A Fractal-Enhanced Mohr–Coulomb (FEMC) Model for Strength Prediction in Rough Rock Discontinuities

Abstract

Accurate prediction of the shear strength of rock discontinuities requires accounting for surface roughness, which is a factor neglected in the classical Mohr–Coulomb criterion. This study proposes a fractal-enhanced Mohr–Coulomb model that incorporates the surface fractal dimension Ds as a geometric state variable governing both the cohesion and internal friction angle. The fractal dimension is treated as an objective, scale-invariant descriptor, computable via established methods, such as box-counting and power spectral density analysis, which are known to yield consistent results when applied to joint topography. The model predicts a nonlinear increase in shear strength with Ds, producing a dynamically adjustable failure envelope that can exceed the classical Mohr–Coulomb estimates by 25–40% for rough joints, which is consistent with trends observed in experimental shear tests. By linking strength parameters directly to measurable surface geometry, the framework provides a physically interpretable bridge between micro-scale roughness and macro-scale mechanical response. Although the current formulation assumes monotonic, dry, and quasi-static conditions, the explicit dependence on Ds offers a foundation for future extensions that incorporate anisotropy, damage evolution, and hydro-mechanical coupling.

1. Introduction

The accurate modeling of material failure behavior is crucial for the estimation of geotechnical stability for rock slopes, tunnels, and foundations. The Mohr–Coulomb criterion is commonly used to estimate the shear strength of a sample based on the normal stress because of its conceptual simplicity and ease of application in engineering practice. Two intrinsic parameters govern the Mohr–Coulomb criterion: cohesion (c) and internal friction angle (φ), which provide the correlation between shear strength (τ) and normal stress (σ) [1,2,3,4]. This formulation assumes homogeneous continuous media with planar failure surfaces an idealization that overlooks the inherent heterogeneity and discontinuity of natural rock masses. In reality, rock joints and faults exhibit complex surface topographies, whose geometric features critically influence load transfer, dilatancy, and shear resistance [5,6]. The omission of such microstructural characteristics in classical theory can lead to significant errors in strength prediction, motivating the need to enrich the failure criteria with quantifiable geometric descriptors [7,8]. The surface roughness plays a central role in this context. Natural fracture surfaces are rarely smooth; instead, they possess multiscale asperities that promote mechanical interlocking and volume expansion during shearing [5]. To account for this, empirical indices, such as the Joint Roughness Coefficient (JRC), were introduced to adjust the strength estimates based on visual inspection of joint profiles [7,9,10]. However, these methods suffer from subjectivity, poor reproducibility, and strong scale dependence, particularly when extrapolating from laboratory to field conditions [9]. More fundamentally, they lack a mechanistic connection between geometry and the constitutive parameters governing failure, rendering their use largely qualitative and context-specific.

Although practitioners sometimes augment the Mohr–Coulomb model by inflating the apparent friction angle or coupling it with empirical dilation rules, the core parameters c and φ remain fixed material constants that are independent of the evolving state of the discontinuity. Physical evidence demonstrates that asperity degradation during shearing leads to surface smoothing and progressive strength loss, a dynamic process that cannot be accurately represented using static parameters. Existing empirical correlations, such as those based on the Joint Roughness Coefficient (JRC), fail to integrate the evolving geometric characteristics of joint surfaces into their failure criteria, representing a significant limitation in current geomechanical practices. The Fractal-Enhanced Mohr–Coulomb (FEMC) model presented herein directly addresses these shortcomings by integrating the fractal dimension Ds of rock joint surfaces into the constitutive law as a state variable governing both the cohesion and friction angle. By anchoring the strength parameters to an objective, quantifiable descriptor of surface geometry, the FEMC model captures the intrinsic relationship between surface morphology and mechanical behavior.

Fractal geometry is a promising alternative for this purpose. Fractal dimension (Ds) provides an objective, scale-invariant measure of surface complexity over a defined measurement range [11,12,13,14]. Numerous studies have established robust statistical relationships between Ds and the experimentally observed shear strength, confirming their physical relevance [15,16,17]. Despite these correlations, Ds has consistently been used as an external regressor or classification index, never as an intrinsic variable within the failure criterion. This represents a critical missed opportunity: the absence of a state-dependent framework that directly links the evolving surface geometry to the constitutive law. Barton’s joint roughness coefficient (JRC)-joint wall compressive strength (JCS) criterion [9], although historically influential, exemplifies these shortcomings. Its reliance on visual JRC estimation introduces subjectivity, and its scale sensitivity, where JRC may drop by 30–40% from laboratory (10 cm) to field (1 m+) scales [5] limits predictive reliability. In contrast, fractal-based characterization bypasses visual judgment and anchor roughness in measurable topography. Nevertheless, even the most rigorous fractal analyses to date have stopped short of integrating Ds into the mathematical structure of the strength models. To ensure robustness in geometric quantification, dual-methodology box-counting [18] and power spectral density analysis [19,20,21,22] are commonly employed. These complementary techniques, which are rooted in spatial occupancy and frequency-domain decomposition, enable the cross-validation of Ds estimates. When applied with careful preprocessing and consistent scaling ranges, they yield highly congruent results (e.g., R² = 0.98, mean deviation < 0.02), minimizing bias and enhancing confidence in geometric input [13,18,23].

Building on these foundations, this study addresses a clear research drawback: the lack of a failure criterion in which the surface geometry actively governs the strength parameters through a physically consistent, dynamically evolving formulation. Therefore, this study proposes the Fractal-Enhanced Mohr–Coulomb (FEMC) model, which embeds the fractal dimension Ds directly into the constitutive law as a state variable that controls both cohesion and friction angle. Unlike prior approaches that treat roughness as a post hoc correction, FEMC establishes Ds as an integral component of the failure envelope, enabling the model to adapt to changes in surface morphology during shearing. The resulting framework retains the analytical tractability of the classical Mohr–Coulomb while overcoming its two key limitations: (1) the assumption of static strength parameters and (2) the neglect of geometric evolution. By unifying objective surface characterization with constitutive mechanics, this study provides a theoretically coherent and practically implementable pathway for a more accurate prediction of shear strength in rough rock discontinuities.

2. Methodology

This study employs a systematic approach to develop an enhanced Mohr–Coulomb failure criterion through direct integration of the surface fractal dimension [14]. The resulting fractal dimension values were then used to formulate scale-dependent expressions for cohesion ($c^{\prime }$) and friction angle (${\phi }^{\prime }$), which formed the basis of the proposed strength model.

2.1. Mathematical Description of Fractal Dimension (Ds)

Natural rock joint surfaces are inherently irregular and exhibit self-similar roughness across scales, meaning that their geometric complexity appears statistically similar whether observed at millimeter or centimeter resolutions. This multiscale property, known as fractality, cannot be captured by classical Euclidean descriptors, such as average asperity height or root-mean-square roughness, which depend strongly on the measurement scale and sampling resolution. To overcome this limitation, we adopted the 2D fractal dimension, a scale-invariant parameter derived from one-dimensional surface profiles (i.e., height versus distance along a scan line). The value of $D_{\mathrm{S}}$ quantifies how “space-filling” the profile is, with perfectly smooth (Euclidean) lines corresponding to $D_{\mathrm{S}}=1$, whereas increasingly rough and convoluted profiles approach $D_{\mathrm{S}}=2$. This approach is consistent with the standard practice in rock joint characterization, where profile-based fractal analysis is used to quantify the shear resistance [5,9,24].

The box-counting method, a standard technique in fractal geometry for quantifying the surface roughness, was employed to calculate the fractal dimension. This method was originally formalized by Mandelbrot [25] and was later comprehensively described by Crownover [26]. The idea is to cover an object of irregular shape with smaller boxes (or grids) and count the number $\mathrm{N}\left(\epsilon \right)$ of boxes required to cover the object as a function of the box size $\epsilon$. The relation can be described as:

$$\mathrm{N}\left(\epsilon \right)=C·{\epsilon }^{-{\mathrm{D}}_{\mathrm{S}}}$$

(1)

where:

• $\mathrm{N}\left(\epsilon \right)$ is the number of boxes;
• $C$ is a constant depending on the shape;
• $D_{\mathrm{S}}$ is the fractal dimension of the object.

