A Thermo-Mechanical Coupled Gradient Damage Model for Heterogeneous Rocks Based on the Weibull Distribution

Abstract

This study develops a thermo-mechanical damage (TMD) model for predicting damage evolution in heterogeneous rock materials after heat treatment. The TMD model employs a Weibull distribution to characterize the spatial heterogeneity of the mechanical properties of rock materials and develops a framework that incorporates thermal effects into a nonlocal gradient damage model, thereby overcoming the mesh dependency issue inherent in homogeneous local damage models. The model is validated by numerical simulations of a notched cruciform specimen subjected to combined mechanical and thermal loading, confirming its capability in thermo-mechanical coupled scenarios. Sensitivity analysis shows increased material heterogeneity promotes localized, X-shaped shear-dominated failure patterns, while lower heterogeneity produces more diffuse, network-like damage distributions. Furthermore, the results demonstrate that thermal loading induces micro-damage that progressively spreads throughout the specimen, resulting in a significant reduction in both overall stiffness and critical strength; this effect becomes increasingly pronounced at higher heating temperatures. These findings demonstrate the model’s ability to predict the mechanical behavior of heterogeneous rock materials under thermal loading, offering valuable insights for safety assessments in high-temperature geotechnical engineering applications.

1. Introduction

In the field of geological engineering, thermally induced damage represents a critical influence on the mechanical behavior of geological materials, especially under extreme temperature conditions. This phenomenon assumes particular importance in high-temperature subsurface applications, including but not limited to deep geothermal energy extraction [1,2,3], in situ conversion of oil shale [4,5], ultra-deep drilling operations, and the geological disposal of high-level radioactive waste [6].

At elevated temperatures, thermal loading induces pronounced microstructural alterations—such as crack initiation, pore expansion, and mineral phase transitions—which collectively compromise the integrity of sandstone matrices [7,8]. These microscopic changes inevitably propagate to the macroscopic scale, manifesting as reductions in strength, stiffness, and long-term stability of granite masses subjected to thermo-mechanical coupling [9,10].

The mechanical response of sandstones has been found to be highly sensitive to thermal environments. Experimental studies have revealed that key mechanical parameters—including compressive strength, elastic modulus, P-wave velocity, cohesion, and internal friction angle—tend to display nonlinear degradation trends at elevated temperatures [11,12,13], typically characterized by gradual decreases or temperature-dependent fluctuations. Fan Zhang [14], based on ultrasonic pulse velocity and triaxial compression tests, reported that when the heating temperature exceeds 500 °C, thermal cracking becomes dominant, resulting in marked reductions in both elastic modulus and compressive strength. Weiqiang Zhang [15], using acoustic emission measurements, observed that the granite density and connectivity of microcracks increase significantly within the range of 200–500 °C. Zhihong Zhao [16], employing particle-based mechanical modeling, demonstrated that thermal stress accumulated during heating is released upon cooling, and that the overall reduction in granite strength is primarily attributed to thermally induced microcrack formation.

In recent years, extensive experimental investigations and numerical simulations have greatly advanced our understanding of the mechanical behavior of rocks under high-temperature conditions. These studies have significantly enriched the theoretical framework and validated key aspects of thermal–mechanical responses. However, experimental methods are often time-consuming and costly, making them less practical for large-scale engineering applications.

In such contexts, numerical approaches offer a more feasible alternative. Nevertheless, existing constitutive models remain inadequate for accurately capturing the distinct evolution of strength degradation and deformation behavior induced by thermal loading [17,18]. This limitation is particularly evident under complex thermo-mechanical coupling conditions, where conventional models often struggle to effectively represent the coupled processes of thermally driven microcrack growth and anisotropic damage evolution [19].

Early numerical modeling efforts focused on thermo-mechanical coupling frameworks but did not incorporate material heterogeneity. In these models, material properties are often assumed to be spatially homogeneous, which fails to reflect the intrinsic variability in shale behavior arising from its complex microstructure. Given the inherently heterogeneous nature of shale, such idealized assumptions are insufficient to capture the spatial diversity in crack initiation sites, propagation paths, and localized failure mechanisms during the thermal damage evolution process [20,21,22,23].

To address these limitations, recent studies have incorporated material heterogeneity into numerical simulations to more accurately reproduce the thermo-mechanical response of rocks. A widely adopted approach involves assigning statistical distributions to key material parameters to represent their spatial variability [24,25]. Among these, the Weibull distribution has emerged as the most commonly used due to its mathematical simplicity, physically interpretable parameters, and ability to control the degree of heterogeneity. The Weibull-based framework has demonstrated notable success in thermal damage modeling [26,27,28,29], offering both robustness and effectiveness. For instance, several studies have employed Weibull-distributed heterogeneous fields to simulate crack propagation and strength evolution under varying thermal loads, thereby enhancing the physical fidelity of numerical predictions.

