CN106844848B

CN106844848B - Consider the construction method of the two-dimentional season cracking model of moment of flexure contribution factor

Info

Publication number: CN106844848B
Application number: CN201611162775.4A
Authority: CN
Inventors: 黄书岭; 丁秀丽; 李欢; 邬爱清; 徐平; 吴勇进; 高源�; 朱良韬
Original assignee: Changjiang River Scientific Research Institute Changjiang Water Resources Commission
Current assignee: Changjiang River Scientific Research Institute Changjiang Water Resources Commission
Priority date: 2016-12-15
Filing date: 2016-12-15
Publication date: 2018-01-16
Anticipated expiration: 2036-12-15
Also published as: CN106844848A

Abstract

The invention discloses the building method for the two-dimentional season cracking model for considering moment of flexure contribution factor, two-dimentional season cracking model includes considering that the rock mass of moment of flexure contribution factor carefully sees particle bond stress two-dimensional model, considers that the thin sight particle of moment of flexure contribution factor bonds the two-dimentional exponential type pattern of timeliness deterioration decay, considers moment of flexure contribution effect and mole coulomb with stretching cut-off limit carefully sees particle and bonds season cracking criterion, considers the thin sight particle linear contact two dimensional model of damping effect.The present invention is adapted to this kind of rock mass of the relation index of coincidence type between stress and crack propagation velocity, and technical support is provided for the prediction of country rock long-time stability, evaluation and the optimization design of this kind of deep rock mass engineering project under flat state.

Description

Consider the construction method of the two-dimentional season cracking model of moment of flexure contribution factor

Technical field

Season cracking analysis technical field is carefully seen the present invention relates to engineering rock mass, in particular to one kind consider moment of flexure contribution because The construction method of the two-dimentional season cracking model of son.

Background technology

Unstability and destruction after deep rock mass engineering project excavation are frequently not to occur at once after excavation, are usually present Obvious deformation fracture is ageing and the hysteresis quality of catastrophe (rock burst, large deformation etc.), seriously endangers the construction safety and length of engineering Phase runs.At present, the timeliness achievements in mechanical research in terms of thin sight is relatively fewer.The time that Liu Ning etc. ruptures to silk screen griotte Effect has carried out experiment and numerical analysis (particle flow simulation of buried griotte Fracture propagation time effect, rock mechanics and work Journey journal, 2011, Vol.30No.10:1989-1996)；The application creep meso mechanical model such as Sun Jinshan is short to silk screen griotte Phase and long-term strength feature have carried out numerically modeling, and (numerical simulation of silk screen griotte creep impairment evolution mesomechanics feature is ground Study carefully, rock-soil mechanics, 2013, Vol.34No.12:3601-3608).These are all with paralleling binding model in discrete element Based on (Parallel-Bonded Model, PBM), rock is described according to the relation between driving stress and crack propagation velocity The stone carefully season cracking in sight aspect.But there are many weak points in this kind of paralleling binding model：First, paralleling binding breaks After splitting, force of sliding friction is only considered between particle, does not account for being broken the intergranular way of contact, after not meeting rock masses fracturing The thin motion modes for seeing particles under external load of difference；Secondly, intergranular shear fracture criterion is one and paralleling binding The parallel horizontal linear of direct stress, namely this shear fracture criterion are unrelated with paralleling binding direct stress state, as long as parallel viscous Tie shear stress and be more than or equal to fixed paralleling binding shear fracture intensity, shear fracture can occur between particle, can not embody Different paralleling binding direct stress have the objective fact of different paralleling binding shear fracture intensity in rock mass；In addition, for rock mass This kind of friction adhesive material, above-mentioned this paralleling binding model do not have the difference for considering to bond torque and acted on to contact failure Influence, influence of the contribution degree to different lithology for bonding torque is accordingly to be regarded as unanimously, exaggerating cohesive torque and breaking rock mass Bad effect.

The content of the invention

It is an object of the invention to for drawbacks described above, it is proposed that a kind of two-dimentional season cracking for considering moment of flexure contribution factor The construction method of model, the present invention are adapted to this kind of rock mass of the relation index of coincidence type between stress and crack propagation velocity, Technology branch is provided for the prediction of country rock long-time stability, evaluation and the optimization design of this kind of deep rock mass engineering project under flat state Hold.

The purpose of the present invention is reached by following measure：A kind of two-dimentional season cracking for considering moment of flexure contribution factor The construction method of model, it is characterized in that, comprises the following steps：

Step 1：Setting rock mass carefully see particle two-dimensional parallel bonded contact geometric parameters quantity include paralleling binding area and Paralleling binding the moment of inertia, R^a、R^bThe particle radius at both ends, under two-dimensional case, paralleling binding unit thickness are contacted for paralleling binding For 1 when paralleling binding area and paralleling binding the moment of inertia determined respectively by formula (2), formula (3)：

Wherein：Particle two-dimensional parallel is carefully seen for rock mass and bonds radius,Particle two-dimensional parallel is carefully seen for rock mass and bonds diameter Multiplier, A are that rock mass carefully sees particle two-dimensional parallel bond area circle, and I is that rock mass carefully sees particle two-dimensional parallel bonding the moment of inertia；

Step 201：The initial time step size increments that particle two-dimensional parallel grain bonds timeliness decay deterioration are carefully seen using rock mass Δ t, the diameter of particle two-dimensional parallel bonding timeliness decay deterioration is carefully seen by exponential type function calculating rock mass, and formula (4) comes really It is fixed：

Wherein：To consider that the two-dimensional parallel of moment of flexure contribution factor bonds normal stress,To judge that rock mass carefully sees particle Two-dimensional parallel bonds stress threshold values when starting timeliness deterioration decay,The stretching that particle two-dimensional parallel bonding is carefully seen for rock mass is strong Degree,To consider the two-dimensional parallel bond stress ratio of moment of flexure contribution factor, β₁、β₂Respectively rock mass carefully sees particles parallel bonding The first control parameter, the second control parameter of timeliness deterioration,Particle two-dimensional parallel bonding is carefully seen for rock mass to decline with time deterioration The diameter subtracted,Diameter when particle two-dimensional parallel bonding does not decay is carefully seen for rock mass, e is natural constant, and Δ t is that rock mass is carefully seen Particle two-dimensional parallel bonds the incremental time of timeliness decay deterioration；

Step 202：According to the formula (4) in step 201, setting rock mass carefully sees the exponential type that particles parallel grain bonds diameter Timeliness decay factor β, see formula (5)：

Wherein：A'、I'、It is expressed as rock mass and carefully sees particle two-dimensional parallel bonding with time deterioration decay Paralleling binding diameter, paralleling binding radius, paralleling binding area, paralleling binding the moment of inertia, paralleling binding diameter multiplier, A、I、Carefully seen for rock mass particles parallel bond paralleling binding diameter when not decaying, paralleling binding radius, Paralleling binding area, paralleling binding the moment of inertia, paralleling binding diameter multiplier；

Step 203：According to the formula (5) in the formula (1) in step 1~formula (3) and step 202, rock mass is set Thin particle two-dimensional parallel of seeing bonds geometric parameter timeliness deterioration evanescent mode；Rock mass carefully see particle two-dimensional parallel bond diameter with Time increases and constantly deteriorates decay, and two-dimensional parallel bonds the paralleling binding area and paralleling binding inertia when unit thickness is 1 Square increases also with the time and constantly deteriorates decay, sees formula (6), formula (7) respectively：

Wherein：A', I' are expressed as rock mass and carefully see particle two-dimensional parallel bonding with the parallel viscous of time deterioration decay Radius, paralleling binding area, paralleling binding the moment of inertia are tied, A, I carefully see parallel viscous when particles parallel bonding does not decay for rock mass Junction area, paralleling binding the moment of inertia；

Step 204：The rock mass for calculating j-th to k-th successively carefully sees the two-dimensional parallel bonding that particle includes time effect Normal direction moment of flexure increment；Under two-dimensional case, by the speed of paralleling binding both ends particle, angular speed and given cycle calculations time Step size increments Δ t, determine that i-th of rock mass carefully sees particle two-dimensional parallel Binder Phase pair by formula (8), formula (9), formula (10) CornerTwo-dimensional parallel bonds normal direction incremental displacementAnd two-dimensional parallel bonds tangential incremental displacementTie again The formula (5) in the formula (7) and step 202 in step 203 is closed, determines that i-th of rock mass carefully sees particle and include time effect Two-dimensional parallel bonds moment of flexure increment, is specifically shown in formula (11)：

Wherein, ff, j, k are natural numbers, and 2≤j≤ff≤k, j are the rock comprising time effect in each cycle calculations Body carefully sees particle two-dimensional parallel and bonds uncracked initial index value after decay, and ff is some middle index value, and k is to follow every time During ring calculates, the rock mass comprising time effect carefully sees particle two-dimensional parallel and bonds uncracked most end index value after decay, i the One to last two-dimensional parallel bonded particulate index value,The carefully sight of respectively i-th of rock mass The grain a ends of two-dimensional parallel bonded contact and the absolute movement speed and angular speed at b ends, n_n、n_sParticle two-dimensional parallel is carefully seen for rock mass The normal direction unit vector and tangential unit vector in bonded contact face,Respectively rock mass carefully sees particle two-dimensional parallel Normal direction displacement increment, tangential displacement increment are bonded,Particle two-dimensional parallel is carefully seen for rock mass and bonds normal stiffness,For rock Body carefully sees particle two-dimensional parallel and bonds moment of flexure increment；

Step 205：Formula (8), formula (9) in the formula (6) and formula (7), step 204 in step 203 and Formula (5) in formula (11) and step 202, successively renewal calculate j-th to k-th rock mass and carefully see particles parallel bonding not Rupture and the two-dimensional parallel comprising time effect bonds normal force, tangential force and tangential moment of flexure；Pass through formula (12), formula (13), formula (14) calculates two-dimensional parallel bonding normal force, tangential force and the tangential moment of flexure that i-th of rock mass carefully sees particle contact； Under two-dimensional case, normal direction moment of flexure is bonded to determine that rock mass carefully sees particles parallel by formula (15)：

Normal force：

Tangential force：

Tangential moment of flexure：

Normal direction moment of flexure：

Wherein：Respectively i-th of rock mass carefully sees particle and includes time effect The paralleling binding normal force answered, paralleling binding tangential force, the paralleling binding normal direction moment of flexure comprising time effect, paralleling binding are tangential Moment of flexure, paralleling binding Normal Displacement increment and paralleling binding tangential displacement increment,Particle two-dimensional parallel bonding is carefully seen for rock mass Shear stiffness, +=is the reflexive operator of addition, and -=is the reflexive operator of subtraction；

Step 206：Moment of flexure contribution factor is setConsider that moment of flexure particles parallel of seeing thin to rock mass bonds normal direction The percentage contribution of direct stress, direct stress calculation formula is bonded according to two-dimensional parallelBonded with two-dimensional parallel Calculation Shear formulaSimultaneously by A, I in the two formula andWith A', I' andReplace, then by step Formula (5) in formula (6) and formula (7) and step 202 in 203 substitutes into, and obtains comprising exponential type time effect and curved The two-dimensional parallel of square contribution factor bonds direct stressShear stress is bonded with two-dimensional parallelCalculation formula, formula is seen respectively And formula (17) (16)；

Step 207：Time effect will be included in step 206Substitute into formula (18), it may be determined that consider moment of flexure Contribution factor and mole-coulomb season cracking criterion limited with stretching cut-off, and j-th to k-th two dimension is calculated successively Paralleling binding stress, whether ruptured and fracture mode for judging that rock mass is carefully seen particles parallel and bonded；In the rock mass of the criterion Thin see contains exponential type time effect and consideration moment of flexure contribution factor in particle two-dimensional parallel bond stress；