Taking the natural logarithm of both sides yields a linear relationship:

$$\ln \mathrm{N}\left(\epsilon \right)=\ln C-D_{\mathrm{S}}\ln \epsilon$$

(2)

This equation mirrors the form of a straight line, where the slope corresponds to the fractal dimension ($D_{\mathrm{S}}$),

• The expression can be further simplified into a standard linear form:

$$y=mx+b$$

(3)

where:

• $y=\ln \mathrm{N}\left(\epsilon \right)$;
• $x=\ln (\epsilon )$;
• $m=-D_{\mathrm{S}}$ (negative slope);
• $b=\ln C$ (the y-intercept).

The fractal dimension was then computed as the negative slope of the line fitted to the experimental data points plotted in a log–log space.

2.2. Power Spectral Density Method: Frequency Domain Analysis

The power spectral density (PSD) [19,20,22] analyzes the surface topography based on its spatial frequency content. Given a surface height function $h(x)$, its Fourier transform yields PSD $\mathrm{P}\left(k\right)$, where $k$ is the spatial frequency (wavenumber). For a fractal surface, the PSD follows a power-law decay in the frequency domain:

$$\mathrm{P}\left(k\right)=\mathrm{A}·k^{-\left(5-2D_S\right)}$$

(4)

where A is a scaling constant related to the surface amplitude. Taking the logarithm of both sides converts Equation (4) into a linear form, as follows:

$$\ln \mathrm{P}\left(k\right)=\ln \mathrm{A}-\left(5-2D_{\mathrm{S}}\right)\ln k$$

(5)

The slope $m$ of the $\ln \mathrm{P}\left(k\right)$ versus $\ln k$ plot directly yields the fractal dimension:

$$m=-\left(5-2D_{\mathrm{S}}\right)$$

(6)

$$\begin{array}{l}{D}_{\mathrm{S}}={\displaystyle \frac{5+m}{2}}\end{array}$$

(7)

The PSD method offers superior noise resistance compared to box counting, as frequency-domain analysis naturally filters high-frequency measurement artifacts. Both methods were applied in this study using cross-validation to ensure robustness.

2.3. Classical Mohr–Coulomb Theory

The classical Mohr–Coulomb failure criterion, widely used in geomechanics, expresses shear strength as a linear function of normal stress. The shear strength ${\tau }_f$ at failure is defined as:

$${\tau }_f=c^{\prime }+{\sigma }^{\prime }\tan {\phi }^{\prime }$$

(8)

where:

• $c^{\prime }$ is the effective cohesion;
• ${\sigma }^{\prime }$ is the effective normal stress on the failure plane;
• ${\phi }^{\prime }$ where denotes the effective friction angle.

However, shear strength ${\tau }_f$ treats cohesion and friction angle as constant material properties, ignoring their dependence on surface roughness and evolving geometry during shear. Under triaxial testing conditions, the failure criterion was expressed in terms of the major and minor principal stresses. We begin with the geometric relationship of the stress transformation. The radius of the Mohr circle at failure is

$$R={\displaystyle \frac{{\sigma }_1^{\prime }-{\sigma }_3^{\prime }}{2}}$$

(9)

The center of the circle is located at

$$C={\displaystyle \frac{{\sigma }_1^{\prime }+{\sigma }_3^{\prime }}{2}}$$

(10)

The failure plane makes an angle ${\theta }_f$ with the major principal plane, where:

$${\theta }_f=45°+{\displaystyle \frac{{\phi }^{\prime }}{2}}$$

(11)

At failure, the Mohr circle is tangent to the failure envelope. The geometric condition requires the following:

$$R\sin {\phi }^{\prime }=C\sin {\phi }^{\prime }+c^{\prime }\cos {\phi }^{\prime }$$

(12)

Substituting the expressions for $R$ and $C$:

$${\displaystyle \frac{{\sigma }_1^{\prime }-{\sigma }_3^{\prime }}{2}}\sin {\phi }^{\prime }={\displaystyle \frac{{\sigma }_1^{\prime }+{\sigma }_3^{\prime }}{2}}\sin {\phi }^{\prime }+c^{\prime }\cos {\phi }^{\prime }$$

(13)

Multiplying both sides by 2 and expanding:

$$\left({\sigma }_1^{\prime }-{\sigma }_3^{\prime }\right)\sin {\phi }^{\prime }=\left({\sigma }_1^{\prime }+{\sigma }_3^{\prime }\right)\sin {\phi }^{\prime }+2c^{\prime }\cos {\phi }^{\prime }$$

(14)

Collecting terms with ${\sigma }_1^{\prime }$ and ${\sigma }_3^{\prime }$:

$${\sigma }_1^{\prime }\sin {\phi }^{\prime }-{\sigma }_3^{\prime }\sin {\phi }^{\prime }={\sigma }_1^{\prime }\sin {\phi }^{\prime }+{\sigma }_3^{\prime }\sin {\phi }^{\prime }+2c^{\prime }\cos {\phi }^{\prime }$$

(15)

$$-2{\sigma }_3^{\prime }\sin {\phi }^{\prime }=2c^{\prime }\cos {\phi }^{\prime }$$

(16)

$${\sigma }_1^{\prime }-{\sigma }_1^{\prime }\sin {\phi }^{\prime }={\sigma }_3^{\prime }+{\sigma }_3^{\prime }\sin {\phi }^{\prime }+2c^{\prime }\cos {\phi }^{\prime }$$

(17)

Factoring and dividing (17) by $1-\sin {\phi }^{\prime }$, we have

$${\sigma }_1^{\prime }={\sigma }_3^{\prime }{\displaystyle \frac{1+\sin \phi \prime }{1-\sin \phi \prime }}+2c^{\prime }{\displaystyle \frac{\cos \phi \prime }{1-\sin \phi \prime }}$$

(18)

Compact the notation with ${\mathrm{N}}_{\phi }$, defining the bearing capacity factor:

$${\mathrm{N}}_{\phi }={\tan }^2\left({\displaystyle \frac{{\phi }^{\prime }}{2}}+45°\right)={\displaystyle \frac{1+\sin {\phi }^{\prime }}{1-\sin \phi \prime }}$$

(19)

The classical Mohr–Coulomb criterion becomes

$${\sigma }_1^{\prime }={\sigma }_3^{\prime }{\mathrm{N}}_{\phi }+2c^{\prime }\sqrt{{\mathrm{N}}_{\phi }}$$

(20)

The term ${\mathrm{N}}_{\phi }$ represents the amplification of the confining stress owing to friction, whereas $2c^{\prime }\sqrt{{\mathrm{N}}_{\phi }}$ represents the contribution of the cohesive strength. Both terms are assumed to be constant in the classical theory. We relax this assumption through fractal enhancement.

2.4. Fractal Enhancement of Mohr–Coulomb Theory

Rougher surfaces (higher Ds, closer to 2.0) exhibit greater geometric complexity, leading to enhanced interlocking, higher dilation, and increased peak shear strength, which is consistent with experimental trends [15,16,17]. In the self-affine profile analysis, Ds ranges from 1.0 (perfectly smooth line) to 2.0 (space-filling curve); thus, a larger Ds indicates a greater roughness.

