CN114169182B - Method and equipment for reconstructing ellipsoid model of rock mass surface crack - Google Patents

Method and equipment for reconstructing ellipsoid model of rock mass surface crack Download PDF

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CN114169182B
CN114169182B CN202210132670.3A CN202210132670A CN114169182B CN 114169182 B CN114169182 B CN 114169182B CN 202210132670 A CN202210132670 A CN 202210132670A CN 114169182 B CN114169182 B CN 114169182B
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王守光
穆鹏宇
王嘉敏
李春元
崔春阳
杨冠宇
张海宽
李海涛
齐庆新
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General Coal Research Institute Co Ltd
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Abstract

The invention provides an ellipsoid model reconstruction method and equipment for a rock mass surface fracture, wherein the method adopts an ellipsoid model to visually express fracture pore distribution characteristics, and simultaneously carries out statistical analysis on the fracture pore distribution characteristics in a sample to realize quantitative description on the fracture pore distribution characteristics; the method adopts the ellipsoid to reconstruct the rock mass surface fracture field, simplifies the surface fracture field in the rock mass into the ellipsoid fracture, is convenient to describe in mathematical geometry, establishes the theoretical relationship between the simplified fracture field and the physical mechanical properties and deformation and damage of the rock mass, and has important significance for the development of the mechanical theory of the fractured rock mass.

Description

Method and equipment for reconstructing ellipsoid model of rock mass surface crack
Technical Field
The invention relates to the technical field of rock mass analysis, in particular to an ellipsoid model reconstruction method and device for rock mass surface cracks, computer equipment and a storage medium.
Background
A rock mass in the nature has a large number of primary fractures, and when geotechnical engineering construction is carried out, the new fractures can be generated in the complete rock mass due to stress field changes caused by increasing or releasing internal stress of the rock mass caused by artificial interference. At present, a great deal of geotechnical engineering is being built at home and abroad, particularly, China is in the rapid development period of infrastructure, and the engineering of traffic, water conservancy, energy, mines, national defense and the like is inseparable from rock masses. In general, geotechnical engineering involves excavation, loading or unloading, and after a rock mass is disturbed, mechanical properties, strength characteristics, failure modes, engineering stability and the like of the rock mass are determined by structural planes such as faults, joints, cracks and the like existing in the rock mass rather than properties of the rock mass. The influence of the weak structures on the properties of the rock mass such as strength and deformation must be fully considered when performing work such as engineering design, construction and stability evaluation of the rock mass. Therefore, the research on the failure mechanism and the strength characteristics of the fractured rock mass is always a hot point problem in the geotechnical engineering industry.
Observation by various devices is an effective means for researching the fracture structure of the rock mass, and currently, optical microscope scanning, electron microscope scanning, CT scanning and the like are mainly adopted. The microscopic fracture structure and the ultramicropore structure of the rock surface can be observed and described by an optical method represented by an optical microscope and an electron microscope scanning method represented by a Scanning Electron Microscope (SEM) and a Transmission Electron Microscope (TEM). The CT scanning experiment can observe and obtain the internal fracture image of the rock mass. However, due to the high complexity of the natural fracture field of the rock mass, the theoretical relationship between the natural fracture structure of the rock mass and the deformation and destruction characteristics of the rock mass obtained by the CT scanning experiment is still not solved so far.
The existing method for simplifying the natural fracture structure of the rock mass mainly comprises a bat model. The basic implementation process of the bat model is as follows: the method comprises the steps of firstly carrying out CT scanning on a rock body to obtain a three-dimensional image, then identifying a pore crack in the gray image based on a gray value or other methods, and finally simplifying the pore crack by using a bat model, wherein the simplification of the bat model is based on the principle of a maximum sphere method, the mutually communicated pore cracks are replaced by spheres with equivalent volumes, the cylindrical rods represent pore throats (the pore diameters of the two are consistent), and the spheres are connected through the cylindrical rods. The model thus created is also called the pore-throat model. In the bat model, the "ball" is considered the aperture, while the "stick" is considered the throat. The model can visually express the pore throat distribution characteristics, and can realize the quantitative description of the pore distribution characteristics by the quantitative measurement and statistics of the sphere, namely, the statistical analysis of the pore distribution characteristics in the sample.
However, there are problems with using a bat model for fracture simplification: the bat model adopts two different geometric bodies, namely a ball body and a rod body, and because of the difficulty in theoretical analysis of geometric characteristics, the bat model simplifies the fracture field, is difficult to establish a relationship between the theory and the solid mechanics theory, and is difficult to research the theoretical relationship of the fracture field to the elastic property and the damage of the rock mass; and the face fracture in the rock mass is dominant, and simultaneously, the face fracture has direct influence on the mechanical property of the rock mass, and the bat model can only simplify the hole fracture.
Disclosure of Invention
The invention provides an ellipsoid model reconstruction method, an ellipsoid model reconstruction device, computer equipment and a storage medium for a rock mass surface fracture, and aims to establish a theoretical relationship of macro-micro mechanical characteristics of a rock mass and provide theoretical support for rock engineering construction such as mining, water conservancy and tunnels.