Wherein：f^s、fⁿRespectively rock mass carefully sees the timeliness shear fracture criterion of particle two-dimensional parallel bonding, timeliness stretching is broken Split criterion,Two-dimensional parallel for the time effect containing exponential type of i-th of contact bonds shear stress,Connect for i-th Tactile time effect containing exponential type and the two-dimensional parallel bonding direct stress of consideration moment of flexure contribution factor,Respectively rock mass It is thin to see tensile strength, the shearing strength that particle two-dimensional parallel bonds,The cohesive strength of particle two-dimensional parallel bonding is carefully seen for rock mass,The internal friction angle of particle two-dimensional parallel bonding is carefully seen for rock mass；f^sParalleling binding shear fracture is represented more than or equal to 0, less than 0 table Show that shear fracture does not occur for paralleling binding；fⁿParalleling binding tensile fracture is represented more than or equal to 0, paralleling binding is represented not less than 0 Generation tensile fracture；

Step 208：If the f in formula (18) in step 207^sOr fⁿMore than or equal to 0, show that rock mass carefully sees particle Paralleling binding is ruptured, and hereafter rock mass carefully sees the motor pattern of particle using the two-dimensional linear contact mould for considering damping effect Type is expressed；If the f in formula (18) in step 207^sAnd fⁿBoth less than 0, show that paralleling binding does not rupture, continue cycling through Step 201 calculated, renewal, judges that rock mass carefully sees the paralleling binding state of particle contact to 207, up to rock mass do not produce it is new Paralleling binding ruptures or paralleling binding ruptures accelerated development and forms macroscopic failure, loop termination.

Preferably, the rock mass carefully sees the initial time step size increments Δ t that particle two-dimensional parallel bonds timeliness decay deterioration Determination, be using consider moment of flexure contribution factor paralleling binding timeliness deterioration decay two-dimentional exponential type pattern, by it is each when Two-dimensional parallel in spacer step bonds decay first and ruptures the time be lost to determine, namely by first paralleling binding by finger Number pattern formula decay time for being lasted of rupture divided by until calculating circulation time required for first paralleling binding rupture Count to estimate initial time step size increments Δ t, see formulaIts In, Particle two-dimensional parallel bonding diameter multiplier, n are carefully seen for the rock mass of i-th of contact_cFor One rock mass carefully sees the number that particle two-dimensional parallel bonds the cycle calculations needed for rupture, β_σ、β_τRespectively rock mass carefully sees particle two Tie up the timeliness deterioration factor corresponding to paralleling binding tensile strength, the timeliness deterioration factor corresponding to two-dimensional parallel bond shear strength, i It is followed successively by first and carefully sees particles parallel bonding number to last rock mass, ∞ is infinity.

Preferably, the rock mass carefully sees particle two-dimensional parallel and bonds timeliness deterioration factor-beta corresponding to tensile strength_σAnd rock mass It is thin to see timeliness deterioration factor-beta corresponding to particle two-dimensional parallel bond shear strength_τDetermination comprise the following steps, wherein, these The formula subscript 1 included in step represents first two-dimensional parallel that timeliness decay deterioration is carried out by exponential type pattern and bonds rupture Label；

Step 211：Under two-dimensional case, carefully seen by rock mass particles parallel bond the speed of both ends particle, angular speed and to Fixed cycle calculations time step increment Delta t_c, pass through formulaDetermine the phase of paralleling binding contact To cornerPass through formulaDetermine paralleling binding normal direction incremental displacementIt is logical Cross formulaDetermine the tangential incremental displacement of paralleling bindingPass through formulaDetermine the moment of flexure increment of paralleling binding contact；

Step 212：According to the formula in step 211Pass through formulaDetermine paralleling binding normal force；According to the formula in step 211Pass through formulaDetermine paralleling binding tangential force；According to step Formula in 211And formulaPass through formulaDetermine the tangential moment of flexure of paralleling binding；Pass through formulaParalleling binding normal direction moment of flexure is determined, Wherein, +=is the reflexive operator of addition, and -=is the reflexive operator of subtraction；

Step 213：Under two-dimensional case, pass through formulaDetermine that paralleling binding just should Power, pass through formulaDetermine paralleling binding shear stress, by A, I in the two formula andWith A', I' andReplace Change, then substitute into the formula (5) in the formula (6) and formula (7) and step 202 in step 203, acquisition includes exponential type The two-dimensional parallel of time effect and moment of flexure contribution factor bonds direct stress calculation formulaWith Two-dimensional parallel comprising exponential type time effect bonds Calculation Shear formula

Step 214：WillSubstitute into formulaAnd make β=β_σ；Will Substitute into formulaAnd make β=β_τ, accordingly, respectively obtain the rock mass and carefully see the stretching of particle two-dimensional parallel bonding by force Timeliness corresponding to degree deteriorates the factorAnd rock mass carefully sees particle two dimension Timeliness corresponding to paralleling binding shear strength deteriorates the factor

Preferably, the rock mass carefully see particles parallel bond rupture after, rock mass carefully see particle motor pattern use Consider that the two-dimensional linear contact model of damping effect is expressed, for describe rock mass carefully see particles parallel bond it is thin after season cracking Stress, deformation and the moving law of particle are seen, considers that the structure of the two-dimensional linear contact model of damping effect comprises the following steps：

Step 301：By Monte Carlo searching algorithms, traversal finds rock mass and carefully sees each two-dimensional linear contact of particle A, two-dimensional linear contact jaw b (particle and particle, particle and wall) centre coordinate are held, under two-dimensional case, is passed through formula (19) Calculate both centre distances：

Wherein：D is the centre distance between two-dimensional linear contact both ends particle and particle or particle and wall,For Two-dimensional linear contact jaw a coordinate,For two-dimensional linear contact jaw b coordinate.

Step 302：Rock mass carefully sees the unit vector of each contact point between particle and passes through formula (20) under two-dimensional plane state Calculate, if the contact between particle and particle, utilize the center point coordinate that two-dimensional linear contact both ends are obtained in step 301 And apart from calculating, if particle contacts with wall, directly calculated using the normal vector equivalence replacement of wall, it is determined that each contact The unit vector of section：

Wherein：n_iFor the unit vector of contact,For contact jaw b direction vector,For contact jaw a direction vector, n_wallTo constrain the direction vector of wall.

Step 303：After rock mass carefully sees particles parallel bonding rupture, the contact lap U of two-dimensional linear contact-segment, pass through step Rapid 301 calculate the particle radius R at grain spacing d and two-dimensional linear contact both ends^a、R^b, recycle formula (21) to determine.It is logical Cross setting particle two-dimensional linear contact reference distance g_r, and formula (22) is combined, determine the distance g of particle two-dimensional linear contact_s：

g_s=| U |-g_r (22)

Step 304：Determine that rock mass carefully sees grain contact point normal direction, tangential equivalent stiffness, using contacting both ends particle entities Or the rigidity k of wall^a, k^bThe equivalent rigidity (general designation of normal stiffness and shear stiffness) instead of contact point, by formula (23) Calculate：

Wherein：K_n、K_sFor equivalent normal stiffness and shear stiffness,For particle and particle or particle and wall Contact a ends normal direction and shear stiffness,Normal direction for particle and the contact b ends of particle or particle with wall and cut To rigidity.

Step 305：Determine to contact the intergranular speed of related movement in both ends in rock mass, utilize formula (24), formula (25) To calculate.Wherein e_pqzFor Ricci index alternators, calculated according to formula (26)：

Wherein：V_pWith V_qEquivalence, V_pWith V_qTo contact the intergranular speed of related movement in both ends in rock mass, p, q are index etc. Valency symbol, p=1, q=1 represent that particle contacts with particle, and expression particle contacts with wall when p=2, q=2,For The speed of particle and the contact b end units of particle or particle with wall,It is particle and particle or particle and wall Contact a end units speed,It is angular speed of the particle with the contact a end units of particle or particle with wall,It is angular speed of the particle with the contact b end units of particle or particle with wall,It is particle and particle or particle The displacement at the contact a ends with wall,It is displacement of the particle with the contact b ends of particle or particle with wall,For drift index The middle transition symbol of conversion,The speed at the contact a ends of pellet-pellet or particle-wall when index symbol is p is represented,The speed at the contact a ends of pellet-pellet or particle-wall when index symbol is q is represented,When expression index symbol is p The speed at the contact b ends of pellet-pellet or particle-wall,Represent pellet-pellet or particle-wall when index symbol is q Contact b ends speed (only a ends and two, b ends contact jaw).

Step 306：For the initial time step size increments Δ t of linear contact model value, estimated by formula (29) Minimum time step Δ t, it is ensured that the calculating time step of constructed model is less than the value, you can ensure that system integral calculates In stable；Determine that the total displacement increment of each linear contact, Normal Displacement increase by formula (30), formula (31), formula (32) Amount and tangential displacement increment：

R=min (R_a,R_b) (27)

ΔU_p1=V_p1Δt (30)

Wherein：R is the equivalent redius that rock mass carefully sees particle, and m is that rock mass carefully sees granular mass, and J1 is that rock mass carefully sees particle Rotary inertia；k^{It is flat}Particle system translational stiffness, k are carefully seen for rock mass^TurnParticle system rotational stiffness is carefully seen for rock mass；ΔU_p1For rock mass The thin total displacement increment for seeing the contact of particle two-dimensional linear, Δ δ_n、Physical significance is identical, represents that rock mass carefully sees particle two dimension The Normal Displacement increment of linear contact, Δ δ_s、Physical significance is identical, represents that rock mass carefully sees the contact of particle two-dimensional linear Tangential displacement increment, V_p1With V_q1The speed of related movement at particle contact both ends is carefully seen for rock mass, n is unit normal vector, p1, Q1 is tensor index figure shift.

Step 307：The ultimate range as existing for formula (22) judgement rock mass carefully sees particle surface contact permission, calculates normal direction With tangential displacement updating factor, in addition, the renewal that rock mass carefully sees particle two-dimensional linear contact normal direction displacement increment is using previous The Normal Displacement increment of step obtains with updating factor α product, and rock mass carefully sees particle two-dimensional linear contact tangential displacement increment Renewal is obtained using the tangential displacement increment of back and updating factor α product.

Wherein：(g_s)₀The surface that initial time is calculated for model contacts distance, g_sThe distance of particle contact is carefully seen for rock mass, α is displacement updating factor.

Step 308：The normal direction linear force that rock mass carefully sees the contact of particle two-dimensional linear takes relative vector to add up (M_l=1) and Absolute vectors adds up (M_l=0) pattern, calculated and obtained by formula (33), (34)；Rock mass carefully sees the contact of particle two-dimensional linear Tangential linear force carefully sees particle contact slide using rock mass to represent, is calculated and obtained by formula (35)：

Wherein：Respectively rock mass carefully sees the two-dimensional linear contact normal direction linear force, tangential of stress deformation between particle Linear force, k_n、k_sRespectively rock mass carefully sees the two-dimensional linear contact normal direction of stress deformation between particle, tangential linear rigidity, Δ δ_n、 Δδ_sRespectively rock mass carefully sees the Normal Displacement increment of particle two-dimensional linear contact, tangential displacement increment,Respectively Initial normal force increment size, the tangential force increment size of particle two-dimensional linear contact are carefully seen for rock mass,Particle is carefully seen for rock mass not Stiction during slip,Particle force of sliding friction is carefully seen for rock mass, its value can by friction coefficient μ withProduct obtains.

