# Development of a Model for Initial Compaction Stage in Rock Based on Energy Dissipation

Authors: Tongqing Wu; Lei Nie; Zhiwei Cai; Yue Wu; Qijun Shi; Junzhi Lin; Kui Wang; Nianchun Xu
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

An accurate model is essential for the numerical simulation of the rock failure occurring under different conditions. However, the expression of the initial compression stage is often overlooked in the widespread constitutive models. In this pursuit, several models considering the initial compression stage were gathered from the previous studies, and a model based on the energy dissipation principle was proposed in the present study. Subsequently, the experimental data of five different types of rock under uniaxial load was collected from the literature to verify the validity of the proposed model, and the performance was compared with the other models. It was found that all the models listed in this study exhibited good performance in expressing the stress-strain curve during the initial compression stage.


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## Body

## 1. Introduction

A rock is a solid mass of material composed of one or more minerals. As a special geological material, it undergoes a long and complex diagenetic process, contains a certain number of micropores and cracks with different shapes and random distribution, and presents the characteristics of heterogeneity and anisotropy. The compression under the loading of rock often proceeds through four stages, and the first stage is always the compression of the initial microcracks, irrespective of the type of rock [1, 2]. The length of the initial microcracks in the compression stage can be long or short due to the variations in the types of rock, pore size [3], and moisture content [4] that significantly affect the establishment of the whole constitutive model of rock for the numerical simulation.Although persuasive, some of the studies on the fracture of rock under different conditions face practical difficulties in their execution and feasibility. Therefore, numerical simulation can be a good choice in such cases, and an appropriate constitutive model plays a critical role in numerical simulation. In recent decades, substantial efforts have been made to establish an accurate constitutive model for describing the deformation and strength characteristics of rock materials related to rock engineering [5–12]. The damage models of rock are currently one of the main trends in the research on rock damage, in which the damage variable is the fatal foundation. Studies on the definition of the damage variables have attracted many scholars at home and abroad after being first proposed by Kachanov [13]. Later, Young’s modulus [14, 15] was employed as one of the main parameters to define the damage variables. With the development of the acoustic emission technique and the relationship between the stress damage of rocks and the characteristics of the acoustic emission [16], the acoustic emission event count is one of the sensitive characteristics that can define the damage variable [17]. In recent years, an increasing number of studies have focused on the analysis of energy in rocks damaged by stress [18], and it was observed that the energy dissipation exhibited a close relationship with the change in the mechanical properties of rocks [19], so the energy dissipation can be taken to define the damage variable. Based on the continuum damage mechanics, the damage models can be deduced from two points of view, viz., the macroscopic and the microscopic view [20–23]. The damage variable in the damage statistical method is deduced from the original definition, but the whole rock is regarded as an ensemble of rock micro-elements, and the destruction of the rock is considered the constant failure of the rock micro-elements. Due to random distribution of the rock micro-element strength, a certain intensity distribution needs to be employed, and generally the log-normal distribution, normal distribution, and Weibull distribution find their application. The Weibull distribution is the most reasonable among the three distributions [21], and if properly applied, the whole deformation process of the rocks, including the statistical damage, strain softening, and damage weakening, can be well described [24]. Unfortunately, the plastic deformation in the initial compression stage cannot be reflected by these models, since for each micro-element, the stress-strain relationship is deemed to obey Hooke’s law when the stress is small, ignoring the fact that the plastic deformation process due to the initial voids is compacted.The fact that the nonlinearity of the stress-strain curve exists during the initial compression stage has attracted the attention of some scholars, and they have proposed many hypotheses about this phenomenon. For instance, based on the parabolic strength criterion and the Weibull distribution, the void compaction stage was taken into consideration to formulate the damage evolution model [25]. Later, because of the upper concave stress-strain curve, which can be expressed as a parabola-like equation through the origin, the stress-strain curve during the initial compression stage was proposed [3]. Furthermore, the exp−F/F0m in the damage statistical constitutive model in the elastic stage and the yield stage was transformed as 1−exp−F/F0m, which could be suitable in the compacted section [26]. Based on the earlier methods, a model for the initial compaction stage in the rock was proposed in the present study using the energy dissipation principle, which considered the initial compaction stage.In the present study, the experimental stress-strain curves under the uniaxial compression in five types of rocks were extracted to study the initial compaction stage, based on the earlier studies [27–31]. Initially, the deformation characteristics and the energy evolution mechanisms during the rock failure process were investigated. Subsequently, a model for the initial compaction stage in the rock was proposed from the view of the energy dissipation, and a comparison between the methods in the literature and the model developed in this study was performed to verify the feasibility of the developed method.


## 2. Energy Characteristics

### 2.1. Energy Theory

The failure of rock under load is always accompanied by an energy transformation. When the rock is under an external load without a heat exchange between the specimen and the surroundings, the external load is totally transformed into the strain energyU that can be calculated according to the first law of thermodynamics, part of which is consumed (the dissipative energy, Ud) and most of it is stored as elastic energy (the releasable elastic strain energy, Ue) [32, 33], as shown in equation (1).(1)U=Ue+Ud.For each part of the rock element energy in the principal stress space, the total absorption energy and elastic strain energy can be calculated as follows:(2)U=∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3,Ue=12σ1ε1e+12σ2ε2e+12σ3ε3e,εie=1Eiσi−μiσj+σk,where σi,σj,σki,j,k=1,2,3 are the principal stresses in each direction, εi is the total strain in the three principal stress directions, εie is the related elastic strain, and Ei,μi is the unloaded elastic modulus and Poisson’s ratio, respectively.According to Xie et al. [32], Huang et al. [33], and Chi et al. [34], when the specimen is under uniaxial load, the expression of U can be expressed as follows:(3)U=∫0ε1σi+σi+12dε1,where the variables ε1 and σi are the strain and the stress at any point of the stress-strain curve, respectively. The expression of Ue is as follows:(4)Ue=12σ1ε1e=12Euσ12,where σ1 is the axial, ε1e is the elastic strain, and Eu is the unloading elastic modulus.The data for this study was collected from the experiments in the literature, and it was difficult to obtainEu from the previous studies due to the absence of the unloading experiment. Also, the unloading elastic modulus, Eu, was unknown and replaced by the initial elastic modulus, E0, which was proposed by Huang et al. [33] for its feasibility. Thus, the expression Ue can be expressed as follows:(5)Ue=12σ1ε1e,=12Euσ12≈12E0σ12.According to equation (1), the dissipated strain energy, Ud, can be obtained as follows:(6)Ud=U−Ue=∫0ε1σi+σi+12dε1−12E0σ12.The shaded area shown in Figure1 represents the elastic strain energy, Ue, stored before the failure point and released to cause the failure and collapse of the rock. The energy is dissipated in the process of closing the native internal microcracks, thereby creating internal fissures and expanding them, and their change satisfies the second law of thermodynamics, i.e., a unidirectional and irreversible process.Figure 1
Relationship between the release of elastic energy and dissipated energy in the rock.


