the fracture process zone length of a propagating crack or the cohesive length ℓ c [8][9][10].
Quasi-brittle materials like rocks pose challenges for analysis using Linear Elastic Fracture Mechanics (LEFM) when examining length scales below the Fracture Process Zone (FPZ).This zone forms around the crack tip and other stress concentrators prior to crack propagation [8,10].Quantifying ℓ pz is required to successfully model quasi-brittle materials failure and to understand the mechanics of fracture and the multiscale physical properties of these materials [11,12].Different schemes are used to quantify ℓ pz ; however, there is no agreement on a robust method [13].
In this context, a roughness analysis utilizing a domain-based multifractal approach is utilized to ascertain ξ, which is subsequently corroborated by two distinct mechanical models necessitating disparate experimental inputs.Through this validation, it is inferred that ξ represents the extent of the fracture process zone associated with a rapidly advancing crack.This elucidates the relationship between fracture toughness and surface roughness.Furthermore, leveraging these findings, a model is formulated to ascertain the intrinsic tensile strength or cohesive stress σ c .This can clarify the effects of geometrical properties [14][15][16], loading rate [17,18], and other extrinsic features on the tensile strength of quasi-brittle materials.

Materials and Methods
In this study, the geometry-dependent length of the fracture process zone (ℓ pz ) of sandstone, marble, and fine-grained (FG) and coarse-grained (CG) granites was quantified (Figure 1).Notched Semi-Circular Bending (NSCB) specimens with a radius of 37.5 mm were prepared and tested as per ISRM for fracture toughness measurement [19].Ring specimens with a radius of 37.5 mm and different central hole radii from 1.5 to 14 mm were made from the studied rocks and failed under quasi-static loading conditions.
In this context, a roughness analysis utilizing a domain-based multifractal approach is utilized to ascertain ξ, which is subsequently corroborated by two distinct mechanical models necessitating disparate experimental inputs.Through this validation, it is inferred that ξ represents the extent of the fracture process zone associated with a rapidly advancing crack.This elucidates the relationship between fracture toughness and surface roughness.Furthermore, leveraging these findings, a model is formulated to ascertain the intrinsic tensile strength or cohesive stress c.This can clarify the effects of geometrical properties [14][15][16], loading rate [17,18], and other extrinsic features on the tensile strength of quasi-brittle materials.

Materials and Methods
In this study, the geometry-dependent length of the fracture process zone (ℓpz) of sandstone, marble, and fine-grained (FG) and coarse-grained (CG) granites was quantified (Figure 1).Notched Semi-Circular Bending (NSCB) specimens with a radius of 37.5 mm were prepared and tested as per ISRM for fracture toughness measurement [19].Ring specimens with a radius of 37.5 mm and different central hole radii from 1.5 to 14 mm were made from the studied rocks and failed under quasi-static loading conditions.Moreover, high-resolution 3D X-ray computed tomography data from fractured areas of some NSCB specimens acquired at the Australian synchrotron were used for reconstructing the roughness of the fractured surfaces (Figure 1).The critical length scale ξ was determined by analyzing the reconstructed fractured surfaces and using multifractal analysis.

Sandstone
Finally, the failure of the NSCB and ring specimens was analyzed to validate that ξ was equal to ℓc.Further details of these procedures are specified in [20].