Step 309：The normal direction damping that rock mass carefully sees particle linear contact uses full normal mode M_d={ 0,2 } and tensionless winkler foundation Pattern M_dTwo kinds of={ 1,3 }, calculated by formula (39), wherein m_cFor equivalent particle quality, calculated by formula (40), rock mass is thin The tangential damping for seeing particle linear contact uses full shear mode M_d={ 0,1 } and cunning-cut-off-die formula M_d={ 2,3 }, according to formula (41) calculate,

Wherein：Respectively rock mass carefully sees the linear damping force of normal direction, the tangential linear damping of particle linear contact Power, β_nThe normal direction damped coefficient of particle linear contact, β are carefully seen for rock mass_sThe tangential damping system of particle linear contact is carefully seen for rock mass Number, k_nThe normal direction linear rigidity of particle linear contact, k are carefully seen for rock mass_sThe tangential linear firm of particle linear contact is carefully seen for rock mass Degree,Respectively rock mass carefully sees the normal direction speed and tangential velocity of particle linear contact, F^*It is linear that particle is carefully seen for rock mass The full normal direction damping force of contact, expression formula arem_cEquivalent particle quality, m are carefully seen for rock mass⁽¹⁾For rock Body carefully sees the thin sight granular mass of particle contact jaw 1, m⁽²⁾The thin sight granular mass of particle contact jaw 2, F are carefully seen for rock mass^dFor rock Body carefully sees the total damping power of particle linear contact.

The construction method of the two-dimentional season cracking model proposed by the present invention for considering moment of flexure contribution factor, in paralleling binding After effect rupture, intergranular contact side is expressed by the embedded two-dimensional linear model for considering damping effect between particles Formula, not only it is contemplated that sliding friction acts on, and it is also conceivable to intergranular deformation characteristic, meets rock mass particle in plane The characteristics of motion under state；Secondly, moment of flexure contribution factor is introduced in paralleling binding Stress calculation, not only considers moment of flexure to flat Row bonds direct stress contribution, and also contemplates influence of the moment of flexure to rock mass long-term strength；In addition, in constructed two-dimentional timeliness It is embedded in Rupture Model to consider moment of flexure contribution effect and put down with mole-coulomb season cracking criterion of stretching cut-off limit, the criterion Exponential type time effect is contained in row bond stress and adds moment of flexure contribution factor, can not only be described with paralleling binding just The difference of stress correlation timeliness shear fracture intensity, can also reasonably be expressed timeliness tensile fracture, and consider curved Influence of the square to paralleling binding season cracking, meet the essential characteristic of rock mass season cracking in a flat state.The present invention for The prediction of deep rock mass engineering project country rock long-time stability, evaluation and optimization design provide direct technical support under flat state.

A kind of construction method of the two-dimentional season cracking model of consideration moment of flexure contribution factor proposed by the invention, its is beneficial Effect and advantage are mainly reflected in：

(1) not only examined in the present invention by setting moment of flexure contribution factor in bonding direct stress calculation formula in two-dimensional parallel Percentage contribution of the moment of flexure to paralleling binding direct stress is considered, and has also contemplated influence of the moment of flexure to rock mass long-term strength, simultaneously Too strong energy impact ripple secondary damage to caused by adjacent domain is produced during reduction rock masses fracturing, is adapted to description plane should The mesomechanics fracture behaviour of rock mass under power or plane strain condition.

(2) consider that the two-dimensional parallel of moment of flexure contribution factor bonds diameter timeliness and deteriorates decay mode in the present invention by building Formula, including set the exponential type two-dimensional parallel related to the paralleling binding stress for considering moment of flexure contribution factor to bond diameter deterioration and decline Size reduction mode, set paralleling binding diameter to increase progressively deteriorates evanescent mode over time, at the same set paralleling binding area and The corresponding timeliness deterioration evanescent mode of cross sectional moment of inertia.This forming types are adapted to describe under plane stress or plane strain condition The mesomechanics season cracking mechanism and response pattern of deep rock mass.

(3) it is embedded to consider moment of flexure contribution effect and with stretching in the present invention in constructed two-dimentional season cracking model End mole-coulomb season cracking criterion of limit.Exponential type time effect and increase are included in the criterion paralleling binding stress Moment of flexure contribution factor, the difference of related to paralleling binding direct stress timeliness shear fracture intensity can not only be described, can be with Timeliness tensile fracture is reasonably expressed, and considers influence of the moment of flexure to paralleling binding season cracking, meets planar strip Rock mass season cracking pattern under part.

(4) in the present invention in constructed two-dimentional season cracking model, the embedded two-dimensional linear for considering damping effect connects Model structure is touched, after rock mass carefully sees particles parallel bonding season cracking, passes through and specifies two dimensional touch reference distance to set rock mass Interparticle contact distance, set consider rock mass particle between stress deformation two dimensional touch pattern and set between rock mass particle Consider the binding mode of two-dimentional sliding friction, while the damping mode of two dimensional touch is set, can rationally describe plane stress or flat Particle motion and stress characteristic of the deep engineering rock mass after season cracking under the strained condition of face.

Brief description of the drawings

Fig. 1 is that rock mass particle contacts schematic diagram with particle in model of the present invention.

Fig. 2 is that rock mass particle contacts schematic diagram with rigid wall in model of the present invention.

Fig. 3 is rock mass particle overlap condition schematic diagram in model of the present invention.

Fig. 4 is rock mass particle Rigidity Calculation schematic diagram in model of the present invention.

Fig. 5 is linear tangential force and tangential displacement schematic diagram in model of the present invention

Fig. 6 is two-dimentional season cracking physical model schematic diagram constructed in the present invention.

Fig. 7 is linear parallel adhesive structure schematic diagram in model of the present invention.

Fig. 8 is the consideration moment of flexure contribution effect in model of the present invention and mole-coulomb fracture criteria limited with stretching cut-off Schematic diagram.

Fig. 9 is diameter timeliness decay schematic diagram in model of the present invention.

Figure 10 is model diameter timeliness attenuation rate logarithm of the present invention and stress curve schematic diagram.

Figure 11 is the normal direction in model two dimensional touch face of the present invention and tangential unit vector schematic diagram.

Figure 12 is that model structure builds schematic flow sheet in the present invention.

Figure 13 is model creep impairment distribution map of the present invention.

Figure 14 is model creeping displacement and time history of the present invention.

Figure 15 is the creeping displacement and time history of model difference torque contribution degree of the present invention.

In figure：1 represents the centre distance d of the particle of rock mass two, and 2 represent the intergranular half contact distance of rock mass two, and 3 represent rock The intergranular half reference distance g of body two_r, 4 represent rock mass particle a coordinate, and 5 represent rock mass particle b coordinate, 6 surface rock mass Particle contacts the centre coordinate of distance, and 7 represent rock mass particle surface contact distance g_s, 8 represent the intergranular contact unit of rock mass Normal vector, 9 represent rock mass particle a radius R^a, 10 represent rock mass particle b radius R^b, 11 represent connecing for rock mass grain contact point Lap U is touched, 12 represent rigidity (normal direction, shear stiffness are referred to as) k of b (rock mass particle or border wall)^b, 13 represent a (rocks Body particle or border wall) rigidity (normal direction, shear stiffness be referred to as) k^a, 14 represent the equivalent stiffness of contact point, and 15 represent Total displacement increment Delta U_i, 16 represent initial normal forceIncrement size, 17 representatives initially contact force vectors and 18 representatives are initially cut Xiang LiIncrement size, 19 represent constructed two-dimentional season cracking model normal direction displacement increment Δ δ_nOr20 represent institute The two-dimentional season cracking model tangential displacement increment Delta δ of structure_sOr21 represent the tensile strength of paralleling binding22 Represent paralleling binding normal stiffness23 represent the normal stiffness K of linear contact point_n, 24 represent paralleling binding shear stiffness25 represent paralleling binding shear strength, and 25.1 representFor the cohesive strength of paralleling binding, 25.2, which represent the interior of paralleling binding, rubs Wipe angle26 represent the shear stiffness K of linear contact point_s, 27 represent the coefficient of sliding friction, and 28 are represented as the resistance of linear contact normal direction Buddhist nun's factor beta_n, 29 represent the tangential damped coefficient β of linear contact_s, 30 represent paralleling binding diameter (radius) multiplier31 representatives are flat Row bonds diameter32 represent consideration moment of flexure contribution effect and end mole-coulomb season cracking criterion of limit with stretching, and 33 Represent the paralleling binding shear stress comprising time effect of i-th of contact34 represent being imitated comprising the time for i-th of contact The paralleling binding direct stress of moment of flexure contribution factor and should be considered35 represent the radius of paralleling binding timeliness decay36 generations The diameter of table paralleling binding timeliness decay37 represent diameter when paralleling binding is not decayed38, which represent paralleling binding, does not decline Radius when subtracting39 are represented as logarithm turnover rate ln (γ), and 40 representatives judge that rock mass particle starts when timeliness deterioration decays Stress threshold values41 represent tensile strength42 represent the paralleling binding stress ratio for considering moment of flexure contribution factor43 Represent control rock mass and carefully see the second control parameter β that particles parallel bonds timeliness deterioration₂, 44 represent the normal direction in two dimensional touch face to Measure n_n, 45 represent the tangential unit vector n in two dimensional touch face_s。

Embodiment

Below in conjunction with the accompanying drawings with specific construction step and embodiment, model of the present invention is explained in detail.Example Illustrate it is only understanding of the auxiliary for the present invention, the practical ranges without limiting the present invention.After the present invention has been read, Modification of the those skilled in the art to the various equivalent form of values of the present invention belongs to the apllied claim of the present invention and limited Fixed scope.

Note：Formula has been write exactly before all labels in specification, is formula label such as formula (1).

As shown in Fig. 1~Figure 11, the present invention considers that the two-dimentional season cracking model of moment of flexure contribution factor is adapted to two dimension Grain discrete element, Particles in Two Dimensions discontinuous deformation analysis, Particles in Two Dimensions manifold member；It is curved that two-dimentional season cracking model includes consideration The two-dimensional parallel bond analysis model of square contribution factor, consider that the two-dimensional parallel of moment of flexure contribution factor bonds the deterioration of diameter timeliness and declined Size reduction mode, consider moment of flexure contribution effect and with mole-coulomb season cracking criterion and rock mass season cracking of stretching cut-off limit The two-dimensional linear contact model of damping effect is considered afterwards.