### 2.2. Energy Analysis of Different Types of Rock

#### 2.2.1. Deformation Characteristics

The uniaxial stress-strain curves in the different rock types, like coal, granite, marble, shale, and sandstone, are shown in Figure2. From these curves, different stages can be clearly identified. In the initial compression stage, the native internal fractures close under the external force, causing a nonlinear deformation and making the rock stronger. The concave degree varies in different types of rock since different rocks have different mineral compositions and structures. After most of the micro-fractures have closed and gotten stable, the rock deformation enters the second stage, i.e., the elastic deformation stage. Some of the elastic parameters, such as Young’s modulus, E, and Poisson’s ratio, μ, can be calculated at this stage, since at this stage the rock is regarded as a linear elastic material. After the elastic deformation stage, the rock enters the yield stage. The closed fractures rebegin to open and extend; meanwhile, new cracks are created as well. The high stress makes them link with each other, as the curve exhibits nonlinear plastic deformation. In the failure stage, the fractures in rock are constantly accumulated and connected, resulting in a qualitative change of internal microcracks to macroscopic cracks, which eventually destroy the rock.Figure 2
Uniaxial stress-strain curves in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)


#### 2.2.2. Energy Conversion Characteristics

There is no doubt that the rock failure process under loading is accompanied by a transformation of energy. As described in Section2.2.1, the destruction of rock does not happen at once; on the other hand, it is the consequence of the accumulation of damage, from partial damage to local megascopic fractures. Such a process is thermodynamically irreversible and involves the conversion of energy. Hence, it is necessary to investigate the energy storage, dissipation, and release in this process, as shown in Figure 1.The curves of the three energy conversion processes are shown in Figure3, in which the total strain energy was obtained by the stress-strain curve integral. The elastic strain energy was calculated using equation (5), and the dissipated strain energy was obtained from equation (6). The E0 in equation (5) was the tangent modulus of approximately 40%–60% of the peak strength of the rock, instead of the Eu.Figure 3
Energy conversion processes under uniaxial load in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)As shown in Figure3, different types of rock have different failure modes. However, three kinds of energy, i.e., the total energy, the elastic energy, and the dissipation energy, and the five different types of rocks have the same law of energy conversion processes.(i)
Initial compression stage: As the stress is small, the numerical values of the total, elastic, and dissipated energies are small as well. The dissipated energy is slightly higher than the elastic energy, which is dissipated by the compaction of the original interstice in the rock.(ii)
Elastic stage: In this stage, the whole rock sample is considered a linear elastic body. The elastic energy accounts for the majority of the absorbed energy, and after the unloading process, it gets released, while the dissipation energy increases slowly due to the stable microcracks in the rock.(iii)
Yield stage: Before the yield point, the primary fractures in the rock begin to develop again, with the secondary pores and fissures sprouting that are stored in the interior. During this process, the elastic energy continues to accumulate at the approximate rate as the elastic stage, which is stored in the interior. The rate of energy dissipation gradually increases as more energy is taken to dissipate for microcrack growth and interpenetration. The closer the stress is to the peak strength, the more energy dissipates and a higher number of macro cracks appear. Meanwhile, the rock almost has no more capacity for elastic energy.(iv)
Failure stage: In this stage, the rock still absorbs negligible energy by the external force, whereas the dissipative energy increases rapidly and the stored elastic energy releases rapidly. In other words, the entire damaged process of rock needs to dissipate enough energy, and the rock sample loses the strength to store the energy. As a result, the whole rock is completely destroyed.From the above analysis, it can be observed that before the peak strength, the elastic energy predominates the total energy, whereas only a bit of energy is used for initial fracture closure. The dissipation energy gradually dominates the total energy from the yield point since the reopening of the initial cracks, creation of new fractures, relative dislocation of a fracture surface, and propagation of cracks under compression need a large amount of energy to dissipate.


