# Evolution Procedure of Multiple Rock Cracks under Seepage Pressure

Authors: Taoying Liu; Ping Cao; Hang Lin
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

In practical geotechnical engineering, most of rock masses with multiple cracks exist in water environment. Under such circumstance, these adjacent cracks could interact with each other. Moreover, the seepage pressure, produced by the high water pressure, can change cracks’ status and have an impact on the stress state of fragile rocks. According to the theory of fracture mechanics, this paper discusses the law of crack initiation and the evolution law of stress intensity factor at the tip of a wing crack caused by compression-shear stress and seepage pressure. Subsequently, considering the interaction of the wing cracks and  the additional stress caused by rock bridge damage, this paper proposes the intensity factor evolution equation under the combined action of compression-shear stress and seepage pressure. In addition, this paper analyzes the propagation of cracks under different seepage pressure which reveals that the existence of seepage pressure facilitates the wing crack’s growth. The result indicates that the high seepage pressure converts wing crack growth from stable form to unstable form. Meanwhile, based on the criterion and mechanism for crack initiation and propagation, this paper puts forward the mechanical model for different fracture transfixion failure modes of the crag bridge under the combined action of seepage pressure and compression-shear stress. At the last part, this paper, through investigating the flexibility tensor of the rock mass’s initial damage and its damage evolution in terms of jointed rock mass's damage mechanics, deduces the damage evolution equation for the rock mass with multiple cracks under the combined action of compression-shear stress and seepage pressure. The achievement of this investigation provides a reliable theoretical principle for quantitative research of the fractured rock mass failure under seepage pressure.


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## Body

## 1. Introduction

In recent years, rock mechanics is supposed to consider the most striking feature of rock masses that their blocky structures are caused by the discontinuity surfaces such as joints, cracks, and faults. As frictional force is overcome by shear stress induced by far-field stresses near the crack surface, the crack surface is inclined to slide over each other, which induces stress concentration on tip of crack and leads to the initiation and splitting propagation of the wing crack at last [1–3]. Worse, the existence of seepage pressure could reinforce the trend of the microfracture and fault degradation to fractured rock masses. With the development of rock mechanics engineering, increasing number of situations is involved in seepage pressure which results in numerous engineering accidents around the world [4, 5]. Hence, researches on rock mechanics referring to seepage pressure are gradually becoming a hot spot in the area of geotechnical engineering. A vast number of scholars have made great effort in developing crack propagation theories [6–8] or developing techniques to calculate the stress intensity factors of crack tips [9–12], or studying the relationship between microdamage development and macrodeformation of rock under uniaxial compression [13–15]. Nevertheless, the former studies mainly focus on mechanical mechanism of special single crack rather than the effect of seepage pressure. Besides, there are few reasonable models for evolution laws of stress intensity factor for the branch crack tip in the multicrack rock mass, as well as studies of the damage and fracture evolution mechanism of the fractured rock mass with multiple cracks under seepage pressure.As for the multicrack rock mass, the mechanical state on the crack surface could be changed by the action of seepage pressure. As the branch crack expands, the interaction of cracks leads to the continuous degradation of the macroscopic mechanical properties of the rock mass [8]. The interaction theory of multiple cracks in rock masses is a key factor in the analysis of microdamage mechanism. Therefore, based on previous studies and the rock fracture mechanics criterion, this paper proposes the mechanical model of the multicrack rock mass under the action of seepage pressure and investigates traits of gradual fracture and damage evolution of the multicrack rock under seepage pressure on the basis of exploring the law of the compression-shear crack initiation, branch crack growth, and rock bridge connection. Then, this paper applies the self-consistent theory to setting up the constitutive and damage evolution equation for the multicrack rock mass under seepage pressure.


## 2. Analysis of Compression-Shear Multicrack Damage Fracture Models

### 2.1. Crack Initiation

The underground rock usually exists in a compression stress state, and a lot of testing results and theoretical calculation prove that cracks expand approximately in the direction perpendicular to the maximum principal stress [16, 17] as shown in Figure 1.Figure 1
Sketch of wing cracks seeding and propagation.Firstly, this paper assumes that the rock mass is categorized as crisp flexible, meeting the theory of linear elastic fracture mechanics, and the seepage pressure is equal in every directions along the crack. The fractured rock mass is under the remote field stressσ1 and σ3, where σ1 is the maximum principal stress, σ1≥σ3. The angle between the crack and vertical stress σ1 is ψ, and there is a seepage pressure p in the crack. The normal stress on the crack surface is compressive stress. The shear stress forces the crack to slide which generates a friction μσne+C owing to the part closure of the crack, where μ is the friction coefficient on the crack surface and C is the cohesion on the crack surface. Meanwhile, introducing the coefficient β which presents the ratio of the connected area to the total area, the seepage pressure contributes βp to the surface. Therefore, the effective shear driving force τeff and effective normal stress σne appear in following formulas [18] (here the compressive stress is positive):(1a)σne=σn-βp=(1-Cn)(σ1sin2ψ+σ3cos2ψ)-βp,(1b)τeff=(1-Cv)σ1-σ32sin2ψ-μσne-C,(1c)Cn=πaπa+(E0/(1-v02)Kn),Cv=πaπa+(E0/(1-v02)Ks),where Cn and Cv are the compression transmitting factor and shearing transmitting factor, respectively, which are resulted from the part closure of the crack.According to the maximum circumferential stress criterion, the initial crack extends along the direction of the maximum normal stress. Thus, the cracking angleθ3 can be obtained (θ3=70.5°) [19]. Then, the stress intensity factor at the wing crack initiation can be concluded as [20]
(2)KI=23τeffπa.
Making KI=KIC in (2), KIC is the fracture toughness, the biggest crack stress intensity factor. It is easy to get the ultimate strength  σ11 of the fracture rock under seepage pressure(3a)σ11=3KIC/πa+2C-2μβp+Bσ3A,(3b)A=-μ(1-Cn)(1-cos2ψ)+(1-Cv)sin2ψ,B=μ(1-Cn)(1+cos2ψ)+(1-Cv)sin2ψ.From  (3a), it is clear to see that the ultimate strength σ11 decreases linearly with the increase of the seepage pressure  p.As for multicrack rock masses, when at the initial stage of crack expansion or the crack space is relatively large, it can be treated as the wing crack propagation of a single crack. Under such circumstance, the stress intensity factorKI at the crack tip could be defined through the revised wing crack calculation model [21]. Considering the additional seepage pressure in the crack, the stress intensity factor KI consists of the stress intensity factor produced by the effective shear driving force Te and the remote field stress σ1 and σ3, and it can be shown as in (4)
(4)KI=3τnealtyπsin-1(1lty)sinθcosθ2-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl+pπl,
where lty is the impact factor which is the function of the wing crack length l, wing crack azimuth θ, and main crack length a, as shown in (5)
(5)lty=[1+9l4acos2(θ2)](1-e-l/a)+0.667sec2(θ2)e-l/a.The wing crack extends approximately in the direction perpendicular to the maximum principal stress untilKI=KIC, and then the wing crack propagation length l under the combined action of compression-shear stress and seepage pressure can be calculated as shown in (6)
(6)l=6τne(alty/π)sin-1(1/lty)sinθcos(θ/2)-2KIC[(σ1+σ3)+(σ1-σ3)cos2(θ+β)-2p]π.


