Evolution Procedure of Multiple Rock Cracks under Seepage Pressure

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 rockmass’s damagemechanics, deduces the damage evolution equation for the rockmass 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.


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][2][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][7][8] or developing techniques to calculate the stress intensity factors of crack tips [9][10][11][12], or studying the relationship between microdamage development and macrodeformation of rock under uniaxial compression [13][14][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.

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.
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 stress  1 and  3 , where  1 is the maximum principal stress,  1 ≥  3 .The angle between the crack and vertical stress  1 is , and there is a seepage pressure  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   +  owing to the part closure of the crack, where  is the friction coefficient on the crack surface and  is the cohesion on the crack surface.Meanwhile, introducing the coefficient  which presents the ratio of the connected area to the total area, the seepage pressure contributes  to the surface.Therefore, the effective shear driving force  eff and effective normal stress   appear in following formulas [18] (here the compressive stress is positive): where   and  V 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 angle  3 can be obtained ( 3 = 70.5 ∘ ) [19].Then, the stress intensity factor at the wing crack initiation can be concluded as [20] Making   =   in (2),   is the fracture toughness, the biggest crack stress intensity factor.It is easy to get the ultimate strength  11 of the fracture rock under seepage pressure From (3a), it is clear to see that the ultimate strength  11 decreases linearly with the increase of the seepage pressure .
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 factor   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   consists of the stress intensity factor produced by the effective shear driving force   and the remote field stress  1 and  3 , and it can be shown as in ( 4) where   is the impact factor which is the function of the wing crack length , wing crack azimuth , and main crack length , as shown in ( 5) The wing crack extends approximately in the direction perpendicular to the maximum principal stress until   =   , and then the wing crack propagation length  under the combined action of compression-shear stress and seepage pressure can be calculated as shown in ( 6)

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.
Assuming that the number of the compression-shear cracks per unit area is   , the length between main crack centers and the rock bridge lengths between wing cracks, respectively, are presented as follows: In Figure 2,  =   sin  is balanced by the tensile stress   3 in the rock bridge together with seepage pressure in the wing crack: where Acting on the wing crack,   3 produces an additional intensity factor at the crack tip [23]: 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: 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 define  0 for the initial damage of the rock mass and  for the damage when the wing crack expands to : Substituting ( 11) into ( 8), the fellow equation can be easily obtained Combining ( 10), (11), and (12), the relation curves between the damage variables  and dimensionless stress intensity factor (  / 1 √) at the wing crack tip with different crack density are shown in Figure 3.
The graph reveals that with the increase of the damage variables, the dimensionless stress intensity factor   / 1 √ at the crack tip under seepage pressure decreases at the very beginning but rises gradually when the equivalent crack length is 0.5 (/).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  varies from  0 to 1.When  = 1 and the wing crack stress intensity factor   ⩾   at the crack tip, the wing crack connects, the rock bridge cracks, and the rock mass loses bearing capacity.
Figure 4 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 ( = 0MPa), the wing crack expands stably (  / < 0).While when the seepage pressure is relatively high ( = 15 MPa), the wing crack tends to expand unstably.Besides, there is the expansion stage that   / > 0 under such condition and the growth rate of wing crack stress intensity factor becomes greater with the increase of seepage pressure .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.

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.

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 Figure 5.When the wing crack reaches the critical length  1 = /sin ,  1 =  1 /, 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  1 = / sin , this paper establishes the failure criteria: When   ( 1 ) ≥   , axial transfixion failure occurs in the rock bridge.