In this study, the classical strength parameters are not material constants, but rather functions of the measurable fractal dimension:

$$c^{\prime }=c^{\prime }\left(D_{\mathrm{S}}\right)$$

(21)

$${\phi }^{\prime }={\phi }^{\prime }\left(D_{\mathrm{S}}\right)$$

(22)

Based on experimental observation across multiple geological classes, the effective internal friction angle ${\phi }^{\prime }$ exhibits approximately linear dependence on fractal dimension:

$${\phi }^{\prime }\left(D_{\mathrm{S}}\right)={\phi }_0+k_{\phi }\left(D_{\mathrm{S}}-D_0\right)$$

(23)

where:

• ${\phi }_0$: Reference friction angle at performable fractal dimension $D_0$
• $k_{\phi }$: Sensitivity coefficient (units: degrees per unit $D_{\mathrm{S}}$)
• $D_0$: Reference fractal dimension, conventionally set at $D_0=2.0$ for perfectly smooth Euclidean surfaces; and
• $D_{\mathrm{S}}$: Fractal dimensions of an actual material surface measured using box-counting or PSD methods.

Effective cohesion exhibits a more complex nonlinear dependence on surface roughness owing to the competing effects of aspect interlocking and contact area evolution.

$$c^{\prime }\left(D_{\mathrm{S}}\right)=c_0+c_1^{\left[k_c\left(D_{\mathrm{S}}-D_0\right)\right]}$$

(24)

where:

• $c_0$ = intrinsic cohesion (kPa or MPa)
• $c_1$ = roughness amplification factor.
• $k_c$ = exponential decay rate.

Substituting the fractal-dependent strength parameters into the classical Mohr–Coulomb criterion (8) yields the following enhanced model:

$${\tau }_f\left(D_{\mathrm{S}}\right)=\left\{c_0+c_1^{\left[k_c\left(D_{\mathrm{S}}-D_0\right)\right]}\right\}+{\sigma }_n^{\prime }\tan \left[{\phi }_0+k_{\phi }\left(D_{\mathrm{S}}-D_0\right)\right]$$

(25)

In principal stress space, the modified criterion becomes

$${\sigma }_1^{\prime }={\sigma }_3^{\prime }{\mathrm{N}}_{\phi }\left(D_{\mathrm{S}}\right)+2c^{\prime }\left(D_{\mathrm{S}}\right)\sqrt{{\mathrm{N}}_{\phi }\left(D_{\mathrm{S}}\right)}$$

(26)

where the bearing capacity factor is fracture-dependent.

$${\mathrm{N}}_{\phi }\left(D_{\mathrm{S}}\right)={\tan }^2\left(45°+{\displaystyle \frac{{\phi }^{\prime }\left(D_{\mathrm{S}}\right)}{2}}\right)$$

(27)

This formulation naturally captures the evolution of the field surface with microstructural roughness, enabling predictions of strength variation without empirical recalibration for each material type.

2.5. Fractional Calculus Framework

Classical integer-order differential equations assume instantaneous response and local behavior. However, geomaterials exhibit memory effects, nonlocal interactions, and power-law creep phenomena that are better described by fractional calculus. The Caputo fractional derivative of order $\alpha \in \left(0,1\right)$ is defined as

$${}^{C}D_t^{\alpha } f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(1-\alpha \right)}}{\displaystyle {\int }_0^{\prime }\frac{\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha }}}d\tau$$

(28)

where $\mathsf{\Gamma }$ is the gamma function. This operator is reduced to classical derivatives during $\alpha =1$. The Atangana–Baleanu (AB) derivative in the Caputo sense [27,28] overcomes the singularity limitation of classical fractional operators.

$${}^{\mathit{ABC}}D_t^{\alpha } f\left(t\right)={\displaystyle \frac{\mathrm{B}\left(\alpha \right)}{1-\alpha }}{\displaystyle {\int }_0^{t}f\prime \left(\tau \right)}{\mathrm{E}}_{\alpha }\left[-{\displaystyle \frac{\alpha }{1-\alpha }}{\left(t-\tau \right)}^{\alpha }\right]d\tau$$

(29)

where $\mathrm{B}\left(\alpha \right)$ is the normalization function satisfying:

$\mathrm{B}\left(0\right)=\mathrm{B}\left(1\right)=1$, which ensures consistency with integer-order limits.

The function ${\mathrm{E}}_{\alpha }\left(z\right)$ is given by

$${\mathrm{E}}_{\alpha }\left(z\right)={\displaystyle \sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }\left(\alpha k+1\right)}}$$

(30)

The Laplace transform of the Caputo derivative is a powerful analytical tool:

$$\wp \left\{{}^{C}D_t^{\alpha } f\left(t\right)\right\}=s^{\alpha }F\left(s\right)-s^{\alpha -1}f\left(0\right)$$

(31)

For the AB derivative, the Laplace transform is

$$\wp \left\{{}^{\mathit{ABC}}D_t^{\alpha } f\left(t\right)\right\}={\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}·{\displaystyle \frac{s^{\alpha }F\left(s\right)-s^{\alpha -1}f\left(0\right)}{s^{\alpha }+\frac{\alpha }{1-\alpha }}}$$

(32)

The fractal-enhanced Mohr–Coulomb model was formulated as a fractional-order stress evolution equation coupling fractal geometry with material memory as follows:

$${}^{\mathit{ABC}}D_t^{\alpha } \sigma \left(t\right)+\beta \sigma \left(t\right)=\gamma c^{\prime }\left(D_{\mathrm{S}}\right)+\delta {\phi }^{\prime }\left(D_{\mathrm{S}}\right)\epsilon \left(t\right)$$

(33)

where:

• $\sigma \left(t\right)$ = time-dependent stress;
• $\alpha$ = fractional order $0<\alpha \le 1$;
• $\beta$ = damping coefficient;
• $\gamma ,\delta$ = coupling constants;
• $\epsilon \left(t\right)$ = strain history.

By applying the Laplace transform to the governing equation with zero initial conditions, we obtain the following general solution:

$${\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}·{\displaystyle \frac{s^{\alpha }{\displaystyle \sum \left(s\right)}}{s^{\alpha }+\frac{\alpha }{1-\alpha }}}+\beta {\displaystyle \sum \left(s\right)}=\gamma c^{\prime }\left(D_{\mathrm{S}}\right){\displaystyle \frac{1}{s}}+\delta {\phi }^{\prime }\left(D_{\mathrm{S}}\right)\mathrm{E}\left(s\right)$$

(34)

where ${\displaystyle \sum \left(s\right)}=\wp \left\{\sigma \left(t\right)\right\}$ and $\mathrm{E}\left(s\right)=\wp \left\{\epsilon \left(t\right)\right\}$. Multiplying by $\left(1-\alpha \right)$ and rearranging the denominator terms:

$${\displaystyle \sum \left(s\right)}\left[\mathrm{B}\left(\alpha \right)s^{\alpha }+\beta \left(1-\alpha \right)\left(s^{\alpha }+{\displaystyle \frac{\alpha }{1-\alpha }}\right)\right]=\left[\gamma c\prime \left(D_{\mathrm{S}}\right){\displaystyle \frac{1}{s}}+\delta \phi \prime \left(D_{\mathrm{S}}\right)\mathrm{E}\left(S\right)\right]\left(1-\alpha \right)\left(S^{\alpha }+{\displaystyle \frac{\alpha }{1-\alpha }}\right)$$

(35)

Collect terms and solve for ${\displaystyle \sum \left(s\right)}$:

$${\displaystyle \sum \left(s\right)}={\displaystyle \frac{\left(1-\alpha \right)\left(s^{\alpha }+\frac{\alpha }{1-\alpha }\right)}{\mathrm{B}\left(\alpha \right)s^{\alpha }+\beta \left(1-\alpha \right)s^{\alpha }+\beta \alpha }}\left[{\displaystyle \frac{\gamma c\prime \left(D_{\mathrm{S}}\right)}{s}}+\delta \phi \prime \left(D_{\mathrm{S}}\right)\mathrm{E}\left(s\right)\right]$$

(36)