## 2.1. Energy Theory

The failure of rock under load is always accompanied by an energy transformation. When the rock is under an external load without a heat exchange between the specimen and the surroundings, the external load is totally transformed into the strain energyU that can be calculated according to the first law of thermodynamics, part of which is consumed (the dissipative energy, Ud) and most of it is stored as elastic energy (the releasable elastic strain energy, Ue) [32, 33], as shown in equation (1).(1)U=Ue+Ud.For each part of the rock element energy in the principal stress space, the total absorption energy and elastic strain energy can be calculated as follows:(2)U=∫0ε1σ1dε1+∫0ε2σ2dε2+∫0ε3σ3dε3,Ue=12σ1ε1e+12σ2ε2e+12σ3ε3e,εie=1Eiσi−μiσj+σk,where σi,σj,σki,j,k=1,2,3 are the principal stresses in each direction, εi is the total strain in the three principal stress directions, εie is the related elastic strain, and Ei,μi is the unloaded elastic modulus and Poisson’s ratio, respectively.According to Xie et al. [32], Huang et al. [33], and Chi et al. [34], when the specimen is under uniaxial load, the expression of U can be expressed as follows:(3)U=∫0ε1σi+σi+12dε1,where the variables ε1 and σi are the strain and the stress at any point of the stress-strain curve, respectively. The expression of Ue is as follows:(4)Ue=12σ1ε1e=12Euσ12,where σ1 is the axial, ε1e is the elastic strain, and Eu is the unloading elastic modulus.The data for this study was collected from the experiments in the literature, and it was difficult to obtainEu from the previous studies due to the absence of the unloading experiment. Also, the unloading elastic modulus, Eu, was unknown and replaced by the initial elastic modulus, E0, which was proposed by Huang et al. [33] for its feasibility. Thus, the expression Ue can be expressed as follows:(5)Ue=12σ1ε1e,=12Euσ12≈12E0σ12.According to equation (1), the dissipated strain energy, Ud, can be obtained as follows:(6)Ud=U−Ue=∫0ε1σi+σi+12dε1−12E0σ12.The shaded area shown in Figure1 represents the elastic strain energy, Ue, stored before the failure point and released to cause the failure and collapse of the rock. The energy is dissipated in the process of closing the native internal microcracks, thereby creating internal fissures and expanding them, and their change satisfies the second law of thermodynamics, i.e., a unidirectional and irreversible process.Figure 1
Relationship between the release of elastic energy and dissipated energy in the rock.


## 2.2. Energy Analysis of Different Types of Rock

### 2.2.1. Deformation Characteristics

The uniaxial stress-strain curves in the different rock types, like coal, granite, marble, shale, and sandstone, are shown in Figure2. From these curves, different stages can be clearly identified. In the initial compression stage, the native internal fractures close under the external force, causing a nonlinear deformation and making the rock stronger. The concave degree varies in different types of rock since different rocks have different mineral compositions and structures. After most of the micro-fractures have closed and gotten stable, the rock deformation enters the second stage, i.e., the elastic deformation stage. Some of the elastic parameters, such as Young’s modulus, E, and Poisson’s ratio, μ, can be calculated at this stage, since at this stage the rock is regarded as a linear elastic material. After the elastic deformation stage, the rock enters the yield stage. The closed fractures rebegin to open and extend; meanwhile, new cracks are created as well. The high stress makes them link with each other, as the curve exhibits nonlinear plastic deformation. In the failure stage, the fractures in rock are constantly accumulated and connected, resulting in a qualitative change of internal microcracks to macroscopic cracks, which eventually destroy the rock.Figure 2
Uniaxial stress-strain curves in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)


### 2.2.2. Energy Conversion Characteristics

There is no doubt that the rock failure process under loading is accompanied by a transformation of energy. As described in Section2.2.1, the destruction of rock does not happen at once; on the other hand, it is the consequence of the accumulation of damage, from partial damage to local megascopic fractures. Such a process is thermodynamically irreversible and involves the conversion of energy. Hence, it is necessary to investigate the energy storage, dissipation, and release in this process, as shown in Figure 1.The curves of the three energy conversion processes are shown in Figure3, in which the total strain energy was obtained by the stress-strain curve integral. The elastic strain energy was calculated using equation (5), and the dissipated strain energy was obtained from equation (6). The E0 in equation (5) was the tangent modulus of approximately 40%–60% of the peak strength of the rock, instead of the Eu.Figure 3
Energy conversion processes under uniaxial load in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)As shown in Figure3, different types of rock have different failure modes. However, three kinds of energy, i.e., the total energy, the elastic energy, and the dissipation energy, and the five different types of rocks have the same law of energy conversion processes.(i)
Initial compression stage: As the stress is small, the numerical values of the total, elastic, and dissipated energies are small as well. The dissipated energy is slightly higher than the elastic energy, which is dissipated by the compaction of the original interstice in the rock.(ii)
Elastic stage: In this stage, the whole rock sample is considered a linear elastic body. The elastic energy accounts for the majority of the absorbed energy, and after the unloading process, it gets released, while the dissipation energy increases slowly due to the stable microcracks in the rock.(iii)
Yield stage: Before the yield point, the primary fractures in the rock begin to develop again, with the secondary pores and fissures sprouting that are stored in the interior. During this process, the elastic energy continues to accumulate at the approximate rate as the elastic stage, which is stored in the interior. The rate of energy dissipation gradually increases as more energy is taken to dissipate for microcrack growth and interpenetration. The closer the stress is to the peak strength, the more energy dissipates and a higher number of macro cracks appear. Meanwhile, the rock almost has no more capacity for elastic energy.(iv)
Failure stage: In this stage, the rock still absorbs negligible energy by the external force, whereas the dissipative energy increases rapidly and the stored elastic energy releases rapidly. In other words, the entire damaged process of rock needs to dissipate enough energy, and the rock sample loses the strength to store the energy. As a result, the whole rock is completely destroyed.From the above analysis, it can be observed that before the peak strength, the elastic energy predominates the total energy, whereas only a bit of energy is used for initial fracture closure. The dissipation energy gradually dominates the total energy from the yield point since the reopening of the initial cracks, creation of new fractures, relative dislocation of a fracture surface, and propagation of cracks under compression need a large amount of energy to dissipate.


