Elastoplastic Constitutive Model for Energy Dissipation and Crack Evolution in Rocks

Abstract

The construction of an elastoplastic constitutive model for energy dissipation and crack evolution in rocks is crucial for accurately predicting their failure processes. This study first constructs a theoretical elastoplastic constitutive model by analyzing the mechanical properties of rocks, energy dissipation, and crack evolution under conventional triaxial compression. Subsequently, a three-dimensional finite difference scheme for the theoretical model is derived to implement a numerical algorithm. Finally, using argillaceous siltstone as an example, the validity of the theoretical model and its algorithmic implementation is verified through experimental testing, result analysis, model construction, secondary development, and numerical simulation. The research indicates that the dissipated energy is equal to the integral of the stress–strain curve minus the elastic strain energy, which can be quantitatively described using strength parameters. The volumetric strain of cracks is equal to the plastic volumetric strain, which can be indirectly quantified using the dilation angle. The simulated stress–strain curves closely align with the experimental data, and the simulated dissipated energy and crack volumetric strain are consistent with the theoretical calculations, confirming that the theoretical model effectively captures the nonlinear mechanical behavior, energy dissipation, and crack evolution of rocks.

1. Introduction

The failure process of rocks subjected to load can be characterized in terms of stress, strain, energy, and crack evolution [1,2]. Stress and strain, which can be directly measured through experimental methods, are commonly employed to develop elastoplastic constitutive models that delineate the mechanical properties of rock materials [3,4]. However, it is essential to recognize that the failure of rock materials is fundamentally an energy-driven instability phenomenon, serving as a comprehensive representation of crack evolution [5,6]. Thus, there is an urgent need to establish a rock elastoplastic constitutive model that accurately captures energy dissipation and crack propagation to enhance our ability to describe and predict the actual failure processes of rocks.

Rocks do not behave as linear elastic materials. Following plastic yielding, their stress–strain curves demonstrate considerable hardening and softening behaviors, while volumetric strains exhibit distinct dilatancy characteristics [7,8]. (1) To quantitatively describe the hardening and softening behaviors of rock materials, researchers model the material as an elastoplastic element with uniform internal strain, reflecting the overall deformation effects of rocks under load. They developed corresponding hardening–softening models aimed at accurately predicting the stress–strain curves of rock materials [9,10]. Currently, these models are deeply integrated with advanced computational capabilities, providing reliable computational tools for rock engineering applications [11,12]. For instance, the strain softening model implemented in FLAC^3D 6.0 allows users to define various mechanical parameters through piecewise linear functions of equivalent plastic strain, thus transforming the challenge of simulating rock hardening and softening into the determination of evolving model parameters. This significantly simplifies the application of these models in engineering numerical simulations [13,14]. (2) To quantitatively characterize the dilatancy characteristics of rock materials, the dilatancy angle is utilized to quantify the nonlinear volumetric strain of rocks. This concept was first introduced by Hansen [15], representing the ratio of plastic volumetric strain to shear strain, which can be derived from flow rules and the corresponding plastic potential function. Subsequently, Vermeer [16,17], building on the non-associated flow rule, derived a calculation formula for the dilatancy angle, stipulating that it should lie between zero and the internal friction angle. They highlighted that the associated flow rule would result in no energy dissipation for plastic strain. This formula has since been extensively applied in the stability analysis of rock engineering. For example, Zhao [18,19,20] systematically examined the significance of the dilatancy angle in predicting rock failure and displacement near excavation boundaries, proposing improved methodologies for the dilatancy angle model, which have been effectively validated through laboratory tests, numerical simulations, and field practices.

The process of rock failure under load fundamentally involves the input, accumulation, dissipation, release, and transformation of energy, along with the dynamics of crack closure, initiation, propagation, and coalescence [21,22,23,24], specifically the following: (1) The failure process of rock materials is invariably associated with energy dissipation, which drives the redistribution of internal stresses and nonlinear strains, subsequently influencing the macroscopic mechanical properties of the rock and resulting in mechanical behaviors such as hardening and softening [25,26]. Excluding the thermal energy generated by environmental temperature fluctuations, the energy input from external work is partially accumulated as elastic strain energy and partially dissipated as dissipative energy. When the accumulated elastic strain energy surpasses its storage capacity, it converts into dissipated energy and is released externally, culminating in rock failure [27,28]. In this context, Xie [29,30] examined the energy transformation laws during rock failure from a thermodynamic perspective, indicating that these laws more accurately reflect the essence of rock failure and asserting that dissipated energy provides a more comprehensive description of the progressive failure process of rock materials. Presently, dissipated energy is computed by integrating stress–strain curves and subtracting the elastic strain energy, thus becoming a crucial parameter for characterizing the transition from deformation to instability and failure in rocks. This methodology has facilitated the development of constitutive models of rock based on energy conservation principles and has enhanced their application in numerical simulations [31,32,33]. (2) The failure process of rock materials is also characterized by crack evolution, which results in stress concentration and uneven deformation within the material, thereby impacting the overall mechanical properties of the rock and causing the volumetric strain to exhibit nonlinear characteristics, such as dilatancy [34,35,36]. For an extended period, the internal failure and overall deformation attributed to crack evolution have been focal points of research, primarily due to the random distribution of cracks and the difficulties associated with predicting their evolution [37,38,39]. To tackle this issue, researchers have introduced the concept of crack volumetric strain as a means to quantitatively characterize crack evolution. This value is determined by calculating the volumetric strain and subtracting the elastic volumetric strain [40,41]. This approach circumvents the complexities associated with random crack distribution by concentrating on the total volumetric change resulting from crack progression, effectively sidestepping the analytical challenges presented by randomness. In recent years, crack volumetric strain has been extensively utilized in experimental research and theoretical analysis. It not only elucidates the mechanical mechanisms underlying rock deformation and failure but also effectively delineates the rules governing crack strain evolution and wing crack length through the establishment of corresponding crack propagation models [42,43,44].