Consider that the two-dimensional parallel bond analysis model of moment of flexure contribution factor refers to that two-dimensional parallel bonds direct stress calculation formulaIn be provided with moment of flexure contribution factorConsider that moment of flexure bonds the contribution journey of direct stress to two-dimensional parallel Degree；

The rock mass of i-th of contact carefully sees particles parallel and bonds normal forceComputational methods be：I-th The rock mass of individual contact carefully sees particles parallel and bonds tangential moment of flexureComputational methods be：

Above-mentionedIn formula,The direct stress of particle two-dimensional parallel bonding is carefully seen for i-th of rock mass,Respectively Particles parallel bonding normal force, tangential moment of flexure are carefully seen for the rock mass of i-th of contact；Particle two-dimensional parallel bonding is carefully seen for rock mass Radius,For moment of flexure contribution factor,I is the moment of inertia that rock mass carefully sees particle two-dimensional parallel bonding, and A is that rock mass is carefully seen Particle two-dimensional parallel bond area, i be followed successively by first to last rock mass carefully see particles parallel bond number,For rock mass Thin particle two-dimensional parallel of seeing bonds normal stiffness,Particle two-dimensional parallel is carefully seen for rock mass and bonds normal direction displacement increment, Particle two-dimensional parallel is carefully seen for rock mass and bonds circumferentially opposite rotating angle increment, +=is the reflexive operator of addition, and -=is that subtraction is reflexive Operator, normal direction moment of flexure

Consider moment of flexure contribution factor two-dimensional parallel bond diameter timeliness deterioration evanescent mode include considering moment of flexure contribution because The two-dimensional parallel of son bonds timeliness deterioration evanescent mode, when rock mass carefully sees particle two-dimensional parallel and bonds timeliness deterioration decay, if The related exponential type pattern of paralleling binding stress to considering moment of flexure contribution factor is put, the two dimension in this exponential type pattern is put down Row bonds diameter and decay is progressively deteriorated with the time, sees paralleling binding diameter formulaFormula In,To consider that the two-dimensional parallel of moment of flexure contribution factor bonds normal stress,Glued to judge that rock mass carefully sees particle two-dimensional parallel Knot starts stress threshold values during timeliness deterioration decay,The tensile strength of particle two-dimensional parallel bonding is carefully seen for rock mass,For Consider the two-dimensional parallel bond stress ratio of moment of flexure contribution factor, β₁、β₂Respectively rock mass carefully sees particles parallel and bonds timeliness deterioration First control parameter, the second control parameter,The diameter that particle two-dimensional parallel bonding deteriorates decay with the time is carefully seen for rock mass, Diameter when particle two-dimensional parallel bonding does not decay is carefully seen for rock mass, e is natural constant, and Δ t carefully sees particle two dimension for rock mass and put down Row bonds the incremental time of timeliness decay deterioration；Particle two-dimensional parallel bond area is carefully seen there is provided rock mass and the moment of inertia timeliness is bad Change evanescent mode, see paralleling binding areal calculation formula when paralleling binding unit thickness is 1 respectivelyIt is parallel Bond the moment of inertia calculation formula when unit thickness is 1Wherein, β carefully sees particle two-dimensional parallel for rock mass and glued The exponential type timeliness decay factor of diameter is tied, its calculation formula is shown inIn formula,A'、I'、It is expressed as rock Body carefully see particle two-dimensional parallel bonding with the time deteriorate decay paralleling binding diameter, paralleling binding radius, paralleling binding area, Paralleling binding the moment of inertia, paralleling binding diameter multiplier, A、I、Particles parallel is carefully seen for rock mass and is bonded and is not declined Paralleling binding diameter, paralleling binding radius, paralleling binding area, paralleling binding the moment of inertia, paralleling binding diameter multiplier when subtracting；

Particle two-dimensional parallel bonding carefully to be seen according to this two-dimentional exponential type timeliness deterioration evanescent mode estimation rock mass broken simultaneously The initial time step-length split, is shown in formulaWherein, Particle two-dimensional parallel bonding diameter multiplier, n are carefully seen for the rock mass of i-th of contact_cFor first Rock mass carefully sees the number that particle two-dimensional parallel bonds the cycle calculations needed for rupture, β_σ、β_τRespectively it is flat carefully to see particle two dimension for rock mass Row bonds the timeliness deterioration factor corresponding to tensile strength, the timeliness deterioration factor corresponding to two-dimensional parallel bond shear strength, and i is successively Particles parallel is carefully seen for first to last rock mass and bonds number, and ∞ is infinity.Rock mass carefully sees particle two-dimensional parallel bonding Timeliness corresponding to tensile strength deteriorates factor-beta_σTimeliness corresponding to particle two-dimensional parallel bond shear strength is carefully seen with rock mass to deteriorate Factor-beta_τCalculation formula be respectively

Wherein,The paralleling binding normal force of respectively i-th particle contact, paralleling binding tangential force, The tangential moment of flexure of paralleling binding,The cohesive strength of particle two-dimensional parallel bonding is carefully seen for rock mass,Particle two-dimensional parallel is carefully seen for rock mass Cohesive internal friction angle.

Consider moment of flexure contribution effect and mole-coulomb season cracking criterion with stretching cut-off limit refers to judging that rock mass is thin When seeing particle two-dimensional parallel bonding season cracking, effect is contributed using embedded consideration moment of flexure and ends rubbing for limit with stretching That-coulomb season cracking criterion of strength judges, sees formula

Wherein, f^s、fⁿRespectively rock mass carefully sees the timeliness shear fracture criterion of particle two-dimensional parallel bonding, timeliness stretching is broken Split criterion,Respectively rock mass carefully sees particle two-dimensional parallel and bonds tensile strength, shearing strength,Point Not Wei i-th of time effect containing exponential type contacted and consider moment of flexure contribution factor rock mass carefully seeing particle two-dimensional parallel bond just Stress, shear stress,

The rock mass of i-th of time effect containing exponential type contacted and consideration moment of flexure contribution factor is carefully seen particle two-dimensional parallel and glued Knot direct stress calculation formula beThe rock of the time effect containing exponential type of i-th of contact Body carefully see particle two-dimensional parallel bond shear stress calculation formula be

Exponential type time effect is contained in the two-dimensional parallel bond stress of the criterion, it is flat to see that rock mass carefully sees particle two dimension Row bonds the exponential type timeliness decay factor calculation formula of diameterβ₁、β₂Point It Wei not control rock mass carefully to see particles parallel and bond the first control parameter of timeliness deterioration, the second control parameter；

f^sParticle two-dimensional parallel is carefully seen more than or equal to 0 expression rock mass and bonds shear fracture, and particle is carefully seen less than 0 expression rock mass Two-dimensional parallel bonds, and shear fracture does not occur；fⁿParticle two-dimensional parallel, which is carefully seen, more than or equal to 0 expression rock mass bonds tensile fracture, it is small Carefully see particle two-dimensional parallel in 0 expression rock mass and bond and tensile fracture does not occur.

Consider that the two-dimensional linear contact model of damping effect refers to that rock mass carefully sees particles parallel bonding after rock mass season cracking After season cracking, reference distance g is contacted by given two-dimensional linear_rParticle two-dimensional linear contact distance g is seen there is provided thin_s, see Rock mass carefully sees the contact of particle two-dimensional linear away from calculation formulaWherein, For rock mass internal particle and particle two Dimensional linear contact jaw a coordinate,For rock mass internal particle and particle two-dimensional linear contact jaw b coordinate, R^a、R^bRespectively Two-dimensional linear contact jaw a particle radius and two-dimensional linear contact jaw b particle radius are carefully seen for rock mass；Set and consider that rock mass is thin The two-dimensional linear contact mode of stress deformation, sets the work for considering two-dimentional sliding friction line power between rock mass particle between sight particle With pattern, rock mass carefully sees the two-dimensional linear contact normal direction linear force calculation formula of stress deformation between particleTake M_l=1 is relative vector accumulation mode, takes M_l=0 is the cumulative mould of absolute vectors Formula, the two-dimensional linear that rock mass carefully sees stress deformation between particle contact tangential linear force calculation formula and areWherein,Respectively rock mass carefully sees the two-dimensional linear contact of stress deformation between particle Normal direction linear force, tangential linear force, k_n、k_sRespectively rock mass carefully sees the two-dimensional linear contact normal direction, tangential of stress deformation between particle Linear rigidity, Δ δ_n、Δδ_sRespectively Normal Displacement increment, tangential displacement increment,Respectively initial normal force Increment size and tangential force increment size,Stiction when not slided for particle, Carefully seen for rock mass Particle force of sliding friction, by friction coefficient μ withProduct obtains；Four kinds of damping modes of two dimensional touch are set simultaneously, wherein Normal direction damping uses full normal mode M_d={ 0,2 } and tensionless winkler foundation pattern M_dTwo kinds of={ 1,3 }, passes through formulaCalculate, wherein m_cFor equivalent particle quality, by formulaCalculate, tangential damping uses full shear mode M_d={ 0,1 } and cunning-cut-off-die formula M_d=2, 3 }, according to formulaTo calculate, wherein：Respectively normal direction damping force, tangential Damping force, β_nFor normal direction damped coefficient, β_sFor tangential damped coefficient,For normal direction speed, tangential velocity, F^*It is thin for rock mass The full normal direction damping force of particle linear contact is seen, expression formula ism_cFor equivalent particle quality, m⁽¹⁾For The rock mass of two-dimensional linear contact jaw 1 carefully sees granular mass, m⁽²⁾Rock mass for two-dimensional linear contact jaw 2 carefully sees granular mass.

The present invention considers the construction method of the two-dimentional season cracking model of moment of flexure contribution factor, comprises the following steps：

Step 1：Setting rock mass carefully see particles parallel bonded contact geometric parameters quantity include paralleling binding area with it is parallel Bond the moment of inertia, R^a、R^bThe respectively particle radius at two-dimensional parallel bonded contact a ends, the particle radius at b ends,Carefully seen for rock mass Particles parallel bonds diameter or radius multiplier, under two-dimensional case, paralleling binding area A when paralleling binding unit thickness is 1 Determined respectively by formula (2), formula (3) with paralleling binding the moment of inertia I：

Step 201：The initial time step size increments Δ that particle two-dimensional parallel bonds timeliness decay deterioration is carefully seen using rock mass T, the diameter that rock mass carefully sees particle two-dimensional parallel and bond timeliness decay deterioration is calculated by exponential type function, formula (4) determines：

Wherein：To consider that the two-dimensional parallel of moment of flexure contribution factor bonds normal stress,To judge rock mass carefully sight Grain two-dimensional parallel bonds stress threshold values when starting timeliness deterioration decay,The stretching of particle two-dimensional parallel bonding is carefully seen for rock mass Intensity,To consider the two-dimensional parallel bond stress ratio of moment of flexure contribution factor, β₁、β₂Respectively rock mass is carefully seen particles parallel and glued The first control parameter, the second control parameter of timeliness deterioration are tied,Particle two-dimensional parallel bonding is carefully seen for rock mass to deteriorate with the time The diameter of decay,Diameter when particle two-dimensional parallel bonding does not decay is carefully seen for rock mass, e is natural constant, and Δ t is that rock mass is thin See the incremental time that particle two-dimensional parallel bonds timeliness decay deterioration；

Step 202：According to the formula (4) in step 201, setting rock mass carefully sees the finger that particle two-dimensional parallel grain bonds diameter Number type timeliness decay factor β, is shown in formula (5)：

Step 204：The rock mass for calculating j-th to k-th successively carefully sees the two-dimensional parallel bonding that particle includes time effect Normal direction moment of flexure increment；Under two-dimensional case, by the speed of paralleling binding both ends particle, angular speed and given cycle calculations time Step size increments Δ t_c, determine that i-th of rock mass carefully sees particle two-dimensional parallel Binder Phase by formula (8), formula (9), formula (10) To cornerTwo-dimensional parallel bonds normal direction incremental displacementAnd two-dimensional parallel bonds tangential incremental displacementAgain With reference to the formula (5) in the formula (7) and step 202 in step 203, determine that i-th of rock mass carefully sees particle and include time effect Two-dimensional parallel bond moment of flexure increment, be specifically shown in formula (11)：

Wherein, ff, j, k are natural numbers, and 2≤j≤ff≤k, j are the rock comprising time effect in each cycle calculations Body carefully sees particle two-dimensional parallel and bonds uncracked initial index value after decay, and ff is some middle index value, and k is to follow every time During ring calculates, the rock mass comprising time effect carefully sees particle two-dimensional parallel and bonds uncracked most end index value after decay, i the One to last paralleling binding particle index value,Respectively i-th of rock mass carefully sees particle two Tie up a ends of paralleling binding contact and the absolute movement speed and angular speed at b ends, n_nAnd n_sParticle two-dimensional parallel is carefully seen for rock mass to glue The normal direction unit vector of contact surface and tangential unit vector are tied,WithRespectively rock mass is carefully seen particle two-dimensional parallel and glued Connection to displacement increment and tangential displacement increment,Particles parallel is carefully seen for rock mass and bonds normal stiffness,Carefully seen for rock mass Particle two-dimensional parallel bonds moment of flexure increment.

