CN106599476B - Rock mass two dimension is thin to see season cracking power function type model method - Google Patents
Rock mass two dimension is thin to see season cracking power function type model method Download PDFInfo
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Abstract
Particle bond stress two-dimensional model is carefully seen the invention discloses the rock mass of the thin sight season cracking power function pattern type of rock mass two dimension, including consideration moment of flexure contribution factor, the thin sight particle of consideration moment of flexure contribution factor bonds the thin sight particle linear contact two dimensional model that two-dimentional power function pattern formula, the consideration moment of flexure that timeliness deterioration decays contribute effect and particle bonding season cracking criterion, consideration damping effect carefully seen with mole coulomb that stretching cut-off limits.The relation that the present invention is adapted between stress and crack propagation velocity meets this kind of rock mass of power function type, 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
Technical field
Season cracking analysis technical field is carefully seen the present invention relates to engineering rock mass, when being seen in particular to a kind of rock mass two dimension is thin
Effect rupture power function type model method.
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.《Buried griotte Fracture propagation time effect
Particle flow simulation》One text has carried out experiment and two-dimensional numerical analysis (rock mechanics to the time effect of silk screen griotte rupture
With engineering journal, 2011, Vol.30No.10:1989-1996)；《Silk screen griotte creep impairment evolution mesomechanics feature
Numerical simulation study》The one text two-dimentional creep meso mechanical model of application is short-term to silk screen griotte and long-term strength feature is carried out
Numerically modeling (rock-soil mechanics, 2013, Vol.34No.12:3601-3608).This class model is to build driving stress with exponential type
Relation between crack propagation velocity, for describing season cracking of the rock carefully in sight aspect, suitable for stress and Crack Extension
This kind of rock mass of index of coincidence expression way between speed.In addition, also there is following weak point in this class model：(1) between particle
Shear fracture criterion be a horizontal linear parallel with paralleling binding direct stress, namely this shear fracture criterion with it is parallel
It is unrelated to bond direct stress state, as long as paralleling binding shear stress is more than or equal to fixed paralleling binding shear fracture intensity,
Shear fracture can occur for intergranular, and can not embody different paralleling binding direct stress in rock mass has different paralleling binding shear fractures
The objective fact of intensity；(2) do not account for bonding influence of the difference effect of torque to contact failure, the contribution of torque will be bonded
The influence to different lithology is spent to be accordingly to be regarded as unanimously；(3) for not conforming to symbol index expression way between stress and crack propagation velocity
Rock mass, this class model lack adaptability.
The content of the invention
It is an object of the invention to for drawbacks described above, it is proposed that a kind of rock mass two dimension is thin to see season cracking power function pattern
Type method, the relation that the present invention is adapted between stress and crack propagation velocity meet this kind of rock mass of power function type, for flat
The prediction of country rock long-time stability, evaluation and the optimization design of this kind of deep rock mass engineering project provide technical support under surface state.
The purpose of the present invention is reached by following measure：Rock mass two dimension is thin to see season cracking power function pattern type side
Method, it is characterized in that, the power function pattern type is adapted to Particles in Two Dimensions discrete element, Particles in Two Dimensions Discontinuous deformation anaysis
Method, Particles in Two Dimensions manifold member；The power function pattern type carefully sees particle bonding including the rock mass based on moment of flexure contribution factor should
Power two-dimensional model, the rock mass based on moment of flexure contribution factor carefully see particle bond timeliness deterioration decay two-dimentional power function pattern formula,
Mole-coulomb contributed effect based on moment of flexure and limited with stretching cut-off carefully sees particle and bonds season cracking criterion and based on damping
The thin sight particle linear contact two dimensional model of effect.
Further, the rock mass based on moment of flexure contribution factor carefully sees particle bond stress two-dimensional model and refers to that rock mass is thin
See particle and bond direct stress two dimension calculation formula
In be provided with moment of flexure contribution factorBased on the moment of flexure contribution journey seen particle two dimension and bond direct stress thin to rock mass
Degree；Above-mentionedIn formula,Particle two dimension, which is carefully seen, for i-th of rock mass bonds direct stress,Respectively i-th contact
Rock mass carefully see particle two dimension bond normal force, tangential moment of flexure,Particle two dimension is carefully seen for rock mass and bonds radius,For moment of flexure tribute
Offer the factor,I is that rock mass carefully sees the moment of inertia that particle two dimension bonds, and A is that rock mass carefully sees particle two dimension bond area, i
Be followed successively by first to last rock mass carefully see particle bond number.
Further, the rock mass based on moment of flexure contribution factor carefully sees the two-dimentional power that particle bonds timeliness deterioration decay
Function type pattern be included in rock mass carefully see particle two dimension bond timeliness deterioration decay when, there is provided with based on moment of flexure contribution factor
Bond stress correlation power function pattern formula, the rock mass in this power function pattern formula carefully see particle two dimension bond diameter with the time by
Step deterioration decay, see cohesive diameter formula,
In formula,Normal stress is bonded for the two dimension based on moment of flexure contribution factor,To judge that rock mass carefully sees particle two dimension
Stress threshold values when starting timeliness deterioration decay is bonded,Particle two dimension is carefully seen for rock mass and bonds tensile strength,For based on
The two-dimentional bond stress ratio of moment of flexure contribution factor, β1、β2Respectively control rock mass carefully sees the first control that particle bonds timeliness deterioration
Parameter, the second control parameter,Particle two dimension is carefully seen for rock mass and bonds the diameter that decay is deteriorated with the time,Carefully seen for rock mass
Grain two dimension bonds diameter when not decaying, and Δ t is that rock mass carefully sees the incremental time that particle two dimension bonds timeliness decay deterioration；Set
Rock mass carefully sees particle bond area and face the moment of inertia timeliness deterioration decay two-dimensional model, when to see cohesive unit thickness respectively be 1
Rock mass carefully see particle with the time deteriorate decay bond area A' calculation formulaWhen bonding unit thickness is 1
Bonding the moment of inertia I' calculation formulaWherein, A, I are respectively that rock mass carefully sees gluing when particle bonding does not decay
Junction area and bonding the moment of inertia,A', I' are respectively that rock mass carefully sees the bonding radius, viscous that particle bonding deteriorates decay with the time
Junction area and bonding the moment of inertia, β are that rock mass carefully sees the power function type timeliness decay factor that particle two dimension bonds diameter, and it calculates public affairs
Formula is shown in
In formula,A'、I'、It is expressed as rock mass and carefully sees the bonding that particle two dimension bonding deteriorates decay with the time
Diameters, radius, bond area are bonded, the moment of inertia is bonded, bonds diameter multiplier,A、I、Carefully seen for rock mass
Particle bonds bonding diameter, bonding radius, bond area, bonding the moment of inertia, the bonding diameter multiplier when not decaying；While according to
This power function type timeliness deterioration evanescent mode estimation rock mass carefully sees the initial time step delta t that particle bonds rupture, sees formula
Wherein, Particle two dimension bonding diameter multiplier, n are carefully seen for the rock mass of i-th of contactc
The number of the cycle calculations needed for particle two dimension bonding rupture, β are carefully seen for first rock massσ、βτRespectively rock mass carefully sees particle two
Dimension bonds the timeliness deterioration factor, the timeliness deterioration factor corresponding to two-dimentional bond shear strength corresponding to tensile strength, and i is followed successively by the
Particle carefully being seen to last rock mass for one and bonding number, ∞ is infinity.