In summary, despite significant advancements in the prediction of hardening–softening behavior and dilatancy characteristics in elastoplastic constitutive models for rocks, there has been limited research on the representation of energy dissipation and crack evolution within these frameworks. Consequently, this paper firstly analyzes the elastoplastic characterization of dissipated energy and crack volumetric strain in rocks subjected to conventional triaxial compression, elucidating their specific representation in a theoretical elastoplastic constitutive model. Subsequently, a three-dimensional finite difference format is derived to implement the numerical algorithm of the theoretical model in FLAC^3D. Finally, using argillaceous siltstone as a case study, the model is validated, demonstrating its capability to accurately predict energy dissipation and crack evolution. The findings of this research provide a theoretical basis for the elastic–plastic modeling of energy dissipation and crack evolution in rocks, as well as important insights for the development and validation of such models.

2. Elastoplastic Representation of Mechanical Properties

This section analyzes the nonlinear mechanical behavior of rocks during conventional triaxial compression and discusses the appropriate representation of these mechanical characteristics within elastoplastic constitutive models.

2.1. Elastic Parameters

As shown in Figure 1a, the conventional triaxial compression stress–strain curve of a rock can be divided into four stages, defined by three characteristic stresses. Initially, during the elastic stage before the dilatational stress ${\sigma }_d$, corresponding to the crack’s initial compaction, linear elastic deformation, and stable propagation stages, the overall deformation can be considered as linear elastic. Next, in the hardening stage after ${\sigma }_d$, corresponding to the accelerated crack propagation stage, the overall deformation exhibits plastic yielding. Following this, in the softening stage after the peak stress ${\sigma }_f$, corresponding to the post-peak crack damage stage, the overall strength rapidly declines. Finally, in the residual stage after the residual stress ${\sigma }_r$, the rock is fully damaged, and the overall deformation can be considered ideally plastic.

Therefore, assuming that the elastoplastic constitutive model of rock exhibits linear elastic deformation before ${\sigma }_d$, it can be expressed as follows:

$$\left\{\begin{array}{c}{\epsilon }_1^e=\left[{\sigma }_1-\mu \left({\sigma }_2+{\sigma }_3\right)\right]/E\\ {\epsilon }_2^e=\left[{\sigma }_2-\mu \left({\sigma }_1+{\sigma }_3\right)\right]/E\\ {\epsilon }_3^e=\left[{\sigma }_3-\mu \left({\sigma }_1+{\sigma }_2\right)\right]/E\end{array}\right.$$

(1)

where ${\epsilon }_1^e$, ${\epsilon }_2^e$, and ${\epsilon }_3^e$ are the first, second, and third principal elastic strains, respectively, and ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ are the first, second, and third principal stresses, respectively. Under conventional triaxial compression, ${\epsilon }_2^e={\epsilon }_3^e$ and ${\sigma }_2={\sigma }_3$ are the constant confining pressures. The elastic parameters ($E,\mu$) are also constants. By differentiating Equation (1), we obtain the following:

$$\left\{\begin{array}{c}E=∆{\sigma }_1/∆{\epsilon }_1^e\\ \mu =-∆{\epsilon }_2^e/∆{\epsilon }_1^e=-∆{\epsilon }_3^e/∆{\epsilon }_1^e\end{array}\right.$$

(2)

where $∆$ denotes the increment. Due to the assumption of linear elastic deformation before ${\sigma }_d$, ($E,\mu$) are taken as the secant modulus and corresponding Poisson’s ratio at ${\sigma }_d$ on the stress–strain curve, respectively.

2.2. Yield Function and Plastic Shear Strain

As shown in Figure 1b, the rock exhibits a shear failure mode with a fracture angle $\theta$ under conventional triaxial compression. This is due to the rock material being strong in compression but weak in tension. The lateral confinement under deviatoric loading counteracts some of the internal tensile stress, causing the rock to undergo plastic deformation along shear fractures. Therefore, Figure 1c assumes that after yielding, the stress state of the rock always remains on the Mohr–Coulomb yield surface. The shear strength $\tau$ is represented by the yield surface tangent line, with the intercept as the cohesion $c$ and the inclination as the internal friction angle $\phi$ ($\phi =2\theta -\pi /2$). The yield surface rises during the hardening stage and falls during the softening stage, expressed by the yield function as follows [3,4]:

$$f={\sigma }_1-{\sigma }_3{N}_{\phi }-2c({\gamma }_p)\sqrt{N_{\phi }}=0$$

(3)

where $N_{\phi }=[1+\sin \phi ({\gamma }_p)]/[1-\sin \phi ({\gamma }_p)]$, ${\gamma }_p$ is the plastic shear strain, and $c\left({\gamma }_p\right)$ and $\phi \left({\gamma }_p\right)$ are functions of ${\gamma }_p$ representing cohesion and internal friction angle, respectively. ${\gamma }_p$ can be expressed as follows:

$${\gamma }_p={\epsilon }_1^p-{\epsilon }_3^p$$

(4)

$${\epsilon }_1^p={\epsilon }_1-{\epsilon }_1^e$$

(5)

$${\epsilon }_3^p={\epsilon }_3-{\epsilon }_3^e$$

(6)

$${\epsilon }_V^p={\epsilon }_1^p+{2\epsilon }_3^p$$

(7)

where ${\epsilon }_1$ and ${\epsilon }_3$ are the first and third principal strains, ${\epsilon }_1^p$ and ${\epsilon }_3^p$ are the first and third principal plastic strains, and ${\epsilon }_V^p$ is the plastic volumetric strain. Substituting Equations (1), (5), and (6) into Equation (4) yields the following:

$${\gamma }_p={\epsilon }_1-{\epsilon }_3-\left(1+\mu \right)\left({\sigma }_1-{\sigma }_3\right)/E$$

(8)

According to Equation (8), the plastic shear strain, as a plastic parameter that incorporates deviatoric stress, not only effectively records the degree of plastic yielding at the Mohr–Coulomb yield surface but also reasonably reflects the stress state in which it exists.

2.3. Potential Function

Figure 1d shows that rock is not a fully elastic material, as its volumetric strain exhibits significant dilatancy characteristics. These characteristics are commonly described using the dilatancy angle $\psi$ in continuum mechanics. Therefore, Figure 1e assumes that after yielding, the plastic strain increment direction of the rock is always perpendicular to the Mohr–Coulomb potential surface. Based on the non-associated flow rule, this potential surface can be expressed by the potential function as follows [9,18]:

$$g={\sigma }_1-{\sigma }_3{N}_{\psi }$$

(9)

where $N_{\psi }=[1+\sin \psi \left({\gamma }_p\right)]/[1-\sin \psi \left({\gamma }_p\right)]$, and $\psi \left({\gamma }_p\right)$ is a function of ${\gamma }_p$ representing the dilatancy angle. The principal plastic strain increments can be calculated as follows:

$${∆\epsilon }_i^p=\lambda {\displaystyle \frac{\partial g}{\partial {\sigma }_i}},i=\mathrm{1,2},3$$

(10)

where $\lambda$ is the plastic multiplier, and ${∆\epsilon }_i^p$ can be expressed as follows:

$$\left\{\begin{array}{c}{∆\epsilon }_1^p=\lambda \\ {∆\epsilon }_2^p=0\\ {∆\epsilon }_3^p=-\lambda N_{\psi }\end{array}\right.$$

(11)

Given that under conventional triaxial compression, ${\sigma }_2={\sigma }_3$, the stress state on the potential function in Equation (9) can be expressed as follows:

$$\left\{\begin{array}{c}{g}_1={\sigma }_1-{\sigma }_2{N}_{\psi }\\ g_2={\sigma }_1-{\sigma }_3{N}_{\psi }\end{array}\right.$$

(12)

According to the non-associated flow rule:

$${∆\epsilon }_i^p={\lambda }_1{\displaystyle \frac{{\partial g}_1}{\partial {\sigma }_i}}+{\lambda }_2{\displaystyle \frac{{\partial g}_2}{\partial {\sigma }_i}},i=1,2,3$$

(13)

where ${\lambda }_1$ and ${\lambda }_2$ are the plastic multipliers. Substituting Equation (11) into Equation (13) yields the following:

$$\left\{\begin{array}{c}{∆\epsilon }_1^p={\lambda }_1+{\lambda }_2\\ {∆\epsilon }_2^p=-{\lambda }_1{N}_{\psi }\\ {∆\epsilon }_3^p=-{\lambda }_2{N}_{\psi }\end{array}\right.$$

(14)

The plastic volumetric strain increment is the sum of the three components in Equation (14):

$${∆\epsilon }_{\mathrm{v}}^p={∆\epsilon }_1^p+{∆\epsilon }_2^p+{∆\epsilon }_3^p=({\lambda }_1+{\lambda }_2)(1-N_{\psi })$$

(15)

From the first part of Equations (14) and (15), the dilation angle can be determined as follows:

$$\psi \left({\gamma }_p\right)=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left[{∆\epsilon }_{\mathrm{v}}^p/\left(-2{∆\epsilon }_1^p+{∆\epsilon }_{\mathrm{v}}^p\right)\right]$$

(16)

Equation (16) can be further simplified using Equation (7):

$$\psi \left({\gamma }_p\right)=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left[\left(2{∆\epsilon }_3^p/{∆\epsilon }_1^p+1\right)/\left(2{∆\epsilon }_3^p/{∆\epsilon }_1^p-1\right)\right]$$

(17)

3. Elastoplastic Representation of Energy Dissipation and Crack Evolution

This section analyzes the appropriate representation of energy dissipation and crack evolution in elastoplastic constitutive models based on the first law of thermodynamics and non-associated flow rules.