Rock mass carefully sees the determination that particle two-dimensional parallel bonds the initial time step size increments Δ t of timeliness decay deterioration, is to adopt The two-dimentional exponential type pattern to be decayed with the paralleling binding timeliness deterioration of moment of flexure contribution factor is considered, by the two dimension in each time step Paralleling binding is decayed the rupture time be lost to determine, namely carried out by first paralleling binding by exponential type pattern first Decay ruptures the lasted time divided by until the calculating cycle-index required for first paralleling binding rupture is initial to estimate Time step increment Delta t, is shown in formulaWherein, Particle two-dimensional parallel bonding diameter multiplier, n are carefully seen for the rock mass of i-th of contact_cFor first Rock mass carefully sees the number that particle two-dimensional parallel bonds the cycle calculations needed for rupture, β_σ、β_τRespectively it is flat carefully to see particle two dimension for rock mass Row bonds the timeliness deterioration factor corresponding to tensile strength, the timeliness deterioration factor corresponding to two-dimensional parallel bond shear strength, and i is successively Particles parallel is carefully seen for first to last rock mass and bonds number, and ∞ is infinity.

Rock mass carefully sees particle two-dimensional parallel and bonds timeliness deterioration factor-beta corresponding to tensile strength_σParticle two is carefully seen with rock mass Tie up timeliness deterioration factor-beta corresponding to paralleling binding shear strength_τDetermination comprise the following steps, wherein, included in these steps Formula subscript 1 represent first by exponential type pattern carry out timeliness decay deterioration two-dimensional parallel bond rupture label：

Step 214：WillSubstitute into formulaAnd make β=β_σ, willSubstitute into formulaAnd make β=β_τ, accordingly, respectively obtain the rock mass and carefully see particle two-dimensional parallel Bond the timeliness deterioration factor corresponding to tensile strengthAnd rock mass is thin See the timeliness deterioration factor corresponding to particle two-dimensional parallel bond shear strength

Normal force：

Tangential force：

Tangential moment of flexure：

Normal direction moment of flexure：

Wherein：Respectively i-th of rock mass carefully sees particle and includes time effect The paralleling binding normal force answered, paralleling binding tangential force, the paralleling binding normal direction moment of flexure comprising time effect, paralleling binding are tangential Moment of flexure, paralleling binding Normal Displacement increment and paralleling binding tangential displacement increment,Particle two-dimensional parallel bonding is carefully seen for rock mass Shear stiffness, +=is the reflexive operator of addition, and -=is the reflexive operator of subtraction.

Step 206：Moment of flexure contribution factor is setConsider that moment of flexure particles parallel of seeing thin to rock mass bonds normal direction The percentage contribution of direct stress, direct stress calculation formula is bonded according to two-dimensional parallelBonded with two-dimensional parallel Calculation Shear formulaSimultaneously by A, I in the two formula andWith A', I' andReplace, then by step Formula (5) in formula (6) and formula (7) and step 202 in 203 substitutes into, and obtains comprising exponential type time effect and curved The two-dimensional parallel of square contribution factor bonds direct stressShear stress is bonded with two-dimensional parallelCalculation formula, formula is seen respectively And formula (17) (16).

Step 207：Time effect will be included in step 206Substitute into formula (18), it may be determined that consider moment of flexure Contribution factor and mole-coulomb season cracking criterion limited with stretching cut-off, and j-th to k-th two dimension is calculated successively Paralleling binding stress, whether ruptured and fracture mode for judging that rock mass is carefully seen particles parallel and bonded；In the rock mass of the criterion Thin see contains exponential type time effect and consideration moment of flexure contribution factor in particles parallel bond stress.

Rock mass is carefully seen after particles parallel bonds and rupture, and rock mass carefully sees the motor pattern of particle using considering damping effect Two-dimensional linear contact model express, for describe rock mass carefully see particles parallel bond after season cracking it is thin see particle should Power, deformation and moving law, consider that the structure of the two-dimensional linear contact model of damping effect comprises the following steps：

g_s=| U |-g_r (22)

Wherein：V_pWith V_qEquivalence, V_pWith V_qTo contact the intergranular speed of related movement in both ends in rock mass, p, q are index etc. Valency symbol, p=1, q=1 represent that particle contacts with particle, and expression particle contacts with wall when p=2, q=2,For The speed of particle and the contact b end units of particle or particle with wall,Be particle with particle or particle with The speed of the contact a end units of wall,It is angular speed of the particle with the contact a end units of particle or particle with wall,It is angular speed of the particle with the contact b end units of particle or particle with wall,Be particle with particle or The displacement at contact a end of the grain with wall,It is displacement of the particle with the contact b ends of particle or particle with wall,Refer to for displacement The middle transition symbol of conversion is marked,The speed at the contact a ends of pellet-pellet or particle-wall when index symbol is p is represented,The speed at the contact a ends of pellet-pellet or particle-wall when index symbol is q is represented,When expression index symbol is p The speed at the contact b ends of pellet-pellet or particle-wall,Represent pellet-pellet or particle-wall when index symbol is q Contact b ends speed (only a ends and two, b ends contact jaw).

R=min (R_a,R_b) (27)

ΔU_p1=V_p1Δt (30)

Wherein：R is the equivalent redius that rock mass carefully sees particle, and m is that rock mass carefully sees granular mass, and J1 is that rock mass carefully sees particle Rotary inertia；k^{It is flat}Particle system translational stiffness, k are carefully seen for rock mass^TurnParticle system rotational stiffness is carefully seen for rock mass；ΔU_p1For rock mass The thin total displacement increment for seeing the contact of particle two-dimensional linear, Δ δ_n、Physical significance is identical, represents that rock mass carefully sees particle two dimension The Normal Displacement increment of linear contact, Δ δ_s、Physical significance is identical, represents that rock mass carefully sees the contact of particle two-dimensional linear Tangential displacement increment, V_p1With V_q1The speed of related movement at particle contact both ends is carefully seen for rock mass, n is unit normal vector, p1, q1 For tensor index figure shift.

Step 308：The normal direction linear force that rock mass carefully sees the contact of particle two-dimensional linear takes relative vector to add up (M_l=1) and Absolute vectors adds up (M_l=0) pattern, calculated and obtained by formula (34), (35)；Rock mass carefully sees the contact of particle two-dimensional linear Tangential linear force carefully sees particle contact slide using rock mass to represent, is calculated and obtained by formula (36)：

Step 309：The normal direction damping that rock mass carefully sees particle linear contact uses full normal mode M_d={ 0,2 } and tensionless winkler foundation Pattern M_dTwo kinds of={ 1,3 }, calculated by formula (37), wherein m_cFor equivalent particle quality, calculated by formula (38), rock mass is thin The tangential damping for seeing particle linear contact uses full shear mode M_d={ 0,1 } and cunning-cut-off-die formula M_d={ 2,3 }, according to formula (39) calculate,

Experiment embodiment

Below using deep rock mass as example, the detailed process of the Numerical Implementation of model of the present invention is described in detail with reference to accompanying drawing, please be join Figure 13 to Figure 15 during example figure illustrates and Fig. 1 to Figure 11 in model brief description of the drawings is read, to understand model of the present invention Numerical Implementation step and effect：

Step 1：Using C++ programming languages, and fish language is combined, flow chart is built according to the model structure of the present invention (Figure 12), the two-dimentional season cracking model for considering moment of flexure contribution factor is realized on numerical value platform.

Step 2：Primarily determine that the rill evolution of strata model

Particle diameter is than Rratio, linear contact normal stiffness kn (Fig. 6), linear contact shear stiffness ks (Fig. 6), grain density Ba_rho, particle contact modulus b_Ec, paralleling binding normal stiffness pb_kn (Fig. 6), paralleling binding shear stiffness pb_ks (figures 6), paralleling binding model pb_Ec, the coefficient of friction ba_fric of particle, the average value pb_sn_ of paralleling binding tensile strength It is mean, the standard deviation pb_sn_sdev of paralleling binding tensile strength, paralleling binding cohesive strength average value pb_coh_mean, parallel Bond cohesive strength standard deviation pb_coh_sdev, paralleling binding radius multiplier gamma (Fig. 7), paralleling binding moment of flexure contribution factor Beta, normal direction damped coefficient Apfan (Fig. 6), tangential damped coefficient Apfas (Fig. 6) and internal friction angle pb_phi (Fig. 8) etc. 19 Individual parameter, parameter occurrence are shown in Table one.

Step 3：Generate rock specimens model

Paralleling binding tensile strength and the cohesive strength distribution of model are determined according to Gaussian Profile or weibull distributions, is passed through Uniformly random function distribution determines the particle diameter distribution of particle；Pass through isotropic stress adjusting method and adaptive dynamic swelling Method, the position of particle is adjusted, reduce particle lap；By suspended particulate elimination method, isolated particle is deleted, improves model sample Globality, reduce the generation of defect model.Cast material paralleling binding intensive parameter is finally assigned, generation has true rock mass The rock specimens model of structure.A diameter of 50mm of strata model, highly it is 100mm (Figure 13).

Step 4：The rill evolution of Accurate Calibration rock specimens

The load-deformation curve obtained by indoor single shaft and triaxial compression test, determine the macroscopic elastic modulus of rockPeak strength σ_p, and Poisson's ratioBy optimization method, make rock mass list, triaxial compressions model stress- The stress-strain and macroscopic deformation Parameters and intensive parameter of strain curve and laboratory test coincide, model constructed by acquisition Meso-damage evolution parameter.

Step 5：Rock mass timeliness mechanics parameter is demarcated

A series of timeliness mechanical test under the conditions of different stress-strength ratios is carried out to rock mass, obtains different stress-strength ratios Under the conditions of rock mass deformation Temporal Evolution curve (Figure 14).By parameter fitting method, the secular distortion of actual rock mass is matched Journey, determine rock mass timeliness old oil β₁、β₂。

Step 6：Rock mass timeliness mechanics numerical experimentation

Under conditions of load is certain, different moment of flexure contribution factors is set respectively, obtains moment of flexure contribution factor to rock Body secular distortion and the affecting laws (Figure 15) destroyed.

Table one：The parameter name and value of model of the present invention

In above-described embodiment, the symbol and the symbol in Fig. 1~Figure 11 and brief description of the drawings of formula are mutually corresponding.

Other unspecified parts are prior art, and all of above parameter can be by consulting handbook or calculating Arrive.The present invention is not strictly limited to above-described embodiment.The particular embodiment of the present invention is the foregoing is only, is not used to limit The system present invention.Any modification, equivalent substitution and improvement for being made within the spirit and principles of the invention etc., all in the present invention Protection domain within.