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  is 1/2 the length of the bottom crack,  is 1/2 the length of the upper crack,  is the rock bridge,  and  are the wing cracks produced by effective shear driving force of the main crack  and ,  is the angle between rock bridge and the maximum principal stress, and   and   are the normal stress and shear stress acting on the rock bridge, respectively.According to the element shown in Figure 6 and the principle of mechanics balance, it could be deduced as follows: where Then, the following equation can be obtained ) Assuming that the rock bridge shear damage follows the Mohr-Coulomb strength criterion, conditions for the damage are Substituting ( 16) into ( 18), the fellow equation can be easily obtained as follows: When angle  meets the rock bridge shear failure condition and the wing crack propagation length reaches critical value  2 , the angle  between the rock bridge and  1 also reaches their critical value, and then As the geometric relations show in Figure 6, it is easy to see that Combined ( 20) and ( 21), the critical length of the wing crack can be defined as: where Meanwhile, this paper gets the stress intensity factor at the crack tip when the wing reaches the critical length as follows:

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 tensor  can be determined by the elastic flexibility tensor  0 and the equivalent damage flexibility tensor   [28] where  is the fourth-order unit tensor and  is a tensor of the fourth order for the 3D anisotropy mode of fractured rock.
In terms of the complete rock, the elastic flexibility tensor  0 can be expressed as [29] where  0 and V 0 represented the elastic modulus and Poisson's ratio of the rock, respectively.The equivalent damage flexibility tensor   can be drawn from Betti energy reciprocal theorem.Hence, based on selfconsistent method and strain energy equivalence in solid mechanics, the equivalent damage flexibility tensor    is acquired through calculating the equivalent elastic strain energy.

The Equivalent Damage Flexibility Tensor Based on Equivalent Elastic Strain Energy.
The flexibility matrix of the crack in compression shear state is [30] 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.

The Equivalent Damage Flexibility Tensor Based on Crack's Elastic Strain Energy.
The strain energy of single crack is assumed as [29] where Γ is the line length of cracks.Integrating over the line length, the stress intensity factors of type  or type  for the crack under seepage pressure are where  ()  is the normal stress of the crack and  ()  is the tangential shear stress.Combining the former equation, then we get By assuming there are  groups of cracks in the fractured rock mass, the strain energy of the fractured rock mass will be Set the equation  = ((1 − ] 2 0 )/ 0 ),  () =  ()2  () , in which  () is the density of cracks, then Rewrite (39) into the tensor expression According to the hydrostatic pressure principle and Cauchy criterion, the remote field stress is denoted as   , and the stress matrix on the surface can be defined as [31] [ Defining     = 1, authors rewrite the equation into the tensor expression Considering the compression transferring coefficient   , the equivalent normal stress on the surface of the crack can be expressed as Considering the shear transferring coefficient   again, the shear stress on the crack surface can be expressed as Then, we get Considering the symmetry of   , the following equations can be deduced: Supposing the proportion coefficient  = /,  is the average stress,  = (1/2)  ,   is the first invariant stress.Then, the seepage pressure can be transformed to Finally, it is clear to conclude the additional flexibility tensor of fractured rock mass under seepage pressure as follows: where  () is the equivalent length for the crack.
Assuming there are  groups of cracks in the fractured rock mass, the additional elastic strain energy density of the fractured rock   is Crack extension reduces the stiffness of rock masses and increases its flexibility.Through the derivative of   with respect to the stress tensor, we can get the additional flexibility tensor of fractured rock masses in the damage evolution process At the same time, by calculating the partial derivative of  () ,  ()  eff , and  ()  eff with respect to   , respectively, and considering the symmetry of    , we can get the following results:  where Then, we can get the flexibility tensor    due to the damage evolution of fractured rock masses as follows: (55)

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: where  0  is the elastic flexibility tensor of the complete rock,    is the initial equivalent damage flexibility tensor of the fractured rock mass under seepage pressure, and    is the additional damage flexibility tensor with crack propagation of the fractured rock mass under seepage pressure.
According to generalized Hooke's law [33]: and this paper sets up the constitutive relation of the multicrack rock mass under seepage pressure.

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.

Figure 1 :
Figure 1: Sketch of wing cracks seeding and propagation.

Figure 3 :
Figure 3: The relationship between damage variables and intensity factor at the crack tip of the branch stress.
Schematic drawing of multiple interacting cracks. is the horizontal stress in the crack.  3 is the horizontal stress in the rock bridge, and  is the length of the rock bridge. ) 2 .