3.1. Energy Dissipation and Strength Parameters

Assuming that there was no heat exchange with the external environment during the loading and failure process of the rock, according to the first law of thermodynamics, the total energy $U$ input by the external work will be converted into elastic strain energy $U^e$ and dissipated energy $U^d$ [29,30]:

$$U=U^e+U^d$$

(18)

$$U^e=\left({\sigma }_1{\epsilon }_1^e+{\sigma }_2{\epsilon }_2^e+{\sigma }_3{\epsilon }_3^e\right)/2$$

(19)

Substituting Equation (1) into Equation (19) gives the following:

$$U^e=\left[{\sigma }_1^2+2{\sigma }_3^2-2\mu \left(2{\sigma }_1{\sigma }_3+{\sigma }_3^2\right)\right]/2$$

(20)

Meanwhile, during the conventional triaxial compression of rocks, the total input energy $U$ can also be expressed as follows:

$$U=U_0+U_1+U_3$$

(21)

$$U_0=3\left(1-2\mu \right){\sigma }_3^2/2E$$

(22)

$$U_1={\int }_0^{{\epsilon }_1}{\sigma }_{1}d{\epsilon }_1$$

(23)

$$U_3=2{\int }_0^{{\epsilon }_3}{\sigma }_{3}d{\epsilon }_3$$

(24)

where $U_0$ is the elastic strain energy stored during the confining pressure loading phase, $U_1$ is the positive work performed by the axial stress, and $U_3$ is the negative work performed by the circumferential stress. Substituting Equation (21) into Equation (18) gives the following:

$$U^d=(U_1+U_3)-(U^e-U_0)$$

(25)

Based on Equations (20) and (22)–(24), Equation (25) indicates that the dissipated energy is equal to the integral of the stress–strain curve minus the elastic strain energy. Therefore, in the elastoplastic constitutive model, the cohesion $c\left({\gamma }_p\right)$ and internal friction angle $\phi \left({\gamma }_p\right)$, which are associated with the stress–strain curve in Equation (3), can be introduced to quantitatively characterize the energy dissipation.

3.2. Crack Evolution and Dilatancy Angle

Assuming that the random distribution of crack evolution is neglected during the conventional triaxial compression of rock and only the macroscopic overall deformation is considered, the progression of crack development can be quantitatively characterized by the volumetric strain associated with the cracks [42,43,44]:

$${\epsilon }_V^c={\epsilon }_V-{\epsilon }_V^e$$

(26)

$${\epsilon }_V={\epsilon }_1+2{\epsilon }_3$$

(27)

$${\epsilon }_V^e={\epsilon }_1^e+{2\epsilon }_3^e$$

(28)

where ${\epsilon }_V^c$ is the crack volumetric strain, ${\epsilon }_V$ is the volumetric strain, and ${\epsilon }_V^e$ is the elastic volumetric strain. By combining Equations (1), (5)–(7), and (26)–(28), we obtain the following:

$${\epsilon }_V^c={\epsilon }_V^p={\epsilon }_1+2{\epsilon }_3-\left(1-2\mu \right)\left({\sigma }_1+2{\sigma }_3\right)/E$$

(29)

According to Equation (29), the crack volumetric strain is equivalent to the plastic volumetric strain. Therefore, in the elastoplastic constitutive model, the dilatancy angle $\psi \left({\gamma }_p\right)$, which is associated with the increment of plastic volumetric strain in Equation (9), can be introduced to indirectly quantify the crack evolution.

4. Construction and Algorithm Implementation of the Theoretical Model

This section constructs the corresponding theoretical elastoplastic constitutive model based on the analyses presented in Section 2 and Section 3, as shown in

Table 1. Theoretical elastoplastic constitutive model.

Model Parameter	Parameter Equation
$\mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \mu$	Equation (2)
$\mathrm{Elastic} \mathrm{modulus} E$	Equation (2)
$\mathrm{Yield} \mathrm{function} f$	Equation (3)
$\mathrm{Internal} \mathrm{friction} \mathrm{angle} \phi$	Equation (3)
$\mathrm{Cohesion} c$	Equation (3)
$\mathrm{Plastic} \mathrm{shear} \mathrm{strain} {\gamma }_p$	Equation (8)
$\mathrm{Potential} \mathrm{function} g$	Equation (9)
$\mathrm{Dilatancy} \mathrm{angle} \psi$	Equation (9)
$\mathrm{Elastic} \mathrm{strain} \mathrm{energy} \mathrm{stored} \mathrm{during} \mathrm{the} \mathrm{deviatoric} \mathrm{loading} \mathrm{phase} U^e-U_0$	Equations (20) and (22)
$\mathrm{Axial} \mathrm{and} \mathrm{circumferential} \mathrm{work} U_1+U_3$	Equations (23) and (24)
$\mathrm{Dissipated} \mathrm{energy} U^d$	Equation (25)
$\mathrm{Volumetric} \mathrm{strain} {\epsilon }_V$	Equation (27)
$\mathrm{Elastic} \mathrm{volumetric} \mathrm{strain} {\epsilon }_V^e$	Equation (28)

. Additionally, the three-dimensional finite difference scheme of the theoretical model is derived, facilitating the implementation of the numerical algorithm within FLAC^3D.