Therefore, the first objective of the present invention is to provide a method for reconstructing an ellipsoid model of a rock surface fracture, which includes:
performing CT scanning on the natural fractured rock mass to obtain the topological form and the space coordinate of the rock mass face fracture which take the dominant effect;
adopting an Ear-Clipping algorithm to carry out integral dispersion on the fracture field of the CT scanning surface of the rock mass, and decomposing to obtain a set of a plurality of space triangular fracture surfaces;
fitting each space triangular fracture surface by adopting a surface ellipsoid, establishing a control equation corresponding to the space triangular fracture surface, and realizing ellipsoid reconstruction fitting of the fracture field of the rock CT scanning surface;
and iteratively solving the control equation of each space triangular fracture surface to obtain the simplified ellipsoid-shaped fracture field corresponding to the natural fracture of the rock mass.
Wherein, the step of establishing the control equation corresponding to the space triangle crack surface comprises the following steps:
establishing a target function for reconstructing a space triangular crack surface ellipsoid; wherein the constraint objective of the objective function is to minimize the number of ellipsoids and maximize the coverage;
establishing a constraint equation of triangular crack surface ellipsoid reconstruction in a two-dimensional space; wherein, the constraint condition is that all ellipsoids are positioned in the triangle and do not cover each other;
establishing a constraint equation for triangular crack surface ellipsoid reconstruction in a three-dimensional space; and the constraint condition is to project the cross section of the surface ellipsoid and the space triangle into the coordinate plane for constraint.
In the step of establishing the objective function of the spatial triangular fracture surface ellipsoid reconstruction, the mathematical expression of the objective function is as follows:
Figure 782735DEST_PATH_IMAGE001
（1）
wherein the variables in two dimensions
Figure 263395DEST_PATH_IMAGE002
Wherein, in the step (A),
Figure DEST_PATH_IMAGE003
、
Figure 759098DEST_PATH_IMAGE004
is a two-dimensional space coordinate of an ellipsoid circle section;
variation in three dimensions
Figure DEST_PATH_IMAGE005
Wherein, in the step (A),
Figure 537567DEST_PATH_IMAGE003
、
Figure 231854DEST_PATH_IMAGE004
、
Figure 148994DEST_PATH_IMAGE006
is a three-dimensional space coordinate of an ellipsoid circular section; the number of ellipsoids isn，KIs the coverage rate;
the coverage K is expressed as:
Figure DEST_PATH_IMAGE007
（2）
wherein the content of the first and second substances,
Figure 866414DEST_PATH_IMAGE008
is the radius of the section of an ellipsoid circle,
Figure DEST_PATH_IMAGE009
the area of the triangle is the same as the area of the triangle,
Figure 448574DEST_PATH_IMAGE010
。
wherein, in the step of establishing a constraint equation for reconstructing the triangular cleft surface ellipsoid in the two-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure DEST_PATH_IMAGE011
（3）
the non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle is as follows:
Figure 262947DEST_PATH_IMAGE012
（4）
wherein ABC is the vertex of the space triangle, and the coordinates of the three points ABC are expressed as follows: a (A)
Figure DEST_PATH_IMAGE013
，
Figure 23092DEST_PATH_IMAGE014
）、B（
Figure DEST_PATH_IMAGE015
，
Figure 477076DEST_PATH_IMAGE016
）、C（
Figure DEST_PATH_IMAGE017
，
Figure 675976DEST_PATH_IMAGE018
）
Figure DEST_PATH_IMAGE019
Is as follows
Figure 282538DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure DEST_PATH_IMAGE021
Distance to triangle side AB;
Figure 275902DEST_PATH_IMAGE022
is a first
Figure 223041DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 960053DEST_PATH_IMAGE021
Distance to triangle side BC;
Figure DEST_PATH_IMAGE023
is as follows
Figure 421122DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 850966DEST_PATH_IMAGE021
Distance to triangle side AC;
Figure 92591DEST_PATH_IMAGE024
（5）
the section of the ellipsoid of the constrained surface is inside the space triangle:
Figure DEST_PATH_IMAGE025
（6）
Figure 554666DEST_PATH_IMAGE026
（7）
Figure DEST_PATH_IMAGE027
（8）
wherein, the first and the second end of the pipe are connected with each other,
Figure 198137DEST_PATH_IMAGE028
the equations of the straight line AB, the straight line BC and the straight line AC are respectively shown as the following expressions:
Figure DEST_PATH_IMAGE029
（9）
Figure 470986DEST_PATH_IMAGE030
（10）
Figure DEST_PATH_IMAGE031
（11）
wherein, in the step of establishing a constraint equation for triangular cleft surface ellipsoid reconstruction in the three-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure 199908DEST_PATH_IMAGE032
（12）
the non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle is as follows:
Figure DEST_PATH_IMAGE033
（13）
wherein ABC is the vertex of the space triangle,
Figure 465673DEST_PATH_IMAGE034
is as follows
Figure 963650DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure DEST_PATH_IMAGE035
Distance to triangle side AB;
Figure 672980DEST_PATH_IMAGE036
is as follows
Figure 623619DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 506124DEST_PATH_IMAGE035
Distance to triangle side BC;
Figure DEST_PATH_IMAGE037
is as follows