Nonetheless, current research efforts remain constrained by notable limitations. First, many existing models focus solely on the heterogeneity of Young’s modulus while neglecting the spatial variability of key thermophysical properties, such as thermal expansion coefficients and thermal conductivity [30,31]. This simplification hinders accurate characterization of the micro-scale thermal stress fields arising from mismatches in thermal expansion behavior. Second, conventional local damage models often exhibit pronounced sensitivity to mesh size, resulting in strong numerical dependence and limited generalizability across different discretizations [16,32].

To address these challenges, this study advances a coupled thermo-mechanical damage (TMD) framework by integrating meso-scale statistical heterogeneity, capturing spatial variability in both elastic modulus and thermal expansion coefficients. A gradient-enhanced nonlocal damage model is also employed to mitigate mesh dependency and improve numerical stability. This enhanced model offers a more comprehensive representation of the micro-to-macro evolution of thermally induced damage in heterogeneous rocks, thereby improving accuracy and robustness. Overall, the approach provides a solid theoretical basis for assessing rock mass stability under high-temperature geological conditions.

2. Methods

2.1. Thermo-Mechanical Coupling Framework

2.1.1. Heat Conduction Equation

Before deriving the coupled equations for the thermo-mechanical damage problem, consider an isotropic linear elastic solid material. Let $\mathsf{\Omega }\subset R^n \left(n=1, 2, 3\right)$ represents the reference configuration of the material body and $\partial \mathsf{\Omega }\subset R^{n-1}$ denotes the boundary, as shown in Figure 1. The displacement is defined as $u\left(\mathit{x},t\right)$ and the temperature change is denoted by $\theta \left(\mathit{x},t\right)=T-T_0$, where $t\in \left[0,t_a\right]$ is the time, $\mathit{x}$ represents the position vector, $T$ and $T_0$ are the absolute temperature and initial temperature, respectively. The nonlocal effect is incorporated in the evolution of damage by the gradient of the micro equivalent strain.

Under high-temperature conditions, the temperature change induced by rock deformation is negligible compared to the effects of heat conduction and external heat sources. According to the principle of energy conservation, the governing equation for the temperature field can be expressed as:

$$\lambda {\nabla }^{2}T+Q=\rho C{\displaystyle \frac{\partial T}{\partial t}}$$

(1)

where $\lambda$ is the thermal conductivity, $Q$ is the internal heat source term, $\rho$ is the material density, and $C$ is the specific heat capacity.

2.1.2. The Stress Equilibrium Equation

For quasi-static problems, the stress equilibrium equation governing the mechanical response of the rock can be expressed as:

$$\nabla \cdot \mathit{\sigma }+{\rho }_b{\mathit{f}}_{\mathit{b}}=0$$

(2)

where $\mathit{\sigma }$ is the total Cauchy stress tensor, and ${\mathit{f}}_{\mathit{b}}$ is the gravitational vector. For a thermoelastic medium, taking into account the effect of thermal expansion, the constitutive relation can be written as:

$${\mathit{\sigma }}_{\mathit{e}}={\mathit{C}}_{\mathit{d}\mathit{r}}:{\mathit{\epsilon }}_{\mathit{e}}={\mathit{C}}_{\mathit{d}\mathit{r}}:\left(\mathit{\epsilon }-{\alpha }_T\theta \mathit{\delta }\right)$$

(3)

where ${\mathit{C}}_{\mathit{d}\mathit{r}}$ represents the fourth-order elasticity tensor, ${\mathit{\epsilon }}_{\mathit{r}}$ is the elastic strain tensor, $\mathit{\epsilon }$ is the total strain tensor, ${\alpha }_T$ represents the linear thermal expansion coefficient, and $\mathit{\delta }$ is the second-order identity tensor. For isotropic materials, the thermoelastic constitutive relation can be expressed as:

$${\sigma }_{ij}=2G{\epsilon }_{ij}+{\displaystyle \frac{2G\nu }{1-2\nu }}{\epsilon }_{kk}{\delta }_{ij}-{\displaystyle \frac{2G\left(1+\nu \right)}{1-2\nu }}{\alpha }_{T}T{\delta }_{ij}$$