Fracture and Phase Transition
Analyzing fracture surfaces provides some information about fracture propagation and how a crack front interacts with the microstructure of a material.Quantitative fractography is considered a powerful candidate for estimating ℓc of quasi-brittle materials [21].According to this approach, two domains with various multiscaling features can be identified on fractured surfaces.These two domains can be distinguished by a statistically determined critical length, suggesting a transition from a mixed-order phase at smaller length scales to a first-order phase at scales larger than this critical length.At scales ϵ smaller than the critical length ϵ ≪ ξ, fractured surfaces exhibit multiaffine fractal characteristics [22], characterized by significant intermittency.Conversely, at scales ϵ ≫ ξ, these surfaces display monoaffine fractal properties with a universal roughness exponent, as reported in [23].Therefore, two completely different mechanisms are in place controlling the fracture roughness in these two regimes.
Fracture propagation can be attributed to various mechanisms, with damage percolation being one of them.This mechanism becomes evident through the universality of the fractal dimension observed in percolation clusters on fractured surfaces at length scales ϵ ≪ ξ [1,24,25].To assess whether damage percolation occurred prior to crack propagation, characterized by the formation and merging of damage clusters, the ωϵ field of the reconstructed fractured surfaces was computed following the methodology outlined in [21]: The ωϵ field was computed for all fractured surfaces at a length scale of ϵ = 2d, which was significantly smaller than the characteristic length ξ ≡ ℓc of the rocks under investigation.Here, d represents the tomograph's resolution, approximately 16.5 µm in size, while Δh denotes the ensemble average of height differences across various directions Θ, computed for all X coordinates.
The fractal dimension of the fractured surfaces was then determined by normalizing the caliper length L and the size S of the clusters formed from different fractions of steep Moreover, high-resolution 3D X-ray computed tomography data from fractured areas of some NSCB specimens acquired at the Australian synchrotron were used for reconstructing the roughness of the fractured surfaces (Figure 1).The critical length scale ξ was determined by analyzing the reconstructed fractured surfaces and using multifractal analysis.
Finally, the failure of the NSCB and ring specimens was analyzed to validate that ξ was equal to ℓ c .Further details of these procedures are specified in [20].

Fracture and Phase Transition
Analyzing fracture surfaces provides some information about fracture propagation and how a crack front interacts with the microstructure of a material.Quantitative fractography is considered a powerful candidate for estimating ℓ c of quasi-brittle materials [21].According to this approach, two domains with various multiscaling features can be identified on fractured surfaces.These two domains can be distinguished by a statistically determined critical length, suggesting a transition from a mixed-order phase at smaller length scales to a first-order phase at scales larger than this critical length.At scales ϵ smaller than the critical length ϵ ≪ ξ, fractured surfaces exhibit multiaffine fractal characteristics [22], characterized by significant intermittency.Conversely, at scales ϵ ≫ ξ, these surfaces display monoaffine fractal properties with a universal roughness exponent, as reported in [23].Therefore, two completely different mechanisms are in place controlling the fracture roughness in these two regimes.
Fracture propagation can be attributed to various mechanisms, with damage percolation being one of them.This mechanism becomes evident through the universality of the fractal dimension observed in percolation clusters on fractured surfaces at length scales ϵ ≪ ξ [1,24,25].To assess whether damage percolation occurred prior to crack propagation, characterized by the formation and merging of damage clusters, the ω ϵ field of the reconstructed fractured surfaces was computed following the methodology outlined in [21]: The ω ϵ field was computed for all fractured surfaces at a length scale of ϵ = 2d, which was significantly smaller than the characteristic length ξ ≡ ℓ c of the rocks under investigation.Here, d represents the tomograph's resolution, approximately 16.5 µm in size, while ∆h denotes the ensemble average of height differences across various directions Θ, computed for all X coordinates.
The fractal dimension of the fractured surfaces was then determined by normalizing the caliper length L and the size S of the clusters formed from different fractions of steep cliffs P th on ω ϵ (X) by ξ and ξ 2 , respectively (Figure 2).A universal fractal dimension of 1.7 ± 0.05 for the percolation clusters was calculated across all fractured surfaces, indicating that damage percolation served as the primary mechanism for fracture initiation at scales ϵ ≪ ξ.Consequently, the initiation of fracture in quasi-brittle materials appeared to be unaffected by the material microstructure.However, grain size emerged as a crucial factor, dictating the cohesive length ℓ c .In the second regime, at larger length scales ϵ ≫ ξ, the crack front interacted randomly with the material's microstructure, and Linear Elastic Fracture Mechanics (LEFM) held validity [26,27].
cliffs Pth on ωϵ(X) by ξ and ξ 2 , respectively (Figure 2).A universal fractal dimension of 1.7 ± 0.05 for the percolation clusters was calculated across all fractured surfaces, indicating that damage percolation served as the primary mechanism for fracture initiation at scales ϵ ≪ ξ.Consequently, the initiation of fracture in quasi-brittle materials appeared to be unaffected by the material microstructure.However, grain size emerged as a crucial factor, dictating the cohesive length ℓc.In the second regime, at larger length scales ϵ ≫ ξ, the crack front interacted randomly with the material's microstructure, and Linear Elastic Fracture Mechanics (LEFM) held validity [26,27].