## 2.2.1. Deformation Characteristics

The uniaxial stress-strain curves in the different rock types, like coal, granite, marble, shale, and sandstone, are shown in Figure2. From these curves, different stages can be clearly identified. In the initial compression stage, the native internal fractures close under the external force, causing a nonlinear deformation and making the rock stronger. The concave degree varies in different types of rock since different rocks have different mineral compositions and structures. After most of the micro-fractures have closed and gotten stable, the rock deformation enters the second stage, i.e., the elastic deformation stage. Some of the elastic parameters, such as Young’s modulus, E, and Poisson’s ratio, μ, can be calculated at this stage, since at this stage the rock is regarded as a linear elastic material. After the elastic deformation stage, the rock enters the yield stage. The closed fractures rebegin to open and extend; meanwhile, new cracks are created as well. The high stress makes them link with each other, as the curve exhibits nonlinear plastic deformation. In the failure stage, the fractures in rock are constantly accumulated and connected, resulting in a qualitative change of internal microcracks to macroscopic cracks, which eventually destroy the rock.Figure 2
Uniaxial stress-strain curves in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)


## 2.2.2. Energy Conversion Characteristics

There is no doubt that the rock failure process under loading is accompanied by a transformation of energy. As described in Section2.2.1, the destruction of rock does not happen at once; on the other hand, it is the consequence of the accumulation of damage, from partial damage to local megascopic fractures. Such a process is thermodynamically irreversible and involves the conversion of energy. Hence, it is necessary to investigate the energy storage, dissipation, and release in this process, as shown in Figure 1.The curves of the three energy conversion processes are shown in Figure3, in which the total strain energy was obtained by the stress-strain curve integral. The elastic strain energy was calculated using equation (5), and the dissipated strain energy was obtained from equation (6). The E0 in equation (5) was the tangent modulus of approximately 40%–60% of the peak strength of the rock, instead of the Eu.Figure 3
Energy conversion processes under uniaxial load in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)As shown in Figure3, different types of rock have different failure modes. However, three kinds of energy, i.e., the total energy, the elastic energy, and the dissipation energy, and the five different types of rocks have the same law of energy conversion processes.(i)
Initial compression stage: As the stress is small, the numerical values of the total, elastic, and dissipated energies are small as well. The dissipated energy is slightly higher than the elastic energy, which is dissipated by the compaction of the original interstice in the rock.(ii)
Elastic stage: In this stage, the whole rock sample is considered a linear elastic body. The elastic energy accounts for the majority of the absorbed energy, and after the unloading process, it gets released, while the dissipation energy increases slowly due to the stable microcracks in the rock.(iii)
Yield stage: Before the yield point, the primary fractures in the rock begin to develop again, with the secondary pores and fissures sprouting that are stored in the interior. During this process, the elastic energy continues to accumulate at the approximate rate as the elastic stage, which is stored in the interior. The rate of energy dissipation gradually increases as more energy is taken to dissipate for microcrack growth and interpenetration. The closer the stress is to the peak strength, the more energy dissipates and a higher number of macro cracks appear. Meanwhile, the rock almost has no more capacity for elastic energy.(iv)
Failure stage: In this stage, the rock still absorbs negligible energy by the external force, whereas the dissipative energy increases rapidly and the stored elastic energy releases rapidly. In other words, the entire damaged process of rock needs to dissipate enough energy, and the rock sample loses the strength to store the energy. As a result, the whole rock is completely destroyed.From the above analysis, it can be observed that before the peak strength, the elastic energy predominates the total energy, whereas only a bit of energy is used for initial fracture closure. The dissipation energy gradually dominates the total energy from the yield point since the reopening of the initial cracks, creation of new fractures, relative dislocation of a fracture surface, and propagation of cracks under compression need a large amount of energy to dissipate.


## 3. Damage Constitutive Model of Rock in Initial Compaction Stage

### 3.1. Damage Statistical Constitutive Model Based on Weibull Distribution

The Weibull distribution was proven reasonable for reflecting the internal defect distribution in the rock, so it was chosen to establish the statistical constitutive damage model. Combined with the rock uniaxial test results, the experimental fitting was determined using the parameters in the model [21, 26].In this model, the rock sample was taken as an aggregation of microscopic elements, and the number is denoted asN0. Under uniaxial load, a certain number of microscopic elements were damaged and this was denoted as n. Thus, the expression of damage variable D is as(7)D=nN0.The probability of damage for each element in the Weibull distribution was adopted here, which can be expressed as(8)Pε=mε0εε0m−1exp−εε0m,where P (ε) is the distribution function of the rock micro-element strength and ε is the random variable of the micro-element strength. Because the strain strength theory was adopted here, ε is the strain of the rock, m and ε0 are the distribution parameters obtained from the experimental data.As described in Section2.2.1, the fracture of rock is a continuous process of creation, expansion, and penetration of the internal cracks. Thus, the relationship between the damage variable D and the distribution of micro-element strength can be expressed as(9)D=1−exp−εε0m.Assuming that for each micro-element, the stress-strain relationship is deemed to obey Hooke’s law, the constitutive law can be written as(10)σ1=Eε11−D,where E and ε1 are the elastic modulus and strain of rocks, respectively. Equation (9) is not suitable for expressing the upper concave of the stress-strain curve during the compaction, and the exp−ε/ε0m is replaced in the expression of D, i.e., D=exp−ε/ε0m [26]. Therefore, equation (10) can be rewritten as(11)σ1=Eε11−exp−εε0m.Transforming equation (11), the following stress-strain relationship can be obtained by(12)σ1Eε1=1−exp−εε0m.Taking natural logarithm operation twice on both sides of equation, equation (12) can be obtained as(13)ln−ln1−σ1Eε1=mlnε−mlnε0Let(14)y=ln−ln1−σ1Eε1,x=lnε,x0=−mlnε0,then(15)y=mx+x0.Obviously, through the manipulation of the experimental data from the previous reports, a series of beelines can be obtained as shown in equation (15), where m and x0 are the slope and the intercept, respectively. The parameter ε0 can be easily back-calculated based on x0.