Claims

A kind of 1. construction method for the two-dimentional season cracking model for considering moment of flexure contribution factor, it is characterised in that：Including following step Suddenly：

Step 1：The geometric parameters quantity that setting rock mass carefully sees particles parallel bonded contact includes paralleling binding area and paralleling binding The moment of inertia, R^a、R^bThe respectively particle radius at two-dimensional parallel bonded contact a ends, the particle radius at b ends,Particle is carefully seen for rock mass Two-dimensional parallel bonds diameter multiplier, and under two-dimensional case, rock mass when paralleling binding unit thickness is 1 carefully sees particle two-dimensional parallel Bond area A and paralleling binding the moment of inertia I are determined by formula (2), formula (3) respectively：

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<mrow> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mi>I</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msup> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> <mn>3</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

Wherein：Particle two-dimensional parallel is carefully seen for rock mass and bonds radius,Particle two-dimensional parallel bonding diameter is carefully seen for rock mass to multiply Number, A are that rock mass carefully sees particle two-dimensional parallel bond area, and I is that rock mass carefully sees particle two-dimensional parallel bonding the moment of inertia；

Step 201：The initial time step size increments Δ t that particle two-dimensional parallel bonds timeliness decay deterioration is carefully seen using rock mass, is led to Cross exponential type function and calculate rock mass and carefully see the diameter that particle two-dimensional parallel bonds timeliness decay deterioration, formula (4) determines：

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Wherein：To consider that the two-dimensional parallel of moment of flexure contribution factor bonds normal stress,To judge that rock mass carefully sees particle two dimension Paralleling binding starts stress threshold values during timeliness deterioration decay,The tensile strength of particle two-dimensional parallel bonding is carefully seen for rock mass,To consider the two-dimensional parallel bond stress ratio of moment of flexure contribution factor, β₁、β₂When respectively rock mass carefully sees particles parallel bonding The first control parameter, the second control parameter of deterioration are imitated,Particle two-dimensional parallel bonding is carefully seen for rock mass and is deteriorated with the time and is decayed Diameter,Paralleling binding diameter when particle two-dimensional parallel bonding does not decay is carefully seen for rock mass, e is natural constant, and Δ t is rock Body carefully sees the initial time step size increments that particle two-dimensional parallel bonds timeliness decay deterioration；

Step 202：According to the formula (4) in step 201, setting rock mass carefully sees the exponential type that particle two-dimensional parallel grain bonds diameter Timeliness decay factor β, see formula (5)：

<mrow> <mi>&beta;</mi> <mo>=</mo> <msup> <mover> <mi>D</mi> <mo>&OverBar;</mo> </mover> <mo>&prime;</mo> </msup> <mo>/</mo> <mover> <mi>D</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <msup> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> <mo>&prime;</mo> </msup> <mo>/</mo> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <msup> <mover> <mi>&lambda;</mi> <mo>&OverBar;</mo> </mover> <mo>&prime;</mo> </msup> <mo>/</mo> <mover> <mi>&lambda;</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mo><</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mrow> <mi>a</mi> <mi>a</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&beta;</mi> <mn>1</mn> </msub> <msup> <mi>e</mi> <mrow> <msub> <mi>&beta;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mo>/</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>c</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mfrac> <mrow> <mi>&Delta;</mi> <mi>t</mi> </mrow> <mover> <mi>D</mi> <mo>&OverBar;</mo> </mover> </mfrac> <mo>,</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mrow> <mi>a</mi> <mi>a</mi> </mrow> </msub> <mo>&le;</mo> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mo><</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>c</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

Wherein：A'、I'、It is expressed as rock mass and carefully sees particle two-dimensional parallel bonding with the parallel of time deterioration decay Diameter, paralleling binding radius, paralleling binding area, paralleling binding the moment of inertia, paralleling binding diameter multiplier are bonded,A、I、Paralleling binding diameter, the paralleling binding when particle two-dimensional parallel bonding does not decay are carefully seen for rock mass Radius, paralleling binding area, paralleling binding the moment of inertia, paralleling binding diameter multiplier；

Step 203：According to the formula (5) in the formula (1) in step 1~formula (3) and step 202, setting rock mass is carefully seen Particle two-dimensional parallel bonds geometric parameter timeliness deterioration evanescent mode；Rock mass carefully sees particle two-dimensional parallel and bonds diameter over time Increase and constantly deteriorate decay, two-dimensional parallel bonds the paralleling binding area and paralleling binding the moment of inertia when unit thickness is 1 Increase over time and constantly deteriorate decay, see formula (6), formula (7) respectively：

<mrow> <msup> <mi>A</mi> <mo>&prime;</mo> </msup> <mo>=</mo> <mn>2</mn> <msup> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> <mo>&prime;</mo> </msup> <mo>=</mo> <mi>&beta;</mi> <mi>A</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msup> <mi>I</mi> <mo>&prime;</mo> </msup> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <msup> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> <mrow> <mo>&prime;</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>&beta;</mi> <mn>3</mn> </msup> <mi>I</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

Wherein：A', I' are expressed as rock mass and carefully see the paralleling binding half that particle two-dimensional parallel bonding deteriorates decay with the time Footpath, paralleling binding area, paralleling binding the moment of inertia, A, I carefully see paralleling binding face when particles parallel bonding does not decay for rock mass Product, paralleling binding the moment of inertia；

Step 204：The rock mass for calculating j-th to k-th successively carefully sees the two-dimensional parallel bonding normal direction that particle includes time effect Moment of flexure increment；Under two-dimensional case, by the speed of paralleling binding both ends particle, angular speed and given cycle calculations time step Increment Delta t_c, determine that i-th of rock mass carefully sees particle two-dimensional parallel Binder Phase to turning by formula (8), formula (9), formula (10) AngleTwo-dimensional parallel bonds normal direction incremental displacementAnd two-dimensional parallel bonds tangential incremental displacementIn conjunction with The formula (5) in formula (7) and step 202 in step 203, determines that i-th of rock mass carefully sees particle includes time effect two Paralleling binding moment of flexure increment is tieed up, is specifically shown in formula (11)：

<mrow> <msub> <mrow> <mo>(</mo> <msubsup> <mi>&Delta;U</mi> <mi>i</mi> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mi>i</mi> <mi>b</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mi>i</mi> <mi>a</mi> </msubsup> <mo>)</mo> </mrow> <msub> <mi>n</mi> <mi>n</mi> </msub> <msub> <mi>&Delta;t</mi> <mi>c</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mrow> <mo>(</mo> <msubsup> <mi>&Delta;U</mi> <mi>i</mi> <mi>s</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mi>i</mi> <mi>b</mi> </msubsup> <mo>-</mo> <msubsup> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mi>i</mi> <mi>a</mi> </msubsup> <mo>)</mo> </mrow> <msub> <mi>n</mi> <mi>s</mi> </msub> <msub> <mi>&Delta;t</mi> <mi>c</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mrow> <mo>(</mo> <msubsup> <mi>&Delta;&theta;</mi> <mi>i</mi> <mi>n</mi> </msubsup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mi>&omega;</mi> <mi>i</mi> <mi>b</mi> </msubsup> <mo>-</mo> <msubsup> <mi>&omega;</mi> <mi>i</mi> <mi>a</mi> </msubsup> <mo>)</mo> </mrow> <msub> <mi>&Delta;t</mi> <mi>c</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mrow> <mo>(</mo> <mi>&Delta;</mi> <msup> <msub> <mover> <mi>M</mi> <mo>&OverBar;</mo> </mover> <mi>i</mi> </msub> <mi>s</mi> </msup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>k</mi> <mo>&OverBar;</mo> </mover> <mi>n</mi> </msub> <msup> <mi>&beta;</mi> <mn>3</mn> </msup> <mi>I</mi> <msub> <mrow> <mo>(</mo> <msup> <msub> <mi>&Delta;&theta;</mi> <mi>i</mi> </msub> <mi>n</mi> </msup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

Wherein, ff, j, k are natural numbers, and 2≤j≤ff≤k, j are that the rock mass comprising time effect is thin in each cycle calculations See particle two-dimensional parallel and bond uncracked initial index value after decay, ff is some middle index value, and k counts for circulation every time In calculation, the rock mass comprising time effect carefully sees particle two-dimensional parallel and bonds uncracked most end index value after decay, and i is first To last paralleling binding particle index value,It is flat that respectively i-th of rock mass carefully sees particle two dimension The a ends of row bonded contact and the absolute movement speed and angular speed at b ends, n_nAnd n_sParticle two-dimensional parallel bonding is carefully seen for rock mass to connect The normal direction unit vector and tangential unit vector of contacting surface,WithRespectively rock mass carefully sees particle two-dimensional parallel mull technique To displacement increment and tangential displacement increment,Particles parallel is carefully seen for rock mass and bonds normal stiffness,Particle is carefully seen for rock mass Two-dimensional parallel bonds moment of flexure and increased；

Step 205：Formula (8), formula (9) and formula in the formula (6) and formula (7), step 204 in step 203 (11) formula (5) and in step 202, successively renewal calculate j-th to k-th rock mass and carefully see particles parallel and bond and do not rupture And the two-dimensional parallel comprising time effect bonds normal force, tangential force and tangential moment of flexure；Pass through formula (12), formula (13), public affairs Formula (14) calculates two-dimensional parallel bonding normal force, tangential force and the tangential moment of flexure that i-th of rock mass carefully sees particle contact；In two-dimentional feelings Under condition, normal direction moment of flexure is bonded to determine that rock mass carefully sees particles parallel by formula (15)：

Normal force：

Tangential force：

Tangential moment of flexure：

Normal direction moment of flexure：

Wherein：Respectively i-th of rock mass carefully sees particle and includes time effect Paralleling binding normal force, paralleling binding tangential force, the paralleling binding normal direction moment of flexure comprising time effect, paralleling binding are tangentially curved Square, paralleling binding Normal Displacement increment and paralleling binding tangential displacement increment,Particle two-dimensional parallel bonding is carefully seen for rock mass to cut To rigidity, +=is the reflexive operator of addition, and -=is the reflexive operator of subtraction；

Step 206：Moment of flexure contribution factor is setConsider that moment of flexure particles parallel bonding normal direction of seeing thin to rock mass just should The percentage contribution of power, direct stress calculation formula is bonded according to two-dimensional parallelBond to cut with two-dimensional parallel and answer Power calculation formulaSimultaneously by A, I in the two formula andWith A', I' andReplace, then by step 203 Formula (6) and formula (7) and step 202 in formula (5) substitute into, obtain and contributed comprising exponential type time effect and moment of flexure The two-dimensional parallel of the factor bonds direct stressShear stress is bonded with the two-dimensional parallel comprising exponential type time effectCalculate public Formula, formula (16) and formula (17) are seen respectively；

<mrow> <msub> <msup> <mrow> <mo>(</mo> <mover> <msub> <mi>&sigma;</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mo>)</mo> </mrow> <mo>&prime;</mo> </msup> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mrow> <mo>(</mo> <msup> <mover> <msub> <mi>F</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>n</mi> </msup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <mover> <mi>&beta;</mi> <mo>&OverBar;</mo> </mover> <mo>|</mo> <msub> <mrow> <mo>(</mo> <msup> <mover> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <msup> <mrow> <mo>(</mo> <mover> <msub> <mi>&tau;</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mo>)</mo> </mrow> <mo>&prime;</mo> </msup> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <msub> <mrow> <mo>(</mo> <msup> <mover> <msub> <mi>F</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msup> <mo>)</mo> </mrow> <mrow> <mi>f</mi> <mi>f</mi> </mrow> </msub> <mo>|</mo> </mrow> <mrow> <mn>2</mn> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