Further, mole-coulomb contributed effect based on moment of flexure and limited with stretching cut-off is carefully seen particle and bonded
Season cracking criterion refers to, when rock mass carefully sees particle two dimension and bonds season cracking, effect be contributed based on moment of flexure using embedded
And mole-coulomb season cracking the criterion with stretching cut-off limit judges, sees formula
Wherein, fs、fnRespectively rock mass carefully sees timeliness shear fracture criterion, the timeliness tensile fracture standard that particle two dimension bonds
Then,Respectively rock mass carefully sees particle two dimension and bonds tensile strength, shearing strength,Respectively i-th
The time effect of type containing power function of individual contact and the rock mass based on moment of flexure contribution factor carefully see particle two dimension and bond direct stress, cut and answer
Power,The internal friction angle that particle two dimension bonds carefully is seen for rock mass,The cohesive strength that particle two dimension bonds carefully is seen for rock mass,For rock
Body carefully sees particle two dimension and bonds radius,The bonding normal force and tangential force of respectively i-th particle contact,
Tangential moment of flexure is bonded,For moment of flexure contribution factor,I be followed successively by first to last rock mass carefully see particle bond
Number；
Power function type time effect is contained in the two-dimentional bond stress of the criterion, sees that rock mass is carefully seen particle two dimension and bonded
The timeliness decay factor β calculation formula of diameterIn formulaParticle is carefully seen for rock mass
Two dimension bonding deteriorates the diameter of decay with the time, and D carefully sees diameter when particle two dimension bonding does not decay for rock mass,For base
In the two-dimentional bond stress ratio of moment of flexure contribution factor,Particle two dimension, which is carefully seen, for i-th of rock mass bonds normal stress,To judge
Rock mass carefully sees particle two dimension and bonds the stress threshold values started when timeliness deterioration decays,The drawing that particle two dimension bonds carefully is seen for rock mass
Intensity is stretched, Δ t is that rock mass carefully sees the incremental time that particle two dimension bonds timeliness decay deterioration, β1、β2Respectively control rock mass is carefully seen
Particle bonds the first control parameter of timeliness deterioration, the second control parameter；fsIt is viscous that particle two dimension is carefully seen more than or equal to 0 expression rock mass
Shear fracture is tied, carefully seeing particle two dimension bonding less than 0 expression rock mass, shear fracture does not occur；fnRepresent that rock mass is thin more than or equal to 0
See particle two dimension and bond tensile fracture, carefully seeing particle two dimension bonding less than 0 expression rock mass, tensile fracture does not occur.
Further, the thin sight particle linear contact two dimensional model based on damping effect refers to carefully see in rock mass
After grain bonds season cracking, pass through two dimensional touch reference distance grParticle two-dimensional linear contact distance g is seen there is provided thins, see rock mass
It is thin to see the contact of particle two-dimensional linear away from calculation formula
Wherein,For rock mass internal particle and particle two-dimensional linear contact jaw a coordinate,For rock mass internal particle with
Grain two-dimensional linear contact jaw b coordinate, Ra、RbRespectively rock mass carefully sees two-dimensional linear contact jaw a particle radius and two-dimensional linear
Contact jaw b particle radius；There is provided carefully seeing the two-dimensional linear contact mode that is deformed between particle based on rock mass, carefully seen in rock mass
The binding mode based on two-dimentional sliding friction line power is provided between grain, and the two-dimensional linear that rock mass carefully sees stress deformation between particle connects
Touch normal direction linear force calculation formulaTake Ml=1 is relative vector accumulation mode, takes Ml
=0 is absolute vectors accumulation mode, and rock mass carefully sees the tangential linear force calculation formula of two-dimensional linear contact of stress deformation between particle
ForWherein,Respectively rock mass is carefully seen the two-dimensional linear of stress deformation between particle and connect
Touch normal direction linear force, tangential linear force, kn、ksRespectively rock mass is carefully seen the two-dimensional linear contact normal direction of stress deformation between particle, cut
To linear rigidity, Δ δn、ΔδsRespectively Normal Displacement increment, tangential displacement increment,Respectively initial normal direction
Power increment size and tangential force increment size,Stiction when not slided for particle, It is thin for rock mass
See particle force of sliding friction, by friction coefficient μ withProduct obtains；
The damping mode of two dimensional touch is set simultaneously, and wherein normal direction damping uses full normal mode Md={ 0,2 } and without drawing
Stretch pattern MdTwo kinds of={ 1,3 }, passes through formulaCalculate, wherein mcFor equivalent particle matter
Amount, by formulaCalculate, tangential damping uses full shear mode Md={ 0,1 } and cunning-cut-off-die
Formula Md={ 2,3 }, according to formulaTo calculate, wherein：Respectively normal direction hinders
Buddhist nun's power, tangential damping force, βnFor normal direction damped coefficient, βsFor tangential damped coefficient, knFor normal direction linear rigidity, ksFor tangential line
Property rigidity,Respectively normal direction speed, tangential velocity, F*The full normal direction damping force of particle linear contact is carefully seen for rock mass,
Expression formula ismcFor equivalent particle quality, m(1)Rock mass for two-dimensional linear contact jaw 1 carefully sees particle
Quality, m(2)Rock mass for two-dimensional linear contact jaw 2 carefully sees granular mass.
Further, the rock mass of i-th of contact carefully sees particle and bonds normal force, tangential moment of flexureMeter
Calculation method is： In formula,Particle two dimension is carefully seen for rock mass and bonds normal stiffness,Particle two dimension is carefully seen for rock mass and bonds normal direction displacement increment,Particle two dimension is carefully seen for rock mass and bonds circumferentially opposite turn
Angle increment, +=is the reflexive operator of addition, and -=is the reflexive operator of subtraction, normal direction moment of flexure
Further, the rock mass carefully sees particle two dimension and bonds timeliness deterioration factor-beta corresponding to tensile strengthσAnd shearing
Timeliness corresponding to intensity deteriorates factor-betaτCalculation formula be respectively
Wherein,Particle two dimension is carefully seen for rock mass and bonds radius,Respectively i-th of particle contact
Normal force is bonded, tangential force is bonded, bonds tangential moment of flexure,For moment of flexure contribution factor, Particle is carefully seen for rock mass
Two dimension bonds tensile strength,The cohesive strength that particle two dimension bonds carefully is seen for rock mass,Carefully seen for rock mass in particle two dimension bonding
Angle of friction,Particle two dimension bonding tangential force is carefully seen for the rock mass of i-th of contact, i is followed successively by first to last rock mass
Thin particle of seeing bonds number.
Further, it is described i-th contact the time effect of type containing power function and based on the rock mass of moment of flexure contribution factor
Carefully the calculation formula of sight particle two dimension bonding direct stress isThe letter containing power of i-th of contact
The rock mass of number type time effects, which carefully sees particle two dimension and bonds the calculation formula of shear stress, is
Rock mass two dimension proposed by the invention is thin to see season cracking power function type model method, its advantage and advantage master
It is embodied in：
(1) rock mass is carefully seen in particle bonding direct stress two dimension calculation formula and is provided with moment of flexure contribution factor in the present invention, no
Only account for moment of flexure and the percentage contribution of direct stress is bonded to thin particle of seeing, and also contemplate shadow of the moment of flexure to rock mass long-term strength
Ring, be adapted to the mesomechanics fracture behaviour of rock mass under description plane stress or plane strain condition.
(2) the two-dimentional power letter that the thin sight particle based on moment of flexure contribution factor bonds timeliness deterioration decay is constructed in the present invention
Number pattern formula, when being included in rock mass and carefully seeing particle and bond timeliness deterioration decay, there is provided power function type with based on moment of flexure contribution because
The thin sight particle of the bond stress correlation of son bonds two dimension deterioration evanescent mode, and there is provided thin sight particle bonding is progressively bad with the time
Change the two-dimensional model of decay, there is provided the thin area and cross sectional moment of inertia timeliness deterioration decay two-dimensional model seen particle and bonded；Together
When according to this timeliness deterioration evanescent mode estimation rock mass carefully see particle bond rupture initial time step-length.This power function type
Forming types are adapted to the mesomechanics season cracking mechanism and response pattern for describing a kind of deep rock mass under flat state.
(3) it is embedded to be contributed based on moment of flexure in the present invention in the constructed thin sight season cracking power function pattern type of two dimension
Effect and mole-coulomb limited with stretching cut-off carefully see particle and bond season cracking criterion.Particle, which is carefully seen, in rock mass bonds timeliness
During rupture, using it is embedded based on moment of flexure contribute effect and with stretching end limit mole-coulomb season cracking criterion come
Judge；, not only can be with comprising power function type time effect and moment of flexure contribution factor is added in the criterion particle bond stress
The difference of timeliness shear fracture intensity related to particle bonding direct stress is described, timeliness tensile fracture can also be carried out rational
Expression, and influence of the moment of flexure to bonding season cracking is considered, meet a kind of rock mass under planar condition and carefully see season cracking pattern.