### 3.2. Constitutive Model considering the Void Characteristics

From the macroscopic view, it is a well-known fact that the rock can be considered as an aggregation of voids and a solid skeleton. At the beginning of the load, the number of initial voids is diminished. Therefore, the constitutive model during the initial compression stage with voids can be established [3, 35].As shown in Figure4, the rock sample was conceptualized as the void part and the solid skeleton part, and lv, ls are defined as the length, respectively. Hence, the initial length l of the rock sample is l = lv + ls.Figure 4
Schematic representation of the compression process in rock voids.When the rock sample is under uniaxial stressσ1, the total deformation Δl is produced, which can be divided into two parts. One is the deformation of the void part, denoted as ∆lv, and the other is the deformation of the solid skeleton, denoted as Δls, i.e., Δl = ∆lv + Δls. Accordingly, the strain expressions for the total strain ε1, the void ε1v, and skeleton parts ε1s of the rock are as follows:(16)ε1=Δll=Δlv+Δlsl,ε1v=Δlvlv,ε1s=Δlsls.The coefficientγ was defined to expediently further study the influence of the existence of voids on the constitutive model, and the expression is as follows:(17)γ=ε1vε1.According to a previous study, the value ofγ varies roughly in a linear negative manner with the axial strain, and the initial value of γ is close to 1 [3]. γcc and εcc can be defined as the values of the proportional coefficient and the axial strain at the critical point of the initial crack compaction, respectively. Thus, in the compression stage (section oa in Figure 1), the ε1v can be calculated by(18)ε1v=γε1=1+γcc−1εccε1ε1,ε1≤εcc.From the stress-strain curve under uniaxial loading, between the critical point of the initial crack compaction and the yield point, the deformation of the rock sample is elastic. In other words, the deformation in the elastic stage is undertaken by the solid skeleton since the initial cracks in the rock sample are almost diminished. Therefore, assuming the relationship between the strain and stress during the elastic stage, the deformation can be expressed as(19)σ1=Eε1s.Then,εccs and γcc can be calculated using equations (20) and (21), respectively:(20)εccs=σccE,(21)γcc=1−εccsεcc=1−σccEεcc.The expression ofε1s during the initial compaction stage is given as(22)ε1s=σccEεcc2ε12,ε1≤εcc.At the point of the peak strength, the corresponding strain on the solid skeleton can be calculated by(23)εcs=εc−εcc+σccEThe previous statistical damage constitutive model can well express the stress-strain curve of the rock after the load enters the elastic stage, as described in Section3.1. The replacement of ε1 by ε1s in equation (11) takes the voids into consideration to reflect the nonlinearity of the statistical damage constitutive model before the voids get totally compacted.(24)σ1=Eε1sexp−1Mε1sεcsM,(25)M=1lnEεcs/σc.It is easy to get the values of all of the parameters in equations (24) and (25) by the uniaxial load tests from the reports cited in this paper.


### 3.3. Constitutive Model Based on the Principle of Energy Dissipation

In continuum damage mechanics, the damage variable is essential and its presentation is necessary to describe the development of joints and cracks in a continuum. Studies show that there is a close relationship between energy dissipation and damage in rocks. Therefore, the dissipative energy method was used in the present study to study the degree of rock damage during loading and damage variables of rocks on the basis of energy dissipation in earlier studies.Using the energy dissipation theory, Jin et al. [36] define the damage variable D as follows:(26)D=UdU,where Ud is the dissipated energy and U is the constitutive energy. Similarly, Geng et al. [37] define the damage variable DU as follows:(27)DU=∑i=0mΔUid∑i=0nΔUid=Ud′Udt,where Ud′ and Udt are the accumulated dissipation energies at a given time and at the end, respectively.As shown in Figure5 and the description in Section 2.2.2, before the yield point, the energy gets dissipated, owing to the compaction and re-expansion of the internal microcracks. However, most of the absorbed energy is stored in rocks in the form of elastic energy. As a result, there is a little damage in the rock before the yield point. When the rock enters the yield stage and failure stage, the dissipation energy increases rapidly because of the quick generation and penetration of the cracks, which indicates the detection of a substantial change in the damage. To better reflect the variation of the damage during the crack compaction due to the closure of initial internal microcracks, referring to the definition of the damage variable by Geng et al. [37], and based on the fact that there exist several microcracks in the natural rock, the rock with damage has a damage variable at the beginning and a value of zero at the critical point of the initial crack closure. Thus, the definition of the damage variable in this study can be expressed by equation (28).(28)Dcc=1−UdUccd,where Dcc is the damage variable of rock, and Ud and Uccd are the dissipation energies at a given strain and the accumulated dissipation energy at the critical point of the initial crack closure, respectively.Figure 5
Variation of Damage variablesD and DU in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)According to equation (12), the exponential function can be used to express the stress-strain curve based on the damage variable Dcc as follows:(29)σ1Eε1=exp−α1−Dccβ.Taking natural logarithm operation twice on both sides of the equation, (29) can be obtained as(30)ln−lnσ1Eε1=lnα+βln1−Dcc.LetY=ln−lnσ1/Eε1, X=ln1−Dcc, and X0=lnα, then(31)Y=βx+X0,where the parameter β is the slope of the beeline processed from the experimental data from the equation (31) and α can be obtained by X0.


### 3.4. Result and Discussion

The statistical evaluation of the damage constitutive model shows good agreement with the stages after the point of the initial crack closure. Thus, it is important to introduce a model that exhibits a good fit with the initial compaction. In Section3, several methods, including our hypothesis that highlights the relationship between stress and strain during the initial compaction stage, were described, and the parameters in these equations were obtained from the experimental data.