### 2.2. The Multicrack Interaction Models

As the crack expands or when the crack space is relatively small, the interaction between cracks leads to a damaged connection and unstable break of the rock bridge [22]. The rock bridge interaction mechanical model of the multicrack rock mass with the wing crack expansion is shown in Figure 2.Figure 2
Schematic drawing of multiple interacting cracks.F is the horizontal stress in the crack. σ3′ is the horizontal stress in the rock bridge, and t is the length of the rock bridge.Assuming that the number of the compression-shear cracks per unit area isNA, the length between main crack centers and the rock bridge lengths between wing cracks,  respectively, are presented as follows:
(7)S=1NA,T=S-2(l+acosψ).In Figure2, F=Tesinψ is balanced by the tensile stress σ3′ in the rock bridge together with seepage pressure pin the wing crack:
(8)σ3′=Tesinψ+2plS-2(l+acosψ),
where  Te=2aτne.Acting on the wing crack,σ3′ produces an additional intensity factor at the crack tip [23]:
(9)KI′=-σ3′πl=a(σ1-σ3)sin2ψsinψ+2plNA-1/2-2(l+acosψ)πl.Considering the damage to the rock bridge caused by the wing crack interaction, the stress intensity factor at the wing crack tip is composed of (4) and (9) as follows:
(10)KI=KI+KI′=3τnealtyπsin-1(1lty)sinθcosθ2-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl+pπl+a(σ1-σ3)sin2ψsinψ+2plNA-1/2-2(l+acosψ)πl.From (10), it is clear to see that the interaction of multiple wing cracks in rock bridge damage makes the stress intensity factor at the crack tip larger than that at a single wing crack.The process of rock fracture is the one of the increasing damage and inherent fracture in rock and is the coupling damage result of the microdamage and macrodamage. There are many ways to define the damage variable, such as the cracks quantity crack length, crack area, and crack density. Here, the cracks quantity is adopted to defineD0 for the initial damage of the rock mass and D for the damage when the wing crack expands to l:
(11)D0=π(acosψ)2NA,D=π(l+acosψ)2NA.
Substituting (11) into (8), the fellow equation can be easily obtained
(12)σ3′=-(τnetanψ+2pl)(D0/π)1/2a(1-2(D/π)1/2).Combining (10), (11), and (12), the relation curves between the damage variables D and dimensionless stress intensity factor (KI/σ1πa) at the wing crack tip with different crack density are shown in Figure 3.Figure 3
The relationship between damage variables and intensity factor at the crack tip of the branch stress.The graph reveals that with the increase of the damage variables, the dimensionless stress intensity factorKI/σ1πa at the crack tip under seepage pressure decreases at the very beginning but rises gradually when the equivalent crack length is 0.5 (l/a). The graph also illustrates that the sparser of the cracks are, the higher of the stress intensity at the wing crack will be when the rock bridge is connected. During the wing crack expansion process, the damage variable D varies from D0 to 1. When D=1 and the wing crack stress intensity factor KI⩾KIC at the crack tip, the wing crack connects, the rock bridge cracks, and the rock mass loses bearing capacity.Figure4 shows the relationship between the stress intensity factor at the crack tip and the equivalent crack propagation length under different seepage pressure. It can be concluded that when the seepage pressure is relatively low (p=0MPa), the wing crack expands stably (∂KI/∂L<0). While when the seepage pressure is relatively high (p=15MPa), the wing crack tends to expand unstably. Besides, there is the expansion stage that ∂KI/∂L>0 under such condition and the growth rate of wing crack stress intensity factor becomes greater with the increase of seepage pressure p. It indicates that under high seepage pressure, the compressive-shear rock crack expands at a high speed as long as it cracks, namely, and the high seepage pressure contributes to unstable expansion.Figure 4
The relationship between intensity factor at the crack tip wing stress and crack propagation length.


## 2.1. Crack Initiation

The underground rock usually exists in a compression stress state, and a lot of testing results and theoretical calculation prove that cracks expand approximately in the direction perpendicular to the maximum principal stress [16, 17] as shown in Figure 1.Figure 1
Sketch of wing cracks seeding and propagation.Firstly, this paper assumes that the rock mass is categorized as crisp flexible, meeting the theory of linear elastic fracture mechanics, and the seepage pressure is equal in every directions along the crack. The fractured rock mass is under the remote field stressσ1 and σ3, where σ1 is the maximum principal stress, σ1≥σ3. The angle between the crack and vertical stress σ1 is ψ, and there is a seepage pressure p in the crack. The normal stress on the crack surface is compressive stress. The shear stress forces the crack to slide which generates a friction μσne+C owing to the part closure of the crack, where μ is the friction coefficient on the crack surface and C is the cohesion on the crack surface. Meanwhile, introducing the coefficient β which presents the ratio of the connected area to the total area, the seepage pressure contributes βp to the surface. Therefore, the effective shear driving force τeff and effective normal stress σne appear in following formulas [18] (here the compressive stress is positive):(1a)σne=σn-βp=(1-Cn)(σ1sin2ψ+σ3cos2ψ)-βp,(1b)τeff=(1-Cv)σ1-σ32sin2ψ-μσne-C,(1c)Cn=πaπa+(E0/(1-v02)Kn),Cv=πaπa+(E0/(1-v02)Ks),where Cn and Cv are the compression transmitting factor and shearing transmitting factor, respectively, which are resulted from the part closure of the crack.According to the maximum circumferential stress criterion, the initial crack extends along the direction of the maximum normal stress. Thus, the cracking angleθ3 can be obtained (θ3=70.5°) [19]. Then, the stress intensity factor at the wing crack initiation can be concluded as [20]
(2)KI=23τeffπa.
Making KI=KIC in (2), KIC is the fracture toughness, the biggest crack stress intensity factor. It is easy to get the ultimate strength  σ11 of the fracture rock under seepage pressure(3a)σ11=3KIC/πa+2C-2μβp+Bσ3A,(3b)A=-μ(1-Cn)(1-cos2ψ)+(1-Cv)sin2ψ,B=μ(1-Cn)(1+cos2ψ)+(1-Cv)sin2ψ.From  (3a), it is clear to see that the ultimate strength σ11 decreases linearly with the increase of the seepage pressure  p.As for multicrack rock masses, when at the initial stage of crack expansion or the crack space is relatively large, it can be treated as the wing crack propagation of a single crack. Under such circumstance, the stress intensity factorKI at the crack tip could be defined through the revised wing crack calculation model [21]. Considering the additional seepage pressure in the crack, the stress intensity factor KI consists of the stress intensity factor produced by the effective shear driving force Te and the remote field stress σ1 and σ3, and it can be shown as in (4)
(4)KI=3τnealtyπsin-1(1lty)sinθcosθ2-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl+pπl,
where lty is the impact factor which is the function of the wing crack length l, wing crack azimuth θ, and main crack length a, as shown in (5)
(5)lty=[1+9l4acos2(θ2)](1-e-l/a)+0.667sec2(θ2)e-l/a.The wing crack extends approximately in the direction perpendicular to the maximum principal stress untilKI=KIC, and then the wing crack propagation length l under the combined action of compression-shear stress and seepage pressure can be calculated as shown in (6)
(6)l=6τne(alty/π)sin-1(1/lty)sinθcos(θ/2)-2KIC[(σ1+σ3)+(σ1-σ3)cos2(θ+β)-2p]π.