Figure 45559DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 988107DEST_PATH_IMAGE035
Distance to triangle side AC;
Figure 426041DEST_PATH_IMAGE038
（14）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE039
，
Figure 784342DEST_PATH_IMAGE040
，
Figure DEST_PATH_IMAGE041
and
Figure 256911DEST_PATH_IMAGE042
are respectively a pointAAnd pointBThe coordinates of (a);
the section of the ellipsoid of the constrained surface is inside the space triangle:
the center of the section circle of the ellipsoid of the constraint surface is in the plane ABC:
Figure DEST_PATH_IMAGE043
（15）
the section of the constraint surface ellipsoid coincides with the plane ABC:
Figure 557311DEST_PATH_IMAGE044
（16）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE045
is the normal direction vector of the section of the ellipsoid circle,
Figure 482542DEST_PATH_IMAGE046
and
Figure DEST_PATH_IMAGE047
are respectively an edgeABAndBCthe direction vector of (a) is,
Figure 644533DEST_PATH_IMAGE048
the focal radii of three main shafts of the ellipsoid are respectively;
the center of the section of the ellipsoid of the constraint surface is positioned in the triangle ABC:
projecting the central point M of the cross section of the ellipsoid and the triangle ABC into a coordinate plane for constraint, wherein the constraint conditions are as follows:
Figure DEST_PATH_IMAGE049
（17）
Figure 706030DEST_PATH_IMAGE050
（18）
Figure DEST_PATH_IMAGE051
（19）
wherein the content of the first and second substances,
Figure 177332DEST_PATH_IMAGE052
are respectively straight line
Figure DEST_PATH_IMAGE053
Straight line, straight line
Figure 589858DEST_PATH_IMAGE054
Straight line, line
Figure DEST_PATH_IMAGE055
The equation of (a) is given,
Figure 555540DEST_PATH_IMAGE056
is the projection of ABC in the coordinate plane, and M' is the projection of the center M in the coordinate plane.
The method comprises the following steps of iteratively solving a control equation of each space triangular fracture surface to obtain the simplified ellipsoid-shaped fracture field corresponding to the natural fracture of the rock mass, wherein the steps comprise:
converting the optimization problem of a two-dimensional space and three-dimensional space dual-target constraint equation into a single-target constraint optimization problem;
solving a single-target constraint optimization problem by adopting a KKT condition (Karush-Kuhn-Tucker conditions);
the system of equations for the single objective constrained optimization problem may be solved using Newton's iteration or other modified Newton's iteration.
The step of solving the single-target constraint optimization problem by adopting the KKT condition comprises the following steps:
constructing a generalized Lagrangian function:
Figure DEST_PATH_IMAGE057
（20）
wherein the content of the first and second substances, nis the number of the ellipsoid bodies,
Figure 737123DEST_PATH_IMAGE058
is the coefficient to be determined and is,
Figure DEST_PATH_IMAGE059
，
Figure 113747DEST_PATH_IMAGE060
and
Figure DEST_PATH_IMAGE061
is shown inPPT，
Figure 13569DEST_PATH_IMAGE062
，
Figure DEST_PATH_IMAGE063
Are all known coefficients;
KKT conditions are listed:
Figure 517363DEST_PATH_IMAGE064
（21）
Figure DEST_PATH_IMAGE065
（22）
Figure 819031DEST_PATH_IMAGE066
（23）
Figure DEST_PATH_IMAGE067
（24）
wherein, still include the step: and analyzing the influence of the rock mass surface crack structure on the macroscopic and microscopic mechanical properties of the rock mass according to the ellipsoidal crack field and the microscopic solid mechanical theory.
The method comprises the following steps of analyzing the influence of a rock mass surface crack structure on the macroscopic and microscopic mechanical properties of a rock mass according to the ellipsoidal crack field and the microscopic solid mechanical theory, wherein the steps comprise:
solving the influence of the single ellipsoidal fracture on the elastic property of the rock mass;
and solving the influence of all the ellipsoidal fractures on the elastic property of the rock mass.
The equivalent elastic modulus formula of the rock mass containing the single fracture is obtained based on an equivalent method of a Mori-Tanaka model:
Figure 366556DEST_PATH_IMAGE068
（25）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE069
is the tensor of the elastic modulus of the rock matrix,
Figure 753675DEST_PATH_IMAGE070
i.e. the equivalent modulus tensor to be solved
Figure DEST_PATH_IMAGE071
Represents the volume fraction of the fracture,V 1 is the volume of the fracture,Vis the volume of the matrix; s is the fourth order Eshelby tensor, whose non-zero components are:
Figure 61160DEST_PATH_IMAGE072
（26）
the remaining components of the tensor S are all 0;
in the step of solving the influence of all the ellipsoidal fractures on the elastic property of the rock body, when a 2 nd ellipsoidal fracture is added in the RVE containing the 1 st ellipsoidal fracture, the REV of the rock containing the 1 st fracture is equivalent to a uniform medium, the 2 nd fracture is used as a new inclusion, and the calculation of the formula (25) is repeated; by analogy, assuming that the coal rock has n ellipsoidal fractures, taking the rock containing n-1 fractures as a lossless equivalent elastic medium each time, and calculating the influence of the nth ellipsoidal fracture on the fractured coal rock to obtain the equivalent elastic modulus of the rock under the influence of all n fractures.