(4) in the present invention in the constructed thin sight season cracking power function pattern type of two dimension, insertion is based on damping effect
Thin sight particle linear contact two dimensional model structure, rock mass carefully see particle bond season cracking after, pass through specify two dimensional touch
Reference distance setting rock mass carefully sees interparticle contact distance, sets the two dimensional touch mould that stress deformation between particle is carefully seen based on rock mass
Formula and binding mode of the setting based on two-dimentional sliding friction between rock mass carefully sees particle, while the damping of two dimensional touch is set
Pattern, particle motion and stress characteristic of a kind of deep engineering rock mass after season cracking under planar condition can rationally be described.
Brief description of the drawings
Fig. 1 is carefully to see particle in model of the present invention to contact schematic diagram with particle.
Fig. 2 is carefully to see particle in model of the present invention to contact schematic diagram with rigid wall.
Fig. 3 is that particle overlap condition schematic diagram is carefully seen in model of the present invention.
Fig. 4 is that particle Rigidity Calculation schematic diagram is carefully seen in model of the present invention.
Fig. 5 is carefully to see particle in model of the present invention to bond linear tangential force and tangential displacement schematic diagram.
Fig. 6 is carefully to see particles stick in model of the present invention to touch this structure physical model schematic diagram.
Fig. 7 is that the linear adhesive structure schematic diagram of particle is carefully seen in model of the present invention.
Fig. 8 is to contribute effect based on moment of flexure in model of the present invention and carefully see particle with mole-coulomb that stretching cut-off limits
Bond season cracking criterion schematic diagram.
Fig. 9 is carefully to see particle in model of the present invention to bond diameter (or radius) timeliness deterioration decay schematic diagram.
Figure 10 is that the normal direction in particle two dimensional touch face and tangential unit vector schematic diagram are carefully seen in model of the present invention.
Figure 11 is the schematic flow sheet of the construction method of model of the present invention.
Figure 12 is model basic model sample of the present invention
Figure 13 is model creeping displacement and time history of the present invention
In figure：1 represents the centre distance d of two particles, and 2, which represent rock mass, carefully sees intergranular half contact distance, and 3 represent rock mass
Carefully see intergranular half reference distance gr, 4 represent the coordinate that rock mass carefully sees particle a, and 5 represent the coordinate that rock mass carefully sees particle b, and 6
Rock mass carefully sees the centre coordinate of particle surface contact distance, and 7, which represent rock mass, carefully sees particle surface contact distance gs, 8 represent rock mass
Intergranular contact unit normal vector is carefully seen, 9 represent the radius R that rock mass carefully sees particle aa, 10 represent half that rock mass carefully sees particle b
Footpath Rb, 11 represent the contact lap U that rock mass carefully sees grain contact point, and 12 represent the firm of b (rock mass particle or border wall)
Degree (normal direction, shear stiffness are referred to as) kb, 13 represent rigidity (normal direction, the shear stiffness system of a (rock mass particle or border wall)
Claim) ka, 14 represent the equivalent stiffness that rock mass carefully sees grain contact point, and 15 represent total displacement increment Delta Ui, 16 represent initial normal forceIncrement size, 17 representatives initially contact force vectors and 18 represent initial tangential forceIncrement size, 19 represent constructed two
Tie up season cracking model normal direction displacement increment Δ δnOr20 represent constructed two-dimentional season cracking model tangential displacement
Increment Delta δsOr21 represent the tensile strength that rock mass carefully sees particle bonding22, which represent rock mass, carefully sees particle mull technique
To rigidity23 represent the normal stiffness K that rock mass carefully sees grain contact pointn, 24, which represent rock mass, carefully sees particle bonding shear stiffness25, which represent rock mass, carefully sees particle bond shear strength, and 25.1 representThe cohesive strength of particle bonding, 25.2 generations are carefully seen for rock mass
Table rock mass carefully sees particle and bonds internal friction angle26 represent the shear stiffness K that rock mass carefully sees grain contact points, 27 to represent rock mass thin
The particle linear contact coefficient of sliding friction is seen, 28, which represent rock mass, carefully sees particle linear contact normal direction damped coefficient βn, 29 represent rock
Body carefully sees the tangential damped coefficient β of particle linear contacts, 30, which are represented as rock mass, carefully sees particle bonding radius multiplier31 represent rock mass
Thin particle of seeing bonds diameter 232 represent based on moment of flexure contribution effect and with mole-coulomb season cracking standard of stretching cut-off limit
Then, the 33 bonding shear stress comprising time effect for representing i-th of contact34 represent being imitated comprising the time for i-th of contact
Should and the bonding direct stress based on moment of flexure contribution factor35, which represent rock mass, carefully sees the radius that particle bonds timeliness decay
36, which represent rock mass, carefully sees the diameter that particle bonds timeliness decay37, which represent rock mass, carefully sees particle and bonds diameter when not decaying38, which represent rock mass, carefully sees particle and bonds radius when not decaying39 represent the normal vector that rock mass carefully sees particle contact surface
nn, 40, which represent rock mass, carefully sees the tangential unit vector n of particle contact surfaces。
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 10, rock mass two dimension of the present invention is thin to see season cracking power function type model method, is adapted to two dimension
Granular discrete-element, Particles in Two Dimensions discontinuous deformation analysis, Particles in Two Dimensions manifold member；Power function pattern type includes being based on moment of flexure
The rock mass of contribution factor carefully sees particle bond stress two-dimensional model, the rock mass based on moment of flexure contribution factor carefully sees particle and bonds timeliness
The two-dimentional power function pattern formula of decay is deteriorated, effect is contributed based on moment of flexure and mole-coulomb with stretching cut-off limit carefully sees particle
Bond season cracking criterion and the thin sight particle linear contact two dimensional model based on damping effect.