#### 3.4.1. Performance of the Models

The uniaxial stress-strain curves of the five types of rock,viz. coal, granite, marble, sandstone, and shale, were used to compare the performance of the methods described in Section 3. The calculated values were compared with the experimental data, and the coefficient of determination (R2) of the different models during the initial compaction stage is shown in Table 1.Table 1
Error analysis of the Damage constitutive model.
No.Rock typeCoefficient of determination (R2)Equation (13)Equation (24)Equation (30)1Coal0.99640.99320.99972Granite0.98870.99960.99793Marble0.99860.99080.99804Sandstone0.99970.99680.99915Shale0.99950.99950.9917It can be seen in Table1 that even though the types of rock were different, all of the models exhibited a high degree of fitting, which means that the models listed in Section 3 can be suitably used to imply the development of the voids in the rock. In other words, the initial compression stage occurs and the stress-strain curve exhibits an upper concave curve. From Figure 6, it can be seen that the calculated values of axial stress that were close to the point of the initial crack closure were higher than the experimental data [equation (13)]. On the contrary, the calculated values using equation (30) were the least among the results, experimental data, and fitted values. At the same time, since equation (24) was derived from the theory based on the characteristics of voids, it exhibited a better degree of curvature even though it deviated from the experimental data. Conversely, the degree of curvature of the other two models was not good enough, but the fitting results were much closer to the experimental data.Figure 6
Comparison of the test data and the fitting curves based on different theories. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)In conclusion, based on the energy principle and the statistical damage theory, the statistical damage constitutive model established in this study performs well in the initial compression stage of porous rock. In addition, the model based on the hypothesis of the dissipated energy should be further studied, especially the constitutive model of the whole stress-strain curve under different states. Since the stress-strain behavior of rocks in the postpeak stage is too complex, it is not considered in this paper. Therefore, the model proposed in this paper still has room for improvement.


#### 3.4.2. Parameters in the Models

Except for the parameters in equation (24), derived from the ‘Extremum Method’, the other two equations were obtained by the fitting method. The parameters had no clear physical meaning. However, they can reflect some characteristics of the rock. Table 2 shows the parameters in these different models, and Figure 7 shows the variation in parameters in different rocks.Table 2
Parameters in the models.
No.Rock typeEquation (13)Equation (24)Equation (30)mMΑβ1Coal1.155814.09480.5849−0.24122Granite2.41811.03020.5941−0.23673Marble0.3343.03890.0119−1.51064Sandstone0.64683.43970.8420−0.15315Shale1.17991.44740.4177−0.2967Figure 7
Variation of values of parameters in different rocks.It can be concluded from Table2 and Figure 6, that the parameter M in the equation (24) exhibited a drastic change because of the composition of different rocks, and m in equation (13) changed negatively. Further, α and β in equation (30) exhibited a meager change, which was due to the similarity in the law of the variation of the dissipated energy, although the composition of the rock varied from each other.


## 3.1. Damage Statistical Constitutive Model Based on Weibull Distribution

The Weibull distribution was proven reasonable for reflecting the internal defect distribution in the rock, so it was chosen to establish the statistical constitutive damage model. Combined with the rock uniaxial test results, the experimental fitting was determined using the parameters in the model [21, 26].In this model, the rock sample was taken as an aggregation of microscopic elements, and the number is denoted asN0. Under uniaxial load, a certain number of microscopic elements were damaged and this was denoted as n. Thus, the expression of damage variable D is as(7)D=nN0.The probability of damage for each element in the Weibull distribution was adopted here, which can be expressed as(8)Pε=mε0εε0m−1exp−εε0m,where P (ε) is the distribution function of the rock micro-element strength and ε is the random variable of the micro-element strength. Because the strain strength theory was adopted here, ε is the strain of the rock, m and ε0 are the distribution parameters obtained from the experimental data.As described in Section2.2.1, the fracture of rock is a continuous process of creation, expansion, and penetration of the internal cracks. Thus, the relationship between the damage variable D and the distribution of micro-element strength can be expressed as(9)D=1−exp−εε0m.Assuming that for each micro-element, the stress-strain relationship is deemed to obey Hooke’s law, the constitutive law can be written as(10)σ1=Eε11−D,where E and ε1 are the elastic modulus and strain of rocks, respectively. Equation (9) is not suitable for expressing the upper concave of the stress-strain curve during the compaction, and the exp−ε/ε0m is replaced in the expression of D, i.e., D=exp−ε/ε0m [26]. Therefore, equation (10) can be rewritten as(11)σ1=Eε11−exp−εε0m.Transforming equation (11), the following stress-strain relationship can be obtained by(12)σ1Eε1=1−exp−εε0m.Taking natural logarithm operation twice on both sides of equation, equation (12) can be obtained as(13)ln−ln1−σ1Eε1=mlnε−mlnε0Let(14)y=ln−ln1−σ1Eε1,x=lnε,x0=−mlnε0,then(15)y=mx+x0.Obviously, through the manipulation of the experimental data from the previous reports, a series of beelines can be obtained as shown in equation (15), where m and x0 are the slope and the intercept, respectively. The parameter ε0 can be easily back-calculated based on x0.