Sandstone Marble
FG granite CG granite

ξ Quantification
Well-known kinetic roughening models cannot capture fracture roughness, and it is challenging to estimate a critical cut-off length ξ on fractured surfaces [28].Here, the qthorder structure functions Sq(r) was employed to find ξ [23]:

ξ Quantification
Well-known kinetic roughening models cannot capture fracture roughness, and it is challenging to estimate a critical cut-off length ξ on fractured surfaces [28].Here, the qth-order structure functions Sq(δr) was employed to find ξ [23]: where ∆h(δr) is the ensemble height difference for a separation δr.It was calculated for different q values from 0.2 to 6. H(q) = ζq/q is constant for monofractals, and multifractals can be identified by a variable H(q).Variation of H(q) or intermittency of the studied rough surfaces has been modelled by power laws [29].Therefore, the slope of such power laws λ indicates intermittency.Aligholi and Khandelwal [30] demonstrated that the intermittency of the fractured surfaces of the studied rocks varies across different analyzed domains [δr min , δr max ].For a fixed value of δr min , λ decreases by increasing δr max .Monofractals are not intermittent, and the cut-off length ξ can be identified as the δr min of a domain showing the same value of H(q).Table 1 outlines the statistically estimated parameters for the monofractal domains of various rock types.A detailed analysis was conducted across different domains to identify the monofractal domain [31].Figure 3 presents the spectral analysis of the identified monofractal and multifractal domains.
where ∆h(r) is the ensemble height difference for a separation r.It was calculated for different q values from 0.2 to 6. H(q) = q/q is constant for monofractals, and multifractals can be identified by a variable H(q).Variation of H(q) or intermittency of the studied rough surfaces has been modelled by power laws [29].Therefore, the slope of such power laws λ indicates intermittency.Aligholi and Khandelwal [30] demonstrated that the intermittency of the fractured surfaces of the studied rocks varies across different analyzed domains [rmin, rmax].For a fixed value of rmin, λ decreases by increasing rmax.Monofractals are not intermittent, and the cut-off length ξ can be identified as the rmin of a domain showing the same value of H(q).Table 1 outlines the statistically estimated parameters for the monofractal domains of various rock types.A detailed analysis was conducted across different domains to identify the monofractal domain [31].Figure 3 presents the spectral analysis of the identified monofractal and multifractal domains.

Sandstone Marble
FG granite CG granite  The identified ξ can be considered as ℓc, as explained above.This parameter was utilized in two mechanical models developed from the Point Method (PM) variant of the theory of critical distances [32], and coupled Finite Fracture Mechanics (FFM) [33,34].This application aimed to confirm that the length scale quantified through quantitative fractography corresponded to the cohesive length.The identified ξ can be considered as ℓ c , as explained above.This parameter was utilized in two mechanical models developed from the Point Method (PM) variant of the theory of critical distances [32], and coupled Finite Fracture Mechanics (FFM) [33,34].This application aimed to confirm that the length scale quantified through quantitative fractography corresponded to the cohesive length.