(4)

where G = E/(2(1 + ν)) is the shear modulus, E is the Young’s modulus, ν is the Poisson’s ratio, and the components of the strain tensor ε are given by:

$${\epsilon }_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}\right)$$

(5)

Finally, by substituting Equations (4) and (5) into Equation (2), the thermo-mechanical stress equilibrium equation is obtained as:

$$G{u}_{i,kk}+{\displaystyle \frac{G}{1-2\nu }}{u}_{k,ki}-{\displaystyle \frac{2G\left(1+\nu \right)}{1-2\nu }}{\alpha }_T{\theta }_{,i}+{\rho }_b{f}_{,i}=0$$

(6)

2.2. Non-Local Damage Model

2.2.1. Localizing Gradient Damage Theory

In classical local gradient damage models, the damage variable is typically defined as a function of equivalent strain or stress. However, this local constitutive formulation often renders the boundary value problem ill-posed, resulting in strong mesh sensitivity in the numerical solution. To address this issue, gradient-enhanced nonlocal damage models incorporate gradient terms of the nonlocal equivalent strain, thereby regularizing the governing equations and effectively mitigating mesh dependency. In conventional implementations, the governing equation for the nonlocal strain field takes the following form:

$$\tilde{e}-l^2{\nabla }^2\tilde{e}=e$$

(7)

In this context, $e$ denotes the local equivalent strain, defined as a scalar function of the strain tensor, while $\tilde{e}$ represents the corresponding nonlocal equivalent strain. Despite its ability to mitigate mesh dependency, the standard gradient damage model with a constant internal length scale enforces a fixed range of nonlocal interaction, which may result in spurious spreading of the damage zone. As a remedy, the present study adopts a localizing gradient damage formulation by introducing a damage-dependent length scale. This approach allows the nonlocal interaction to diminish progressively with increasing damage, thereby ensuring convergence and physical consistency of the damage profile. The governing equation for the nonlocal equivalent strain in the localizing gradient damage model is expressed as:

$$\tilde{e}-e=\nabla \cdot \left(g{l}^2\nabla \tilde{e}\right)$$

(8)

The interaction function $g$ decays progressively with increasing damage, and is defined as [33]:

$$g={\displaystyle \frac{\left(1-R\right)\exp \left(-\eta D\right)+R-\exp \left(-\eta \right)}{1-\exp \left(-\eta \right)}}$$

(9)

where $\eta$ is the parameter of the interaction attenuation rate, and $R$ is the parameter of the residual interaction. In this paper, $\eta =5$ and $R=0.05$ are taken. The relationship between the damage variable $D$and the non-local equivalent strain adopts the exponential damage evolution law, and its form is as follows [34]:

$$D=\left\{\begin{array}{c}0, if \kappa <{\kappa }_0\\ 1-{\displaystyle \frac{{\kappa }_0}{\kappa }}\left\{1-\alpha +\alpha \exp \left[-\beta \left(\kappa -{\kappa }_0\right)\right]\right\}, \mathrm{OTHERWISE}\end{array}\right.$$

(10)

where $\alpha$ and $\beta$ are material parameters, ${\kappa }_0=f_t/E$ is the strain threshold for damage initiation, $f_t$ denotes the uniaxial tensile strength of the rock, and $\kappa$ represents the historical maximum value of equivalent strain:

$$\kappa \left(t\right)=\max \left\{\tilde{e}\left(\tau \right)|0\le \tau \le t\right\}$$

(11)

where t indicates the current loading time step

The constitutive equation of the localizing gradient damage model can be further expressed as:

$$\mathit{\sigma }=\left(1-D\right)\mathit{C}:\mathit{\epsilon }+h\left(e-\tilde{e}\right){\displaystyle \frac{\partial e}{\partial \mathit{\epsilon }}}$$

(12)

The second term on the right-hand side is the gradient enhancement term, which incorporates the gradient of a non-local damage-related variable to regularize strain localization and introduce an internal length scale into the constitutive model.