## 3.2. Constitutive Model considering the Void Characteristics

From the macroscopic view, it is a well-known fact that the rock can be considered as an aggregation of voids and a solid skeleton. At the beginning of the load, the number of initial voids is diminished. Therefore, the constitutive model during the initial compression stage with voids can be established [3, 35].As shown in Figure4, the rock sample was conceptualized as the void part and the solid skeleton part, and lv, ls are defined as the length, respectively. Hence, the initial length l of the rock sample is l = lv + ls.Figure 4
Schematic representation of the compression process in rock voids.When the rock sample is under uniaxial stressσ1, the total deformation Δl is produced, which can be divided into two parts. One is the deformation of the void part, denoted as ∆lv, and the other is the deformation of the solid skeleton, denoted as Δls, i.e., Δl = ∆lv + Δls. Accordingly, the strain expressions for the total strain ε1, the void ε1v, and skeleton parts ε1s of the rock are as follows:(16)ε1=Δll=Δlv+Δlsl,ε1v=Δlvlv,ε1s=Δlsls.The coefficientγ was defined to expediently further study the influence of the existence of voids on the constitutive model, and the expression is as follows:(17)γ=ε1vε1.According to a previous study, the value ofγ varies roughly in a linear negative manner with the axial strain, and the initial value of γ is close to 1 [3]. γcc and εcc can be defined as the values of the proportional coefficient and the axial strain at the critical point of the initial crack compaction, respectively. Thus, in the compression stage (section oa in Figure 1), the ε1v can be calculated by(18)ε1v=γε1=1+γcc−1εccε1ε1,ε1≤εcc.From the stress-strain curve under uniaxial loading, between the critical point of the initial crack compaction and the yield point, the deformation of the rock sample is elastic. In other words, the deformation in the elastic stage is undertaken by the solid skeleton since the initial cracks in the rock sample are almost diminished. Therefore, assuming the relationship between the strain and stress during the elastic stage, the deformation can be expressed as(19)σ1=Eε1s.Then,εccs and γcc can be calculated using equations (20) and (21), respectively:(20)εccs=σccE,(21)γcc=1−εccsεcc=1−σccEεcc.The expression ofε1s during the initial compaction stage is given as(22)ε1s=σccEεcc2ε12,ε1≤εcc.At the point of the peak strength, the corresponding strain on the solid skeleton can be calculated by(23)εcs=εc−εcc+σccEThe previous statistical damage constitutive model can well express the stress-strain curve of the rock after the load enters the elastic stage, as described in Section3.1. The replacement of ε1 by ε1s in equation (11) takes the voids into consideration to reflect the nonlinearity of the statistical damage constitutive model before the voids get totally compacted.(24)σ1=Eε1sexp−1Mε1sεcsM,(25)M=1lnEεcs/σc.It is easy to get the values of all of the parameters in equations (24) and (25) by the uniaxial load tests from the reports cited in this paper.


## 3.3. Constitutive Model Based on the Principle of Energy Dissipation

In continuum damage mechanics, the damage variable is essential and its presentation is necessary to describe the development of joints and cracks in a continuum. Studies show that there is a close relationship between energy dissipation and damage in rocks. Therefore, the dissipative energy method was used in the present study to study the degree of rock damage during loading and damage variables of rocks on the basis of energy dissipation in earlier studies.Using the energy dissipation theory, Jin et al. [36] define the damage variable D as follows:(26)D=UdU,where Ud is the dissipated energy and U is the constitutive energy. Similarly, Geng et al. [37] define the damage variable DU as follows:(27)DU=∑i=0mΔUid∑i=0nΔUid=Ud′Udt,where Ud′ and Udt are the accumulated dissipation energies at a given time and at the end, respectively.As shown in Figure5 and the description in Section 2.2.2, before the yield point, the energy gets dissipated, owing to the compaction and re-expansion of the internal microcracks. However, most of the absorbed energy is stored in rocks in the form of elastic energy. As a result, there is a little damage in the rock before the yield point. When the rock enters the yield stage and failure stage, the dissipation energy increases rapidly because of the quick generation and penetration of the cracks, which indicates the detection of a substantial change in the damage. To better reflect the variation of the damage during the crack compaction due to the closure of initial internal microcracks, referring to the definition of the damage variable by Geng et al. [37], and based on the fact that there exist several microcracks in the natural rock, the rock with damage has a damage variable at the beginning and a value of zero at the critical point of the initial crack closure. Thus, the definition of the damage variable in this study can be expressed by equation (28).(28)Dcc=1−UdUccd,where Dcc is the damage variable of rock, and Ud and Uccd are the dissipation energies at a given strain and the accumulated dissipation energy at the critical point of the initial crack closure, respectively.Figure 5
Variation of Damage variablesD and DU in the different types of rocks. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)According to equation (12), the exponential function can be used to express the stress-strain curve based on the damage variable Dcc as follows:(29)σ1Eε1=exp−α1−Dccβ.Taking natural logarithm operation twice on both sides of the equation, (29) can be obtained as(30)ln−lnσ1Eε1=lnα+βln1−Dcc.LetY=ln−lnσ1/Eε1, X=ln1−Dcc, and X0=lnα, then(31)Y=βx+X0,where the parameter β is the slope of the beeline processed from the experimental data from the equation (31) and α can be obtained by X0.


## 3.4. Result and Discussion

The statistical evaluation of the damage constitutive model shows good agreement with the stages after the point of the initial crack closure. Thus, it is important to introduce a model that exhibits a good fit with the initial compaction. In Section3, several methods, including our hypothesis that highlights the relationship between stress and strain during the initial compaction stage, were described, and the parameters in these equations were obtained from the experimental data.


### 3.4.1. Performance of the Models

The uniaxial stress-strain curves of the five types of rock,viz. coal, granite, marble, sandstone, and shale, were used to compare the performance of the methods described in Section 3. The calculated values were compared with the experimental data, and the coefficient of determination (R2) of the different models during the initial compaction stage is shown in Table 1.Table 1
Error analysis of the Damage constitutive model.
No.Rock typeCoefficient of determination (R2)Equation (13)Equation (24)Equation (30)1Coal0.99640.99320.99972Granite0.98870.99960.99793Marble0.99860.99080.99804Sandstone0.99970.99680.99915Shale0.99950.99950.9917It can be seen in Table1 that even though the types of rock were different, all of the models exhibited a high degree of fitting, which means that the models listed in Section 3 can be suitably used to imply the development of the voids in the rock. In other words, the initial compression stage occurs and the stress-strain curve exhibits an upper concave curve. From Figure 6, it can be seen that the calculated values of axial stress that were close to the point of the initial crack closure were higher than the experimental data [equation (13)]. On the contrary, the calculated values using equation (30) were the least among the results, experimental data, and fitted values. At the same time, since equation (24) was derived from the theory based on the characteristics of voids, it exhibited a better degree of curvature even though it deviated from the experimental data. Conversely, the degree of curvature of the other two models was not good enough, but the fitting results were much closer to the experimental data.Figure 6
Comparison of the test data and the fitting curves based on different theories. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)In conclusion, based on the energy principle and the statistical damage theory, the statistical damage constitutive model established in this study performs well in the initial compression stage of porous rock. In addition, the model based on the hypothesis of the dissipated energy should be further studied, especially the constitutive model of the whole stress-strain curve under different states. Since the stress-strain behavior of rocks in the postpeak stage is too complex, it is not considered in this paper. Therefore, the model proposed in this paper still has room for improvement.