Rock mass based on moment of flexure contribution factor carefully sees particle bond stress two-dimensional model and refers to that rock mass is carefully seen particle and bonded just
Stress two dimension calculation formulaIn be provided with moment of flexure contribution factorBased on moment of flexure to the thin sight particle of rock mass
Two dimension bonds the percentage contribution of direct stress；Above-mentionedIn formula,Particle two dimension, which is carefully seen, for i-th of rock mass bonds direct stress,The rock mass of respectively i-th contact carefully sees particle two dimension and bonds normal force, tangential moment of flexure,Particle is carefully seen for rock mass
Two dimension bonds radius,For moment of flexure contribution factor,I is that rock mass carefully sees the moment of inertia that particle two dimension bonds, and A is rock mass
Thin to see particle two dimension bond area, i, which is followed successively by first and carefully sees particle to last rock mass, bonds number.The rock of i-th of contact
Body carefully sees particle two dimension and bonds normal forceBond tangential moment of flexureComputational methods be：Normal force
In formula,Particle two dimension is carefully seen for rock mass and bonds normal direction displacement increment,Particle two dimension is carefully seen for rock mass and bonds normal stiffness,
Tangential moment of flexureIn formula,Particle two dimension is carefully seen for rock mass and bonds circumferentially opposite rotating angle increment, +=is to add
The reflexive operator of method, -=is the reflexive operator of subtraction, normal direction moment of flexure
The two-dimentional power function pattern formula that rock mass based on moment of flexure contribution factor carefully sees particle bonding timeliness deterioration decay includes
When rock mass carefully sees particle two dimension and bonds timeliness deterioration decay, there is provided related to the bond stress based on moment of flexure contribution factor
Power function pattern formula, the thin sight particle two dimension in this power function pattern formula bond diameter and decay are progressively deteriorated with the time, see viscous
Diameter formula is tied,
In formula,Particle two dimension is carefully seen for rock mass and bonds the diameter that decay is deteriorated with the time,Particle two is carefully seen for rock mass
Dimension bonds diameter when not decaying,Normal stress is bonded for the two dimension based on moment of flexure contribution factor,To judge that rock mass is carefully seen
Particle two dimension bonds stress threshold values when starting timeliness deterioration decay,The tensile strength that particle two dimension bonds carefully is seen for rock mass,For the two-dimentional bond stress ratio based on moment of flexure contribution factor, β1、β2It is bad that particle two dimension bonding timeliness is carefully seen for control rock mass
Two designated indexs changed, Δ t are that rock mass carefully sees the incremental time that particle two dimension bonds timeliness decay deterioration；It is thin there is provided rock mass
Particle bond area and face the moment of inertia timeliness deterioration decay two-dimensional model are seen, sees that rock mass when cohesive unit thickness is 1 is thin respectively
See the bond area calculation formula that particle deteriorates decay with the timeBond the moment of inertia I' when unit thickness is 1
Calculation formulaWherein, β is that rock mass carefully sees the timeliness decay factor that particle two dimension bonds diameter, and it is calculated
Formula is shown in
Wherein,A'、I'、It is expressed as rock mass and carefully sees particle two dimension bonding with the viscous of time deterioration decay
Diameter is tied, radius, bond area is bonded, bonds the moment of inertia, bonds diameter multiplier,A、I、It is thin for rock mass
Bonding diameter when particle bonding does not decay is seen, radius, bond area is bonded, bonds the moment of inertia, bonds diameter multiplier；Press simultaneously
The initial time step delta t of particle bonding rupture is carefully seen according to this power function type timeliness deterioration evanescent mode estimation rock mass, sees public affairs
Formula
Wherein, Particle two dimension bonding diameter multiplier, n are carefully seen for the rock mass of i-th of contactc
The number of the cycle calculations needed for particle two dimension bonding rupture, β are carefully seen for first rock massσ、βτRespectively rock mass carefully sees particle two
Dimension bonds the timeliness deterioration factor, the timeliness deterioration factor corresponding to two-dimentional bond shear strength corresponding to tensile strength, and i is followed successively by the
Particle carefully being seen to last rock mass for one and bonding number, ∞ is infinity.
Rock mass carefully sees particle two dimension and bonds timeliness deterioration factor-beta corresponding to tensile strengthσWith two-dimentional bond shear strength pair
The timeliness deterioration factor-beta answeredτCalculation formula be respectively
Wherein,Bonding normal force, tangential force and the bonding that respectively i-th of particle contacts are tangential curved
Square,Particle two dimension is carefully seen for rock mass and bonds tensile strength,The cohesive strength that particle two dimension bonds carefully is seen for rock mass,For rock mass
The thin internal friction angle seen particle two dimension and bonded.
Contribute effect based on moment of flexure and carefully see particle bonding season cracking criterion with mole-coulomb for stretching cut-off limit and refer to
When rock mass carefully sees particle two dimension and bonds season cracking, effect is contributed and with stretching cut-off limit based on moment of flexure using embedded
Mole-coulomb season cracking criterion judges, sees formula
Its
In, fs、fnRespectively rock mass carefully sees timeliness shear fracture criterion, the timeliness tensile fracture criterion that particle two dimension bonds,
Respectively rock mass carefully sees particle two dimension and bonds tensile strength, shearing strength,Respectively i-th contact contains
Power function type time effect and rock mass based on moment of flexure contribution factor carefully see particle two dimension and bond direct stress, shear stress.
The time effect of type containing power function of i-th of contact and the rock mass based on moment of flexure contribution factor are carefully seen particle two dimension and bonded
The calculation formula of direct stress is
I-th contact the time effect of type containing power function rock mass carefully see particle two dimension bond shear stress calculation formula be
Power function type time effect is contained in the two-dimentional bond stress of the criterion, sees that rock mass is carefully seen particle two dimension and bonded
The timeliness decay factor β calculation formula of diameterβ1、β2Respectively control rock mass thin
See particle and bond the first control parameter of timeliness deterioration, the second control parameter；fsParticle two dimension is carefully seen more than or equal to 0 expression rock mass
Shear fracture is bonded, carefully seeing particle two dimension bonding less than 0 expression rock mass, shear fracture does not occur；fnRock mass is represented more than or equal to 0
Thin particle two dimension of seeing bonds tensile fracture, and carefully seeing particle two dimension bonding less than 0 expression rock mass, tensile fracture does not occur.
Thin sight particle linear contact two dimensional model based on damping effect refers to that carefully seeing particle in rock mass bonds season cracking
Afterwards, reference distance g is contacted by given two-dimensional linearrParticle two-dimensional linear contact distance g is seen there is provided thins, see that rock mass is carefully seen
Particle two-dimensional linear is contacted away from calculation formula
Wherein,For rock mass internal particle and particle two-dimensional linear contact jaw a coordinate,For rock mass inside
The coordinate of particle and particle two-dimensional linear contact jaw b, Ra、RbRespectively rock mass carefully see two-dimensional linear contact jaw a particle radius and
Two-dimensional linear contact jaw b particle radius；There is provided the two-dimensional linear contact mode that is deformed between particle is carefully seen based on rock mass, in rock
Body, which is carefully seen, is provided with the binding mode based on two-dimentional sliding friction line power between particle, and rock mass carefully sees two of stress deformation between particle
Dimensional linear contacts normal direction linear force calculation formulaTake Ml=1 adds up for relative vector
Pattern, take Ml=0 is absolute vectors accumulation mode, and rock mass carefully sees the tangential linear force of two-dimensional linear contact of stress deformation between particle
Calculation formula isWherein,Respectively rock mass carefully sees two of stress deformation between particle
Dimensional linear contact normal direction linear force, tangential linear force, kn、ksRespectively rock mass is carefully seen the two-dimensional linear of stress deformation between particle and connect
Touch normal direction, tangential linear rigidity, Δ δn、ΔδsRespectively Normal Displacement increment, tangential displacement increment,Respectively
Initial normal force increment size and tangential force increment size,Stiction when not slided for particle,
Particle force of sliding friction is carefully seen for rock mass, by friction coefficient μ withProduct obtains；
The damping of the damping mode for setting two-dimensional linear to contact simultaneously, wherein normal direction uses full normal mode Md={ 0,2 } and
Tensionless winkler foundation pattern MdTwo kinds of={ 1,3 }, passes through formulaWherein mcTo be equivalent
Granular mass, by formulaCalculate, tangential damping uses full shear mode Md={ 0,1 } and
Cunning-cut-off-die formula Md={ 2,3 }, according to formulaTo calculate, wherein：Respectively
For normal direction damping force, tangential damping force, βnFor normal direction damped coefficient, βsFor tangential damped coefficient, knFor normal direction linear rigidity, ks
For tangential linear rigidity,For normal direction speed, tangential velocity, mcFor equivalent particle quality.F*Particle line is carefully seen for rock mass
Property contact full normal direction damping force, expression formula ism(1)For the granular mass of two-dimensional linear contact jaw 1,
m(2)For the granular mass of two-dimensional linear contact jaw 2.