## 2.2. The Multicrack Interaction Models

As the crack expands or when the crack space is relatively small, the interaction between cracks leads to a damaged connection and unstable break of the rock bridge [22]. The rock bridge interaction mechanical model of the multicrack rock mass with the wing crack expansion is shown in Figure 2.Figure 2
Schematic drawing of multiple interacting cracks.F is the horizontal stress in the crack. σ3′ is the horizontal stress in the rock bridge, and t is the length of the rock bridge.Assuming that the number of the compression-shear cracks per unit area isNA, the length between main crack centers and the rock bridge lengths between wing cracks,  respectively, are presented as follows:
(7)S=1NA,T=S-2(l+acosψ).In Figure2, F=Tesinψ is balanced by the tensile stress σ3′ in the rock bridge together with seepage pressure pin the wing crack:
(8)σ3′=Tesinψ+2plS-2(l+acosψ),
where  Te=2aτne.Acting on the wing crack,σ3′ produces an additional intensity factor at the crack tip [23]:
(9)KI′=-σ3′πl=a(σ1-σ3)sin2ψsinψ+2plNA-1/2-2(l+acosψ)πl.Considering the damage to the rock bridge caused by the wing crack interaction, the stress intensity factor at the wing crack tip is composed of (4) and (9) as follows:
(10)KI=KI+KI′=3τnealtyπsin-1(1lty)sinθcosθ2-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl+pπl+a(σ1-σ3)sin2ψsinψ+2plNA-1/2-2(l+acosψ)πl.From (10), it is clear to see that the interaction of multiple wing cracks in rock bridge damage makes the stress intensity factor at the crack tip larger than that at a single wing crack.The process of rock fracture is the one of the increasing damage and inherent fracture in rock and is the coupling damage result of the microdamage and macrodamage. There are many ways to define the damage variable, such as the cracks quantity crack length, crack area, and crack density. Here, the cracks quantity is adopted to defineD0 for the initial damage of the rock mass and D for the damage when the wing crack expands to l:
(11)D0=π(acosψ)2NA,D=π(l+acosψ)2NA.
Substituting (11) into (8), the fellow equation can be easily obtained
(12)σ3′=-(τnetanψ+2pl)(D0/π)1/2a(1-2(D/π)1/2).Combining (10), (11), and (12), the relation curves between the damage variables D and dimensionless stress intensity factor (KI/σ1πa) at the wing crack tip with different crack density are shown in Figure 3.Figure 3
The relationship between damage variables and intensity factor at the crack tip of the branch stress.The graph reveals that with the increase of the damage variables, the dimensionless stress intensity factorKI/σ1πa at the crack tip under seepage pressure decreases at the very beginning but rises gradually when the equivalent crack length is 0.5 (l/a). The graph also illustrates that the sparser of the cracks are, the higher of the stress intensity at the wing crack will be when the rock bridge is connected. During the wing crack expansion process, the damage variable D varies from D0 to 1. When D=1 and the wing crack stress intensity factor KI⩾KIC at the crack tip, the wing crack connects, the rock bridge cracks, and the rock mass loses bearing capacity.Figure4 shows the relationship between the stress intensity factor at the crack tip and the equivalent crack propagation length under different seepage pressure. It can be concluded that when the seepage pressure is relatively low (p=0MPa), the wing crack expands stably (∂KI/∂L<0). While when the seepage pressure is relatively high (p=15MPa), the wing crack tends to expand unstably. Besides, there is the expansion stage that ∂KI/∂L>0 under such condition and the growth rate of wing crack stress intensity factor becomes greater with the increase of seepage pressure p. It indicates that under high seepage pressure, the compressive-shear rock crack expands at a high speed as long as it cracks, namely, and the high seepage pressure contributes to unstable expansion.Figure 4
The relationship between intensity factor at the crack tip wing stress and crack propagation length.


## 3. Wing Crack Connection Model and Damage Criteria

As the branch crack expands, the interaction of the crack results in the continuous degradation of the macroscopic mechanical properties of the rock mass. The wing crack initiation from microcracks to the completely damage of the rock mass is a process of evolution and accumulation of the rock mass damage, as well as a process of the fracture of the connection between cracks [24]. It has been proved by many experimental researches that the gradual damage process of the cracks generally has two forms [25, 26]: (1) the rock bridge axial transfixion failure and (2) the tension-shear compound failure. According to the branch crack propagation evolution mechanism, this paper studies the gradual damage features of those damage models.


### 3.1. The Axial Transfixion Failure

Whenever the wing crack extends stably or unstably, the tension wing crack connects the main crack in another row, as shown in Figure5. When the wing crack reaches the critical length l1c=D/sinψ, L1C=l1c/a, the rock bridge starts to damage in the form of axial transfixion failure. Taking the stress intensity at the crack tip as the criterion when the wing crack reaches the critical length l1c=D/sinψ, this paper establishes the failure criteria:
(13)KI=3τnealtyπsin-1(1lty)sinθcosθ2-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl1c+pπl1c+a(σ1-σ3)sin2ψsinψ+2pl1cNA-1/2-2(l1c+acosψ)πl1c.
When KI(L1C)≥KIC, axial transfixion failure occurs in the rock bridge.Figure 5
Failure characteristic of crag bridge shearing.