By incorporating the thermo-mechanical coupling relation given in Equation (3), the final stress–strain formulation is obtained as:

$$\mathit{\sigma }=\left(1-D\right)\mathit{C}:\left(\mathit{\epsilon }-\alpha \theta \mathit{\delta }\right)+h\left(e-\tilde{e}\right){\displaystyle \frac{\partial e}{\partial \mathit{\epsilon }}}$$

(13)

2.2.2. Local Equivalent Strain

As shown in Equation (11), the local equivalent strain serves as the driving force in the damage evolution process. According to Equation (3), the stress is directly related to the elastic strain tensor, rather than the total strain; hence, the equivalent strain must be defined as a scalar function of the elastic strain tensor. Importantly, the choice of equivalent strain definition influences the resulting damage behavior. Geomaterials such as rock exhibit pronounced tension–compression asymmetry, with compressive strength substantially exceeding tensile strength. To account for this feature, the present study adopts a local equivalent strain formulation derived from the decomposition of elastic strain energy, allowing for a physically consistent description of the asymmetric damage response. The decomposition form of the elastic strain energy density is given by:

$${\mathsf{\Psi }}_e^{\pm }={\displaystyle \frac{1}{2}}{\lambda }_L{⟨tr\left({\mathit{\epsilon }}_{\mathit{e}\pm }\right)⟩}^2+{\mu }_{L}tr\left({\mathit{\epsilon }}_{e\pm }^2\right)$$

(14)

Here, ${\lambda }_L$ and ${\mu }_L$ denote the Lamé constants, and $⟨\cdot ⟩$ represents the Macaulay brackets. The elastic strain energy density is decomposed into its tensile and compressive components, ${\mathsf{\Psi }}_e^{+}$ and ${\mathsf{\Psi }}_e^{-}$, respectively. The corresponding positive and negative strain tensors ${\epsilon }_{\pm }$ are defined as:

$${\epsilon }_{\pm }={\displaystyle \sum }_{i=1}^n{⟨{\mathit{\epsilon }}_{\mathit{i}}⟩}_{\pm }{\mathit{n}}_{\mathit{i}}⨂{\mathit{n}}_{\mathit{i}}$$

(15)

For quasi-brittle materials, to account for the inherent tension–compression asymmetry, the equivalent strain is defined based on the regularized tensile part of the elastic strain energy density.

$$e={\displaystyle \frac{1}{\sqrt{E}}}\sqrt{{\lambda }_L{⟨tr\left({\mathit{\epsilon }}_{\mathit{e}\pm }\right)⟩}^2+{\mu }_{L}tr\left({\mathit{\epsilon }}_{e\pm }^2\right)}$$

(16)

So far, the governing equations for the coupled thermo-mechanical localizing gradient damage model have been established, which are summarized as follows:

$$\left\{\begin{array}{c}{\epsilon }_{ij}={\displaystyle \frac{1}{2}}\left(u_{i,j}+u_{j,i}\right)\\ \mathit{\sigma }=\left(1-D\right)\mathit{C}:\left(\mathit{\epsilon }-\alpha \theta \mathit{\delta }\right)+h\left(e-\tilde{e}\right){\displaystyle \frac{\partial e}{\partial \mathit{\epsilon }}}\\ \nabla \cdot \mathit{\sigma }+{\rho }_b{\mathit{f}}_{\mathit{b}}=0\\ G{u}_{i,kk}+{\displaystyle \frac{G}{1-2\nu }}{u}_{k,ki}-{\displaystyle \frac{2G\left(1+\nu \right)}{1-2\nu }}{\alpha }_T{\theta }_{,i}+{\rho }_b{f}_{,i}=0\\ \tilde{e}-e=\nabla \cdot \left(g{l}^2\nabla \tilde{e}\right)\end{array}\right.$$

(17)

The boundary conditions are set as:

$$\left\{\begin{array}{c}\mathit{u}\left(\mathit{x},t\right)={\mathit{u}}^{*}\left(\mathit{x},t\right)\\ \nabla \overline{\epsilon }\cdot \mathit{n}=0\\ \theta \left(\mathit{x},t\right)={\theta }^{*}\left(\mathit{x},\mathit{t}\right)\end{array}\right.\begin{array}{c}on \partial {\mathsf{\Omega }}_u\\ on \partial \mathsf{\Omega }\\ on \partial {\mathsf{\Omega }}_{\mathsf{\theta }} \end{array}$$

(18)

In addition, the initial conditions are expressed as:

$$\left\{\begin{array}{c}\mathit{u}\left(\mathit{x},0\right)={\mathit{u}}_{\mathbf{0}}\left(\mathit{x}\right)\\ \overline{\epsilon }\left(\mathit{x},0\right)={\overline{\epsilon }}_0\left(\mathit{x}\right)\\ \theta \left(\mathit{x},0\right)={\theta }_0\left(\mathit{x}\right)\end{array}\right.\begin{array}{c}in \mathsf{\Omega }\\ in \mathsf{\Omega }\\ in \mathsf{\Omega } \end{array}$$

(19)

2.3. Weibull Distribution Model

The inherent heterogeneity of rock plays a crucial role in its damage evolution behavior. To capture this heterogeneity at the mesoscale, the present model assigns key mechanical properties based on a Weibull statistical distribution.