The second objective of the invention is to provide an ellipsoid model reconstruction device for rock mass surface fracture, comprising:
the scanning module is used for carrying out CT scanning on the natural fractured rock mass to obtain the topological form and the space coordinates of the rock mass face cracks which take the dominant effect;
the dispersion module is used for integrally dispersing the fracture field of the CT scanning surface of the rock mass by adopting an Ear-Clipping algorithm and decomposing to obtain a set of a plurality of space triangular fracture surfaces;
the control equation building module is used for fitting each space triangle crack surface by adopting a surface ellipsoid, building a control equation corresponding to the space triangle crack surface and realizing ellipsoid reconstruction fitting of the crack field of the rock CT scanning surface;
and the ellipsoidal fracture field construction module is used for iteratively solving the control equation of each space triangular fracture surface to obtain the simplified ellipsoidal fracture field corresponding to the natural fracture of the rock body.
A third object of the present invention is to provide a computer device, which includes a memory, a processor and a computer program stored in the memory and running on the processor, wherein the processor executes the computer program to implement the method according to the foregoing technical solution.
A fourth object of the invention is to propose a non-transitory computer-readable storage medium on which a computer program is stored, which computer program, when executed by a processor, implements the method of the aforementioned technical solution.
Compared with the prior art, the ellipsoid model reconstruction method for the rock mass surface fracture provided by the invention has the advantages that the ellipsoid model is adopted to visually express the fracture pore distribution characteristics, and meanwhile, the statistical analysis is carried out on the fracture pore distribution characteristics in a sample, so that the quantitative description on the fracture pore distribution characteristics is realized; the method adopts the ellipsoid to reconstruct the rock mass surface fracture field, simplifies the surface fracture field in the rock mass into the ellipsoid fracture, is convenient to describe in mathematical geometry, establishes the theoretical relationship between the simplified fracture field and the physical mechanical properties and deformation and damage of the rock mass, and has important significance for the development of the mechanical theory of the fractured rock mass.
Drawings
The invention and/or additional aspects and advantages will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a schematic flow chart of an ellipsoid model reconstruction method for a rock mass surface crack provided by the invention.
FIG. 2 is a schematic diagram of reconstruction of a two-dimensional triangular fracture surface ellipsoid in the method for reconstructing an ellipsoid model of a rock mass surface fracture provided by the invention.
FIG. 3 is a schematic diagram of the projection of the center of a cross section of a surface ellipsoid and a triangle to a coordinate plane in the method for reconstructing an ellipsoid model of a rock mass surface crack provided by the invention.
FIG. 4 is a schematic structural diagram of an ellipsoid model reconstruction device for a rock mass surface crack provided by the invention.
Fig. 5 is a schematic structural diagram of a non-transitory computer-readable storage medium according to the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention.
Fig. 1 is an ellipsoid model reconstruction method for a rock mass surface crack provided by an embodiment of the present invention. The method comprises the following steps:
step 101, performing CT scanning on the natural fractured rock mass to obtain the topological form and the space coordinates of the rock mass face cracks which take the dominant effect.
102, integrally dispersing the fracture field of the CT scanning surface of the rock mass by adopting an Ear-Clipping algorithm, and decomposing to obtain a set of a plurality of space triangular fracture surfaces.
Aiming at the face crack, the Ear-Clipping algorithm is to decompose a simple polygon into a triangle set and obtain a plurality of space triangle cracks after dispersion.
And 103, fitting each space triangular fracture surface by adopting a surface ellipsoid, establishing a control equation corresponding to the space triangular fracture surface, and realizing ellipsoid reconstruction fitting of the fracture field of the rock CT scanning surface.
For the surface crack, a surface ellipsoid is adopted to fit each triangle, and for convenient calculation and consideration, the lengths of two main shafts of the surface ellipsoid are set to be equal, namely the cross section of the ellipsoid is circular.
The control equation for the reconstruction of the fracture ellipsoid of the building face is as follows:
and building an objective function of the surface fracture ellipsoid reconstruction. The triangular fracture surface ellipsoid reconstruction is a dual-target constraint optimization problem, and aims to minimize the number of ellipsoids and maximize the coverage rate. The mathematical expression listing the objective function is:
Figure DEST_PATH_IMAGE073
（1）
wherein the variables in two dimensions
Figure 951755DEST_PATH_IMAGE002
Wherein, in the step (A),
Figure 748810DEST_PATH_IMAGE003
、
Figure 538737DEST_PATH_IMAGE004
is a two-dimensional space coordinate of an ellipsoid circle section;
variation in three dimensions
Figure 712230DEST_PATH_IMAGE005
Wherein, in the step (A),
Figure 722911DEST_PATH_IMAGE003
、
Figure 690867DEST_PATH_IMAGE074
、
Figure 724682DEST_PATH_IMAGE006
is a three-dimensional space coordinate of an ellipsoid circular section; the number of ellipsoids isn，KIs the coverage rate;
the coverage K is expressed as:
Figure 436286DEST_PATH_IMAGE007
（2）
wherein the content of the first and second substances,
Figure 301474DEST_PATH_IMAGE008
is the radius of the section of an ellipsoid circle,
Figure 440331DEST_PATH_IMAGE009
the area of the triangle is the same as the area of the triangle,
Figure 210710DEST_PATH_IMAGE010
。
the two-dimensional spatial triangular slit surface ellipsoid reconstruction is shown in figure 2.