### 3.2. The Tension-Shear Compound Failure

With the expansion of wing cracks under seepage pressure, the cut-resistant capacity of the rock bridge is increasingly weakened. When the wing crack expands to a certain degree; the rock bridge at the crack tip between adjacent wing cracks is cut off by the shear stress, consequently, the crack is connected in the shear direction [27]. The mechanical analyzing diagram for the element of the rock bridge composite failure is shown in Figure 6, where AB is 1/2 the length of the bottom crack, EF is 1/2 the length of the upper crack, CD is the rock bridge, BC and ED are the wing cracks produced by effective shear driving force of the main crack AB and EF, θ is the angle between rock bridge and the maximum principal stress, and σCD and τCD are the normal stress and shear stress acting on the rock bridge, respectively.Figure 6
Failure characteristic of crag bridge shearing.According to the element shown in Figure6 and the principle of mechanics balance, it could be deduced as follows:
(14)∑Fx=0,∑Fy=0,2r3σ3-2a(τnesinψ+σnecosψ)-2r2(τCDsinθ+σnCDcosθ)-2pl=0,2r1σ1+2a(τnecosψ-σnesinψ)+2r2(τCDcosθ-σnCDsinθ)=0,
where
(15)r1=asinψ+12d2+h2sinθ,r2=d2+h2cosθ-2l2cosθ,r3=acosψ+12d2+h2cosθ,θ=ψ-arctandh.
Then, the following equation can be obtained
(16)τCD=2A1tanθ-2B1(d2+h2cosθ-2l)(1+tan2θ),σCD=2A1+2B1tanθ(d2+h2cosθ-2l)(1+tan2θ),(17)A1=r3σ3-a(τnesinψ+σnecosψ)-pl,B1=r1σ1+a(τnecosψ-σnesinψ).Assuming that the rock bridge shear damage follows the Mohr-Coulomb strength criterion, conditions for the damage are(18)τCD-c-σCDtanφ⩾0.Substituting (16) into (18), the fellow equation can be easily obtained as follows:
(19)(12d2+h2cosθ-l)ctan2θ+(B1tanφ-A1)tanθ+A1tanφ+B+12cd2+h2cosθ-cl⩽0.When angleθ meets the rock bridge shear failure condition and the wing crack propagation length reaches critical value l2c, the angle θ between the rock bridge and σ1 also reaches their critical value, and then
(20)θc=arctanA1-B1tanφ+Δ(d2+h2cosθ-2l2c)c.As the geometric relations show in Figure6, it is easy to see that
(21)tanθc=d2+h2sinθ(d2+h2cosθ-2l2c).Combined (20) and (21), the critical length of the wing crack can be defined as:
(22)l2c=A2-A22-4(c+ptanφ)B22(c+ptanφ),
where
(23)A2=r3σ3tanφ-a(τnesinψ+σnecosψ)tanφ+B1+12d2+h2[(2c-ptanφ)cosθ+psinθ],B2=14(d2+h2)c+12d2+h2×[(r3σ3-aτnesinψ-aσnecosψ)(cosθtanφ-sinθ)+B1(sinθtanφ+cosθ)].Meanwhile, this paper gets the stress intensity factor at the crack tip when the wing reaches the critical length as follows:(24)KI(l2c)=KI+KI′=3τeffaltyπsin-1(1lty)sin(θ)cos(θ)-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl2c+pπl2c+a(σ1-σ3)sin2ψsinψ+2pl2c2(l2c+acosψ)-NA-1/2πl2c.


## 3.1. The Axial Transfixion Failure

Whenever the wing crack extends stably or unstably, the tension wing crack connects the main crack in another row, as shown in Figure5. When the wing crack reaches the critical length l1c=D/sinψ, L1C=l1c/a, the rock bridge starts to damage in the form of axial transfixion failure. Taking the stress intensity at the crack tip as the criterion when the wing crack reaches the critical length l1c=D/sinψ, this paper establishes the failure criteria:
(13)KI=3τnealtyπsin-1(1lty)sinθcosθ2-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl1c+pπl1c+a(σ1-σ3)sin2ψsinψ+2pl1cNA-1/2-2(l1c+acosψ)πl1c.
When KI(L1C)≥KIC, axial transfixion failure occurs in the rock bridge.Figure 5
Failure characteristic of crag bridge shearing.


## 3.2. The Tension-Shear Compound Failure

With the expansion of wing cracks under seepage pressure, the cut-resistant capacity of the rock bridge is increasingly weakened. When the wing crack expands to a certain degree; the rock bridge at the crack tip between adjacent wing cracks is cut off by the shear stress, consequently, the crack is connected in the shear direction [27]. The mechanical analyzing diagram for the element of the rock bridge composite failure is shown in Figure 6, where AB is 1/2 the length of the bottom crack, EF is 1/2 the length of the upper crack, CD is the rock bridge, BC and ED are the wing cracks produced by effective shear driving force of the main crack AB and EF, θ is the angle between rock bridge and the maximum principal stress, and σCD and τCD are the normal stress and shear stress acting on the rock bridge, respectively.Figure 6
Failure characteristic of crag bridge shearing.According to the element shown in Figure6 and the principle of mechanics balance, it could be deduced as follows:
(14)∑Fx=0,∑Fy=0,2r3σ3-2a(τnesinψ+σnecosψ)-2r2(τCDsinθ+σnCDcosθ)-2pl=0,2r1σ1+2a(τnecosψ-σnesinψ)+2r2(τCDcosθ-σnCDsinθ)=0,
where
(15)r1=asinψ+12d2+h2sinθ,r2=d2+h2cosθ-2l2cosθ,r3=acosψ+12d2+h2cosθ,θ=ψ-arctandh.
Then, the following equation can be obtained
(16)τCD=2A1tanθ-2B1(d2+h2cosθ-2l)(1+tan2θ),σCD=2A1+2B1tanθ(d2+h2cosθ-2l)(1+tan2θ),(17)A1=r3σ3-a(τnesinψ+σnecosψ)-pl,B1=r1σ1+a(τnecosψ-σnesinψ).Assuming that the rock bridge shear damage follows the Mohr-Coulomb strength criterion, conditions for the damage are(18)τCD-c-σCDtanφ⩾0.Substituting (16) into (18), the fellow equation can be easily obtained as follows:
(19)(12d2+h2cosθ-l)ctan2θ+(B1tanφ-A1)tanθ+A1tanφ+B+12cd2+h2cosθ-cl⩽0.When angleθ meets the rock bridge shear failure condition and the wing crack propagation length reaches critical value l2c, the angle θ between the rock bridge and σ1 also reaches their critical value, and then
(20)θc=arctanA1-B1tanφ+Δ(d2+h2cosθ-2l2c)c.As the geometric relations show in Figure6, it is easy to see that
(21)tanθc=d2+h2sinθ(d2+h2cosθ-2l2c).Combined (20) and (21), the critical length of the wing crack can be defined as:
(22)l2c=A2-A22-4(c+ptanφ)B22(c+ptanφ),
where
(23)A2=r3σ3tanφ-a(τnesinψ+σnecosψ)tanφ+B1+12d2+h2[(2c-ptanφ)cosθ+psinθ],B2=14(d2+h2)c+12d2+h2×[(r3σ3-aτnesinψ-aσnecosψ)(cosθtanφ-sinθ)+B1(sinθtanφ+cosθ)].Meanwhile, this paper gets the stress intensity factor at the crack tip when the wing reaches the critical length as follows:(24)KI(l2c)=KI+KI′=3τeffaltyπsin-1(1lty)sin(θ)cos(θ)-12[(σ1+σ3)+(σ1-σ3)cos2(θ+β)]πl2c+pπl2c+a(σ1-σ3)sin2ψsinψ+2pl2c2(l2c+acosψ)-NA-1/2πl2c.