### 3.4.2. Parameters in the Models

Except for the parameters in equation (24), derived from the ‘Extremum Method’, the other two equations were obtained by the fitting method. The parameters had no clear physical meaning. However, they can reflect some characteristics of the rock. Table 2 shows the parameters in these different models, and Figure 7 shows the variation in parameters in different rocks.Table 2
Parameters in the models.
No.Rock typeEquation (13)Equation (24)Equation (30)mMΑβ1Coal1.155814.09480.5849−0.24122Granite2.41811.03020.5941−0.23673Marble0.3343.03890.0119−1.51064Sandstone0.64683.43970.8420−0.15315Shale1.17991.44740.4177−0.2967Figure 7
Variation of values of parameters in different rocks.It can be concluded from Table2 and Figure 6, that the parameter M in the equation (24) exhibited a drastic change because of the composition of different rocks, and m in equation (13) changed negatively. Further, α and β in equation (30) exhibited a meager change, which was due to the similarity in the law of the variation of the dissipated energy, although the composition of the rock varied from each other.


## 3.4.1. Performance of the Models

The uniaxial stress-strain curves of the five types of rock,viz. coal, granite, marble, sandstone, and shale, were used to compare the performance of the methods described in Section 3. The calculated values were compared with the experimental data, and the coefficient of determination (R2) of the different models during the initial compaction stage is shown in Table 1.Table 1
Error analysis of the Damage constitutive model.
No.Rock typeCoefficient of determination (R2)Equation (13)Equation (24)Equation (30)1Coal0.99640.99320.99972Granite0.98870.99960.99793Marble0.99860.99080.99804Sandstone0.99970.99680.99915Shale0.99950.99950.9917It can be seen in Table1 that even though the types of rock were different, all of the models exhibited a high degree of fitting, which means that the models listed in Section 3 can be suitably used to imply the development of the voids in the rock. In other words, the initial compression stage occurs and the stress-strain curve exhibits an upper concave curve. From Figure 6, it can be seen that the calculated values of axial stress that were close to the point of the initial crack closure were higher than the experimental data [equation (13)]. On the contrary, the calculated values using equation (30) were the least among the results, experimental data, and fitted values. At the same time, since equation (24) was derived from the theory based on the characteristics of voids, it exhibited a better degree of curvature even though it deviated from the experimental data. Conversely, the degree of curvature of the other two models was not good enough, but the fitting results were much closer to the experimental data.Figure 6
Comparison of the test data and the fitting curves based on different theories. (a) Coal. (b) Granite. (c) Marble. (d) Shale. (e) Sandstone.
(a)(b)(c)(d)(e)In conclusion, based on the energy principle and the statistical damage theory, the statistical damage constitutive model established in this study performs well in the initial compression stage of porous rock. In addition, the model based on the hypothesis of the dissipated energy should be further studied, especially the constitutive model of the whole stress-strain curve under different states. Since the stress-strain behavior of rocks in the postpeak stage is too complex, it is not considered in this paper. Therefore, the model proposed in this paper still has room for improvement.


## 3.4.2. Parameters in the Models

Except for the parameters in equation (24), derived from the ‘Extremum Method’, the other two equations were obtained by the fitting method. The parameters had no clear physical meaning. However, they can reflect some characteristics of the rock. Table 2 shows the parameters in these different models, and Figure 7 shows the variation in parameters in different rocks.Table 2
Parameters in the models.
No.Rock typeEquation (13)Equation (24)Equation (30)mMΑβ1Coal1.155814.09480.5849−0.24122Granite2.41811.03020.5941−0.23673Marble0.3343.03890.0119−1.51064Sandstone0.64683.43970.8420−0.15315Shale1.17991.44740.4177−0.2967Figure 7
Variation of values of parameters in different rocks.It can be concluded from Table2 and Figure 6, that the parameter M in the equation (24) exhibited a drastic change because of the composition of different rocks, and m in equation (13) changed negatively. Further, α and β in equation (30) exhibited a meager change, which was due to the similarity in the law of the variation of the dissipated energy, although the composition of the rock varied from each other.


## 4. Conclusions

Based on the energy principle and the statistical damage theory, the nonlinear deformation characteristics of rocks with an initial void during the initial void compaction stage were evaluated in the present study, and the following conclusions can be drawn:(1)
The deformation characteristics of the rocks during compression were studied in five different types of rocks. Based on the energy principle, the energy evolution characteristics for different types of rocks cited from previous studies show that although the energy under external force was mostly converted to elastic energy before the yield stage, the dissipated energy was greater than the elastic energy at the beginning of the loading, and the dissipation of energy was mainly caused by the closure of the initial microcracks in the rock.(2)
Due to the complexity of the diagenetic environment, micro-fractures existed in most of the rocks, the rock material was divided into two parts: solid skeleton and voids, which makes the stress-strain curves of rocks highly nonlinear. The basic formexp−ε/ε0m of the statistical damage constitutive model based on Weibull distribution was changed to 1−exp−ε/ε0m, which could better reflect the deformation characteristics of the voids in rocks during the initial compaction stage.(3)
A statistical damage constitutive model considering energy dissipation was proposed based on the analysis of the energy evolution characteristics and the statistical damage model in the initial compaction stage of porous rock. The rationality and feasibility of the model were verified. However, this study considered specifically the initial void compaction stage in the porous rock and focused on the improvement of the simulation method in the whole process of rock deformation and failure based on the energy dissipation principle and statistical damage theory.



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*Source: 7186594-2022-07-13.xml*