In the step of establishing a constraint equation for triangular fracture surface ellipsoid reconstruction in a two-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure DEST_PATH_IMAGE075
（3）
the non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle is as follows:
Figure 460426DEST_PATH_IMAGE012
（4）
wherein ABC is the vertex of the space triangle, and the coordinates of the three points ABC are expressed as follows: a (A)
Figure 445699DEST_PATH_IMAGE013
，
Figure 693141DEST_PATH_IMAGE014
）、B（
Figure 763865DEST_PATH_IMAGE015
，
Figure 817272DEST_PATH_IMAGE016
）、C（
Figure 657052DEST_PATH_IMAGE017
，
Figure 324663DEST_PATH_IMAGE018
）
Figure 617104DEST_PATH_IMAGE019
Is as follows
Figure 474201DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 168488DEST_PATH_IMAGE021
Distance to triangle side AB;
Figure 757732DEST_PATH_IMAGE022
is as follows
Figure 803048DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 198258DEST_PATH_IMAGE021
Distance to triangle side BC;
Figure 747051DEST_PATH_IMAGE023
is as follows
Figure 22043DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 289076DEST_PATH_IMAGE021
Distance to triangle side AC;
Figure 487977DEST_PATH_IMAGE024
（5）
the section of the ellipsoid of the constrained surface is inside the space triangle:
Figure 891276DEST_PATH_IMAGE025
（6）
Figure 87902DEST_PATH_IMAGE026
（7）
Figure 842232DEST_PATH_IMAGE027
（8）
wherein the content of the first and second substances,
Figure 579243DEST_PATH_IMAGE028
the equations of the straight line AB, the straight line BC and the straight line AC are respectively shown as the following expressions:
Figure 102629DEST_PATH_IMAGE029
（9）
Figure 719424DEST_PATH_IMAGE030
（10）
Figure 695470DEST_PATH_IMAGE031
（11）
in the step of establishing a constraint equation for triangular fracture surface ellipsoid reconstruction in three-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure 236173DEST_PATH_IMAGE032
（12）
the non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle is as follows:
Figure 879644DEST_PATH_IMAGE033
（13）
wherein ABC is the vertex of the space triangle,
Figure 152493DEST_PATH_IMAGE034
is as follows
Figure 881415DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 960229DEST_PATH_IMAGE035
Distance to triangle side AB;
Figure 458207DEST_PATH_IMAGE036
is as follows
Figure 151225DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 367443DEST_PATH_IMAGE035
Distance to triangle side BC;
Figure 249948DEST_PATH_IMAGE037
is as follows
Figure 602432DEST_PATH_IMAGE020
Center of section of ellipsoid
Figure 482663DEST_PATH_IMAGE035
Distance to triangle side AC;
Figure 920598DEST_PATH_IMAGE038
（14）
wherein the content of the first and second substances,
Figure 341215DEST_PATH_IMAGE039
，
Figure 813785DEST_PATH_IMAGE040
，
Figure 848606DEST_PATH_IMAGE041
and
Figure 773836DEST_PATH_IMAGE042
are respectively a pointAAnd pointBThe coordinates of (a);
the section of the ellipsoid of the constrained surface is inside the space triangle:
the center of the section circle of the ellipsoid of the constraint surface is in the plane ABC:
Figure 998144DEST_PATH_IMAGE043
（15）
the section of the constraint surface ellipsoid coincides with the plane ABC:
Figure 262903DEST_PATH_IMAGE044
（16）
wherein the content of the first and second substances,
Figure 547254DEST_PATH_IMAGE045
is the normal direction vector of the section of the ellipsoid circle,
Figure 959781DEST_PATH_IMAGE046
and
Figure 722201DEST_PATH_IMAGE047
are respectively an edgeABAndBCthe direction vector of (a) is,
Figure 90734DEST_PATH_IMAGE048
the focal radii of three main shafts of the ellipsoid are respectively;
the center of the section circle of the ellipsoid of the constraint surface is positioned inside the triangle ABC:
projecting the central point M of the cross section of the ellipsoid and the triangle ABC into a coordinate plane for constraint, wherein the constraint conditions are as follows:
Figure 545986DEST_PATH_IMAGE049
（17）
Figure 445809DEST_PATH_IMAGE050
（18）
Figure 11919DEST_PATH_IMAGE051
（19）
wherein the content of the first and second substances,
Figure 985692DEST_PATH_IMAGE052
are respectively straight line
Figure 611845DEST_PATH_IMAGE053
Straight line, straight line
Figure 998964DEST_PATH_IMAGE054
Straight line, straight line
Figure 103186DEST_PATH_IMAGE055
The equation of (a) is given,
Figure 452171DEST_PATH_IMAGE056
is the projection of ABC in the coordinate plane, and M' is the projection of the center M in the coordinate plane.
The plane ABC may be perpendicular to the xOy plane, and at this time, projection to other coordinate planes is required, and a judgment statement should be added:
if ABC⊥𝑥𝑂𝑦
if ABC⊥y𝑂z
projecting the point A, B, C, M to the coordinate plane & #119909 & #119874 &z
else
Projecting the point A, B, C, M to the coordinate plane y & #119874, z
end if
else
Projecting the point A, B, C, M to the coordinate plane & #119909 & #119874; y
end if
And 104, iteratively solving a control equation of each space triangular fracture surface to obtain an ellipsoidal fracture field simplified corresponding to the rock natural fracture.