The three-dimensional finite difference format of the theoretical model is as follows: Firstly, the next step stress ${\sigma }_n^N$ is obtained from the current step stress ${\sigma }_n^c$ and strain increment $∆{\epsilon }_i$. The elastic guess stress tensor ${\sigma }_n^I$ can be expressed as follows [45]:

$${\sigma }_n^I={\sigma }_n^c+{\mathrm{S}}_n(∆{\epsilon }_i),n=1,2,3,i=1,2,3$$

(30)

The components of ${\mathrm{S}}_n(∆{\epsilon }_i)$ are as follows:

$$\left\{\begin{array}{c}{\mathrm{S}}_1(∆{\epsilon }_1,∆{\epsilon }_2,∆{\epsilon }_3)={\delta }_1∆{\epsilon }_1+{\delta }_2\left(∆{\epsilon }_2+∆{\epsilon }_3\right)\\ {\mathrm{S}}_2(∆{\epsilon }_1,∆{\epsilon }_2,∆{\epsilon }_3)={\delta }_1∆{\epsilon }_2+{\delta }_2\left(∆{\epsilon }_1+∆{\epsilon }_3\right)\\ {\mathrm{S}}_3(∆{\epsilon }_1,∆{\epsilon }_2,∆{\epsilon }_3)={\delta }_1∆{\epsilon }_3+{\delta }_2\left(∆{\epsilon }_1+∆{\epsilon }_2\right)\end{array}\right.$$

(31)

where ${\delta }_1=E(1-\mu )/\left[\left(1-2\mu \right)\left(1+\mu \right)\right]$, ${\delta }_2=E\mu /\left[\left(1-2\mu \right)\left(1+\mu \right)\right]$. The values of ${\delta }_1$ and ${\delta }_2$ are determined by Equation (2). Meanwhile, the plastic shear strain ${\gamma }_p$ in its total form is given by Equation (8), and, numerically, ${\gamma }_p$ is computed in an incremental form:

$${\gamma }_p={\gamma }_p^c+{∆\gamma }_p$$

(32)

$${∆\gamma }_p=\sqrt{3}{∆\epsilon }^p\left(1+N_{\psi }\right)/\sqrt{1+N_{\psi }+{N_{\psi }}^2}$$

(33)

$${∆\epsilon }^p=\sqrt{\left[{\left({∆\epsilon }_1^p-{∆\epsilon }_m^p\right)}^2+{\left({∆\epsilon }_m^p\right)}^2+{\left({∆\epsilon }_3^p-{∆\epsilon }_m^p\right)}^2\right]/2}$$

(34)

$${∆\epsilon }_m^p=({∆\epsilon }_1^p+{∆\epsilon }_3^p)/3$$

(35)

where ${\gamma }_p^c$ and ${∆\gamma }_p$ are the current step plastic shear strain and plastic shear strain increment, respectively. The total and incremental forms of ${\gamma }_p$ are numerically equal. The internal friction angle $\phi$ and cohesion $c$ within the yield function $f$ are determined by $c\left({\gamma }_p\right)$ and $\left({\gamma }_p\right)$, as given in Equation (3). The dilatancy angle $\psi$ in the potential function $g$ is determined by $\left({\gamma }_p\right)$, as given in Equation (9).

Secondly, substituting ${\sigma }_n^I$ into Equation (3), if $f\left({\sigma }_n^I\right)<0$, the material is in the elastic stage and ${\sigma }_n^N={\sigma }_n^I$; if $f\left({\sigma }_n^I\right)\ge 0$, the material reaches the yield stage, and the strain increment $∆{\epsilon }_i$ splits into elastic and plastic parts:

$$∆{\epsilon }_i={∆\epsilon }_i^e+{∆\epsilon }_i^p$$

(36)

The stress increment for the current step $∆{\sigma }_n$ can be expressed as follows:

$$∆{\sigma }_n={\mathrm{S}}_n{(∆\epsilon }_i^e)$$

(37)

The next step stress ${\sigma }_n^N$ can be expressed as follows:

$${\sigma }_n^N={\sigma }_n^c+∆{\sigma }_n$$

(38)

By substituting Equations (10), (30), (36), and (37) into Equation (38), we obtain the following:

$${\sigma }_n^N={\sigma }_n^I-{\mathrm{S}}_n\left(\lambda {\displaystyle \frac{\partial g}{\partial {\sigma }_i}}\right)$$

(39)

Substituting Equations (11) and (31) into Equation (39), we obtain the following:

$$\left\{\begin{array}{c}{\sigma }_1^N={\sigma }_1^I-\lambda \left({\alpha }_1-{\alpha }_2{N}_{\psi }\right)\\ {\sigma }_2^N={\sigma }_2^I-\lambda \left({\alpha }_2-{\alpha }_2{N}_{\psi }\right)\\ {\sigma }_3^N={\sigma }_3^I-\lambda \left({\alpha }_2-{\alpha }_1{N}_{\psi }\right)\end{array}\right.$$

(40)

At this point, ${\sigma }_n^N$ is on the Mohr–Coulomb yield surface, and according to Equation (3):

$$f\left({\sigma }_n^N\right)=0$$

(41)

Equation (39) can be written as the complete differential form of the yield function $f$:

$$f\left({\sigma }_n^I\right)-f\left({\sigma }_n^N\right)={\displaystyle \frac{\partial f}{\partial {\sigma }_n}}{\mathrm{S}}_n\left(\lambda {\displaystyle \frac{\partial g}{\partial {\sigma }_i}}\right)$$

(42)

Substituting Equations (3), (11), (31), and (41) into Equation (42), we obtain the plastic multiplier $\lambda$:

$$\lambda ={\displaystyle \frac{f\left({\sigma }_n^I\right)-f\left({\sigma }_n^N\right)}{f\left[{\mathrm{S}}_n\left(\frac{\partial g}{\partial {\sigma }_i}\right)\right]-f(0)}}={\displaystyle \frac{f\left({\sigma }_n^I\right)}{\left({\alpha }_1-{\alpha }_2{N}_{\psi }\right)+({\alpha }_1{N}_{\psi }-{\alpha }_2)N_{\psi }}}$$

(43)

Thirdly, the elastic strain energy stored during the deviatoric loading phase $U^e-U_0$, the axial and circumferential work $U_1+U_3$, and the dissipated energy $U^d$ are calculated according to Equations (20) and (22)–(25). The volumetric strain ${\epsilon }_V$, elastic volumetric strain ${\epsilon }_V^e$, and crack volumetric strain ${\epsilon }_V^c$ are determined according to Equations (27)–(29). The detailed computational workflow of the theoretical model is illustrated in Figure 2.