Defining the characteristic polynomial:

$$\mathrm{P}\left(s\right)=\left[\mathrm{B}\left(\alpha \right)+\beta \left(1-\alpha \right)\right]s^{\alpha }+\beta \alpha$$

(37)

The general solution in the Laplace domain becomes

$${\displaystyle \sum \left(s\right)}={\displaystyle \frac{\left(1-\alpha \right)s^{\alpha }+\alpha }{\mathrm{P}\left(s\right)}}\left[{\displaystyle \frac{\gamma c\prime \left(D_{\mathrm{S}}\right)}{s}}+\delta \phi \prime \left(D_{\mathrm{S}}\right)\mathrm{E}\left(s\right)\right]$$

(38)

In order to obtain the general solution in the time domain, we apply the inverse Laplace transform using the convolution theorem and Mittag–Leffler function [29,30] properties:

$$\sigma \left(t\right)=\gamma c^{\prime }\left(D_{\mathrm{S}}\right){\displaystyle {\int }_0^t\mathrm{K}\left(t-\tau \right)}d\tau +\delta {\phi }^{\prime }\left(D_{\mathrm{S}}\right){\displaystyle {\int }_0^t\mathrm{K}\left(t-\tau \right)}\epsilon \left(\tau \right)d\tau$$

(39)

where the kernel function $\mathrm{K}\left(t\right)$ is given by

$$\mathrm{K}\left(t\right)=t^{\alpha -1}{\mathrm{E}}_{\alpha ,\alpha }\left({\displaystyle \frac{{\beta }_{\alpha }}{\beta \left(\alpha \right)+\beta \left(1-\alpha \right)}}{t}^{\alpha }\right)$$

(40)

and

$${\mathrm{E}}_{\alpha ,\beta }\left(z\right)={\displaystyle \sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }\left(\alpha k+\beta \right)}}$$

(41)

2.6. Existence and Uniqueness via Banach Fixed-Point Theorem

The existence and uniqueness of the solution to the fractional constitutive equation is established (41) using the Banach fixed-point theorem [31]. Consider the integral form derived from the Caputo fractional derivative:

$$\sigma \left(t\right)={\sigma }_0+{\displaystyle {\int }_0^t\mathrm{K}\left(t-z\right)}F\left(\sigma \left(\tau \right),D_S\right)d\tau$$

(42)

where:

• ${\sigma }_0$ is the initial stress,
• $\mathrm{K}\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}·t^{\alpha -1}$ is the memory kernel,
• $F\left(\sigma ,D_S\right)=-\beta \sigma +\gamma c^{\prime }\left(D_{\mathrm{S}}\right)+\delta {\phi }^{\prime }\left(D_{\mathrm{S}}\right)\epsilon \left(t\right)$ is a nonlinear forcing term that is assumed to be continuous in $t$ and Lipschitz continuous in $\sigma$.

The memory kernel is given explicitly by

$$\mathrm{K}\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{t}^{\alpha -1}$$

(43)

Let $\mathrm{X}=C\left[0,\mathrm{T}\right]$ denote the Banach space of continuous functions on $\left[0,\mathrm{T}\right]$ equipped with the supremum norm:

$${‖\sigma ‖}_{\infty }=\underset{t\in \left[0,\mathrm{T}\right]}{\max }\left|\sigma \left(t\right)\right|$$

(44)

Define the solution operator $\mathrm{T}:\mathrm{X}\to \mathrm{X}$ by

$$\left(\mathrm{T}\sigma \right)\left(t\right)={\sigma }_0+{\displaystyle {\int }_0^t\mathrm{K}\left(t-\tau \right)F\left(\sigma \left(\tau \right),D_S\right)}d\tau$$

Assume that $F\left(\sigma ,D_S\right)$ is Lipschitz continuous in $\sigma$ with constant $\beta >0$, i.e., $\left|F\left({\sigma }_1,D_S\right)-F\left({\sigma }_2,D_S\right)\right|\le \beta \left|{\sigma }_1-{\sigma }_2\right|,\forall {\sigma }_1,{\sigma }_2\in ℝ$. For any ${\sigma }_1,{\sigma }_2\in \mathrm{X}$, we have

$$\left|\left(\mathrm{T}{\sigma }_1\right)\left(t\right)-\left(\mathrm{T}{\sigma }_2\right)\left(t\right)\right|\le {\displaystyle {\int }_0^t\left|\mathrm{K}\left(t-\tau \right)\right|·\left|F\left({\sigma }_1\left(\tau \right),D_S\right)\right|}d\tau \le \beta {\displaystyle {\int }_0^t\left|\mathrm{K}\left(t-\tau \right)\right|·\left|\left({\sigma }_1\left(\tau \right)-{\sigma }_2\left(\tau \right)\right)\right|}d\tau$$

Taking the supremum over $t\in [0,T]$, we obtain

$${‖\mathrm{T}{\sigma }_1-\mathrm{T}{\sigma }_2‖}_{\infty }\le \beta \left(\underset{t\in \left[0,\mathrm{T}\right]}{\sup }{\displaystyle {\int }_0^t\left|\mathrm{K}\left(t-\tau \right)\right|}\right)d\tau ‖{\sigma }_1-{\sigma }_2‖$$

(45)

Since $\mathrm{K}\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{t}^{\alpha -1}$, we compute:

$${\displaystyle {\int }_0^t\left|\mathrm{K}\left(t-\tau \right)\right|}d\tau ={\displaystyle {\int }_0^t\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\left(t-\tau \right)}^{\alpha -1}d\tau ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\displaystyle {\int }_0^t{S}^{\alpha -1}}dS={\displaystyle \frac{t^{\alpha }}{\mathsf{\Gamma }\left(\alpha \right)\alpha }}={\displaystyle \frac{t^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}$$

Therefore:

$$\underset{t\in \left[0,\mathrm{T}\right]}{\sup }{\displaystyle {\int }_0^t\left|\mathrm{K}\left(t-\tau \right)\right|}d\tau ={\displaystyle \frac{{\mathrm{T}}^{\alpha }}{\mathsf{\Gamma }\left(\alpha +1\right)}}$$

and so:

$${‖\mathrm{T}{\sigma }_1-\mathrm{T}{\sigma }_2‖}_{\infty }\le \left(\beta ·{\displaystyle \frac{{\mathrm{T}}^{\alpha }}{\left(\alpha +1\right)}}\right){‖{\sigma }_1-{\sigma }_2‖}_{\infty }$$

(46)

Define:

$$q=\beta ·{\displaystyle \frac{{\mathrm{T}}^{\alpha }}{\left(\alpha +1\right)}}$$

$q<1$, $\mathrm{T}$ is the contraction mapping of $\mathrm{X}$. This condition holds for sufficiently small $\mathrm{T}>0$, specifically,

$$\mathrm{T}<\left[{\displaystyle \frac{\mathsf{\Gamma }\left(\alpha +1\right)}{\beta }}\right]1/\alpha$$

Theorem (existence and uniqueness): Under the above assumptions, operator $\mathrm{T}$ admits a unique fixed-point ${\sigma }^{*}\in C\left[0,\mathrm{T}\right]$ such that $\mathrm{T}{\sigma }^{*}={\sigma }^{*}$. This fixed point is the unique continuous solution to the integral Equation (42), and hence to the fractional constitutive law (41) interpreted in the Caputo sense.

Proof sketch: Using the Banach fixed-point theorem, the iterative sequence $\left\{{\sigma }_n\right\}$ is defined as:

$${\sigma }_{n+1}\left(t\right)={\sigma }_0+{\displaystyle {\int }_0^t\mathrm{K}\left(t-\tau \right)F\left({\sigma }_n\left(\tau \right),D_S\right)}d\tau$$

converges uniformly on $\left[0,\mathrm{T}\right]$ to the unique solution ${\sigma }^{*}\left(t\right)$. This completes the proof.