Converting the dual-target constraint optimization problem shown in the formula (1) into a single-target constraint optimization problem;
solving a single-target constraint optimization problem by adopting a KKT condition (Karush-Kuhn-Tucker conditions);
the system of equations for the single objective constrained optimization problem may be solved using Newton's iterative method or other modified Newton's iterative method.
The step of solving the single-target constraint optimization problem by adopting the KKT condition comprises the following steps:
constructing a generalized Lagrangian function:
Figure 249226DEST_PATH_IMAGE057
（20）
wherein the content of the first and second substances, nis the number of the ellipsoid bodies,
Figure 858062DEST_PATH_IMAGE058
is the coefficient to be determined and is,
Figure 31554DEST_PATH_IMAGE059
；
Figure 979919DEST_PATH_IMAGE062
，
Figure 947875DEST_PATH_IMAGE076
are all known coefficients;
KKT conditions are listed:
Figure 44007DEST_PATH_IMAGE064
（21）
Figure 490031DEST_PATH_IMAGE065
（22）
Figure 181651DEST_PATH_IMAGE066
（23）
Figure 320508DEST_PATH_IMAGE067
（24）
Figure 841619DEST_PATH_IMAGE060
and
Figure 91335DEST_PATH_IMAGE061
the expression of (c) is as follows:
Figure DEST_PATH_IMAGE077
furthermore, it comprises step 105: and analyzing the influence of the rock mass surface crack structure on the macroscopic and microscopic mechanical properties of the rock mass according to the ellipsoidal crack field and the microscopic solid mechanical theory.
Obtaining an equivalent elastic modulus formula of a rock mass containing a single fracture based on an equivalent method of a Mori-Tanaka model:
Figure 263559DEST_PATH_IMAGE078
（25）
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE079
is the tensor of the elastic modulus of the rock matrix,
Figure 511001DEST_PATH_IMAGE080
i.e. the equivalent modulus tensor to be solved
Figure DEST_PATH_IMAGE081
Represents the volume fraction of the fracture,V 1 is the volume of the fracture,Vis the volume of the matrix; s is the fourth order Eshelby tensor, whose non-zero components are:
Figure 768676DEST_PATH_IMAGE082
（26）
the remaining components of the tensor S are all 0;
when a 2 nd ellipsoidal fracture is added in the RVE containing the 1 st ellipsoidal fracture, the REV of the rock containing the 1 st fracture is equivalent to a uniform medium, the 2 nd fracture is taken as a new inclusion, and the calculation of the formula (25) is repeated; by analogy, assuming that the coal rock has n ellipsoidal fractures, taking the rock containing n-1 fractures as a lossless equivalent elastic medium each time, calculating the influence of the nth ellipsoidal fracture on fractured coal rock, and obtaining the equivalent elastic modulus of the rock under the influence of all n fractures.
In order to implement the embodiment, the invention further provides an ellipsoid model reconstruction device for a rock surface fracture, as shown in fig. 4, including:
the scanning module 310 is used for performing CT scanning on the natural fractured rock mass to obtain the topological form and the spatial coordinates of the dominant rock mass face fracture;
the dispersion module 320 is used for performing integral dispersion on the fracture field of the CT scanning surface of the rock mass by adopting an Ear-Clipping algorithm and decomposing to obtain a set of a plurality of space triangular fracture surfaces;
the control equation building module 330 is configured to fit each space triangle fracture surface by using a surface ellipsoid, build a control equation corresponding to the space triangle fracture surface, and implement ellipsoid reconstruction fitting on a fracture field of a rock body CT scanning surface;
and the ellipsoidal fracture field construction module 340 is used for iteratively solving the control equation of each space triangular fracture surface to obtain the simplified ellipsoidal fracture field corresponding to the natural fracture of the rock body.
In order to implement the embodiment, the present invention further provides another computer device, including: the device comprises a memory, a processor and a computer program stored on the memory and capable of running on the processor, wherein the processor executes the computer program to realize the reconstruction of the ellipsoid model of the rock surface fracture according to the embodiment of the invention.
As shown in fig. 5, the non-transitory computer readable storage medium comprises a memory 810 of instructions executable by a processor 820 of an ellipsoid model reconstruction device from a rock face fracture to perform the method, an interface 830. Alternatively, the storage medium may be a non-transitory computer readable storage medium, for example, the non-transitory computer readable storage medium may be a ROM, a Random Access Memory (RAM), a CD-ROM, a magnetic tape, a floppy disk, an optical data storage device, and the like.
To achieve the described embodiments, the invention also proposes a non-transitory computer-readable storage medium having stored thereon a computer program which, when executed by a processor, enables an ellipsoid model reconstruction of a rock face fracture as an embodiment of the invention.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an example," "a specific example," or "some examples," etc., mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, a schematic representation of the terms does not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples. Furthermore, various embodiments or examples and features of different embodiments or examples described in this specification can be combined and combined by one skilled in the art without contradiction.
Furthermore, the terms "first", "second" and "first" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one such feature. In the description of the present invention, "a plurality" means at least two, e.g., two, three, etc., unless explicitly specified otherwise.
Any process or method descriptions in flow charts or otherwise described herein may be understood as representing modules, segments, or portions of code which include one or more executable instructions for implementing steps of a custom logic function or process, and alternate implementations are included within the scope of the preferred embodiment of the present invention in which functions may be executed out of order from that shown or discussed, including substantially concurrently or in reverse order, depending on the functionality involved, as would be understood by those reasonably skilled in the art of the present invention.