The corresponding probability density function is defined as:

$$f\left(v\right)={\displaystyle \frac{m}{v_0}}{\left({\displaystyle \frac{v}{v_0}}\right)}^{m-1}{e}^{-{\left({\displaystyle \frac{v}{v_0}}\right)}^m}$$

(20)

In this expression, $v$ denotes the relevant mechanical parameter. In this study, both the elastic modulus and thermal expansion coefficient are modeled using a Weibull distribution, whereas the remaining material parameters are assumed to be spatially uniform. The parameter $v_0$ represents the mean value of the property, while $m$ is the Weibull shape parameter controlling the degree of heterogeneity. Various values of $m$ yield different probability density functions, as illustrated in Figure 2, and the associated cumulative distribution functions are shown in Figure 3. It is evident from the figures that increasing $m$ leads to a narrower distribution, implying enhanced material uniformity. In the limit as $m$ approaches infinity, the heterogeneity vanishes, and the material behavior converges to that of a perfectly homogeneous medium.

2.4. Finite Element Solution Algorithm

To investigate the mechanical response of rock materials subjected to thermally induced damage, a heterogeneous thermo-mechanical gradient damage model is developed. The overall simulation workflow is illustrated in Figure 4. Initially, spatially varying distributions of elastic modulus and heat capacity are generated. These distributions, along with the geometric domain and relevant material parameters, are incorporated into the simulation framework. The modeling incorporates the heat conduction equation, mechanical constitutive relation, and damage evolution law, resulting in a fully coupled thermo-mechanical damage simulation model.

To realistically reproduce the thermal treatment process and uniaxial compressive failure of rock, a boundary temperature of $T_0$ is applied and maintained for 200 s, followed by a cooling stage at a temperature $T_1$ for an additional 200 s. Subsequently, a displacement-controlled loading is imposed at the upper boundary until complete material failure occurs. The simulation employs a segregated solution strategy, with the MUMPS solver used to sequentially compute the displacement, temperature, and damage fields. Throughout the simulation, key physical quantities—including displacement, temperature, damage variable, stress, and boundary reaction force—are monitored and recorded for further analysis.

In the present simulation framework, several key assumptions are made to balance model complexity and computational efficiency. First, the heterogeneity of the rock material is represented by statistical distributions of material properties, specifically employing Weibull distributions to characterize spatial variability in elastic modulus and heat capacity. Second, the model neglects transient heat conduction effects within each heating or cooling stage, effectively assuming quasi-steady temperature distributions during the relatively long hold times at boundary temperatures $T_0$ and $T_1$. Thus, while the overall thermal treatment is transient, the internal temperature gradients evolve slowly enough that transient heat conduction effects have a limited influence on the development of thermal stresses and damage during each stage. Additionally, the model assumes isotropic and rate-independent mechanical behavior, focusing on quasi-static loading conditions. While these assumptions limit the model’s applicability to scenarios involving rapid thermal shocks or dynamic loading, they are consistent with the experimental setup and objectives of this study.

3. TMD Model Validation

In this subsection, we simulated the damage behavior of a notched cruciform specimen under displacement loading and thermal loading conditions. By comparing our results with those from previous studies, we demonstrated that the proposed TMD model reliably simulates quasi-static mechanical damage as well as thermally induced damage.

The geometric configuration of the cruciform specimen is shown in Figure 5. The initial temperature is maintained at $T_1=273 \mathrm{K}$, with normal constraints applied at the left, right, and bottom edges, as shown in Figure 5. A plane stress assumption is adopted in this case. The applied loads are as follows:

• The top edge is pulled vertically upward to u = 0.05 mm while the bottom edge is fixed, with no thermal load applied.
• Thermal loading consists of gradually cooling the bottom edge to T₂ = 263 K, and heating the top edge to T₃ = 283 K. The top edge is free, and the bottom edge is fixed throughout.

The material parameters are mainly taken from [35,36] and are summarized in

Table 1. Parameters for the Cruciform Specimen.