Validating ξ ≡ ℓ c by PM
The point method failure criterion can be used to quantify the inherent tensile strength σ 0 .In this method, σ 0 is considered as the stress value at CL/2 (half of a characteristic dis-tance from a stress concentrator).Creager and Paris [35] proposed a solution for calculating the stress distribution in front of a blunted notch: where x is the distance from the notch tip in the direction of the crack propagation, ρ is the notch tip radius, and K U is the apparent stress intensity factor and can be determined experimentally [19].
The PM can be used for calculating the apparent fracture toughness K Ic : CL is an approximation of ℓ pz , and half of this length estimates the plastic zone radius [8].This criterion involves two unknown variables: CL and σ 0 .Another form of the PM was recently developed that follows a cohesive zone modelling of a fracture and uses ℓ c /2 and σ c instead of CL/2 and σ 0 [36].
This version of the PM was employed for predicting σ c by quantifying the ℓ c values with the aid of roughness analysis.In this method, σ c corresponds to the tensile stress value at ℓ c /2 from the stress concentrator.Figure 4a shows the σ c approximation using the developed PM for different rock specimens.Subsequently, the pair of ℓ c and σ c was incorporated into the Dugdale-Barenblatt (D-B) formula (Equation ( 5)) to forecast the critical stress intensity factor or the fracture toughness K Ic .This prediction demonstrated excellent agreement with the experimental measurements obtained for the same specimen following the ISRM guidelines [19]:

Validating ξ ≡ ℓc by PM
The point method failure criterion can be used to quantify the inherent tensile strength 0.In this method, 0 is considered as the stress value at CL/2 (half of a characteristic distance from a stress concentrator).Creager and Paris [35] proposed a solution for calculating the stress distribution in front of a blunted notch: where x is the distance from the notch tip in the direction of the crack propagation,  is the notch tip radius, and K U is the apparent stress intensity factor and can be determined experimentally [19].
The PM can be used for calculating the apparent fracture toughness KIc: CL is an approximation of ℓpz, and half of this length estimates the plastic zone radius [8].This criterion involves two unknown variables: CL and 0.Another form of the PM was recently developed that follows a cohesive zone modelling of a fracture and uses ℓc/2 and c instead of CL/2 and 0 [36].
This version of the PM was employed for predicting c by quantifying the ℓc values with the aid of roughness analysis.In this method, c corresponds to the tensile stress value at ℓc/2 from the stress concentrator.Figure 4a shows the c approximation using the developed PM for different rock specimens.Subsequently, the pair of ℓc and c was incorporated into the Dugdale-Barenblatt (D-B) formula (Equation ( 5)) to forecast the critical stress intensity factor or the fracture toughness KIc.This prediction demonstrated excellent agreement with the experimental measurements obtained for the same specimen following the ISRM guidelines [19]: The experimental tests were performed at a 5 mm/min rate, since at slower rates, the specimens were not completely separated, and roughness reconstruction for such samples was challenging.Table 1 reports this agreement for all rock types, which verified the quantified ℓc by fractography as well as determined c by the developed PM.ℓ pz of different ring geometries are calculated using the energy criterion, then the perfect power law emerging from LEFM is used to approximate σ c as σ tc (ℓ c ).There is a very good agreement between the σ c values approximated by PM and CFFM (Table 1).
The experimental tests were performed at a 5 mm/min rate, since at slower rates, the specimens were not completely separated, and roughness reconstruction for such samples was challenging.Table 1 reports this agreement for all rock types, which verified the quantified ℓ c by fractography as well as determined σ c by the developed PM.

Validating ξ ≡ ℓ c by CFFM
The CFFM represents a thorough failure criterion necessitating the satisfaction of both stress and energy criteria prior to the initiation of fracture propagation [34]: where (x, y) is the Cartesian coordinate system (centered at the ring center in our experiments, as shown in Figure 5).These two equations integrate the stress σ x (y) and crack driving force G over a critical crack advance ∆.The stress criterion needs an average critical tensile stress σ u over ∆, while the energy criterion ensures that the available energy G c ∆ can create the new fracture surface.The energy criterion can also be expressed in relation to the stress intensity factor K I and the intrinsic toughness K Ii (Equation ( 7)).