Step 207：Time effect will be included in step 206Substitute into formula (18), it may be determined that consider moment of flexure contribution The factor and mole-coulomb season cracking criterion limited with stretching cut-off, and j-th to k-th two-dimensional parallel is calculated successively Bond stress, whether ruptured and fracture mode for judging that rock mass is carefully seen particles parallel and bonded；Carefully seen in the rock mass of the criterion Exponential type time effect is contained in particle two-dimensional parallel bond stress and considers moment of flexure contribution factor；

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mi>f</mi> <mi>s</mi> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>&tau;</mi> <mo>&OverBar;</mo> </mover> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mover> <mi>&tau;</mi> <mo>&OverBar;</mo> </mover> <mi>c</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>&tau;</mi> <mo>&OverBar;</mo> </mover> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&prime;</mo> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&prime;</mo> </msup> <mi>tan</mi> <mover> <mi>&phi;</mi> <mo>&OverBar;</mo> </mover> <mo>-</mo> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <msup> <mover> <msub> <mi>F</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msup> <mo>|</mo> </mrow> <mrow> <mn>2</mn> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mrow> <mo>(</mo> <mo>-</mo> <msup> <mover> <msub> <mi>F</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msup> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <mover> <mi>&beta;</mi> <mo>&OverBar;</mo> </mover> <mo>|</mo> <msup> <mover> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msup> <mo>|</mo> </mrow> <mrow> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mo>)</mo> </mrow> <mi>tan</mi> <mover> <mi>&phi;</mi> <mo>&OverBar;</mo> </mover> <mo>-</mo> <mover> <mi>c</mi> <mo>&OverBar;</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mi>f</mi> <mi>n</mi> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&prime;</mo> </msup> <mo>-</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>c</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mrow> <mo>(</mo> <mo>-</mo> <msup> <mover> <msub> <mi>F</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>n</mi> </msup> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <mover> <mi>&beta;</mi> <mo>&OverBar;</mo> </mover> <mo>|</mo> <msup> <mover> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo>&OverBar;</mo> </mover> <mi>s</mi> </msup> <mo>|</mo> </mrow> <mrow> <mi>&beta;</mi> <mover> <mi>R</mi> <mo>&OverBar;</mo> </mover> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>&sigma;</mi> <mo>&OverBar;</mo> </mover> <mi>c</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>

Wherein：f^s、fⁿRespectively it is accurate carefully to see the timeliness shear fracture criterion of particle two-dimensional parallel bonding, timeliness tensile fracture for rock mass Then,Two-dimensional parallel for the time effect containing exponential type of i-th of contact bonds shear stress,For i-th of contact Time effect containing exponential type and the two-dimensional parallel bonding direct stress for considering moment of flexure contribution factor,Respectively rock mass is carefully seen Tensile strength, the shearing strength of particle two-dimensional parallel bonding,The cohesive strength of particle two-dimensional parallel bonding is carefully seen for rock mass,For Rock mass carefully sees the internal friction angle of particle two-dimensional parallel bonding；f^sParalleling binding shear fracture is represented more than or equal to 0, is represented less than 0 Shear fracture does not occur for paralleling binding；fⁿParalleling binding tensile fracture is represented more than or equal to 0, is not sent out less than 0 expression paralleling binding Raw tensile fracture；

Step 208：If the f in formula (18) in step 207^sOr fⁿMore than or equal to 0, show that rock mass carefully sees the parallel of particle Bonding is ruptured, hereafter rock mass carefully see the motor pattern of particle using the two-dimensional linear contact model of consideration damping effect come Expression；If the f in formula (18) in step 207^sAnd fⁿBoth less than 0, show that paralleling binding does not rupture, continue cycling through step 201 to 207, calculate, renewal, judge that rock mass carefully sees the paralleling binding state of particle contact, up to rock mass do not produce it is new parallel Bond rupture or paralleling binding rupture accelerated development and form macroscopic failure, loop termination.
2. the construction method of the two-dimentional season cracking model according to claim 1 for considering moment of flexure contribution factor, its feature It is：The rock mass carefully sees the determination that particle two-dimensional parallel bonds the initial time step size increments Δ t of timeliness decay deterioration, is to adopt The two-dimentional exponential type pattern to be decayed with the paralleling binding timeliness deterioration of moment of flexure contribution factor is considered, by the two dimension in each time step Paralleling binding is decayed the rupture time be lost to determine, namely carried out by first paralleling binding by exponential type pattern first Decay ruptures the lasted time divided by until the calculating cycle-index required for first paralleling binding rupture is initial to estimate Time step increment Delta t, is shown in formulaWherein, Particle two-dimensional parallel bonding diameter multiplier, n are carefully seen for the rock mass of i-th of contact_cFor first Rock mass carefully sees the number that particle two-dimensional parallel bonds the cycle calculations needed for rupture, β_σ、β_τRespectively it is flat carefully to see particle two dimension for rock mass Row bonds the timeliness deterioration factor corresponding to tensile strength, the timeliness deterioration factor corresponding to two-dimensional parallel bond shear strength, and i is successively Particles parallel is carefully seen for first to last rock mass and bonds number, and ∞ is infinity.
3. the construction method of the two-dimentional season cracking model according to claim 2 for considering moment of flexure contribution factor, its feature It is：The rock mass carefully sees particle two-dimensional parallel and bonds timeliness deterioration factor-beta corresponding to tensile strength_σParticle two is carefully seen with rock mass Tie up timeliness deterioration factor-beta corresponding to paralleling binding shear strength_τDetermination comprise the following steps, wherein, included in these steps Formula subscript 1 represent first by exponential type pattern carry out timeliness decay deterioration two-dimensional parallel bond rupture label；

Step 211：Under two-dimensional case, particles parallel is carefully seen by rock mass and bonds the speed of both ends particle, angular speed and given Cycle calculations time step increment Delta t_c, pass through formulaDetermine that the relative of paralleling binding contact turns AnglePass through formulaDetermine paralleling binding normal direction incremental displacementPass through public affairs FormulaDetermine the tangential incremental displacement of paralleling bindingPass through formulaDetermine the moment of flexure increment of paralleling binding contact；

Step 212：According to the formula in step 211Pass through formulaDetermine paralleling binding normal force；According to the formula in step 211Pass through formulaDetermine paralleling binding tangential force；According to step Formula in 211And formulaPass through formulaDetermine the tangential moment of flexure of paralleling binding；Pass through formulaParalleling binding normal direction moment of flexure is determined, Wherein, +=is the reflexive operator of addition, and -=is the reflexive operator of subtraction；

Step 213：Under two-dimensional case, pass through formulaParalleling binding direct stress is determined, is led to Cross formulaDetermine paralleling binding shear stress, by A, I in the two formula andWith A', I' andReplace, so The formula (5) in the formula (6) and formula (7) and step 202 in step 203 is substituted into afterwards, acquisition includes the exponential type time The two-dimensional parallel of effect and moment of flexure contribution factor bonds direct stress calculation formulaWith comprising The two-dimensional parallel of exponential type time effect bonds Calculation Shear formula

Step 214：WillSubstitute into formulaAnd make β=β_σ, willSubstitute into formulaAnd make β=β_τ, accordingly, respectively obtain the rock mass and carefully see particle two-dimensional parallel Bond the timeliness deterioration factor corresponding to tensile strengthAnd rock mass is thin See the timeliness deterioration factor corresponding to particle two-dimensional parallel bond shear strength
4. the construction method of the two-dimentional season cracking model according to claim 1 for considering moment of flexure contribution factor, its feature It is：The rock mass is carefully seen after particles parallel bonds and rupture, and rock mass carefully sees the motor pattern of particle using considering damping effect The two-dimensional linear contact model answered is expressed, for describe rock mass carefully see particles parallel bond after season cracking it is thin see particle should Power, deformation and moving law, consider that the structure of the two-dimensional linear contact model of damping effect comprises the following steps：

Step 301：By Monte Carlo searching algorithms, traversal find rock mass carefully see each two-dimensional linear contact jaw a of particle, Two-dimensional linear contact jaw b, particle and particle or the centre coordinate of particle and wall, under two-dimensional case, counted by formula (19) Calculate both centre distances：

<mrow> <mi>d</mi> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mi>b</mi> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mi>a</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>y</mi> <mi>i</mi> <mi>b</mi> </msubsup> <mo>-</mo> <msubsup> <mi>y</mi> <mi>i</mi> <mi>a</mi> </msubsup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>

Wherein：D is the centre distance between two-dimensional linear contact both ends particle and particle or particle and wall,For two dimension Linear contact end a coordinate,For two-dimensional linear contact jaw b coordinate；

Step 302：Rock mass is carefully seen the unit vector of each contact point between particle and calculated by formula (20) under two-dimensional plane state, If the contact between particle and particle, using obtained in step 301 two-dimensional linear contact both ends center point coordinate and away from From calculating, if particle contacts with wall, directly calculated using the normal vector equivalence replacement of wall, it is determined that each contact-segment Unit vector：

Wherein：n_iFor the unit vector of contact,For contact jaw b direction vector,For contact jaw a direction vector, n_wall To constrain the direction vector of wall；

Step 303：After rock mass carefully sees particles parallel bonding rupture, the contact lap U of two-dimensional linear contact-segment, pass through step 301 calculate the particle radius R at grain spacing d and two-dimensional linear contact both ends^a、R^b, recycle formula (21) to determine；Pass through Set particle two-dimensional linear contact reference distance g_r, and formula (22) is combined, determine the distance g of particle two-dimensional linear contact_s：

g_s=| U |-g_r (22)

Step 304：Determine that rock mass carefully sees grain contact point normal direction, tangential equivalent stiffness, using contact both ends particle entities or The rigidity k of wall^a, k^bThe equivalent rigidity instead of contact point, calculated by formula (23)：

<mrow> <msub> <mi>K</mi> <mi>n</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>k</mi> <mi>n</mi> <mi>a</mi> </msubsup> <msubsup> <mi>k</mi> <mi>n</mi> <mi>b</mi> </msubsup> </mrow> <mrow> <msubsup> <mi>k</mi> <mi>n</mi> <mi>a</mi> </msubsup> <mo>+</mo> <msubsup> <mi>k</mi> <mi>n</mi> <mi>b</mi> </msubsup> </mrow> </mfrac> <mo>,</mo> <msub> <mi>K</mi> <mi>s</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>k</mi> <mi>s</mi> <mi>a</mi> </msubsup> <msubsup> <mi>k</mi> <mi>s</mi> <mi>b</mi> </msubsup> </mrow> <mrow> <msubsup> <mi>k</mi> <mi>s</mi> <mi>a</mi> </msubsup> <mo>+</mo> <msubsup> <mi>k</mi> <mi>s</mi> <mi>b</mi> </msubsup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>

Wherein：K_n、K_sFor equivalent normal stiffness and shear stiffness,For particle and the contact of particle or particle with wall The normal direction and shear stiffness at a ends,Normal direction and shear stiffness for particle with the contact b ends of particle or particle with wall；

Step 305：Determine to contact the intergranular speed of related movement in both ends in rock mass, counted using formula (24), formula (25) Calculate；Wherein e_pqzFor Ricci index alternators, calculated according to formula (26)：

<mrow> <msub> <mi>V</mi> <mi>p</mi> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>p</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mi>b</mi> </msub> <mo>-</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>p</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mi>a</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>p</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <msub> <mi>e</mi> <mrow> <mi>p</mi> <mi>q</mi> <mi>z</mi> </mrow> </msub> <msubsup> <mi>&omega;</mi> <mi>q</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>p</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <msub> <mi>e</mi> <mrow> <mi>p</mi> <mi>q</mi> <mi>z</mi> </mrow> </msub> <msubsup> <mi>&omega;</mi> <mi>q</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>V</mi> <mi>q</mi> </msub> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>q</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mi>b</mi> </msub> <mo>-</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>q</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mi>a</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>q</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <msub> <mi>e</mi> <mrow> <mi>q</mi> <mi>p</mi> <mi>z</mi> </mrow> </msub> <msubsup> <mi>&omega;</mi> <mi>p</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <msubsup> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mi>q</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </msubsup> <mo>+</mo> <msub> <mi>e</mi> <mrow> <mi>q</mi> <mi>p</mi> <mi>z</mi> </mrow> </msub> <msubsup> <mi>&omega;</mi> <mi>p</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </msubsup> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mi>z</mi> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>