## 4. The Damage Evolution Mechanism of Multicrack Rock Masses

The macroscopic mechanical effect of fractured rock would be reflected by the change of its flexibility, and then the damage tensorD can be determined by the elastic flexibility tensor C0 and the equivalent damage flexibility tensor Cd [28]
(25)D=I-(Cd)-1C0,
where I is the fourth-order unit tensor and D is a tensor of the fourth order for the 3D anisotropy mode of fractured rock.In terms of the complete rock, the elastic flexibility tensorC0 can be expressed as [29]
(26)Cijkl0=1+v0E0δikδjl-v0E0δijδkl,
where E0 and v0 represented the elastic modulus and Poisson’s ratio of the rock, respectively.The equivalent damage flexibility tensorCd can be drawn from Betti energy reciprocal theorem. Hence, based on self-consistent method and strain energy equivalence in solid mechanics, the equivalent damage flexibility tensor Cijkld is acquired through calculating the equivalent elastic strain energy.


### 4.1. The Equivalent Damage Flexibility Tensor Based on Equivalent Elastic Strain Energy

The flexibility matrix of the crack in compression shear state is [30]
(27)[C0]=[C110C1100C210C220000C330].Based on Betti’s theorem, that is, Maxwell-Betti reciprocal work theorem, the work done by one set of forces through the displacements produced by another set of forces is equal to the work done by the latter set of forces through the displacements produced by the former set of forces.(
1
) When σx=τxy=0, σy≠0,
(28)W12=(2bσy-2ap)(C22σy′2d)=4(bσy-ap)σ′dC22,W21=(2bσy′)(σy-p)C2202d+ΔWc,
where  ΔWc=(2aσy′)(σy-p)Cn/Kn.Then, we get(29)W21=4σy′dbC220(σy-p)+2aσy′(σy-p)CnKn.According toW21=W12, whilst, the crack size can be neglected compared to the size of the rock mass,
(30)C22=(1-R)(C220+aCn2Knbd),
where R=p/σy.(
2
) When σx=σy=0, τxy≠0,
(31)W12=(τ2d)(C33τ′2b)=4C33τ′bd,W21=(τ′2d)(C33τ2b)+ΔWc=4τC33τ′bd+ΔWc,ΔWc=τ′2aτCsKs.
Then, we obtain the following equation:
(32)W21=4τ′τdbC330+2aτ′τCsKs.According toW21=W12, we get
(33)C33=C330+aCs2Ksbd.Thus, the parameters of the equivalent damage flexibility matrix[Cd] can be expressed as follows:
(34)[Cd]=[0000aCn(1-R)2Knbd-RC220000aCsKs2bd].