Geometry and Rate Effects on 𝜎c
The results presented in Figure 4b suggest a novel powerful method for analyzing the effect of geometry including shape and size on material tensile strength by taking into account the ℓpz following CZM.The experimental measurements and results of the developed CFFM are summarized in Table 2.According to these results, if the radius of the ring central hole R is larger than the ℓpz quantified by the energy criterion ℓpz ≪ R, then tc~ℓpz −0.5 .Otherwise, if ℓpz ≫ R, then the stress will not be concentrated on the expected parts of the ring but will distribute in a larger area, and analytical solutions will fail.The power exponent 0.5 was in agreement with other proposed size effect laws emerged from linear elasticity [14,15], since tc was calculated as a function of failure load and linear elastic stress distribution x (y), which is totally acceptable for y ≥ ℓpz.c, however, has nothing to do with the applied stress and its geometry-dependent distribution.Cohesive stress is purely determined by the material properties, causing the material to fail once it reaches this stress level.In fact, the cohesive stress of a material, together with the geometry of the sample or structure, environmental conditions, and loading conditions, sets the ℓpz and the apparent tensile strength tc.The ℓpz values calculated by the energy criterion call for generalizing the D-B formula in order to approximate the length of FPZ by adding a new factor f that is a function of geometry as well as of the environmental and loading conditions: According to our experimental observations after a threshold, the apparent fracture toughness increased with the loading rate.For the studied NSCB specimens, the KIc values measured at 0.05 and 5 mm/min experimental rates are reported in Table 1 and Table 2,  The developed CFFM in this study replaced the finite crack advance ∆ with the length of the process zone ℓ pz and determined this quantity for different ring specimens according to the energy criterion and the intrinsic fracture toughness K Ii determined experimentally.
K Ii was determined as per ISRM suggested method [19] at a slow experimental rate of 0.05 min/mm to avoid dynamic effects.All ring tests were also carried out at the same rate to minimize the dynamic effects.Then, ℓ pz was imported into the stress criterion to determine the geometry-dependent tensile strength σ tc of the rings using Kirsch's solution together with Hobbs' correction [20,37]: where P max and B are applied failure load and specimen thickness, respectively.Finally, by plotting ℓ pz against σ tc for different rings of the same rock type, a perfect power law σ tc ~ℓpz −0.5 emerging from linear elasticity determined the intrinsic tensile strength or the cohesive stress of the crack, σc, corresponding to ℓ pz of the crack or the cohesive length quantified by roughness analysis ξ ≡ ℓ c (Figure 4b).Table 1 summarizes these results.The determined σ c following the developed CFFM was identical to the one determined using the developed PM, within experimental error.This remarkable agreement further validated the precision of the cohesive length ℓ c or the length of the process zone of a propagating crack determined through quantitative fractography on the post-mortem fracture surfaces of the analyzed rock materials.

Geometry and Rate Effects on σ c
The results presented in Figure 4b suggest a novel powerful method for analyzing the effect of geometry including shape and size on material tensile strength by taking into account the ℓ pz following CZM.The experimental measurements and results of the developed CFFM are summarized in Table 2.According to these results, if the radius of the ring central hole R is larger than the ℓ pz quantified by the energy criterion ℓ pz ≪ R, then σtc~ℓ pz −0.5 .Otherwise, if ℓ pz ≫ R, then the stress will not be concentrated on the expected parts of the ring but will distribute in a larger area, and analytical solutions will fail.The power exponent 0.5 was in agreement with other proposed size effect laws emerged from linear elasticity [14,15], since σtc was calculated as a function of failure load and linear elastic stress distribution σ x (y), which is totally acceptable for y ≥ ℓ pz .σ c , however, has nothing to do with the applied stress and its geometry-dependent distribution.Cohesive stress is purely determined by the material properties, causing the material to fail once it reaches this stress level.In fact, the cohesive stress of a material, together with the geometry of the sample or structure, environmental conditions, and loading conditions, sets the ℓ pz and the apparent tensile strength σ tc .The ℓ pz values calculated by the energy criterion call for generalizing the D-B formula in order to approximate the length of FPZ by adding a new factor f that is a function of geometry as well as of the environmental and loading conditions: According to our experimental observations after a threshold, the apparent fracture toughness increased with the loading rate.For the studied NSCB specimens, the K Ic values measured at 0.05 and 5 mm/min experimental rates are reported in Tables 1 and 2, respectively.As it indicated by its name, K Ic or critical stress intensity factor is a measure of stress concentration that can change as a function of geometry or loading rate and changes the behavior of a material in the K dominance region where ℓ pz is.
It was shown that at quasi-statics loading rates, cohesive stress is material-dependent and independent of the loading rate.However, it was suggested that at dynamic loading rates, the properties of the material will change.Zhao [38] suggested that the cohesion parameter in the Mohr-Coulomb strength criterion or the so-called cohesive strength is the cause of the Dynamic Increase Factor (DIF) of material strength.Therefore, in future research, it will be investigated by determining the cohesive stress σ c of fractured surfaces of the same rock types broken at dynamic loading rates using roughness analysis and the developed multifractal method.