Parameter	Symbol	Unit	Value
Young’s modulus	$E_0$	KPa	218.4
Poisson’s rate	${\nu }_0$	-	0.2
Length scale parameter	$l$	mm	1
Material parameters 1	$\alpha$	-	0.99
Material parameters 2	$\beta$	-	1500
Interaction decay rate parameter	$\eta$	-	1
Residual interaction parameter	$R$	-	0.05
Initial damage strain	${\kappa }_i$	-	$5\times {10}^{-4}$

. By setting $\rho =0$, the transient effects of heat conduction are neglected. A free triangular mesh is used along the anticipated crack path with a maximum mesh size of $h=l/4$. The remainder of the domain is discretized using a free quadrilateral mesh with a maximum element size of 3 mm. Approximately 180,000 elements, generated using an unstructured free quadrilateral mesh, are employed for both case 1 and case 2. In both cases, 100 incremental steps are performed to reach the final state, with a displacement increment of Δu = 5 × 10⁻⁴ mm and a temperature increment of ΔT = 0.1 °C. The system is solved using the MUMPS solver with a segregated method.

Figure 6 shows the final damage morphology. Under pure mechanical loading, damage initiates from the tip of the defect and extends almost horizontally to the right. Under thermal loading, the bottom of the cruciform specimen contracts while the top expands; damage initially extends upward from the defect tip and then gradually deflects to the right. In both loading conditions, the damage morphology agrees well with the reference crack path (indicated by the red line in Figure 5 reported in the literature [37], thereby validating the TMD model’s applicability under both pure mechanical and thermal loads.

4. Results

This chapter investigates the uniaxial compressive mechanical behavior of granite after high-temperature thermal treatment. Consistent with previous studies [31,32], the model assigns Young’s modulus and thermal expansion coefficient using Weibull distributions to capture their spatial variability, as illustrated in Figure 7. In contrast, other key mechanical parameters, such as internal friction angle and Poisson’s ratio, are assumed spatially uniform due to their relatively minor heterogeneity [38,39,40]. The damage process of the rock specimen can be divided into two stages.

The first stage corresponds to the high-temperature thermal treatment, as illustrated in Figure 7a. The specimen initially held a temperature of $T_0=20 °\mathrm{C}$, and a Dirichlet temperature boundary condition of $T=T_1$ is applied to all four boundaries to simulate the heating process. The specimen is then gradually cooled back to the initial temperature $T_0$. The heating duration is $t=200 \mathrm{s}$ and the cooling duration is $t=200 \mathrm{s}$. During this process, microscopic damage is induced in the rock specimen due to its heterogeneity.

The second stage simulates a uniaxial compression test, as shown in Figure 7b. In this stage, the vertical displacement at the bottom of the rock is fixed at 0, with the lower left corner being restrained, and a vertical downward displacement load $u$ is applied at the top until failure. The geometric parameters of the model are provided in Figure 7, while the remaining physical parameters are listed in

Table 2. Parameters for rock thermally induced damage loading.

Parameter	Symbol	Unit	Value
Young’s modulus	$E_0$	GPa	30
Poisson’s rate	${\nu }_0$	-	0.384
Length scale parameter	$l$	mm	0.5
Material parameters 1	$\alpha$	-	0.99
Material parameters 2	$\beta$	-	2000
Density	$\rho$	$\mathrm{kg}/{\mathrm{m}}^3$	2576
Specific heat	$C_p$	$\mathrm{J}/\left(\mathrm{kg}\cdot \mathrm{K}\right)$	800
Initial damage stress	St	MPa	116.4
Interaction decay rate parameter	$\eta$	-	5
Residual interaction parameter	$R$	-	0.05

. Due to the heterogeneous distribution of Young’s modulus in the model, a uniform initial damage strain ${\kappa }_0$ does not exist; instead, the actual initial damage strain is determined by the following equation:

$${\kappa }_0={\displaystyle \frac{St}{E}}$$

(21)

This formulation accounts for the spatial variability of material properties, resulting in different initial damage strains across the heterogeneous rock domain.

4.1. Effects of Heterogeneity on Mechanical Behavior

As outlined in Section 2.3, varying the Weibull distribution parameter $m$ corresponds to different degrees of material heterogeneity; the smaller the value of $m$, the higher the degree of material heterogeneity. This subsection examines the thermal damage and uniaxial compressive behavior of the rock under different heterogeneity levels.

Figure 8 illustrates the relationship between the maximum stress and the heterogeneity parameter $m$ under a constant heating temperature of $600 °\mathrm{C}$. It is evident that the maximum stress increases with $m$. In particular, when $m=20$, the maximum stress is approximately 50% higher than that at $m=1.5$. The physical basis of this phenomenon is that a larger $m$ indicates that the distribution of Young’s modulus is more concentrated around its mean value, indicating a more homogenized material. Conversely, with a smaller $m$, the fluctuations in the internal Young’s modulus are larger, causing some points within the material to deviate significantly from the mean. This makes stress concentrations more likely to occur in these “weak regions,” which can prematurely trigger local damage and significantly reduce the overall load-bearing capacity. Consequently, materials characterized by a low m are more susceptible to early-stage local instabilities during loading.