5. Validation of the Theoretical Model

This section validates the theoretical elastoplastic constitutive model and its algorithmic implementation using argillaceous siltstone as a case study. The validation process encompasses experimental testing, result analysis, model construction, secondary development, and numerical simulation.

5.1. Conventional Triaxial Compression Test

As shown in Figure 3a, a DZSZ-15 multi-field coupled triaxial rock experimental apparatus was utilized at the Highway Engineering Experimental Center of Changsha University of Science and Technology to conduct conventional triaxial compression tests on argillaceous siltstone. The experimental apparatus comprises an axial compression loading system, a confining pressure loading system, a data acquisition system, and a control panel. The maximum axial load capacity is 50 kN, with an accuracy of ±0.001 kN; the maximum confining pressure is ±0.001 kN, with an accuracy of ±0.0001 MPa; and the maximum measurable deformation is 25 mm, with an accuracy of ±0.001 mm.

As illustrated in Figure 3b, rock samples of argillaceous siltstone collected from the Zhuzhou train station in Hunan Province were processed into standard cylindrical specimens (height: 100 mm, diameter: 50 mm), ensuring that the flatness of the end surfaces was less than 0.05 cm. The samples exhibiting significant discrepancies in acoustic velocity were excluded. Conventional triaxial compression tests were then conducted at the confining pressures of 1, 2, 3, 4, and 5 MPa. During the testing process, the control panel applied the hydrostatic pressure at a rate of 0.02 MPa/s until the desired confining pressure was reached and stabilized, followed by an axial pressure application at a rate of 0.002 mm/s until failure occurred. The data acquisition system recorded data every 0.2 s, from which the principal stresses ${\sigma }_1$, ${\sigma }_3$ and principal strains ${\epsilon }_1$, ${\epsilon }_3$ were derived. The stress–strain curves are depicted in Figure 3c.

5.2. Analysis of Elastic Parameters

In the absence of elastoplastic coupling, the elastic parameters ($E,\mu$) are determined from the secant modulus of the dilatational stress ${\sigma }_d$ and the corresponding Poisson’s ratio, as indicated in Equation (2) and illustrated in the stress–strain curve in Figure 3c. By neglecting the confining pressure effects, the average elastic modulus is calculated as $\overline{E}=6.51 \mathrm{G}\mathrm{P}\mathrm{a}$, and the average Poisson’s ratio is determined to be $\overline{\mu }=0.244$.

5.3. Analysis of Strength Parameters

Based on the yield function $f$ in Equation (3), the strength parameters ($c,\phi$) were inversely determined by processing the stress–strain relationships in Figure 3c. As illustrated in Figure 4a, with $\overline{E}=6.51 \mathrm{G}\mathrm{P}\mathrm{a}$ and $\overline{\mu }=0.244$, the plastic shear strain ${\gamma }_p$ was determined using Equation (8). For ${\gamma }_p$ ranging from 0 to 2.0% at intervals of 0.1%, the subsequent yield strengths $({\sigma }_1-{\sigma }_3)$ were inversely determined. As shown in Figure 4b, five Mohr circles were drawn at the same ${\gamma }_p$. Tangents between adjacent circles provided eight tangent points, which were then used to fit the tangent line and derive the strength parameters ($c,\phi$). The specific findings are illustrated in Figure 4c. The cohesion $c$ and internal friction angle $\phi$ can be fitted as follows [3]:

$$c\left({\gamma }_p\right)={\alpha }_1+{\alpha }_2/({\gamma }_p+{\alpha }_3)-{\alpha }_4\mathrm{e}\mathrm{x}\mathrm{p}(-{\alpha }_5{\gamma }_p)$$

(44)

$$\phi \left({\gamma }_p\right)={\beta }_1+({\gamma }_p+{\beta }_2)\mathrm{e}\mathrm{x}\mathrm{p}[{({\gamma }_p-{\beta }_3)}^2/{\beta }_4]$$

(45)

where ${\alpha }_1~{\alpha }_5$ and ${\beta }_1~{\beta }_4$ are the fitting parameters, and the fitting results are shown in Figure 4c.

5.4. Analysis of Dilatancy Angle

Based on the potential function $g$ in Equation (9), the dilatancy angle $\psi$ is inversely derived through processing the stress–strain relationships in Figure 3c. Since the relationship between $({\sigma }_1-{\sigma }_3)$ and ${\gamma }_p$ is already provided, as shown in Figure 4a, the first and third principal plastic strain increments, ${∆\epsilon }_1^p$ and ${∆\epsilon }_3^p$, can be calculated first. Subsequently, the dilatancy angle can be obtained using Equation (17). The calculation results are shown in Figure 5. The average dilatancy angle can be fitted using an exponential difference function [18]:

$$\psi \left({\gamma }_p\right)={\eta }_1[\mathrm{e}\mathrm{x}\mathrm{p}(-{\eta }_2{\gamma }_p)-\mathrm{e}\mathrm{x}\mathrm{p}(-{\eta }_3{\gamma }_p\left)\right]+{\eta }_4$$

(46)

where ${\eta }_1~{\eta }_4$ are the fitting parameters, and the fitting results are shown in Figure 5.