### 4.2. The Equivalent Damage Flexibility Tensor Based on Crack’s Elastic Strain Energy

The strain energy of single crack is assumed as [29]
(35)Ud(i)=2∫0aGdΓ=21-ν02E0∫0a(KI2+KII2)da,
where  Γ is the line length of cracks.Integrating over the line length, the stress intensity factors of typeI or type II for the crack under seepage pressure are
(36)KI(i)=σf(i)πa(i),KII(i)=τq(i)πa(i),
where σf(i) is the normal stress of the crack and τq(i) is the tangential shear stress. Combining the former equation, then we get
(37)Ud(i)=1-ν02E0πa(i)2[σf(i)2+τq(i)2].By assuming there aren groups of cracks in the fractured rock mass, the strain energy of the fractured rock mass will be
(38)Ud=∑i=1nUd(i)=1-ν02E0π∑i=1na(i)2ρ(i)[σf(i)2+τq(i)2].Set the equationH=((1-ν02)/E0)π, A(i)=a(i)2ρ(i), in which ρ(i) is the density of cracks, then
(39)Ud=H∑i=1nA(i)[σf(i)2+τq(i)2].
Rewrite (39) into the tensor expression
(40)Ud=12σijCijkldσkl=H∑i=1nA(i)[σf(i)σf(i)+τq(i)τq(i)].According to the hydrostatic pressure principle and Cauchy criterion, the remote field stress is denoted asσij, and the stress matrix on the surface can be defined as [31]
(41)[TxvTyv]=[σxx-pσxyσyxσxx-p][νxνy].Definingnini=1, authors rewrite the equation into the tensor expression
(42)Ti(i)=(σij(i)-δijp)nj(i)=σij(i)nj(i)-pni(i).Considering the compression transferring coefficientCn, the equivalent normal stress on the surface of the crack can be expressed as
(43)σf(i)=Ti(i)nj(i)=(1-Cn(i))σij(i)nj(i)ni(i)-β(i)p,(σf(i))i=(1-Cn)σjk(i)nj(i)nk(i)ni(i)-β(i)pni(i).Considering the shear transferring coefficientCs again, the shear stress on the crack surface can be expressed as
(44)(τf(i))i=Ti(i)-(σf(i))i=(1-Cn(i))σjk(i)nj(i)(δkl-nk(i)ni(i)).Then, we get(45)(σf(i))i(σf(i))i=((1-Cn(i))σjk(i)nj(i)nk(i)ni(i)-β(i)pni(i))×((1-Cn(i))σst(i)ns(i)nt(i)ni(i)-β(i)pni(i)).Considering the symmetry ofσij, the following equations can be deduced:
(46)(σf(i))i(σf(i))i=(1-Cn(i))2σjk(i)nj(i)nk(i)ns(i)nt(i)σst(i)-(1-Cn(i))σjk(i)nj(i)nk(i)p-(1-Cn(i))σst(i)ns(i)nt(i)p+β(i)2p2,(τf(i))i(τf(i))i=(1-Cs(i))2σjk(i)nj(i)(δkl-nk(i)ni(i))σst(i)ns(i)×(δti-ni(i)ni(i))=(1-Cs(i))2σjk(i)×(δklnj(i)ns(i)-nj(i)nk(i)ns(i)nt(i))σst(i).Supposing the proportion coefficientR=p/σ¯, σ¯ is the average stress, σ¯=(1/2)σii,σii is the first invariant stress. Then, the seepage pressure can be transformed to
(47)p=pσssσss=σstδstp2σ¯=12σstδstR,p2=p2σkkσkkσssσss=σkjδkjσstδtsp2σ¯p2σ¯=14σkjδkjσstδtsR2.Then, we get(48)(σf(i))i(σf(i))i=(1-Cn(i))2σjk(i)nj(i)nk(i)ns(i)nt(i)σst(i)-12(1-Cn(i))Rσjk(i)×(nj(i)nk(i)σst(i)+δjkns(i)nt(i))+14(β(i)R)2σjk(i)σst(i)σts,Ud=12σjkCjkstdσsi=H∑i=1nA(i)σjk(i)[(1-Cn(i))2nj(i)nk(i)ns(i)nt(i)H∑i=1nA(i)σjk(i)-12(1-Cn(i))β(i)Rσjk(i)H∑i=1nA(i)σjk(i)×(nj(i)nk(i)σst(i)+δjkns(i)nt(i))H∑i=1nA(i)σjk(i)+14(Rβ(i))2σjk(i)σst(i)σts+(1-Cs(i))2H∑i=1nA(i)σjk(i)×(δktnj(i)ns(i)-nj(i)nk(i)ns(i)nt(i))(1-Cn(i))2]σst(i).Finally, it is clear to conclude the additional flexibility tensor of fractured rock mass under seepage pressure as follows:(49)Cijkld=(1-Cn(i))2nj(i)nk(i)ns(i)nt(i)-12(1-Cn(i))β(i)Rσjk(i)(nj(i)nk(i)σst(i)+δjkns(i)nt(i))+14(Rβ(i))2σjk(i)σst(i)σts+(1-Cs(i))2(δktnj(i)ns(i)-nj(i)nk(i)ns(i)nt(i)).


### 4.3. The Fracture Damage Evolution Equation

The additional elastic strain energy densityukzi produced by the wing crack is [32]
(50)ukz(i)=4E0ρv(i)a(i)2(52σeff(i)L(i)+23τeff(i))2,
where L(i) is the equivalent length for the crack.Assuming there aren groups of cracks in the fractured rock mass, the additional elastic strain energy density of the fractured rock Ukz is
(51)Ukz=∑i=1nukz(i)=12σijCijklkzσkl.Crack extension reduces the stiffness of rock masses and increases its flexibility. Through the derivative ofUkz with respect to the stress tensor, we can get the additional flexibility tensor of fractured rock masses in the damage evolution process
(52)∂Ukz∂σij=Cijklkzσkl.At the same time, by calculating the partial derivative ofL(i), σeff(i), and τeff(i) with respect to σij, respectively, and considering the symmetry of Cijklkz, we can get the following results:
(53)∂Ukz∂σij=1E0∑i=1n{M3(i)3a(i)2ρv(i)[M3(i)3nj(i)nk(i)nk(i)nt(i)M1(i)nj(i)nk(i)nk(i)nt(i)-M2(i)×(nj(i)nk(i)δkj+δkinj(i)nt(i)+nj(i)nk(i)δij+δitnj(i)nk(i))-M3(i)3βRδijδkl]}σkl,
where
(54)M1(i)=(20σeff(i)L(i)+163τeff(i))×[1σn(i)(52L(i)+23f)-1τn(i)(52cotθ(i)L(i)+23f)+52σeff(i)N1(i)1σn(i)],M2(i)=(5σeff(i)L(i)+43τeff(i))×[1τn(i)(52cotθ(i)L(i)+23f)+52σeff(i)N2(i)],M3(i)=(20σeff(i)L(i)+163τeff(i))×[23f+52(σeff(i)N3(i)+L(i))],N1(i)=L(i)[-4a(i)cosθ(i)σeff(i)2(fτn(i)-σn(i))-(KIC(i)2-4a(i)cosθ(i)σeff(i)-2πa(i)A(i)KIC(i)σeff(i)πa(i)A(i)KIC(i)σeff(i))×(τn(i)-σn(i)cotθ(i))]×(2πa(i)A(i)σeff(i)σn(i)τn(i))-1,N2(i)=L(i)[-4a(i)cosθ(i)σeff(i)2-(KIC(i)2-4a(i)cosθ(i)σeff(i)τeff(i)-2πa(i)A(i)KIC(i)σeff(i))cotθ(i)]×(2πa(i)A(i)σeff(i)σn(i)τn(i))-1,N3(i)=[-(-2(KIC(i)πa(i))2σeff(i)3-2πcosθ(i)(fσeff(i)-τeff(i)σeff(i)2))]×L(i)A(i)-2KIC(i)L(i)σeff(i)22πa(i),A(i)=KIC(i)2σeff(i)22πa(i)-2πcosθ(i)τeff(i)σeff(i)2.Then, we can get the flexibility tensorCijklkz due to the damage evolution of fractured rock masses as follows:
(55)Cijklkz=1E0∑i=1n{M3(i)3a(i)2ρv(i)[M3(i)3M1(i)nj(i)nk(i)nk(i)nt(i)-M2(i)×(nj(i)nk(i)δkj+δkinj(i)nt(i)+nj(i)nk(i)δij+δitnj(i)nk(i))-M3(i)3βRδijδkl]}.