Conclusions
The correlation between the physics and the mechanics of brittle fracture was investigated through the quantification of the cohesive length across four distinct rock types.Employing statistical physics and an innovative domain-based multifractal analysis, this critical length scale was assessed, revealing a transition from multifractality at smaller scales to mono-fractality at larger scales, suitable for modeling fractures using LEFM.It was confirmed that a quasi-brittle fracture can be viewed as a transformative phase transition, where continuous damage percolation dominates, accompanied by occasional avalanches at smaller scales.The transition to a first-order phase occurs at a critical length scale, established as the cohesive length and beyond.
The quantification of the cohesive length holds significant importance in fracture mechanics.According to the theory of critical distances and cohesive zone modeling, the stress corresponding to the length of the fracture process zone of a rapidly propagating crack equates to the cohesive stress or intrinsic tensile strength of the material-a pivotal parameter in modeling the fracture and failure of quasi-brittle materials.Thus, knowledge of the cohesive length facilitates the measurement of the intrinsic tensile strength of such materials.
For the determination of cohesive stress or intrinsic tensile strength, two distinct mechanical approaches based on the Point Method (PM) formulation of the theory of critical distances and CFFM were employed.Remarkably, the cohesive stress values obtained from both methods exhibited perfect alignment, underscoring the successful determination of the length of the fracture process zone of a rapidly propagating crack, the cohesive length, and the intrinsic tensile strength of the studied rock materials.
Finally, the utilization of the formulated CFFM methodologies for determining the length of the fracture process zone in quasi-brittle materials across varied stress concentrations was examined.Additionally, the developed approach holds promise for elucidating the variations in tensile strength of identical quasi-brittle materials under diverse conditions.

Figure 1 .
Figure 1.Reconstructed fractured surfaces of the studied rock materials.The x-and z-axes are in the directions of crack propagation and crack front, respectively.

Figure 1 .
Figure 1.Reconstructed fractured surfaces of the studied rock materials.The x-and z-axes are in the directions of crack propagation and crack front, respectively.

Figure 2 .
Figure 2. Results of the percolation analysis on fractured surfaces of the studied rocks.

Figure 2 .
Figure 2. Results of the percolation analysis on fractured surfaces of the studied rocks.

Figure 3 .
Figure 3. Identified monofractal and multifractal domains of the studied rock fractured surfaces.

Figure 3 .
Figure 3. Identified monofractal and multifractal domains of the studied rock fractured surfaces.

Figure 4 .
Figure 4. Cohesive stress c approximation as a function of cohesive length ℓc quantified by roughness analysis.(a) PM: the stress distribution is calculated using the C-P solution and the experimentally measured failure load, then c is approximated as  (ℓc) on this curve.(b) CFFM: ℓpz of different ring geometries are calculated using the energy criterion, then the perfect power law emerging from

Figure 4 .
Figure 4. Cohesive stress σ c approximation as a function of cohesive length ℓ c quantified by roughness analysis.(a)PM: the stress distribution is calculated using the C-P solution and the experimentally measured failure load, then σ c is approximated as σ (ℓ c ) on this curve.(b) CFFM: ℓ pz of different ring geometries are calculated using the energy criterion, then the perfect power law emerging from LEFM is used to approximate σ c as σ tc (ℓ c ).There is a very good agreement between the σ c values approximated by PM and CFFM (Table1).

Table 1 .
Overview of the computed statistical physics parameters alongside the mechanical properties of the investigated rock materials.

Table 1 .
Overview of the computed statistical physics parameters alongside the mechanical properties of the investigated rock materials.

Table 2 .
Results of the ring tests and the developed CFFM for different geometries and rock types.