5.5. Argillaceous Siltstone Model

As shown in

Table 2. Argillaceous siltstone model.

Model Parameter	Parameter Equation
$\mathrm{Average} \mathrm{elastic} \mathrm{modulus} \overline{E}$	6.51 GPa
$\mathrm{Average} \mathrm{Poisson}’\mathrm{s} \mathrm{ratio} \overline{\mu }$	0.244
$\mathrm{Yield} \mathrm{function} f$	Equation (3)
$\mathrm{Cohesion} c$	Equation (44)
$\mathrm{Internal} \mathrm{friction} \mathrm{angle} \phi$	Equation (45)
$\mathrm{Plastic} \mathrm{shear} \mathrm{strain} {\gamma }_p$	Equation (8)
$\mathrm{Potential} \mathrm{function} g$	Equation (9)
$\mathrm{Dilatancy} \mathrm{angle} \psi$	Equation (46)
$\mathrm{Elastic} \mathrm{strain} \mathrm{energy} \mathrm{stored} \mathrm{during} \mathrm{the} \mathrm{deviatoric} \mathrm{loading} \mathrm{phase} U^e-U_0$	Equations (20) and (22)
$\mathrm{Axial} \mathrm{and} \mathrm{circumferential} \mathrm{work} U_1+U_3$	Equations (23) and (24)
$\mathrm{Dissipated} \mathrm{energy} U^d$	Equation (25)
$\mathrm{Volumetric} \mathrm{strain} {\epsilon }_V$	Equation (27)
$\mathrm{Elastic} \mathrm{volumetric} \mathrm{strain} {\epsilon }_V^e$	Equation (28)

, the constructed elastoplastic constitutive model for argillaceous siltstone can simultaneously characterize the energy dissipation and crack evolution. Moreover, when the plastic shear strain ${\gamma }_p$ is greater than or equal to 2%, ${\gamma }_p$ remains constant at 2%.

5.6. Model Development and Numerical Simulation

Based on the algorithm implementation of the theoretical elastoplastic constitutive model, as outlined in Table 1, a secondary development of the argillaceous siltstone model, as presented in Table 2, was conducted using C++ in the Microsoft Visual Studio 2010 development environment. The developed model was subsequently integrated into FLAC^3D 6.0 for a numerical simulation.

To ensure the internal variables of the developed model are strictly consistent with the theoretical calculations [9], a single cubic element is established for the numerical simulations of the conventional triaxial compression at confining pressures of 1, 2, 3, 4, and 5 MPa, as shown in Figure 6. During the confining pressure loading phase, the element is subjected to equal confining pressures ${\sigma }_3$ in the X, Y, and Z directions until stabilization. During the deviatoric loading phase, the bottom face is fixed, while the top face is subjected to a downward rate of 0.002 mm/s until failure. The entire loading process records the following parameters: ${\gamma }_p$, $c$, $\phi$, $\psi$, ${\sigma }_1$, ${\sigma }_3$, ${\epsilon }_1$, ${\epsilon }_3$, $U_1+U_3$, $U^e-U_0$, $U^d$, ${\epsilon }_V$, ${\epsilon }_V^e$, and ${\epsilon }_V^c$. The processed data are illustrated in Figure 7, Figure 8, Figure 9 and Figure 10.

Figure 7 demonstrates that the simulated degradation of $c$, $\phi$, and $\psi$ with ${\gamma }_p$ is consistent with the theoretical values of Equations (44)–(46). This consistency is because the degradation of parameters $c$, $\phi$, and $\psi$ in the developed model is programmed based on $c\left({\gamma }_p\right)$, $\phi \left({\gamma }_p\right)$, and $\psi \left({\gamma }_p\right)$, indirectly confirming the successful integration of these parameters into the model.

Figure 8 shows that the simulated stress–strain curves at various confining pressures closely match the experimental data. This concordance is due to the variable ${\gamma }_p$ in Equation (8), which incorporates deviatoric stress (${\sigma }_1-{\sigma }_3$) and effectively captures both the plastic strain extent and the confining pressure effect. The dependent variables $c\left({\gamma }_p\right)$, $\phi \left({\gamma }_p\right)$, and $\psi \left({\gamma }_p\right)$ adjust the entire yielding process of the rock. Although there are slight deviations between the simulated and test data, they objectively represent the overall yield behavior of the rock under varying levels of confining pressure. This validates that the theoretical elastoplastic constitutive model effectively characterizes its nonlinear mechanical behavior.

As depicted in Figure 9 and Figure 10a, during the elastic phase (${\gamma }_p=0$), $U^d=0$, $U_1+U_3$ overlaps entirely with $U^e-U_0$. During yielding, $U^d=\left(U_1+U_3\right)-(U^e-U_0)$. This is because the work performed by external forces is entirely stored as elastic strain energy during the elastic phase and converted into both elastic strain energy and dissipative energy during yielding. Equation (25) in Figure 10a employs the simulated stress–strain curve values for theoretical calculations. The dissipated energy of the rock under various confining pressures aligns with the theoretical predictions, thereby validating that the theoretical elastoplastic constitutive model accurately represents its energy dissipation behavior.

