Determining the Cohesive Length of Rock Materials by Roughness Analysis

Abstract

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 variations under diverse stress concentrations and loading conditions.

1. Introduction

Mandelbrot et al. (1984) 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.

2. 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.

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].

3. Results and Discussions

3.1. 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]:

$${\omega }_{ϵ}\left(\mathrm{X}\right)=\frac{1}{2}\mathrm{l}\mathrm{o}\mathrm{g}\left({〈{\Delta h\left(ϵ\right)}^2〉}_{\mathsf{\Theta }}\right)$$

(1)

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 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 cliffs P_th on ω_ϵ(X) by ξ and ξ², 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].

3.2. 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]:

$${\mathrm{S}}_{\mathrm{q}}\left(\mathsf{\delta }\mathrm{r}\right)={〈{\left|\Delta \mathrm{h}\left(\mathsf{\delta }\mathrm{r}\right)\right|}^{\mathrm{q}}〉}^{1/\mathrm{q}}={〈{\left|\mathrm{h}\left(\mathrm{X}+\mathsf{\delta }\mathrm{r}\right)-\mathrm{h}\left(\mathrm{X}\right)\right|}^{\mathrm{q}}〉}_{\mathrm{X}}^{1/\mathrm{q}}\propto {\mathsf{\delta }\mathrm{r}}^{{\mathsf{\zeta }}_{\mathrm{q}}\mathrm{*}1/\mathrm{q}}$$

(2)

where Δh(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. Overview of the computed statistical physics parameters alongside the mechanical properties of the investigated rock materials.

Rock Type	𝓁c [mm]	λ (δr < 𝓁c)	H (δr > 𝓁c)	D (ϵ = 33 μm)	σc [MPa] PM	KIc [MPa.m0.5] PM	KIc [MPa.m0.5] ISRM	σc [MPa] CFFM
Sandstone	0.4	0.34	0.53	1.68 ± 0.06	15.57	0.497	0.498	15.95
Marble	1.1	0.23	0.60	1.69 ± 0.06	18.44	0.976	1.002	17.11
FG granite	0.5	0.29	0.52	1.68 ± 0.05	35.90	1.281	1.314	34.48
CG granite	0.9	0.21	0.53	1.72 ± 0.06	24.88	1.191	1.218	24.11

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.

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.

3.3. Validating ξ ≡ 𝓁_c by PM

The point method failure criterion can be used to quantify the inherent tensile strength σ₀. In this method, σ₀ 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:

$$\sigma \left(x,0\right)=\frac{2K^{\mathrm{U}}}{\sqrt{\pi }}\frac{x+\rho }{{\left(2x+\rho \right)}^{3/2}}$$

(3)

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:

$$CL=\frac{1}{\pi }{\left(\frac{K_{\mathrm{I}\mathrm{c}}}{{\sigma }_0}\right)}^2$$

(4)

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 σ₀. 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 σ₀ [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]:

$${\mathcal{l}}_{\mathrm{c}}=\frac{\pi }{8}{\left(\frac{K_{\mathrm{I}\mathrm{c}}}{{\sigma }_{\mathrm{c}}}\right)}^2$$

(5)

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.

3.4. 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]:

$$\left\{\begin{array}{c}{\int }_0^{\Delta }{\sigma }_x\left(y\right)dy\ge {\sigma }_{\mathrm{u}}\Delta \\ {\int }_0^{\Delta }G\left(c\right)dc\ge G_{\mathrm{c}}\Delta \end{array}\right.$$

(6)

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 Δ. The stress criterion needs an average critical tensile stress σ_u over Δ, while the energy criterion ensures that the available energy G_cΔ 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)).

$$\left\{\begin{array}{c}{\int }_0^{{\mathcal{l}}_{\mathrm{p}\mathrm{z}}}{\sigma }_x\left(y\right)dy\ge {\sigma }_{\mathrm{t}\mathrm{c}}{\mathcal{l}}_{\mathrm{p}\mathrm{z}}\\ {\int }_0^{{\mathcal{l}}_{\mathrm{p}\mathrm{z}}}{K}_{\mathrm{I}}^2\left(c\right)dc\ge K_{\mathrm{I}\mathrm{i}}^2{\mathcal{l}}_{\mathrm{p}\mathrm{z}}\end{array}\right.$$

(7)

The developed CFFM in this study replaced the finite crack advance Δ 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]:

$${\sigma }_{\mathrm{t}\mathrm{c}}=1+\frac{19}{3}{\left(\frac{R}{R_0}\right)}^2\left(\frac{P_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\pi B{R}_0}\right)\frac{\left(-2R^4+R^2{\left(R+{\mathcal{l}}_{\mathrm{p}\mathrm{z}}\right)}^2+{\left(R+{\mathcal{l}}_{\mathrm{p}\mathrm{z}}\right)}^4\right)}{{\mathcal{l}}_{\mathrm{p}\mathrm{z}}{\left(R+{\mathcal{l}}_{\mathrm{p}\mathrm{z}}\right)}^3}$$

(8)

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~𝓁_pz^−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.

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

Rock Type	R [mm]	R0 [mm]	B [mm]	Pmax [MPa]	σmax [MPa]	KIc [MPa.m0.5]	𝓁pz [mm]	σtc [MPa]
Sandstone	1.46	37.43	30.08	11.57	3.30	0.45	7.14	4.09
	3.03	37.50	30.23	11.55	3.38	0.45	0.37	16.49
	6.07	37.42	30.10	8.08	2.66	0.45	0.54	13.72
	14.96	37.44	29.98	2.61	1.49	0.45	1.98	7.17
Marble	1.49	37.50	30.00	21.26	6.08	0.80	5.98	7.87
	3.09	37.41	29.88	12.39	3.68	0.80	21.22	4.28
	6.05	37.41	29.85	11.73	3.90	0.80	1.01	17.89
	14.76	37.48	29.98	4.82	2.71	0.80	1.85	13.19
CG granite	3.05	37.69	29.85	23.57	6.95	1.02	0.52	31.66
	6.04	37.74	30.03	16.88	5.51	1.02	0.71	27.18
	13.00	37.72	30.07	7.27	3.58	1.02	1.78	17.12
FG granite	3.04	37.70	30.01	27.96	8.19	1.08	0.37	40.13
	6.00	37.78	30.10	19.33	6.28	1.08	0.58	31.93
	12.92	37.80	30.10	8.66	4.22	1.08	1.31	21.32

. 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:

$${\mathcal{l}}_{\mathrm{p}\mathrm{z}}=\frac{{\mathcal{l}}_{\mathrm{c}}}{{\mathrm{f}}^2}=\frac{\mathsf{\pi }}{8}{\left(\frac{{\mathrm{K}}_{\mathrm{I}\mathrm{c}}}{{\mathsf{\sigma }}_{\mathrm{c}}}\right)}^2$$

(9)

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 Table 1 and Table 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.

4. 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.