As shown in figure 11, the thin structure side for seeing season cracking power function type model method of rock mass two dimension of the present invention
Method, comprise the following steps：
Step 1：Set the thin geometric parameters quantity for seeing particle bonded contact inside rock mass, including particle bond area and particle
Bond the moment of inertia, Ra、RbParticle radius, the particle radius at b ends at respectively two-dimentional bonded contact a ends,Particle is carefully seen for rock mass
Diameter multiplier or radius multiplier are bonded, under two-dimensional case, particle bond area A when unit thickness is 1 is bonded and bonds used
Property square I is determined by formula (2), formula (3) respectively：
Wherein：Particle two dimension is carefully seen for rock mass and bonds radius,Particle two dimension is carefully seen for rock mass and bonds diameter multiplier or half
Footpath multiplier, A are that rock mass carefully sees particle two dimension bond area, and I is that rock mass carefully sees particle two dimension bonding the moment of inertia；
Step 201：The initial time step size increments Δ t that particle two dimension bonds timeliness decay deterioration is carefully seen using rock mass, is led to
Cross power function form calculus rock mass and carefully see the diameter that particle two dimension bonds timeliness decay deteriorationDetermined by formula (4)；
Wherein：Normal stress is bonded for the two dimension based on moment of flexure contribution factor,To judge that rock mass carefully sees particle two dimension
Stress threshold values when starting timeliness deterioration decay is bonded,The tensile strength that particle two dimension bonds carefully is seen for rock mass,For base
In the two-dimentional bond stress ratio of moment of flexure contribution factor, β1、β2Respectively rock mass carefully sees the first control ginseng that particle bonds timeliness deterioration
Number, the second control parameter,Particle two dimension is carefully seen for rock mass and bonds the diameter that decay is deteriorated with the time,Particle is carefully seen for rock mass
Two dimension bonds diameter when not decaying, and Δ t is that rock mass carefully sees the incremental time that particle two dimension bonds timeliness decay deterioration；
Step 202：According to the formula (4) in step 201, the thin timeliness decay factor seen particle two dimension and bond diameter is set
β, see formula (5)：
Wherein：A'、I'、It is expressed as rock mass and carefully sees particle two dimension bonding with the viscous of time deterioration decay
Diameter is tied, radius, bond area is bonded, bonds the moment of inertia, bonds diameter multiplier,A、I、It is thin for rock mass
Bonding diameter when particle two dimension bonding does not decay is seen, radius, bond area is bonded, bonds the moment of inertia, bonds 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 dimension of seeing bonds geometric parameter timeliness deterioration evanescent mode, and in the case of planar, it is viscous that rock mass carefully sees particle two dimension
Knot diameter increases and constantly deteriorates decay over time, bonds area and the moment of inertia that particle when unit thickness is 1 bonds
Increase over time and constantly deteriorate decay, see formula (6), formula (7) respectively；
Wherein：A', I' be expressed as rock mass carefully see particle two dimension bond with the time deteriorate decay bonding radius,
Bond area, the moment of inertia is bonded, A, I are that rock mass carefully sees the two-dimentional bond area bonded when not decaying of particle, bonds the moment of inertia；
Step 204：The rock mass for calculating j-th to k-th successively carefully sees the two dimension bonding normal direction that particle includes time effect
Moment of flexure increment, under two-dimensional case, by the speed, angular speed and the given cycle calculations time step increment that bond both ends particle
Δ t, determine that i-th of rock mass carefully sees particle two dimension and bond relative rotation by formula (8), formula (9), formula (10)Rock
Body carefully sees particle two dimension and bonds normal direction incremental displacementAnd rock mass carefully sees particle two dimension and bonds tangential incremental displacement
In conjunction with the formula (5) in the formula (7) and step 202 in step 203, determine that i-th of rock mass is carefully seen particle and imitated comprising the time
The two dimension answered bonds moment of flexure increment, is specifically shown in formula (11)；
Wherein：Ff, j, k, i are natural numbers, and 2≤j≤ff≤k, j are in each cycle calculations, include time effect
Rock mass carefully sees particle two dimension and bonds uncracked initial index value after decay, and ff is middle index value, and k is each cycle calculations
In, the rock mass comprising time effect carefully sees particle two dimension and bonds uncracked most end index value after decay, and i is first to last
One two-dimentional bonded particulate index value,Respectively i-th of rock mass carefully sees particle two dimension bonded contact
A ends and b ends absolute movement speed and angular speed, nn、nsRespectively rock mass carefully sees the normal direction list in particle two dimension bonded contact face
Bit vector and tangential unit vector, Respectively rock mass carefully sees particle two dimension and bonds normal direction displacement increment and tangential position
Move increment,Particle two dimension is carefully seen for rock mass and bonds normal stiffness,Particle two dimension is carefully seen for rock mass and bonds moment of flexure increment.
Wherein, rock mass carefully sees the determination that particle two dimension bonds the initial time step size increments Δ t of timeliness decay deterioration, is to adopt
The two-dimentional power function pattern formula of timeliness deterioration decay is bonded with the thin sight particle based on moment of flexure contribution factor, by each time step
Thin sight particle two dimension bond decay first and rupture time be lost to determine, i.e., by it is first thin see particle and bond press power
Function type pattern carries out time divided by until the first thin calculating seen required for particle bonding ruptures that decay rupture is lasted
Cycle-index estimates initial time step size increments Δ t, sees formulaWherein,For i-th of contact
Rock mass carefully see particle two dimension and bond diameter multiplier, ncParticle two dimension, which is carefully seen, for first rock mass bonds the required circulation meter of rupture
The number of calculation, βσ、βτRespectively rock mass is carefully seen the timeliness deterioration factor, two dimension bonding corresponding to particle two dimension bonding tensile strength and cut
Timeliness corresponding to shearing stress deteriorates the factor, and i is followed successively by first and carefully sees particle bonding number to last rock mass, and ∞ is infinite
Greatly.
Rock mass carefully sees particle two dimension and bonds timeliness deterioration factor-beta corresponding to tensile strengthσIt is viscous that particle two dimension is carefully seen with rock mass
Tie timeliness deterioration factor-beta corresponding to shear strengthτDetermination comprise the following steps, the formula subscript wherein included in these steps
1, which represents first thin sight particle two dimension that timeliness decay deterioration is carried out by power function pattern formula, bonds rupture label：
Step 211：Under two-dimensional case, particle is carefully seen by rock mass and bonds the speed of both ends particle, angular speed and given
Cycle calculations time step increment Delta tc, pass through formulaIt is determined that carefully see the phase of particle bonded contact
To cornerPass through formulaIt is determined that carefully seeing particle bonds normal direction incremental displacement
Pass through formulaDetermine that particle bonds tangential incremental displacementPass through formulaDetermine the moment of flexure increment of particle bonded contact；
Step 212：According to the formula in step 211Pass through formulaIt is determined that carefully see the bonding normal force of particle contact；According to the formula in step 211Pass through formulaIt is determined that the bonding for carefully seeing particle contact is tangential
Power；According to the formula in step 211And formulaPass through formulaIt is determined that carefully see the tangential moment of flexure of bonding of particle contact；Pass through formulaIt is determined that carefully see particle
The bonding normal direction moment of flexure of contact, wherein, +=is the reflexive operator of addition, and -=is the reflexive operator of subtraction；
Step 213：Under two-dimensional case, pass through formulaIt is determined that carefully see particle contact
Direct stress is bonded, passes through formulaIt is determined that carefully see particle contact bonding shear stress, by A, I in the two formula with
AndWith A', I' andReplace, then by formula (5) generation in the formula (6) and formula (7) and step 202 in step 203
Enter, obtain the two dimension that thin sight particle is bonded comprising power function type time effect and based on moment of flexure contribution factor and bond direct stress calculating
FormulaCalculation Shear formula is bonded with the two dimension comprising power function type time effect
Step 214：WillSubstitute into formulaAnd make β=βσ；WillSubstitute into formulaAnd make β=βτ, accordingly, respectively obtain the rock mass and carefully see particle two dimension bonding drawing
Stretch timeliness corresponding to intensity and deteriorate the factorAnd rock mass carefully sees particle
Timeliness corresponding to two-dimentional bond shear strength deteriorates the factor
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 particle and bond and do not rupture
Bonded contact and two dimension comprising time effect bond normal force, tangential force and tangential moment of flexure, pass through formula (12), formula
(13), formula (14) calculates the bonding normal force, tangential force and tangential moment of flexure that i-th of rock mass carefully sees particle two dimension bonded contact,
Under two-dimensional case, normal direction moment of flexure is bonded to determine that rock mass carefully sees particle by formula (15),
Wherein：Respectively i-th of rock mass carefully sees particle and includes time effect
The bonding normal force answered, tangential force, the bonding normal direction moment of flexure comprising time effect are bonded, tangential moment of flexure is bonded, bonds normal direction position
Move increment and bond tangential displacement increment,Particle two dimension is carefully seen for rock mass and bonds shear stiffness, +=is addition from inverse operation
Symbol, -=is the reflexive operator of subtraction.