The logic and/or steps represented in the flowcharts or otherwise described herein, e.g., an ordered listing of executable instructions that can be considered to implement logical functions, can be embodied in any computer-readable medium for use by or in connection with an instruction execution system, apparatus, or device, such as a computer-based system, processor-containing system, or other system that can fetch the instructions from the instruction execution system, apparatus, or device and execute the instructions. For the purposes of this description, a "computer-readable medium" can be any means that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. More specific examples (a non-exhaustive list) of the computer-readable medium would include the following: an electrical connection (electronic device) having one or more wires, a portable computer diskette (magnetic device), a Random Access Memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber device, and a portable compact disc read-only memory (CDROM). Additionally, the computer-readable medium could even be paper or another suitable medium upon which the program is printed, as the program can be electronically captured, via for instance optical scanning of the paper or other medium, then compiled, interpreted or otherwise processed in a suitable manner if necessary, and then stored in a computer memory.
It should be understood that portions of the present invention may be implemented in hardware, software, firmware, or a combination thereof. In the described embodiments, various steps or methods may be implemented in software or firmware stored in a memory and executed by a suitable instruction execution system. If implemented in hardware, as in another embodiment, any one or combination of the following techniques, which are well known in the art, may be used: a discrete logic circuit having a logic gate circuit for implementing a logic function on a data signal, an application specific integrated circuit having an appropriate combinational logic gate circuit, a Programmable Gate Array (PGA), a Field Programmable Gate Array (FPGA), or the like.
One of ordinary skill in the art will appreciate that all or part of the steps carried by the method implementing the embodiments may be implemented by hardware related to instructions of a program, which may be stored in a computer readable storage medium, and the program, when executed, includes one or a combination of the steps of the method embodiments.
In addition, functional units in the embodiments of the present invention may be integrated into one processing module, or each unit may exist alone physically, or two or more units are integrated into one module. The integrated module can be realized in a hardware mode, and can also be realized in a software functional module mode. The integrated module, if implemented in the form of a software functional module and sold or used as a stand-alone product, may also be stored in a computer readable storage medium.
The mentioned storage medium may be a read-only memory, a magnetic or optical disk, etc. Although embodiments of the present invention have been shown and described above, it is understood that the embodiments are illustrative and not restrictive, and that those skilled in the art may make changes, modifications, substitutions and alterations to the embodiments described herein without departing from the scope of the invention.

Claims (6)

1. An ellipsoid model reconstruction method for rock mass surface cracks is characterized by comprising the following steps:
performing CT scanning on the natural fractured rock mass to obtain the topological form and the space coordinate of the rock mass face fracture which take the dominant effect;
adopting an Ear-Clipping algorithm to carry out integral dispersion on the fracture field of the CT scanning surface of the rock mass, and decomposing to obtain a set of a plurality of space triangular fracture surfaces;
fitting each space triangular fracture surface by adopting a surface ellipsoid, establishing a control equation corresponding to the space triangular fracture surface, and realizing ellipsoid reconstruction fitting of the fracture field of the rock CT scanning surface;
iteratively solving a control equation of each space triangular fracture surface to obtain an ellipsoid-shaped fracture field corresponding to the simplification of the rock natural fracture;
the step of establishing a control equation corresponding to the spatial triangular fracture surface comprises:
establishing a target function for reconstructing a space triangular crack surface ellipsoid; wherein the constraint objective of the objective function is to minimize the number of ellipsoids and maximize the coverage;
establishing a constraint equation of triangular crack surface ellipsoid reconstruction in a two-dimensional space; wherein, the constraint condition is that all ellipsoids are positioned in the triangle and do not cover each other;
establishing a constraint equation for triangular crack surface ellipsoid reconstruction in a three-dimensional space; the constraint condition is that the cross section of the surface ellipsoid and the space triangle are projected into a coordinate plane for constraint;
in the step of establishing the objective function of the spatial triangular slit surface ellipsoid reconstruction, the mathematical expression of the objective function is as follows:
Figure 212015DEST_PATH_IMAGE001
（1）
wherein the variables in two dimensions
Figure 775588DEST_PATH_IMAGE002
Wherein, in the step (A),
Figure 974488DEST_PATH_IMAGE003
、
Figure 174525DEST_PATH_IMAGE004
is a two-dimensional space coordinate of an ellipsoid circle section;
variation in three dimensions
Figure 964626DEST_PATH_IMAGE005
Wherein, in the step (A),