Wherein：V_pWith V_qEquivalence, V_pWith V_qTo contact the intergranular speed of related movement in both ends in rock mass, p, q are index symbol of equal value Number, p=1, q=1 represent that particle contacts with particle, and expression particle contacts with wall when p=2, q=2,For particle With the speed of the contact b end units of particle or particle with wall,It is particle and particle or particle and wall The speed of a end units is contacted,It is angular speed of the particle with the contact a end units of particle or particle with wall,It is angular speed of the particle with the contact b end units of particle or particle with wall,It is particle and particle or particle The displacement at the contact a ends with wall,It is displacement of the particle with the contact b ends of particle or particle with wall,For drift index The middle transition symbol of conversion,The speed at the contact a ends of pellet-pellet or particle-wall when index symbol is p is represented,The speed at the contact a ends of pellet-pellet or particle-wall when index symbol is q is represented,When expression index symbol is p The speed at the contact b ends of pellet-pellet or particle-wall,Represent pellet-pellet or particle-wall when index symbol is q Contact b ends speed, only a ends and two, b ends contact jaw；

Step 306：For the initial time step size increments Δ t of linear contact model value, estimated by formula (29) minimum Time step Δ t, it is ensured that the calculating time step of constructed model is less than the value, you can ensure system integral calculate tend to be steady It is fixed；By formula (30), formula (31), formula (32) determine the total displacement increment of each linear contact, Normal Displacement increment and Tangential displacement increment：

R=min (R^a,R^b) (27)

<mrow> <mi>J</mi> <mn>1</mn> <mo>=</mo> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> <msup> <mi>&pi;R</mi> <mn>5</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>

ΔU_p1=V_p1Δt (30)

<mrow> <msub> <mi>&Delta;&delta;</mi> <mi>n</mi> </msub> <mo>=</mo> <msubsup> <mi>&Delta;U</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mi>l</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>V</mi> <mrow> <mi>q</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow> <mi>q</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msub> <mi>&Delta;</mi> <mi>t</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>&Delta;&delta;</mi> <mi>s</mi> </msub> <mo>=</mo> <msubsup> <mi>&Delta;U</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> <mrow> <mi>s</mi> <mi>l</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>&Delta;U</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msubsup> <mi>&Delta;U</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mi>l</mi> </mrow> </msubsup> <mo>=</mo> <msub> <mi>V</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msub> <mi>&Delta;</mi> <mi>t</mi> <mo>-</mo> <msub> <mi>V</mi> <mrow> <mi>q</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow> <mi>q</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow> <mi>p</mi> <mn>1</mn> </mrow> </msub> <mi>&Delta;</mi> <mi>t</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>

Wherein：R carefully sees the equivalent redius of particle, R for rock mass^a、R^bRespectively particle radius, the b at two-dimensional parallel bonded contact a ends The particle radius at end, m are that rock mass carefully sees granular mass, and J1 is the rotary inertia that rock mass carefully sees particle；k^{It is flat}Particle is carefully seen for rock mass System translational stiffness, k^TurnParticle system rotational stiffness is carefully seen for rock mass；ΔU_p1The total of particle two-dimensional linear contact is carefully seen for rock mass Displacement increment, Δ δ_n、Physical significance is identical, represents that rock mass carefully sees the Normal Displacement increment of particle two-dimensional linear contact, Δδ_s、Physical significance is identical, represents that rock mass carefully sees the tangential displacement increment of particle two-dimensional linear contact, V_p1With V_q1For Rock mass carefully sees the speed of related movement at particle contact both ends, and n is unit normal vector, and p1, q1 are tensor index figure shift；

Step 307：The ultimate range as existing for formula (22) judgement rock mass carefully sees particle surface contact permission, calculates normal direction and cuts To displacement updating factor, in addition, rock mass carefully sees the renewal of particle two-dimensional linear contact normal direction displacement increment using back The product of Normal Displacement increment and updating factor α obtains, and rock mass carefully sees the renewal of particle two-dimensional linear contact tangential displacement increment It is to be obtained using the tangential displacement increment of back and updating factor α product；

<mrow> <mi>&alpha;</mi> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <msub> <mi>g</mi> <mi>s</mi> </msub> <mrow> <msub> <mi>g</mi> <mi>s</mi> </msub> <mo>-</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>g</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mn>0</mn> </msub> </mrow> </mfrac> <mo>,</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>g</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> <mo>,</mo> <msub> <mi>g</mi> <mi>s</mi> </msub> <mo><</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>1</mn> <mo>,</mo> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mi>w</mi> <mi>i</mi> <mi>s</mi> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mrow>

Wherein：(g_s)₀The surface that initial time is calculated for model contacts distance, g_sThe distance of particle contact is carefully seen for rock mass, α is Displacement updating factor；

Step 308：The normal direction linear force that rock mass carefully sees the contact of particle two-dimensional linear takes relative vector to add up (M_l=1) and definitely Vector adds up (M_l=0) pattern, calculated and obtained by formula (34), (35)；Rock mass carefully sees the tangential of particle two-dimensional linear contact Linear force carefully sees particle contact slide using rock mass to represent, is calculated and obtained by formula (36)：

<mrow> <msubsup> <mi>F</mi> <mi>n</mi> <mi>l</mi> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mi>n</mi> </msub> <msub> <mi>g</mi> <mi>s</mi> </msub> <mo>,</mo> <msub> <mi>g</mi> <mi>s</mi> </msub> <mo>&le;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mi>w</mi> <mi>i</mi> <mi>s</mi> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>M</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mi>F</mi> <mi>n</mi> <mi>l</mi> </msubsup> <mo>)</mo> </mrow> <mi>o</mi> </msub> <mo>+</mo> <msub> <mi>k</mi> <mi>n</mi> </msub> <msub> <mi>&Delta;&delta;</mi> <mi>n</mi> </msub> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>M</mi> <mi>l</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msubsup> <mi>F</mi> <mi>s</mi> <mo>*</mo> </msubsup> <mo>=</mo> <msub> <mrow> <mo>(</mo> <msubsup> <mi>F</mi> <mi>s</mi> <mi>l</mi> </msubsup> <mo>)</mo> </mrow> <mi>o</mi> </msub> <mo>-</mo> <msub> <mi>k</mi> <mi>s</mi> </msub> <msub> <mi>&Delta;&delta;</mi> <mi>s</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msubsup> <mi>F</mi> <mi>s</mi> <mi>l</mi> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>F</mi> <mi>s</mi> <mo>*</mo> </msubsup> <mo>,</mo> <mo>|</mo> <mo>|</mo> <msubsup> <mi>F</mi> <mi>s</mi> <mo>*</mo> </msubsup> <mo>|</mo> <mo>|</mo> <mo>&le;</mo> <msubsup> <mi>F</mi> <mi>s</mi> <mi>&mu;</mi> </msubsup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msubsup> <mi>F</mi> <mi>s</mi> <mi>&mu;</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>F</mi> <mi>s</mi> <mo>*</mo> </msubsup> <mo>/</mo> <mo>|</mo> <mo>|</mo> <msubsup> <mi>F</mi> <mi>s</mi> <mo>*</mo> </msubsup> <mo>|</mo> <mo>|</mo> <mo>)</mo> </mrow> <mo>,</mo> <mi>o</mi> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mi>r</mi> <mi>w</mi> <mi>i</mi> <mi>s</mi> <mi>e</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>

Wherein：Respectively rock mass carefully sees the two-dimensional linear contact normal direction linear force, tangential linear of stress deformation between particle Power, k_n、k_sRespectively rock mass carefully sees the two-dimensional linear contact normal direction of stress deformation between particle, tangential linear rigidity, Δ δ_n、Δδ_s Respectively rock mass carefully sees the Normal Displacement increment of particle two-dimensional linear contact, tangential displacement increment,Respectively rock Body carefully sees the initial normal force increment size of particle two-dimensional linear contact, tangential force increment size,Particle is carefully seen for rock mass not slide When stiction,Particle force of sliding friction is carefully seen for rock mass, its value can by friction coefficient μ withProduct obtains；

Step 309：The normal direction damping that rock mass carefully sees particle linear contact uses full normal mode M_d={ 0,2 } and tensionless winkler foundation pattern M_dTwo kinds of={ 1,3 }, calculated by formula (37), wherein m_cFor equivalent particle quality, calculated by formula (38), rock mass carefully sight The tangential damping of grain linear contact uses full shear mode M_d={ 0,1 } and cunning-cut-off-die formula M_d={ 2,3 }, come according to formula (39) Calculate,

<mrow> <msubsup> <mi>F</mi> <mi>n</mi> <mi>d</mi> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>&beta;</mi> <mi>n</mi> </msub> <msqrt> <mrow> <msub> <mi>m</mi> <mi>c</mi> </msub> <msub> <mi>k</mi> <mi>n</mi> </msub> </mrow> </msqrt> <mo>)</mo> <msub> <mover> <mi>&delta;</mi> <mo>&CenterDot;</mo> </mover> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>M</mi> <mi>d</mi> </msub> <mo>=</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>(</mo> <msup> <mi>F</mi> <mo>*</mo> </msup> <mo>,</mo> <mo>-</mo> <msubsup> <mi>F</mi> <mi>n</mi> <mi>l</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>M</mi> <mi>d</mi> </msub> <mo>=</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msubsup> <mi>F</mi> <mi>s</mi> <mi>d</mi> </msubsup> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>&beta;</mi> <mi>s</mi> </msub> <msqrt> <mrow> <msub> <mi>m</mi> <mi>c</mi> </msub> <msub> <mi>k</mi> <mi>s</mi> </msub> </mrow> </msqrt> <mo>)</mo> <msub> <mover> <mi>&delta;</mi> <mo>&CenterDot;</mo> </mover> <mi>s</mi> </msub> <mo>,</mo> <msub> <mi>M</mi> <mi>d</mi> </msub> <mo>=</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> <msub> <mi>M</mi> <mi>d</mi> </msub> <mo>=</mo> <mo>{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>39</mn> <mo>)</mo> </mrow> </mrow>

Wherein：Respectively rock mass carefully sees the linear damping force of normal direction of particle linear contact, tangential linear damping power, β_n The normal direction damped coefficient of particle linear contact, β are carefully seen for rock mass_sThe tangential damped coefficient of particle linear contact, k are carefully seen for rock mass_n The normal direction linear rigidity of particle linear contact, k are carefully seen for rock mass_sThe tangential linear rigidity of particle linear contact is carefully seen for rock mass, Respectively rock mass carefully sees the normal direction speed and tangential velocity of particle linear contact, F^*Particle linear contact is carefully seen for rock mass Full normal direction damping force, expression formula ism_cEquivalent particle quality, m are carefully seen for rock mass⁽¹⁾Carefully seen for rock mass The thin sight granular mass of particle contact jaw 1, m⁽²⁾The thin sight granular mass of particle contact jaw 2, F are carefully seen for rock mass^dCarefully seen for rock mass The total damping power of particle linear contact.