Step 206：Moment of flexure contribution factor is setAnswered based on moment of flexure particle bonding normal direction of seeing thin to rock mass
The percentage contribution of power, particle is carefully seen according to rock mass and bonds two-dimentional direct stress calculation formulaBonded with two dimension
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, obtain comprising power function type time effect and
Thin two dimension of the particle bonding based on moment of flexure contribution factor of seeing bonds direct stressCalculation formula and two dimension bond shear stressMeter
Formula is calculated, sees formula (16) and formula (17) respectively,
Step 207：Time effect will be included in step 206Formula (18) is substituted into, it is determined that being based on moment of flexure tribute
Offer the factor and mole-coulomb with stretching cut-off limit carefully sees particle and bond season cracking criterion, and calculate successively j-th to the
Whether k rock mass carefully sees particle two dimension bond stress, ruptured and fracture mode for judging that rock mass is carefully seen particle and bonded,
The rock mass of the criterion, which is carefully seen, contains power function type time effect and based on moment of flexure contribution factor in particle bond stress,
Wherein：fs、fnRespectively rock mass carefully sees timeliness shear fracture criterion, the timeliness tensile fracture standard that particle two dimension bonds
Then,Two dimension for the time effect of type containing power function of i-th of contact bonds shear stress,For containing for i-th of contact
Power function type time effect and the two dimension bonding direct stress based on moment of flexure contribution factor,Respectively rock mass carefully sees particle
Tensile strength, the shearing strength of two dimension bonding,The cohesive strength that particle two dimension bonds carefully is seen for rock mass,Particle is carefully seen for rock mass
The internal friction angle that two dimension bonds；fsParticle is carefully seen more than or equal to 0 expression rock mass and bonds shear fracture, is carefully seen less than 0 expression rock mass
Particle bonds, and shear fracture does not occur；fnParticle is carefully seen more than or equal to 0 expression rock mass and bonds tensile fracture, and rock mass is represented less than 0
Tensile fracture does not occur for thin particle bonding of seeing；
Step 208：F in the formula (18) in step 207sOr fnDuring more than or equal to 0, show that rock mass is carefully seen particle and glued
Bind up one's hair raw rupture, now rock mass carefully sees the motor pattern of particle using the two-dimensional linear contact model based on damping effect come table
Reach；F in the formula (18) in step 207sAnd fnWhen both less than 0, show that rock mass is carefully seen particle and bonded and do not rupture, continue to follow
Ring step 201 calculated, renewal, judges that rock mass carefully sees the bond state of particle contact to 207, up to rock mass do not produce it is new thin
See the rupture of particle bonding or rock mass carefully sees particle and bonds rupture accelerated development and form macroscopic failure, loop termination.
The rock mass is carefully seen after particle bonds and rupture, and rock mass carefully sees the motor pattern of particle using being based on damping effect
Two-dimensional linear contact model express, carefully seeing particle for describing rock mass bonds the thin stress for seeing particle after season cracking, becomes
Shape and moving law, the structure of the two-dimensional linear contact model based on damping effect comprise 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 contact point a ends, the centre distance at contact point b ends：
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 at end：
Wherein：niFor the unit vector of contact,For contact jaw b direction vector,For contact jaw a direction vector,
nwallTo constrain the direction vector of wall；
Step 303：After rock mass carefully sees particle bonding rupture, the contact lap U of each two-dimensional linear contact point, pass through
Step 301 calculates the grain spacing d and particle radius R at two-dimensional linear contact both endsa、Rb, recycle formula (21) to determine；
Reference distance g is contacted by setting particle two-dimensional linearr, and formula (22) is combined, determine the distance of particle two-dimensional linear contact
gs：
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 walla, kbThe equivalent rigidity instead of contact point, calculated by formula (23)：
Wherein：Kn、KsGrain contact point equivalent normal stiffness and shear stiffness are carefully seen for rock mass, For particle with
The normal stiffness and shear stiffness at the contact a ends of particle or particle with wall,For particle and particle or particle and wall
Contact b ends normal stiffness and shear stiffness.
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 epqzFor Ricci index alternators, calculated according to formula (26)：
Wherein：VpWith VqEquivalence, VpWith VqTo 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 thin initial time step size increments Δ t for seeing particle linear contact model of rock mass value, pass through
The minimum time step Δ t of formula (29) estimation, it is ensured that the calculating time step of constructed model is less than the value, you can ensures system
System integral and calculating tends towards stability；The total displacement for determining each linear contact by formula (30), formula (31), formula (32) increases
Amount, Normal Displacement increment and tangential displacement increment：
R=min (Ra,Rb) (27)
ΔUp1=Vp1Δt (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；kIt is flatParticle system translational stiffness, k are carefully seen for rock massTurnParticle system rotational stiffness is carefully seen for rock mass；ΔUp1For rock mass
The thin total displacement increment for seeing the contact of particle two-dimensional linear, Δ δn、Physical significance is identical, represents that rock mass carefully sees particle two
The Normal Displacement increment of dimensional linear contact, Δ δs、Physical significance is identical, represents that rock mass carefully sees the contact of particle two-dimensional linear
Tangential displacement increment, Vp1With Vq1The 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：(gs)0The surface that initial time is calculated for model contacts distance, gsThe 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 (Ml=1) and
Absolute vectors adds up (Ml=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, kn、ksRespectively rock mass carefully sees the two-dimensional linear contact normal direction of stress deformation between particle, tangential linear rigidity, Δ δn、
ΔδsRespectively 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 Md={ 0,2 } and tensionless winkler foundation
Pattern MdTwo kinds of={ 1,3 }, calculated by formula (36), wherein mcFor equivalent particle quality, calculated by formula (37), rock mass is thin
The tangential damping for seeing particle linear contact uses full shear mode Md={ 0,1 } and cunning-cut-off-die formula Md={ 2,3 }, according to formula
(38) calculate,
Wherein：Respectively rock mass carefully sees the linear damping force of normal direction, the tangential linear damping of particle linear contact
Power, βnThe normal direction damped coefficient of particle linear contact, β are carefully seen for rock masssThe tangential damping system of particle linear contact is carefully seen for rock mass
Number, knThe normal direction linear rigidity of particle linear contact, k are carefully seen for rock masssThe 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 aremcEquivalent 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 massdFor rock
Body carefully sees the total damping power of particle linear contact.
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 12 to Figure 13 during example figure illustrates and Fig. 1 to Figure 10 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 11), the thin sight season cracking power function type model method of rock mass two dimension is realized on numerical value platform.
Step 2：Primarily determine that the rill evolution of rock mass season cracking 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, normal stiffness pb_kn (Fig. 6) is bonded, shear stiffness pb_ks (Fig. 6) is bonded, bonds mould
Type pb_Ec, the coefficient of friction ba_fric of particle, the average value pb_sn_mean for bonding tensile strength, the mark of bonding tensile strength
Accurate poor pb_sn_sdev, cohesive strength average value pb_coh_mean, cohesive strength standard deviation pb_coh_sdev, bonding radius multiplier
Gamma (Fig. 7), bond moment of flexure devotion factor beta, normal direction damped coefficient Apfan (Fig. 6), tangential damped coefficient Apfas (Fig. 6)
And 19 parameters such as internal friction angle pb_phi (Fig. 8), parameter occurrence are shown in Table one.
Step 3：Generate strata model
Bonding tensile strength and the cohesive strength distribution of model are determined according to Gaussian Profile or weibull distributions, by uniform
Distribution of random function method determines the particle diameter distribution of particle；By isotropic stress adjusting method and adaptive dynamic swelling method, adjust
The position of whole particle, reduce particle lap；By suspended particulate elimination method, isolated particle is deleted, improves the whole of model sample
Body, reduce the generation of defect model.Finally assign cast material adhesion strength parameter, rock of the generation with true rock mass structure
Body Model.A diameter of 50mm of strata model, highly it is 100mm (Figure 12).