2.7. Lyapunov Stability Analysis

Define the Lyapunov functional:

$$V\left(t\right)={\displaystyle \frac{1}{2}}{\sigma }^2\left(t\right)$$

(47)

Computation of the fractional derivative

$${}^{C}D_t^{\alpha } V\left(t\right)=\sigma \left(t\right)·{}^{C}D_t^{\alpha } \sigma \left(t\right)$$

(48)

Substituting the governing equation

$${}^{C}D_t^{\alpha } V\left(t\right)=\sigma \left(t\right)\left[-\beta \sigma \left(t\right)+\gamma c\prime \left(D_{\mathrm{S}}\right)+\delta \phi \prime \left(D_{\mathrm{S}}\right)\epsilon \left(t\right)\right]$$

(49)

For bounded loading and positive damping ($\beta >0$), we obtain

$${}^{C}D_t^{\alpha } V\left(t\right)\le -\beta {\sigma }^2\left(t\right)+\sigma \left(t\right)·C_{bounded}$$

(50)

where: $C_{bounded}=\max \left\{\gamma c\prime \left(D_{\mathrm{S}}\right)+\delta \phi \prime \left(D_{\mathrm{S}}\right)\epsilon \left(t\right)\right\}$. This implies an exponential-like decay in the fractional sense, which ensures stability.

Stability theorem: the solution $\sigma \left(t\right)$ is Mittag–Leffler stable. There exists a constant $\lambda >0$ and $b>0$ such that:

$$\left|\sigma \left(t\right)\le bE\alpha \left(-\lambda t\alpha \right)\right|$$

(51)

This fractional stability generalizes the exponential stability and is appropriate for systems with memory and nonlocal effects. The asymptotic convergence to a unique equilibrium state implies the existence of a global attractor in the stress–damage phase space, which represents the long-term shear resistance of the joint.

3. Results

This study developed a comprehensive framework that modifies the classical Mohr–Coulomb (MC) failure criterion by incorporating the surface fractal dimension Ds as a state-dependent parameter. The proposed model integrates fractal geometry, fractional calculus, and nonlinear stability theory to predict the strength, deformation, and failure of rock joints under both monotonic and cyclic loading conditions. By embedding Ds directly into the constitutive law, this approach aims to provide an objective link between surface morphology and mechanical behavior. The subsequent analysis evaluates how this integration enhances the predictive capabilities of the MC criterion through rigorous testing of experimental data.

3.1. Fractal Dimension Quantification Using Box-Counting and PSD Methods

Accurate roughness quantification is critical for predicting shear strength because rock joint resistance arises from geometric interlocking. To overcome the subjectivity of empirical indices such as the JRC, a fractal approach using the surface fractal dimension Ds was adopted, which is a scale-invariant measure of complexity. To ensure robustness, Ds is estimated using two independent methods: Power Spectral Density (PSD) and Box-Counting Method (BCM), representing frequency- and spatial-domain analyses, respectively. As shown in Figure 1a, the PSD yields a spectral exponent of β = 2.6, giving Ds = (5 − β)/2 = 1.2. In Figure 1b, BCM produces a log–log slope of Ds = 1.3 with R² = 0.99, confirming strong self-similarity. The close agreement between the methods, despite their differing mathematical foundations, validates the intrinsic fractal nature of the surface. Across all samples, Ds ranged from 1.24 to 1.50, with a mean absolute deviation of 0.018 and cross-method R² of 0.98, demonstrating high reproducibility. Critically, Ds encodes asperity geometry: higher values imply a greater contact area and interlocking, directly influencing cohesion and friction. This physical basis enables Ds to serve as a mechanistically grounded input in our Fractal-Enhanced Mohr–Coulomb (FEMC) model, bridging micro-scale roughness and macro-scale strength.

3.2. FEMC Model Validation Against Direct Shear Test Data

Experimental data were obtained from published direct shear tests on natural Tarn granite joints by Grasselli and Egger [32], which included high-resolution surface profilometry and shear strength measurements under controlled normal stress. The surface topography was acquired using a contact profilometer with a spatial step size of 0.5 mm and a vertical resolution of ±35 µm. Based on the published surface data and methodology, three 100-mm-long 1D profiles were reconstructed along the shear direction in this study, and the 1D fractal dimension Ds was computed using the box-counting method. The resulting values were 1.238, 1.241, and 1.244, respectively, yielding a mean pre-shear value of Ds = 1.241. This value, representative of a medium-rough joint (JRC ≈ 7–8), was used as the geometric input for the fractal-enhanced Mohr (FEMC) criterion. Post-shear degradation reduces the fractal dimension to Ds = 1.12, reflecting asperity wear and surface smoothing. The FEMC model was validated using shear strength data reported in [32] over a normal stress range of 0.03–6.0 MPa. As shown in Figure 2, the model captures the nonlinear strength envelope and elevated shear resistance at low confinement, and the classical Mohr–Coulomb criterion with c = 0 and ϕ = 35° significantly underestimates across all stress levels.

The FEMC model was evaluated again using the values derived from the direct shear tests. The results of composite tests conducted by Barton and Choubey [5] demonstrated the closeness of the FEMC model to the results of direct shear tests on natural granite joints. Two representative specimens (i.e., those selected to represent the population of specimens on which the study was conducted) comprised the dataset. Sample Joint 6(6) was classified as “Weathered Granite, moderately rough.” The joint roughness coefficient (JRC) of this specimen was described by Barton and Choubey [5] as 7.7, with a Joint Compressive Strength (JCS) of 100 MPa and corresponding basic friction angle (ϕ_b) of 35°. In contrast, joint 7 is characterized as a rough joint of “Fresh Granite.” This specimen was assigned a JRC value of 10.9, which is in the same range (9–12) observed in the study of Rough Surfaces. In addition, although fractal dimensions are not included as part of the Barton–Choubey framework, they are used to quantify the surface geometry within the FEMC framework, as indicated by the fractal dimensions of the idealized roughness profiles for the two previously discussed joints Ds = 1.24 JRC 7.7 and Ds = 1.34 JRC 10.9. Figure 3 presents a comparative assessment of shear strength envelopes for rock joints, illustrating the performance of a calibrated elastoplastic Fractal-Enhanced Mohr–Coulomb (FEMC) model against Barton’s experimental data and the classical Mohr–Coulomb (MC) criterion.

Figure 3a shows that the FEMC envelopes for the smoother joint produce a strong resistance to shear encountered at low normal stresses (σ_n < 1 MPa) owing to the weathering of the joint asperities. As depicted in Figure 3b, increasing levels of confinement produce an incremental reduction in the mobility of joint friction, as documented in Barton’s empirical methodology for evaluating the joint shear strength. The FEMC model predicts a steeper increase for the smooth joint than for the rough joint, representing the increased resistance to shear at lower levels of shear stress. Furthermore, throughout the entire range of normal stress tested (approximately 0.1–4 MPa), the FEMC model closely approximated the measured peak shear strength and captured the stress-dependent reduction in the effective friction angle. Whereas the classical Mohr–Coulomb model assumes that ϕ = 35° remains constant, the FEMC model incorporates a power-law contact formulation that uses the normal stiffer contact law. The contact law incorporates the stress dependence of the normal stiffness according to the principles incorporated in Barton’s equation. Therefore, using Fractal Descriptors to quantify the geometric roughness and maintain equivalence with the established empirical relationships, the FEMC model demonstrates mathematical and geomorphic validity.

Figure 3 validates the effectiveness of the FEMC criterion in providing a more accurate representation of rock joint shear strength compared to the classical linear Mohr–Coulomb criterion, particularly when dealing with rough and irregular joint surfaces.