Figure 9 further validates this behavior from a macroscopic loading response perspective. As $m$ decreases, the peak of the load–displacement curve is significantly lowered. In Figure 10, the initial slopes of the curves (i.e., the equivalent elastic modulus) under different $m$ conditions are listed; as $m$ increases, the equivalent elastic modulus becomes larger. This indicates that stronger heterogeneity results in lower stiffness during the elastic phase, earlier damage initiation, and lower stiffness.

In addition, Figure 11 presents the damage morphology under different heterogeneity parameters. It is observed that under high $m$ conditions (e.g., $m=10, 20$), the damage exhibits a more dispersed pattern, with local damage zones gradually connecting to form a network-like failure pattern. In contrast, under low $m$ conditions (e.g., $m=1.5, 2$), the damage band shows distinct X-shaped failure pattern, exhibiting typical shear-dominated failure characteristics. These results confirm that greater heterogeneity (i.e., smaller m) results in a clearer crack propagation path and a more confined failure region.

In summary, Figure 8, Figure 9, Figure 10 and Figure 11 clearly demonstrate that the heterogeneity parameter not only affects the macroscopic mechanical responses (such as maximum stress and stiffness) but also fundamentally alters the microscopic damage evolution mechanism and failure morphology. It is a crucial parameter controlling the transition of the fracture pattern from a concentrated X-shaped crack to a dispersed crack network.

Figure 12 shows the evolution of maximum stress and average stress for different Weibull parameters m, which control the degree of heterogeneity in Young’s modulus. When m is small (e.g., $m=1.5$), the material is highly heterogeneous, leading to strong stress concentrations and a relatively small gap between maximum and average stress. This condition favors the formation of localized failure zones. As m increases, the material becomes more uniform, and the difference between maximum and average stress grows larger, reflecting a more even redistribution of the stress field. This behavior explains why higher m values tend to promote distributed cracking, whereas lower m values result in pronounced localization.

4.2. Effects of Heating Temperature on Mechanical Behavior

To further investigate the effect of heating temperature on damage evolution and mechanical behavior, all subsequent simulations adopt a fixed Weibull shape parameter $m=10$, which is chosen based on values reported in relevant literature [31,36] for granite, over a temperature range of 300 °C to 800 °C.

Taking the case of heating to 800 °C as a representative example, the evolution of microscale damage during the thermal loading process is illustrated. A significant temperature gradient initially develops between the model boundaries and interior, resulting in pronounced thermally induced damage evolution. Figure 13a presents the spatial distribution of the damage variable at different time points, while Figure 13b corresponds to the evolution of the temperature field. Damage begins to appear in the boundary regions from 20 s onward, where thermal stresses rapidly increase along the edges, triggering the nucleation of microcracks. As time progresses to 50 s and 100 s, although the temperature continues to rise, the expansion of the damage zones gradually slows down, indicating that the early temperature gradient predominantly governs the damage distribution. By 200 s, the temperature throughout the model has essentially become uniform at $T_1$, confirming that the designated heating duration is sufficient to reach thermodynamic equilibrium.

Figure 14 further illustrates the damage morphology at the end of the heating phase under different target temperatures. As the temperature increases from 300 °C to 800 °C, damage zones become increasingly dense and widespread. This pattern aligns with the well-established understanding that higher temperatures increase the probability of local failure in the material, thereby exacerbating microstructural instability and inducing a broader spread of microcrack propagation.

To evaluate the extent of high-temperature-induced stiffness degradation, Figure 15 presents the spatial distribution of Young’s modulus under different thermal conditions. At lower temperatures (300–400 °C), most regions maintain relatively high modulus values. In contrast, following exposure to higher temperatures (700–800 °C), a significant reduction in overall stiffness is evident, characterized by large zones with diminished modulus values—indicating a pronounced softening effect due to elevated thermal exposure.