Step 4：The meso-damage evolution parameter of model in the Accurate Calibration present invention
The load-deformation curve obtained by indoor single shaft and triaxial compression test, determine the macroscopic elastic modulus of rock massPeak strength σp, 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, and obtain constructed by the present invention
The meso-damage evolution parameter of model.
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.By parameter fitting method, the secular distortion process of actual rock mass is matched, it is determined that
Rock mass carefully sees the first control parameter β that particle bonds timeliness deterioration1, the second control parameter 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 rock mass secular distortion destruction
Evolution (Figure 13).
The parameter name and value of model of the present invention are as shown in Table 1.
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 10 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 (8)
1. a kind of rock mass two dimension is thin to see season cracking power function type model method, it is characterised in that：The power function pattern type is fitted
Should be in Particles in Two Dimensions discrete element, Particles in Two Dimensions discontinuous deformation analysis, Particles in Two Dimensions manifold member；The power function pattern type
Particle bond stress two-dimensional model is carefully seen including the rock mass based on moment of flexure contribution factor, the rock mass based on moment of flexure contribution factor is carefully seen
Particle bond timeliness deterioration decay two-dimentional power function pattern formula, based on moment of flexure contribute effect and with stretching cut-off limit mole-
The thin particle of seeing of coulomb bonds season cracking criterion and the thin sight particle linear contact two dimensional model based on damping effect.
2. rock mass two dimension according to claim 1 is thin to see season cracking power function type model method, it is characterised in that：It is described
Rock mass based on moment of flexure contribution factor carefully sees particle bond stress two-dimensional model and refers to that rock mass carefully sees particle two dimension and bonds direct stress
Calculation formula
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In be provided with moment of flexure contribution factorBased on the moment of flexure percentage contribution seen particle two dimension and bond direct stress thin to rock mass；Upper
StateIn formula,Particle two dimension, which is carefully seen, for i-th of rock mass bonds normal stress,The rock of respectively i-th contact
Body carefully sees particle two dimension and bonds normal force, tangential moment of flexure,Particle two dimension is carefully seen for rock mass and bonds radius,For moment of flexure contribution because
Son,I is that rock mass carefully sees the moment of inertia that particle two dimension bonds, and A is that rock mass carefully sees particle two dimension bond area, and i is successively
Particle, which is carefully seen, for first to last rock mass bonds number.
3. rock mass two dimension according to claim 1 is thin to see season cracking power function type model method, it is characterised in that：It is described
Rock mass based on moment of flexure contribution factor carefully sees particle and bonds the two-dimentional power function pattern formula of timeliness deterioration decay to be included in rock mass thin
When seeing particle two dimension bonding timeliness deterioration decay, there is provided power function pattern related to the bond stress based on moment of flexure contribution factor
Formula, the rock mass in this power function pattern formula, which carefully sees particle and bonds diameter, progressively deteriorates decay with the time, sees cohesive diameter formula
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In formula,Normal stress is bonded for the two dimension based on moment of flexure contribution factor,Bonded to judge that rock mass carefully sees particle two dimension
Start stress threshold values during timeliness deterioration decay,Particle two dimension is carefully seen for rock mass and bonds tensile strength,For based on moment of flexure
The two-dimentional bond stress ratio of contribution factor, β1、β2Carefully see two-dimentional bond timeliness deterioration two of particle for control rock mass and specify and refer to
Number,Particle two dimension is carefully seen for rock mass and bonds the diameter that decay is deteriorated with the time,Particle two dimension bonding is carefully seen for rock mass not decline
Diameter when subtracting, Δ t are that rock mass carefully sees the incremental time that particle two dimension bonds timeliness decay deterioration；There is provided rock mass carefully to see particle
Bond area and face the moment of inertia timeliness deterioration decay two-dimensional model, are shown in that rock mass when cohesive unit thickness is 1 carefully sees particle respectively
With the bond area A' of time deterioration decay, the moment of inertia I' calculation formula
<mrow>
<msup>
<mi>A</mi>
<mo>&prime;</mo>
</msup>
<mo>=</mo>
<mn>2</mn>
<msup>
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<mi>R</mi>
<mo>&OverBar;</mo>
</mover>
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</msup>
<mo>=</mo>
<mi>&beta;</mi>
<mi>A</mi>
</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>
</mrow>
Wherein, β is that rock mass carefully sees the timeliness decay factor that particle two dimension bonds diameter, and its calculation formula is shown in
<mrow>
<mi>&beta;</mi>
<mo>=</mo>
<msup>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&prime;</mo>
</msup>
<mo>/</mo>
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<mo>&OverBar;</mo>
</mover>
<mo>=</mo>
<msup>
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<mi>R</mi>
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</mover>
<mo>&prime;</mo>
</msup>
<mo>/</mo>
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<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>
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<mtable>
<mtr>
<mtd>
<mrow>
<mn>0</mn>
<mo>,</mo>
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<mover>
<mi>&sigma;</mi>
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</mover>
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<mi>a</mi>
<mi>a</mi>
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</mtd>
</mtr>
<mtr>
<mtd>
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<mi>&beta;</mi>
<mn>1</mn>
</msub>
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<mo>(</mo>
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<mover>
<mi>&sigma;</mi>
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<mi>&sigma;</mi>
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<mi>c</mi>
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</mrow>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
</msup>
<mfrac>
<mrow>
<mi>&Delta;</mi>
<mi>t</mi>
</mrow>
<mover>
<mi>D</mi>
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</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>
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<mi>&sigma;</mi>
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</mover>
<mi>c</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
</mrow>
In formula,A'、I'、Be expressed as rock mass carefully see particle two dimension bond with the time deteriorate decay bonding it is straight
Footpaths, radius, bond area are bonded, the moment of inertia is bonded, bonds diameter multiplier, A、I、Carefully seen for rock mass
Grain bonds bonding diameter, bonding radius, bond area, bonding the moment of inertia, the bonding diameter multiplier when not decaying；Simultaneously according to this
Kind power function type timeliness deterioration evanescent mode estimation rock mass carefully sees the initial time step delta t that particle bonds rupture, sees formula
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<mi>&Delta;</mi>
<mi>t</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>/</mo>
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<mi>n</mi>
<mi>c</mi>
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<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
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<mtable>
<mtr>
<mtd>
<mrow>
<mi>&infin;</mi>
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<mi>&sigma;</mi>
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</mover>
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<msub>
<mover>
<mi>&sigma;</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>a</mi>
<mi>a</mi>
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</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mo>-</mo>
<msup>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&prime;</mo>
</msup>
</mrow>
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<mi>&beta;</mi>
<mn>1</mn>
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</mfrac>
<msup>
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<mi>c</mi>
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<mn>2</mn>
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</msup>
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<mover>
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<mi>a</mi>
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<mi>&sigma;</mi>
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<mi>c</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
</mrow>
Wherein, it is the rock of i-th of contact
Body carefully sees particle two dimension and bonds diameter multiplier, ncParticle two dimension, which is carefully seen, for first rock mass bonds the required cycle calculations of rupture
Number, βσ、βτRespectively rock mass carefully sees the timeliness deterioration factor, two dimension corresponding to particle two dimension bonding tensile strength and bonds shearing by force
Timeliness corresponding to degree deteriorates the factor, and i is followed successively by first and carefully sees particle bonding number to last rock mass, and ∞ is infinity.