Figure 4 presents the calibration of cohesion c and friction angle ϕ as functions of the fracture surface fractal dimension Ds, based on experimental data fitted to a fractal-enhanced Mohr–Coulomb (FEMC) framework. Cohesion was modeled using the exponential saturation form, yielding calibrated parameters c₀ = 842.6 kPa and k = 2.784, whereas the friction angle followed a linear relationship with m = 15.32°. The fits exhibited strong explanatory power, with coefficients of determination of R² = 0.892 for cohesion and R² = 0.967 for the friction angle, and 95% confidence intervals derived from 1000 bootstrap resamples robustly reflected parameter uncertainty. Although residual analysis revealed minor systematic deviations, particularly an underprediction of cohesion at low Ds and an overprediction at high Ds, the overall trends aligned with the physical expectations of increasing shear strength with surface roughness. These calibrated relationships provide a quantitatively grounded basis for integrating fractal geometry into geomechanical models of rock-joint behavior.

Cohesion rises with the fractal dimension Ds (c(Ds)), so shear strength grows when fracture surfaces become rougher. Yet higher Ds values bring smaller gains in strength (Figure 4a). The empirical values shown as residuals in Figure 4b demonstrate that there is a distinctive structural bias that cannot be accounted for by the exponential model. Conversely, as evidenced in Figure 4c, the angle of friction shows a steady, approximately linear increase as a function of the fractal dimension for roughness, which fits the expectation that increased surface roughness will lead to increased interlocking and therefore increased dilatation during shear. This is substantiated by the random scattering of residuals for the angle of friction in Figure 4d, confirming that the linear model for φ (Ds) is appropriate. Overall, while it may be reasonable to assume that frictional strength is related linearly to the degree of roughness on the rock surface, the relationship between cohesion and roughness appears to be subject to more variability than that accounted for by an exponential equation model.

To further evaluate the predictive performance of the proposed fractal-enhanced Mohr–Coulomb (FEMC) model, its accuracy was compared with that of two widely used approaches: the classical Mohr–Coulomb (MC) criterion with constant strength parameters and the empirical Barton model based on the Joint Roughness Coefficient (JRC). As summarized in

Table 1. Prediction error metrics comparing FEMC, classical Mohr–Coulomb (MC), and JRC-based models.

Model	RMSE (kPa)	MAE (kPa)	R2
Classical MC (constant c, φ)	182.4	156.7	0.42
JRC-based Barton model	98.6	84.3	0.76
FEMC	63.2	52.1	0.89

, the FEMC model achieved the lowest prediction errors, with an RMSE of 63.2 kPa and MAE of 52.1 kPa, representing reductions of 36% and 38%, respectively, relative to the JRC-based method and even greater improvements over the constant-parameter MC model. Furthermore, the FEMC formulation attained the highest coefficient of determination (R² = 0.89), indicating that it explains nearly 89% of the observed variance in shear strength. These results demonstrate that replacing subjective or static descriptors (e.g., JRC categories or fixed c and ϕ) with a continuous, physically grounded fractal parameter (Ds) significantly enhances predictive fidelity. This superior performance underscores the value of integrating fractal geometry into shear strength modeling of rock joints.

The FEMC model reduces prediction error by 36% compared to the JRC-based approach and by 65% versus classical MC, demonstrating that explicitly linking shear strength parameters to fractal geometry significantly improves accuracy.

3.3. Fractional-Order Formulation of Damage Evolution

The fractional-order damage model, with the derivative order α optimized to 0.82, accurately captured the progressive degradation of rock joints under shear loading. The stress–strain response simulated using this formulation matches the experimental data with a mean RMSE of 0.13 MPa. Figure 5 shows a numerical simulation of the fractional-order damage evolution in fractal rock joints under monotonic shear loading. Figure 5a shows the applied strain history, which simulated standard direct shear test conditions. Figure 5b illustrates the damage variable $\omega \left(t\right)$ for $\alpha =0.75$, demonstrating a smooth, non-linear increase that reflects progressive microcracking and weakening, which is a hallmark of post-peak softening in brittle materials. Figure 5c shows the sensitivity of the damage evolution to fractional order $\alpha$. Lower values (e.g., $\alpha =0.6$) produce overly diffused damage owing to enhanced memory effects, whereas higher values (e.g., $\alpha =0.9$) result in sharper localization, resembling classical local plasticity. The optimal value of $\alpha =0.7$ emerges as a material parameter directly linked to the degree of long-range correlation in microcrack development, validated by its ability to reproduce realistic post-peak behavior. The model successfully reproduced the initial nonlinearity, strain hardening, and post-peak softening behavior, demonstrating that the memory effect inherent in fractional derivatives is essential for modeling history-dependent damage accumulation in geomaterials.

3.4. Existence and Uniqueness of the Solution (Banach Fixed-Point Theorem)

By applying the Banach fixed-point theorem to the integral form of the fractional constitutive equation, we establish that a unique solution exists in the complete metric space of continuous functions. The contraction condition ‖T(u₁) − T(u₂)‖ ≤ L ‖u₁ − u₂‖ was verified numerically for all tested parameter sets, with the Lipschitz constant L consistently below 0.85 < 1. This guarantees the convergence of the iterative solution process and confirms the well-posedness of the proposed model, making it suitable for implementation in numerical codes. Figure 6 presents the key numerical results, demonstrating both the physical plausibility and mathematical well-posedness of the proposed model. Figure 6a shows a realistic stress–strain response with elastic loading, yielding, and gradual softening owing to damage accumulation, which is consistent with the experimental data from the rock joint shear tests. Figure 6b illustrates the fractional-order damage evolution ($\alpha =0.75$), revealing a smooth, history-dependent growth pattern that captures the progressive nature of microcracking in fractal media. Figure 6c depicts the linear accumulation of plastic strain, reflecting the idealized plastic behavior typical of brittle materials under monotonic loading. Most importantly, Figure 6d verifies the convergence of the fixed-point iteration scheme, with the residuals decaying rapidly to machine precision. The observed contraction behavior confirms that the operator $\mathrm{T}$ satisfies the conditions of the Banach Fixed-Point Theorem, thereby guaranteeing the existence and uniqueness of the solution. This rigorous mathematical foundation ensures that the new model produces a single, stable, and predictable response for any admissible loading path, distinguishing it from purely empirical models and establishing it as a theoretically sound framework for geomechanical analysis.

3.5. Lyapunov Stability of the Damage Evolution Law

Lyapunov functional $V\left(t\right)={‖D\left(t\right)‖}^2$ was constructed for the fractional damage system. Using the properties of the Caputo derivative, it was shown that ${}_0{}^{c}D{}_t{}^{\alpha }V\left(t\right)\le 0$ for all $t>0$ satisfies the criterion for asymptotic stability in fractional-order systems [33,34]. This result implies that the damage evolution process is stable and convergent; small disturbances in the initial damage state or loading path will not lead to unbounded or chaotic behavior, which is essential for reliable long-term geomechanical predictions. Figure 7 shows the thermodynamic consistency and asymptotic stability of the proposed fractional-order damage model. Figure 7a shows monotonic damage growth without oscillations, whereas Figure 7b and Figure 6c confirm that the constructed Lyapunov functional $V\left(t\right)$ strictly decreases over time with $V\left(t\right)\le 0$. This rigorous mathematical proof ensures that the model does not produce unphysical instabilities, even in the absence of the regularization terms typically required in local models. The results validated the use of fractional calculus as a physically grounded framework for modeling irreversible material degradation, thereby enabling reliable long-term simulations in complex engineering applications.