### 4.4. The Constitutive Equation of the Multicrack Rock Mass

To sum up, combining the initial damage and damage evolution tensor, we can get the flexibility tensor of the fractured rock mass under seepage pressureas follows:(56)Cijkl=Cijkl0+Cijkld+Cijklkz,
where Cijkl0 is the elastic flexibility tensor of the complete rock, Cijkld is the initial equivalent damage flexibility tensor of the fractured rock mass under seepage pressure, and Cijklkz is the additional damage flexibility tensor with crack propagation of the fractured rock mass under seepage pressure.According to generalized Hooke’s law [33]:
(57)εij=Cijklσkl,σij=Eijklεkl=(Cijkl)-1εkl,
and this paper sets up the constitutive relation of the multicrack rock mass under seepage pressure.


## 4.1. The Equivalent Damage Flexibility Tensor Based on Equivalent Elastic Strain Energy

The flexibility matrix of the crack in compression shear state is [30]
(27)[C0]=[C110C1100C210C220000C330].Based on Betti’s theorem, that is, Maxwell-Betti reciprocal work theorem, the work done by one set of forces through the displacements produced by another set of forces is equal to the work done by the latter set of forces through the displacements produced by the former set of forces.(
1
) When σx=τxy=0, σy≠0,
(28)W12=(2bσy-2ap)(C22σy′2d)=4(bσy-ap)σ′dC22,W21=(2bσy′)(σy-p)C2202d+ΔWc,
where  ΔWc=(2aσy′)(σy-p)Cn/Kn.Then, we get(29)W21=4σy′dbC220(σy-p)+2aσy′(σy-p)CnKn.According toW21=W12, whilst, the crack size can be neglected compared to the size of the rock mass,
(30)C22=(1-R)(C220+aCn2Knbd),
where R=p/σy.(
2
) When σx=σy=0, τxy≠0,
(31)W12=(τ2d)(C33τ′2b)=4C33τ′bd,W21=(τ′2d)(C33τ2b)+ΔWc=4τC33τ′bd+ΔWc,ΔWc=τ′2aτCsKs.
Then, we obtain the following equation:
(32)W21=4τ′τdbC330+2aτ′τCsKs.According toW21=W12, we get
(33)C33=C330+aCs2Ksbd.Thus, the parameters of the equivalent damage flexibility matrix[Cd] can be expressed as follows:
(34)[Cd]=[0000aCn(1-R)2Knbd-RC220000aCsKs2bd].


## 4.2. The Equivalent Damage Flexibility Tensor Based on Crack’s Elastic Strain Energy

The strain energy of single crack is assumed as [29]
(35)Ud(i)=2∫0aGdΓ=21-ν02E0∫0a(KI2+KII2)da,
where  Γ is the line length of cracks.Integrating over the line length, the stress intensity factors of typeI or type II for the crack under seepage pressure are
(36)KI(i)=σf(i)πa(i),KII(i)=τq(i)πa(i),
where σf(i) is the normal stress of the crack and τq(i) is the tangential shear stress. Combining the former equation, then we get
(37)Ud(i)=1-ν02E0πa(i)2[σf(i)2+τq(i)2].By assuming there aren groups of cracks in the fractured rock mass, the strain energy of the fractured rock mass will be
(38)Ud=∑i=1nUd(i)=1-ν02E0π∑i=1na(i)2ρ(i)[σf(i)2+τq(i)2].Set the equationH=((1-ν02)/E0)π, A(i)=a(i)2ρ(i), in which ρ(i) is the density of cracks, then
(39)Ud=H∑i=1nA(i)[σf(i)2+τq(i)2].
Rewrite (39) into the tensor expression
(40)Ud=12σijCijkldσkl=H∑i=1nA(i)[σf(i)σf(i)+τq(i)τq(i)].According to the hydrostatic pressure principle and Cauchy criterion, the remote field stress is denoted asσij, and the stress matrix on the surface can be defined as [31]
(41)[TxvTyv]=[σxx-pσxyσyxσxx-p][νxνy].Definingnini=1, authors rewrite the equation into the tensor expression
(42)Ti(i)=(σij(i)-δijp)nj(i)=σij(i)nj(i)-pni(i).Considering the compression transferring coefficientCn, the equivalent normal stress on the surface of the crack can be expressed as
(43)σf(i)=Ti(i)nj(i)=(1-Cn(i))σij(i)nj(i)ni(i)-β(i)p,(σf(i))i=(1-Cn)σjk(i)nj(i)nk(i)ni(i)-β(i)pni(i).Considering the shear transferring coefficientCs again, the shear stress on the crack surface can be expressed as
(44)(τf(i))i=Ti(i)-(σf(i))i=(1-Cn(i))σjk(i)nj(i)(δkl-nk(i)ni(i)).Then, we get(45)(σf(i))i(σf(i))i=((1-Cn(i))σjk(i)nj(i)nk(i)ni(i)-β(i)pni(i))×((1-Cn(i))σst(i)ns(i)nt(i)ni(i)-β(i)pni(i)).Considering the symmetry ofσij, the following equations can be deduced:
(46)(σf(i))i(σf(i))i=(1-Cn(i))2σjk(i)nj(i)nk(i)ns(i)nt(i)σst(i)-(1-Cn(i))σjk(i)nj(i)nk(i)p-(1-Cn(i))σst(i)ns(i)nt(i)p+β(i)2p2,(τf(i))i(τf(i))i=(1-Cs(i))2σjk(i)nj(i)(δkl-nk(i)ni(i))σst(i)ns(i)×(δti-ni(i)ni(i))=(1-Cs(i))2σjk(i)×(δklnj(i)ns(i)-nj(i)nk(i)ns(i)nt(i))σst(i).Supposing the proportion coefficientR=p/σ¯, σ¯ is the average stress, σ¯=(1/2)σii,σii is the first invariant stress. Then, the seepage pressure can be transformed to
(47)p=pσssσss=σstδstp2σ¯=12σstδstR,p2=p2σkkσkkσssσss=σkjδkjσstδtsp2σ¯p2σ¯=14σkjδkjσstδtsR2.Then, we get(48)(σf(i))i(σf(i))i=(1-Cn(i))2σjk(i)nj(i)nk(i)ns(i)nt(i)σst(i)-12(1-Cn(i))Rσjk(i)×(nj(i)nk(i)σst(i)+δjkns(i)nt(i))+14(β(i)R)2σjk(i)σst(i)σts,Ud=12σjkCjkstdσsi=H∑i=1nA(i)σjk(i)[(1-Cn(i))2nj(i)nk(i)ns(i)nt(i)H∑i=1nA(i)σjk(i)-12(1-Cn(i))β(i)Rσjk(i)H∑i=1nA(i)σjk(i)×(nj(i)nk(i)σst(i)+δjkns(i)nt(i))H∑i=1nA(i)σjk(i)+14(Rβ(i))2σjk(i)σst(i)σts+(1-Cs(i))2H∑i=1nA(i)σjk(i)×(δktnj(i)ns(i)-nj(i)nk(i)ns(i)nt(i))(1-Cn(i))2]σst(i).Finally, it is clear to conclude the additional flexibility tensor of fractured rock mass under seepage pressure as follows:(49)Cijkld=(1-Cn(i))2nj(i)nk(i)ns(i)nt(i)-12(1-Cn(i))β(i)Rσjk(i)(nj(i)nk(i)σst(i)+δjkns(i)nt(i))+14(Rβ(i))2σjk(i)σst(i)σts+(1-Cs(i))2(δktnj(i)ns(i)-nj(i)nk(i)ns(i)nt(i)).