4. rock mass two dimension according to claim 1 is thin to see season cracking power function type model method, it is characterised in that：It is described
Contribute effect based on moment of flexure and carefully see particle bonding season cracking criterion with mole-coulomb for stretching cut-off limit and refer to judging rock
When body carefully sees particle two dimension bonding season cracking, effect and rubbing with stretching cut-off limit are contributed based on moment of flexure using embedded
That-coulomb season cracking criterion judges, sees formula
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<mi>f</mi>
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<mi>t</mi>
<mi>a</mi>
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<mi>&phi;</mi>
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<mn>2</mn>
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<mi>c</mi>
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</mrow>
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<mi>n</mi>
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<mover>
<mi>&sigma;</mi>
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</msub>
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<mn>1</mn>
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<mn>2</mn>
<mi>&beta;</mi>
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<mi>R</mi>
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</mover>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mo>-</mo>
<msubsup>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>+</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mover>
<mi>&beta;</mi>
<mo>&OverBar;</mo>
</mover>
<mo>|</mo>
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<msub>
<mi>M</mi>
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<mi>&sigma;</mi>
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<mi>c</mi>
</msub>
</mrow>
</mtd>
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</mtable>
</mfenced>
<mo>,</mo>
</mrow>
Wherein, fs、fnRespectively rock mass carefully sees timeliness shear fracture criterion, the timeliness tensile fracture criterion that particle two dimension bonds,Respectively rock mass carefully sees particle two dimension and bonds tensile strength, shearing strength,Connect for respectively i-th
The tactile time effect of type containing power function and the rock mass based on moment of flexure contribution factor carefully see particle two dimension and bond direct stress, shear stress,The internal friction angle that particle two dimension bonds carefully is seen for rock mass,The cohesive strength that particle two dimension bonds carefully is seen for rock mass,It is thin for rock mass
See particle two dimension and bond radius,The bonding normal force and tangential force, bonding of respectively i-th particle contact
Tangential moment of flexure,For moment of flexure contribution factor,I be followed successively by first to last rock mass carefully see particle bond number；
Power function type time effect is contained in the two-dimentional bond stress of the criterion, sees that rock mass carefully sees particle two dimension and bonds diameter
Timeliness decay factor β calculation formula
<mrow>
<mi>&beta;</mi>
<mo>=</mo>
<mfrac>
<msup>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
<mo>&prime;</mo>
</msup>
<mover>
<mi>D</mi>
<mo>&OverBar;</mo>
</mover>
</mfrac>
<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>
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</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>
<mrow>
<mo>(</mo>
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<mover>
<mi>&sigma;</mi>
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</mover>
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<mover>
<mi>&sigma;</mi>
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</mover>
<mi>c</mi>
</msub>
</mfrac>
<mo>)</mo>
</mrow>
<msub>
<mi>&beta;</mi>
<mn>2</mn>
</msub>
</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>
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</mover>
<mi>c</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>,</mo>
</mrow>
In formulaParticle two dimension is carefully seen for rock mass and bonds the diameter that decay is deteriorated with the time,Particle two dimension is carefully seen for rock mass to bond
Diameter when not decaying,For the two-dimentional bond stress ratio based on moment of flexure contribution factor,For based on moment of flexure contribution factor
Two dimension bonds normal stress,The stress threshold values started when timeliness deterioration decays is bonded to judge that rock mass carefully sees particle two dimension,
Particle two dimension is carefully seen for rock mass and bonds tensile strength, Δ t carefully sees the time increasing of particle two dimension bonding timeliness decay deterioration for rock mass
Amount, β1、β2Respectively control rock mass carefully sees particle and bonds the first control parameter of timeliness deterioration, the second control parameter；
fsParticle two dimension is carefully seen more than or equal to 0 expression rock mass and bonds shear fracture, and carefully seeing particle two dimension less than 0 expression rock mass bonds
Shear fracture does not occur；fnParticle two dimension is carefully seen more than or equal to 0 expression rock mass and bonds tensile fracture, is carefully seen less than 0 expression rock mass
Tensile fracture does not occur for particle two dimension bonding.
5. rock mass two dimension according to claim 1 is thin to see season cracking power function type model method, it is characterised in that：It is described
Thin sight particle linear contact two dimensional model based on damping effect refers to after rock mass carefully sees particle bonding season cracking, passes through two
Dimensional linear contact reference distance grParticle two-dimensional linear contact distance g is seen there is provided thins, see that rock mass is carefully seen particle two-dimensional linear and connect
Touch away from calculation formulaWherein, For rock mass inside
The coordinate of particle and particle two-dimensional linear contact jaw a,For rock mass internal particle and particle two-dimensional linear contact jaw b seat
Mark, Ra、RbRespectively rock mass carefully sees two-dimensional linear contact jaw a particle radius and two-dimensional linear contact jaw b particle radius；If
The two-dimensional linear contact mode that stress deformation between particle is carefully seen based on rock mass is put, sets between rock mass particle and is slided based on two dimension
The binding mode for the line power that rubs, rock mass carefully see the two-dimensional linear contact normal direction linear force calculation formula of stress deformation between particleTake Ml=1 is relative vector accumulation mode, takes Ml=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, kn、ksRespectively rock mass carefully sees the two-dimensional linear contact normal direction, tangential of stress deformation between particle
Linear rigidity, Δ δn、ΔδsRespectively 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；
The damping of the damping mode for setting two-dimensional linear to contact simultaneously, wherein normal direction uses full normal mode Md={ 0,2 } and tensionless winkler foundation
Pattern MdTwo kinds of={ 1,3 }, passes through formulaCalculate, wherein mcFor equivalent particle quality,
By formulaCalculate, tangential damping uses full shear mode Md={ 0,1 } and cunning-cut-off-die formula Md
={ 2,3 }, according to formulaTo calculate, wherein：Respectively normal direction damps
Power, tangential damping force, βnFor normal direction damped coefficient, βsFor tangential damped coefficient,For normal direction speed, tangential velocity, F*
The full normal direction damping force of particle linear contact is carefully seen for rock mass, expression formula ismcFor equivalent particle matter
Amount, m(1)Granular mass, m are carefully seen for the rock mass of two-dimensional linear contact jaw 1(2)Rock mass for two-dimensional linear contact jaw 2 carefully sees particle
Quality.
6. rock mass two dimension according to claim 2 is thin to see season cracking power function type model method, it is characterised in that：It is described
The rock mass of i-th of contact carefully sees particle two dimension and bonds normal forceTangential moment of flexureComputational methods be： In formula,Particle two dimension is carefully seen for rock mass and bonds normal stiffness,For rock mass
Thin particle two dimension of seeing bonds normal direction displacement increment,Particle two dimension is carefully seen for rock mass and bonds circumferentially opposite rotating angle increment, +=is
The reflexive operator of addition, -=is the reflexive operator of subtraction.
7. rock mass two dimension according to claim 3 is thin to see season cracking power function type model method, it is characterised in that：It is described
Rock mass carefully sees particle two dimension and bonds timeliness deterioration factor-beta corresponding to tensile strengthσParticle two dimension is carefully seen with rock mass and bonds shearing by force
Timeliness corresponding to degree deteriorates factor-betaτCalculation formula be respectively
<mrow>
<msub>
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</msub>
<mo>=</mo>
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Wherein,Particle two dimension is carefully seen for rock mass and bonds radius,The mull technique of respectively i-th particle contact
Xiang Li, bond tangential force, bond tangential moment of flexure,For moment of flexure contribution factor, It is viscous that particle two dimension is carefully seen for rock mass
Tie tensile strength,The cohesive strength that particle two dimension bonds carefully is seen for rock mass,The interior friction that particle two dimension bonds carefully is seen for rock mass
Angle, i be followed successively by first to last rock mass carefully see particle bond number.
8. rock mass two dimension according to claim 4 is thin to see season cracking power function type model method, it is characterised in that：It is described
The time effect of type containing power function of i-th of contact and the rock mass based on moment of flexure contribution factor carefully see particle two dimension and bond direct stress
Calculation formula be
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The rock mass of the time effect of type containing power function of i-th of contact carefully sees particle two dimension and bonds shear stressCalculation formula
For
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| US20030144796A1 (en) * | 2002-01-04 | 2003-07-31 | Anstey Nigel A. | Method of distinguishing types of geologic sedmentation |
| CN103425899A (en) * | 2013-09-10 | 2013-12-04 | 南京大学 | Method for modeling and simulating three-dimensional discrete element for shale pneumatic and hydraulic fracturing |
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| US20030144796A1 (en) * | 2002-01-04 | 2003-07-31 | Anstey Nigel A. | Method of distinguishing types of geologic sedmentation |
| CN103425899A (en) * | 2013-09-10 | 2013-12-04 | 南京大学 | Method for modeling and simulating three-dimensional discrete element for shale pneumatic and hydraulic fracturing |
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