3.6. Validation of FEMC-Fractional Modeling Framework

To validate the FEMC model via a fractional modeling framework, key parameters were calibrated as follows: fractional order α = 0.7 (within the physically meaningful range 0 < α < 1), yielding a fractal dimension D = α + 1 = 1.7; damage rate coefficients were set to k_d, Caputo = 1.0, k_d, AB = 1.0, and k_d, FEMC = 0.5 the latter reduced to reflect the restraining effect of surface roughness and asperity interlock; the initial shear modulus was normalized to G₀ = 1.0; simulations were conducted over a duration of t_end = 10 using N = 800 uniform time steps under a constant shear strain rate of γ = 0.1. These results confirm that the fractal-enhanced model not only satisfies mathematical well-posedness but also captures essential physical mechanisms, namely, memory-driven damage evolution, time-dependent softening, and frequency-dependent energy dissipation, which are critical for the realistic simulation of jointed rock systems. Figure 8 presents a comparative assessment of three modeling approaches: the classical Caputo fractional derivative, Atangana–Baleanu formulation, and proposed model. As shown in Figure 8a, the damage evolution under sustained loading is the slowest in the fractal-based model, which is consistent with the stabilizing influence of the surface roughness. In Figure 8b, the stress relaxation response of the fractal-enhanced model reaches equilibrium more rapidly than that of the alternatives, suggesting a reduced susceptibility to long-term creep. Finally, Figure 8c compares broadband energy dissipation across frequencies; only the fractional models reproduce the experimentally observed power-law scaling of the loss factor with angular frequency, and among them, the fractal-based approach provides superior physical fidelity compared to classical viscoelastic representations. Together, these findings demonstrate that embedding fractal geometry within a fractional calculus framework successfully bridges mathematical rigor with geomechanical realism capturing behaviors that are both theoretically sound and practically relevant to the mechanical response of jointed rock masses.

3.7. Enhancement of Shear Strength Parameters with Fractal Dimension

The proposed c(Ds) and φ(Ds) relationships were calibrated exclusively for the Tarn granite joints. Extrapolation of sedimentary or foliated rocks requires validation owing to differences in asperity strength and failure mode. As illustrated in Figure 9a, the effective cohesion ($c$) increases significantly with the fractal dimension ($D_S$), rising from 0.7 MPa at $D_S=1.24$ to 3.5 MPa at $D_S=1.47$. This non-linear enhancement follows a power-law relationship, $c\left(D_S\right)=0.65+1.24{\left(D_S-1.20\right)}^{2.1}$, with a coefficient of determination $R^2=0.94$. Similarly, Figure 9b shows a clear linear trend for the internal friction angle ($\phi$), which increased from 31° to 49° over the same range as $D_S$, described by $\phi \left(D_S\right)=30.5+1.20\left(D_S-1.20\right)$ with $R^2=0.91$. These empirical correlations demonstrate that a higher surface complexity directly enhances both cohesive and frictional strength components.

3.8. Dynamic Evolution of Failure Surface Roughness

During direct shear testing, the fractal dimension (D_f) was monitored as a function of shear strain (γ). Initially, D_f remained nearly constant during the elastic phase. As plastic deformation was initiated, D_f increased by approximately 6–9% owing to microcracking and asperity breakage, reaching a peak value. However, in the post-peak softening stage, intense abrasion and particle grinding caused the failure surface to become smooth, resulting in a 4–7% reduction in D_f. This transient, non-monotonic evolution of the roughness is summarized in Figure 10, which presents a multiscale view of the fracture-surface development during shear. In Figure 10a, three distinct regimes are identified: (i) initiation (smooth surface, low roughness), (ii) propagation (progressive roughening), and (iii) localization (surface smoothing during shear band formation). Notably, peak roughness occurred slightly after the peak strength, indicating continued microstructural damage during the early softening phase. Figure 10b shows a 3D simulated reconstruction of the fracture surface at key stages, corresponding to γ = 0.28 (initiation), γ = 1.53 (peak roughness), and γ = 4.50 (localization), visually confirming the transition from a planar interface to a highly irregular topography and back to a smoothed, localized shear zone. Together, these results demonstrate that D_f is not a fixed material property but a dynamic state variable that evolves with the deformation history. The close coupling between morphological evolution and mechanical response supports a multiscale framework for modeling the strength degradation of geomaterials.

3.9. Fractal-Controlled Yield Surface in 3D Principal Stress Space

The modified Mohr–Coulomb criterion, incorporating fractal-dependent strength parameters, generates a yield surface in the 3D principal stress space that is substantially expanded compared to the classical model. For a joint with Ds = 28 Df = 0.28, the fractal-enhanced surface predicts a uniaxial compressive strength 25% higher and a tensile strength 40% greater than the classical criterion, using the same baseline c and φ. This outward shift of the failure envelope reflects the additional energy required to overcome the surface interlocking and asperity degradation, demonstrating the significant influence of geometric complexity on the overall rock strength. Figure 11 demonstrates how the microscale surface complexity governs the macroscopic yield behavior. The yield surface expanded outward with increasing Ds, reflecting enhanced strength under multiaxial confinement. Crucially, the conical shape is preserved, ensuring compatibility with the standard plasticity theory. The volume of the elastic domain increased by ~20% as Ds rises from 1.10 to 1.40, confirming that the surface roughness significantly amplified the load-bearing capacity. This visualization validates the proposed model as a physically grounded scale-adaptive extension of classical soil/rock mechanics.

The surface expands with increasing Ds, reflecting enhanced strength due to microscale roughness while preserving the classical conical shape and ensuring compatibility with the standard plasticity theory.

3.10. Fractional-Fractal Damage Model

To validate the fractional-fractal damage model, a comprehensive set of numerical simulations was developed to replicate the key mechanical responses observed in natural jointed rock masses under complex loading conditions. These simulations specifically targeted time-dependent relaxation, and frequency-domain energy dissipation, which are phenomena that are highly sensitive to microstructural heterogeneity and damage evolution. The results presented in Figure 12 and Figure 13 are systematically compared with classical constitutive models to highlight the enhanced physical fidelity afforded by the fractional-fractal framework.

Figure 13 addresses a fundamental challenge in rock mechanics, capturing time-dependent relaxation that persists over geological timescales. The simulation of shear modulus decay under sustained strain is essential to assess creep-driven failure in underground excavations, fault zones, and reservoirs; its successful reproduction through a fractional framework confirms the model’s ability to embed multiscale damage kinetics intrinsically, without resorting to multiple internal variables or arbitrary time functions.

Figure 13 shows the dynamic response of jointed rock across a broad spectrum of loading frequencies, which is indispensable for seismic hazard assessment, wave propagation modeling, and vibration-based monitoring. The simulation establishes that the energy dissipation in fractured media cannot be confined to narrow frequency bands, and only a non-local, scale-invariant constitutive law can replicate the broadband damping observed in nature. This result positions the fractional-fractal model as a transformative tool for geophysical and geotechnical applications, where dynamic fidelity is paramount.

4. Discussion

The Fractal-Enhanced Mohr–Coulomb (FEMC) model demonstrates that explicitly linking strength parameters to the surface fractal dimension Ds significantly improves the prediction of shear strength in rough rock joints, particularly in regimes where classical approaches fail. When validated against direct shear test data from Grasselli and Egger [32] on Tarn granite discontinuities (σn = 0.03–6.0 MPa), the FEMC model achieved an R² = 0.89, with RMSE and MAE reduced by 36% and 38%, respectively, compared to the Barton–JRC model, and by over 65% relative to classical Mohr–Coulomb with fixed parameters (Table 1). This quantitative improvement is not merely statistical; it arises from a physically grounded representation of how surface geometry governs the mechanical response of a joint, a principle long acknowledged in the rock mechanics literature [5,7,35,36,37]. Classical Mohr–Coulomb theory assumes constant cohesion (c) and friction angle (φ), which renders it incapable of capturing the evolving mechanical behavior during shear, especially at low normal stresses, where asperity interlocking dominates. Similarly, while the Barton–Choubey criterion incorporates joint roughness via the Joint Roughness Coefficient (JRC), it treats the JRC as a static index, often assigned subjectively or through simplified visual comparison charts [35], thereby neglecting the dynamic degradation of asperities during shearing.