Similarly, Figure 9 and Figure 10b indicate that during the elastic phase (${\gamma }_p=0$), ${\epsilon }_V^c=0$, and ${\epsilon }_V$ overlaps with ${\epsilon }_V^e$. During yielding, ${\epsilon }_V^c={\epsilon }_V-{\epsilon }_V^e$. This is because volumetric strain transforms entirely into elastic volumetric strain during the elastic phase and into both elastic and crack volumetric strains during yielding. The initial values of ${\epsilon }_V$ and ${\epsilon }_V^e$ in Figure 9 are set to zero by subtracting the elastic volumetric strain during the confining pressure loading phase. In Figure 10b, the overlap of ${\epsilon }_V^c$ at various confining pressures occurs because Section 3.2 theoretically derives that ${\epsilon }_V^c$ can be indirectly quantified using $\psi$, which is developed based on $\psi \left({\gamma }_p\right)$ in Equation (46). Thus, at the same ${\gamma }_p$, the simulated values of $\psi$ and ${\epsilon }_V^c$ remain unchanged with varying confining pressures, further validating the theoretical rationale. Equation (29) in Figure 10b also utilizes simulated stress–strain curve values for theoretical calculations. The volume strain of the cracks in the rock under various confining pressures is consistent with the theoretical predictions, thereby validating that the theoretical elastoplastic constitutive model effectively characterizes crack evolution.

6. Discussion

This study successfully constructed an elastoplastic constitutive model for rock energy dissipation and crack evolution, validating the model’s effectiveness through numerical simulations and comparisons with experimental data on argillaceous siltstone. Our research indicates that dissipated energy equals the integral of the stress–strain curve minus the elastic strain energy, which can be quantitatively described using strength parameters. Additionally, the crack volumetric strain is equal to the plastic volumetric strain and can be indirectly quantified using the dilatancy angle. This finding provides a solid theoretical basis for the elastoplastic modeling of rock energy dissipation and crack evolution.

Compared to existing elastoplastic constitutive models in the literature, while previous models have made significant progress in characterizing the nonlinear mechanical behavior of rocks [9,10,11,12], they generally overlook the crucial role of energy dissipation and crack evolution in the strain failure process of rocks. Our proposed model effectively describes the energy dissipation and crack evolution process by introducing the concepts of dissipated energy [29,30,31,32,33] and plastic volumetric strain [40,41,43,44], providing a stronger advantage in describing and predicting actual rock failure processes. Furthermore, existing studies often limit themselves to secondary developments based on single material models [4,7,8], whereas our research derives a three-dimensional finite difference format of the model, clarifying the incremental iterative relationships among various mechanical parameters and their detailed computational process in FLAC^3D, thus facilitating effective numerical algorithm implementation and providing theoretical support for the widespread application of the model.

The results of this study have significant implications for rock mechanics and related engineering fields. Firstly, the constructed model can effectively predict the mechanical response of rocks under different loading conditions, providing theoretical guidance for mining, underground engineering, and other domains. Secondly, the quantitative relationship between energy dissipation and crack evolution proposed in the model offers new insights for the design and assessment of rock materials. In the future, this model could be applied to more complex rock engineering projects to enhance the prediction accuracy and reliability.

Despite the achievements of this study, several limitations remain. Firstly, the analysis of the model primarily focuses on shear failure during the triaxial compression of rocks, without considering the effects of tensile failure. Secondly, the model has shortcomings in accounting for the influences of strain rate, temperature, humidity, and other environmental factors on rock behavior. Additionally, the model was verified only using argillaceous siltstone and did not fully leverage the information on dissipated energy and crack volumetric strain to enhance the physical understanding of experimental phenomena. Future research should aim to overcome these limitations to further expand the model’s applicability and practical use.

In summary, this study provides a theoretical foundation, algorithm implementation, and experimental validation for the elastoplastic constitutive model of rock energy dissipation and crack evolution, with hopes for validation and development in broader applications in the future.

7. Conclusions

This study focuses on the construction of an elastoplastic constitutive model for rock energy dissipation and crack evolution, encompassing systematic theoretical, algorithmic, and validation research. The key findings include the following:

(1)

The elastic parameters can be derived from the secant modulus of the stress–strain curve at the expansion stress and the corresponding Poisson’s ratio. The yield function and potential function can be based on the Mohr–Coulomb strength criterion. The plastic shear strain, as a plastic parameter incorporating deviatoric stress, quantifies the degree of plastic strain and reflects the current stress state. These findings lay a foundational basis for the elastoplastic modeling of rock mechanical properties.

(2)

The dissipated energy is defined as the integral of the stress–strain curve minus the elastic strain energy, with energy dissipation quantitatively described in terms of cohesion and internal friction angle related to the stress–strain curve. The crack volumetric strain is equivalent to plastic volumetric strain, and crack evolution can be indirectly quantified using the dilatancy angle associated with increments in plastic volumetric strain. These insights provide a theoretical basis for the elastoplastic modeling of rock energy dissipation and crack evolution.

(3)

A corresponding theoretical elastoplastic constitutive model was established, and its three-dimensional finite difference format was derived. The incremental iterative relationships among the various mechanical parameters and their detailed computational processes within FLAC^3D were clarified, thus facilitating the implementation of the numerical algorithm for the theoretical model.

(4)

The simulated stress–strain curves closely align with the experimental data, and the simulated dissipated energy and crack volumetric strain are consistent with the theoretical calculations. This validates that the theoretical model effectively predicts the mechanical response, energy dissipation, and crack evolution of rocks, thus providing computational support for a more accurate description and prediction of the actual failure processes of rocks.