This temperature-induced degradation in stiffness directly impacts the mechanical response. Figure 16a compares the axial response after different thermal pre-treatments. Specimens preheated to lower temperatures exhibit a steeper initial slope—indicating higher elastic stiffness—and attain markedly larger peak loads. As $T_0$ increases, the curves shift downward and leftward: both stiffness and peak load decrease, the onset of softening occurs at smaller displacement, and the residual load capacity diminishes. These trends are consistent with temperature-induced microcracking and matrix degradation, and they agree qualitatively with the reference experiments in Figure 16b. This observation is further quantified in Figure 17, which correlates the equivalent Young’s modulus the equivalent Young’s modulus at the initial loading stage and the temperature. Notably, the equivalent modulus at $T_1=300 °\mathrm{C}$ is nearly twice that at $T_1=800 °\mathrm{C}$, which clearly illustrates the weakening trend in the material’s elastic properties with rising temperature.

Correspondingly, Figure 18 shows the variation in maximum stress with the heating temperature. The trend shown in this curve aligns closely with the modulus variations in Figure 16, reinforcing that the reduction in load-bearing capacity at elevated temperatures primarily stems from the combined effects of stiffness degradation and thermally induced microdamage.

Finally, Figure 19 provides the final damage morphology under different heating temperatures. It is evident that under low-temperature conditions, the damage zones are concentrated, whereas under high-temperature conditions, the damage manifests as a more dispersed crack network. This trend closely corresponds to the previously described spatial distribution of Young’s modulus, further confirming that temperature plays a dominant role in governing damage evolution and failure modes under thermo-mechanical coupling. Figure 20 compares the damage patterns obtained in this study with those reported in the previous literature. Although the color bars differ between the two images, the spatial distribution and overall morphology of the damage show strong agreement.

To further examine the influence of dimensionality, we compared the 3D simulation with the refined 2D result. As shown in Figure 21 the mid-plane section of the 3D model still develops two inclined shear bands intersecting in an “X” shape, consistent with the pattern observed in the 2D simulations. The refined 2D mesh captures a dense microcrack network extending over the whole specimen, whereas the 3D section shows shear bands that are relatively shorter in length. This indicates that the characteristic “X-shaped” and “network-like” failure modes are not artifacts of the 2D assumption, though the 2D model tends to produce more extended cracking than the 3D case.

5. Conclusions

This work focuses on the damage evolution of heterogeneous rocks after heat treatments. By incorporating a Weibull distribution to characterize the spatial heterogeneity of the mechanical properties of rock materials and integrating thermal effects into a localizing gradient damage model, this work systematically develops a thermo-mechanical coupled damage model, which effectively reveals the effects of heterogeneity and thermal loading on the mechanical behavior of rock materials.

First, the accuracy and effectiveness of the proposed model are validated through numerical simulations of a notched cruciform specimen subjected to both mechanical loading and thermal loading. The model successfully reproduces crack paths that closely align with reference failure patterns, demonstrating its reliability and robustness in thermo-mechanical damage analysis.

Then, sensitivity analysis of the heterogeneity parameter m reveals its significant impact on damage morphology and failure mode. A lower value of m (i.e., higher heterogeneity) tends to induce localized crack propagation, resulting in a distinct X-shaped failure pattern dominated by shear stress. In contrast, higher m values (i.e., lower heterogeneity) promote damage evolution in multiple points, which presents a network-like pattern. Additionally, both maximum stress and stiffness increase with m, highlighting the crucial role of heterogeneity in governing overall mechanical properties.

Furthermore, systematic analysis highlights the impact of thermal loading on damage evolution. During the heating process, thermal stress caused by temperature gradients and material heterogeneity first triggers damage in boundary regions of the rock specimen. As heating goes on, micro-damage gradually develops throughout the entire specimen. The final damage distribution and Young’s modulus curves indicate that higher temperatures lead to more severe damage and a significant reduction in overall stiffness. Additionally, load–displacement curves indicate that elevated temperatures not only reduce stiffness but also significantly weaken critical strength. These results confirm that thermal loading promotes micro-damage evolution, thereby diminishing the overall mechanical performance of rock materials.

The findings of this work underscore that material heterogeneity and thermal loading jointly dictate damage evolution and failure mechanisms, offering valuable insights for structural safety assessments and optimization in geotechnical engineering applications. Understanding how spatial variability in rock properties influences thermo-mechanical damage enables more accurate prediction of failure patterns in engineering projects such as geothermal energy extraction, underground excavations, and nuclear waste storage. In particular, during geothermal energy extraction, model parameters can be adjusted to control and promote crack propagation, which may enhance reservoir permeability and improve energy recovery efficiency. Looking forward, future work will focus on extending the current two-dimensional model to fully three-dimensional simulations and integrating additional physical phenomena such as fluid–rock interactions. Moreover, the model will be further developed to consider the phase transitions of different rock components at high temperatures.