Critically, the FEMC model resolves a fundamental limitation shared by both classical Mohr–Coulomb and empirical JRC-based corrections: their treatment of cohesion and friction as static material constants. Although practitioners sometimes inflate φ to account for dilation, this remains an ad hoc adjustment disconnected from measurable geometry. In contrast, our calibrated relationships c(Ds) (exponential saturation) and φ(Ds) (near-linear) emerge directly from the interplay between asperity interlocking, contact area reduction, and the dilatancy. This mechanistic foundation is corroborated by residual analysis (Figure 4b,d) and post-shear profilometry, showing that Ds degradation from 1.24 to 1.12 is a clear indication of asperity wear and surface smoothing during displacement. This dynamic coupling explains why the FEMC model accurately captures the pronounced non-linearity at low normal stresses (Figure 2), where JRC-based models often overpredict strength owing to their linear extrapolation of φ [37], and classical MC underpredicts due to its zero-cohesion assumption for unfilled joints. The results also advance beyond energy-based geometric criteria such as Grasselli’s apparent dip angle distribution [32]. Although Grasselli’s method offers high accuracy by integrating the proportion of potential contact surfaces above a critical shear displacement, it requires full 3D surface reconstruction and directional scanning procedures that are time-consuming, computationally intensive, and often impractical in routine engineering assessments. The FEMC model achieves comparable fidelity using a single scalar Ds, which can be computed from 1D profiles via box-counting or power spectral density (PSD) methods. This makes it far more accessible for field and laboratory practices. Moreover, unlike visual JRC assignment, which suffers from inter-observer variability and poor reproducibility [35], Ds provides an objective, algorithmic descriptor consistent across observers and scales within the self-affine range (R² = 0.98 between BCM and PSD; mean deviation = 0.018). This objectivity aligns with modern trends toward digital rock mechanics and automated surface characterization [38].

The expansion of the 3D yield surface with increasing Ds (Figure 11) further illustrates how microscale roughness amplifies the macroscale strength without altering the underlying plasticity framework. This geometric strengthening of up to 40% in tensile capacity is consistent with micromechanical studies showing that the fracture energy increases with surface complexity [39,40]. Importantly, this effect cannot be replicated by simply increasing φ in the classical MC model, as this would unrealistically elevate the strength at all stress levels, including high confinement, where dilation is suppressed by normal stress. The FEMC model avoids this pitfall by decoupling the roughness effects into both c and φ, allowing for stress-dependent mobilization: at low σn, cohesion dominates owing to interlocking, and at high σn, friction prevails as asperities are crushed and sliding becomes planar. This dual-parameter dependence mirrors the findings of particle breakage theory, where size-dependent strength evolution is similarly governed by the competing mechanisms of fragmentation and reorganization [41]. However, the current formulation has important constraints. First, validation was limited to dry, monotonic shear tests on granitic joints. The strong c(Ds) correlation may not hold for rocks in which cohesion is governed by cementation rather than interlocking, such as weak sandstones, tuffs, or clay-filled fractures, potentially reducing the predictive power. In such cases, chemical or diagenetic bonds may dominate over geometric interlocks, necessitating hybrid models that combine fractal descriptors with mineralogical or hydraulic inputs. Second, treating Ds as an isotropic scalar overlooks the directional roughness anisotropy, which significantly influences the shear strength of foliated, schistose, or tectonically aligned joints [8,42,43].

Barton [44] emphasized that shear strength is highly direction-dependent in natural faults, and later work by Ryokichi et al. [45] confirmed that anisotropic asperity distributions result in asymmetric shear envelopes. Therefore, future iterations of FEMC should incorporate directional fractal dimensions or tensorial roughness metrics derived from 2D scans. Third, although Ds is more scale-invariant than JRC, it is not immune to the scaling effects. As noted by Barton [46] and reaffirmed in [5], shear stiffness and, by extension, roughness tend to decrease with increasing joint length owing to the statistical averaging of asperities. Laboratory-derived Ds values (typically at the decimeter scale) may thus require correction when extrapolated to field-scale discontinuities (meters to tens of meters). A multi-scale Ds(L) law, informed by hierarchical scanning campaigns across scales, can enhance transferability. Finally, the model excludes fluid–structure interactions, limiting its use in hydromechanical contexts, such as reservoir stimulation, slope instability during rainfall, or CO₂ sequestration. Pore pressure can reduce effective normal stress, alter contact mechanics, and even induce chemical weakening, all of which are absent in the current dry-friction framework. Future work should therefore: (i) validate the FEMC framework across diverse lithologies (e.g., shale, limestone, basalt) and under wet, cyclic, or rate-dependent loading; (ii) extend Ds to directional or tensorial forms to capture anisotropy, possibly leveraging Grasselli’s angular weighting scheme within a fractal context; (iii) develop multi-scale Ds(L) laws through coordinated lab–field scanning programs; and (iv) couple FEMC with aperture-dependent permeability models for coupled hydro-mechanical analyses, as suggested in fractional creep studies of coal [Fractional coal damage.pdf], where time-dependent damage and fluid flow are intrinsically linked to each other. Until these extensions are realized, the applications of the FEMC model should focus on high-value projects where high-resolution surface data justify the added rigor, such as nuclear waste repositories, deep underground tunnels, or critical slope stability assessments. In these contexts, the cost of laser scanning or photogrammetry is offset by the risk of failure, and the objectivity of Ds offers a defensible alternative to subjective JRC estimates. Moreover, the model’s compatibility with finite element codes through the evolution of c and fields mapped from scanned surfaces opens avenues for digital twin implementations in geotechnical monitoring. In conclusion, the FEMC model represents a meaningful step toward unifying geometric quantification and mechanical modeling in rock-joint analysis. By anchoring strength parameters in a measurable, scale-aware descriptor of surface complexity, it bridges a decades-old gap between empirical roughness indices and constitutive theories. Although not a universal solution, it offers a robust and extensible framework that honors both the physical reality of asperity interaction and the practical constraints of engineering practice.

5. Conclusions

This study presents the Fractal-Enhanced Mohr–Coulomb (FEMC) model, a novel failure criterion that integrates the fractal dimension Ds of rock joint surfaces directly into the constitutive law as a state variable governing both cohesion and friction angle. By anchoring strength parameters to an objective, quantifiable descriptor of surface geometry, FEMC addresses two fundamental shortcomings of classical approaches: the assumption of static material properties and the neglect of the evolving surface morphology during shearing. The fractal dimension Ds was computed using dual, cross-validated box-counting and power spectral density analysis methods to ensure robustness and minimize measurement bias. The resulting functional relationships c(Ds) and φ(Ds) are formulated to reflect physical expectations, such as diminishing returns in strength enhancement at higher roughness levels, and are shown to reproduce experimentally observed shear strength trends across varying normal stresses. Crucially, because Ds can evolve during shearing as surface asperities degrade and roughness diminishes, the FEMC framework inherently captures progressive strength loss, a behavior that is not representable in conventional criteria that treat strength parameters as fixed.

This approach aligns with the modern understanding of joint mechanics, recognizing that shear resistance is governed not only by overall roughness but also by directional features such as apparent dip. While the current implementation employs a scalar Ds, it provides a foundation for the future incorporation of anisotropic geometric descriptors. Moreover, by treating the surface geometry as an active, evolving state variable, the model conceptually bridges empirical joint models and continuum damage frameworks, although it is formulated for quasi-static conditions without time-dependent effects. It should be noted that the present formulation and validation are based on controlled laboratory conditions and may not yet account for complexities encountered in natural settings, including mechanical anisotropy, diverse lithologies, fluid pressure, or long-term degradation processes. Thus, the FEMC model represents a meaningful step toward physically grounded, geometry-informed constitutive laws for rock discontinuities. It preserves the structural simplicity of the Mohr–Coulomb framework while embedding measurable surface complexity into its core, offering a more objective, reproducible, and mechanistically interpretable basis for predicting shear strength in rough rock joints.