## 4.3. The Fracture Damage Evolution Equation

The additional elastic strain energy densityukzi produced by the wing crack is [32]
(50)ukz(i)=4E0ρv(i)a(i)2(52σeff(i)L(i)+23τeff(i))2,
where L(i) is the equivalent length for the crack.Assuming there aren groups of cracks in the fractured rock mass, the additional elastic strain energy density of the fractured rock Ukz is
(51)Ukz=∑i=1nukz(i)=12σijCijklkzσkl.Crack extension reduces the stiffness of rock masses and increases its flexibility. Through the derivative ofUkz with respect to the stress tensor, we can get the additional flexibility tensor of fractured rock masses in the damage evolution process
(52)∂Ukz∂σij=Cijklkzσkl.At the same time, by calculating the partial derivative ofL(i), σeff(i), and τeff(i) with respect to σij, respectively, and considering the symmetry of Cijklkz, we can get the following results:
(53)∂Ukz∂σij=1E0∑i=1n{M3(i)3a(i)2ρv(i)[M3(i)3nj(i)nk(i)nk(i)nt(i)M1(i)nj(i)nk(i)nk(i)nt(i)-M2(i)×(nj(i)nk(i)δkj+δkinj(i)nt(i)+nj(i)nk(i)δij+δitnj(i)nk(i))-M3(i)3βRδijδkl]}σkl,
where
(54)M1(i)=(20σeff(i)L(i)+163τeff(i))×[1σn(i)(52L(i)+23f)-1τn(i)(52cotθ(i)L(i)+23f)+52σeff(i)N1(i)1σn(i)],M2(i)=(5σeff(i)L(i)+43τeff(i))×[1τn(i)(52cotθ(i)L(i)+23f)+52σeff(i)N2(i)],M3(i)=(20σeff(i)L(i)+163τeff(i))×[23f+52(σeff(i)N3(i)+L(i))],N1(i)=L(i)[-4a(i)cosθ(i)σeff(i)2(fτn(i)-σn(i))-(KIC(i)2-4a(i)cosθ(i)σeff(i)-2πa(i)A(i)KIC(i)σeff(i)πa(i)A(i)KIC(i)σeff(i))×(τn(i)-σn(i)cotθ(i))]×(2πa(i)A(i)σeff(i)σn(i)τn(i))-1,N2(i)=L(i)[-4a(i)cosθ(i)σeff(i)2-(KIC(i)2-4a(i)cosθ(i)σeff(i)τeff(i)-2πa(i)A(i)KIC(i)σeff(i))cotθ(i)]×(2πa(i)A(i)σeff(i)σn(i)τn(i))-1,N3(i)=[-(-2(KIC(i)πa(i))2σeff(i)3-2πcosθ(i)(fσeff(i)-τeff(i)σeff(i)2))]×L(i)A(i)-2KIC(i)L(i)σeff(i)22πa(i),A(i)=KIC(i)2σeff(i)22πa(i)-2πcosθ(i)τeff(i)σeff(i)2.Then, we can get the flexibility tensorCijklkz due to the damage evolution of fractured rock masses as follows:
(55)Cijklkz=1E0∑i=1n{M3(i)3a(i)2ρv(i)[M3(i)3M1(i)nj(i)nk(i)nk(i)nt(i)-M2(i)×(nj(i)nk(i)δkj+δkinj(i)nt(i)+nj(i)nk(i)δij+δitnj(i)nk(i))-M3(i)3βRδijδkl]}.


## 4.4. The Constitutive Equation of the Multicrack Rock Mass

To sum up, combining the initial damage and damage evolution tensor, we can get the flexibility tensor of the fractured rock mass under seepage pressureas follows:(56)Cijkl=Cijkl0+Cijkld+Cijklkz,
where Cijkl0 is the elastic flexibility tensor of the complete rock, Cijkld is the initial equivalent damage flexibility tensor of the fractured rock mass under seepage pressure, and Cijklkz is the additional damage flexibility tensor with crack propagation of the fractured rock mass under seepage pressure.According to generalized Hooke’s law [33]:
(57)εij=Cijklσkl,σij=Eijklεkl=(Cijkl)-1εkl,
and this paper sets up the constitutive relation of the multicrack rock mass under seepage pressure.


## 5. Conclusions

After discussing the damage and fracture evolution mechanism of the multicrack rock mass under compression-shear stress and seepage pressure, the following main conclusions are drawn.(1)
Through proposing the fractured damage model of the multicrack rock mass under the combined action of compression-shear stress and seepage pressure, this paper discusses the law of crack initiation and the evolution law of stress intensity factor at the tip of a wing crack. The interaction of multiple cracks makes the stress intensity factor at the crack tip larger than that of a single wing crack. Regarding the seepage pressure, the existence of that reinforces the wing crack’s propagation. Moreover, the high seepage pressure converts wing crack growth from stable form to unstable form.(2)
The wing crack initiation from microcracks to the completely damage of the rock mass is a process of evolution and accumulation of the rock mass damage, as well as a process of the fracture of the connection between cracks. Based on the criterion and mechanism for crack initiation and propagation, this paper puts forward the mechanical model for different fracture transfixion failure modes of the crag bridge under the combined action of seepage pressure and compression-shear stress.(3)
According to the strain energy equivalent principle, this paper applies Betti’s reciprocal work theorem of the fractured rock mass to study the flexibility tensor of the rock mass’s initial damage and its damage evolution and deduces the damage constitutive equations for the elastic-plastic fracture and damage evolution. The theory provides a reliable theoretical principle for quantitative research of the fractured rock mass failure under seepage pressure.



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*Source: 738013-2013-05-23.xml*