Figure 453377DEST_PATH_IMAGE006
、
Figure 987126DEST_PATH_IMAGE007
、
Figure 510511DEST_PATH_IMAGE008
is a three-dimensional space coordinate of an ellipsoid circular section; the number of ellipsoids isn，KIs the coverage rate;
the coverage K is expressed as:
Figure 674776DEST_PATH_IMAGE009
（2）
wherein the content of the first and second substances,
Figure 198293DEST_PATH_IMAGE010
is the radius of the section of an ellipsoid circle,
Figure 473416DEST_PATH_IMAGE011
is a triangular area, 0<K<1；
In the step of establishing the constraint equation for reconstructing the ellipsoid of the triangular fracture surface in the two-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure DEST_PATH_IMAGE002
（3）
the non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle is as follows:
Figure 983212DEST_PATH_IMAGE013
（4）
wherein ABC is the vertex of the space triangle, and the coordinates of the three points ABC are expressed as follows:
Figure 508871DEST_PATH_IMAGE014
、
Figure 322107DEST_PATH_IMAGE015
is as followsiSurface ellipseCenter of sphere cross section
Figure 820084DEST_PATH_IMAGE016
Distance to triangle side AB;
Figure 122889DEST_PATH_IMAGE017
is as followsiCenter of section of ellipsoid
Figure 73528DEST_PATH_IMAGE018
Distance to triangle side BC;
Figure 503503DEST_PATH_IMAGE019
is as followsiCenter of section of ellipsoid
Figure 590408DEST_PATH_IMAGE020
Distance to triangle side AC;
Figure 329694DEST_PATH_IMAGE021
（5）
the section of the ellipsoid of the constrained surface is inside the space triangle:
Figure 767628DEST_PATH_IMAGE022
（6）
Figure 922666DEST_PATH_IMAGE023
（7）
Figure 191974DEST_PATH_IMAGE024
（8）
wherein the content of the first and second substances,
Figure 774265DEST_PATH_IMAGE025
the equations of the straight line AB, the straight line BC and the straight line AC are respectively shown as the following expressions:
Figure 496233DEST_PATH_IMAGE026
（9）
Figure 720541DEST_PATH_IMAGE027
（10）
Figure 782038DEST_PATH_IMAGE028
（11）
in the step of establishing a constraint equation for triangular fracture surface ellipsoid reconstruction in three-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure DEST_PATH_IMAGE004
（12）
the non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle is as follows:
Figure 760806DEST_PATH_IMAGE030
（13）
wherein ABC is the vertex of the space triangle,
Figure 319964DEST_PATH_IMAGE031
is as followsiCenter of section of ellipsoid
Figure 235967DEST_PATH_IMAGE032
Distance to triangle side AB;
Figure 691219DEST_PATH_IMAGE033
is as followsiCenter of section of ellipsoid
Figure 387780DEST_PATH_IMAGE034
Distance to triangle side BC;
Figure 688311DEST_PATH_IMAGE035
is as followsiCenter of section of ellipsoid
Figure 521138DEST_PATH_IMAGE036
Distance to triangle side AC;
Figure 147291DEST_PATH_IMAGE037
（14）
wherein the content of the first and second substances,
Figure 268831DEST_PATH_IMAGE038
，
Figure 917594DEST_PATH_IMAGE039
，
Figure 808189DEST_PATH_IMAGE040
and respectively
Figure 401982DEST_PATH_IMAGE041
Is a pointAAnd pointBThe coordinates of (a);
the section of the ellipsoid of the constrained surface is inside the space triangle:
the center of the section of the ellipsoid of the constraint surface is in the plane ABC:
Figure 745238DEST_PATH_IMAGE042
（15）
the section of the constraint surface ellipsoid coincides with the plane ABC:
Figure 715468DEST_PATH_IMAGE043
（16）
wherein the content of the first and second substances,
Figure 460571DEST_PATH_IMAGE044
is the normal direction vector of the section of the ellipsoid circle,
Figure 428527DEST_PATH_IMAGE045
and
Figure 321396DEST_PATH_IMAGE046
are respectively an edgeABAndBCthe direction vector of (a) is,
Figure 501842DEST_PATH_IMAGE047
the focal radii of three main shafts of the ellipsoid are respectively;
the center of the section circle of the ellipsoid of the constraint surface is positioned inside the triangle ABC:
projecting the central point M of the cross section of the ellipsoid and the triangle ABC into a coordinate plane for constraint, wherein the constraint conditions are as follows:
Figure 180079DEST_PATH_IMAGE048
（17）
Figure 53357DEST_PATH_IMAGE049
（18）
Figure 636785DEST_PATH_IMAGE050
（19）
wherein the content of the first and second substances,
Figure 683239DEST_PATH_IMAGE051
are respectively straight line
Figure 402933DEST_PATH_IMAGE052
Straight line, straight line
Figure 509429DEST_PATH_IMAGE053
Straight line, straight line
Figure 314574DEST_PATH_IMAGE054
The equation of (a) is given,
Figure 367981DEST_PATH_IMAGE055
is the projection of ABC in the coordinate plane, and M' is the projection of the center M in the coordinate plane.
2. The method for reconstructing the ellipsoid model of the rock mass surface fracture as claimed in claim 1, wherein the step of iteratively solving the control equation of each space triangular fracture surface to obtain the simplified ellipsoid-type fracture field corresponding to the natural fracture of the rock mass comprises the steps of:
converting the optimization problem of a two-dimensional space and three-dimensional space dual-target constraint equation into a single-target constraint optimization problem;
solving a single-target constraint optimization problem by adopting a KKT condition (Karush-Kuhn-Tucker conditions);
the equation set of the single-objective constraint optimization problem can be solved by using a Newton iteration method or other modified Newton iteration methods;
the step of solving the single-target constraint optimization problem by adopting the KKT condition comprises the following steps:
constructing a generalized Lagrangian function:
Figure 4498DEST_PATH_IMAGE056
（20）
wherein the content of the first and second substances, nis the number of the ellipsoid bodies,
Figure 219579DEST_PATH_IMAGE057
is the coefficient to be determined and is,
Figure 59490DEST_PATH_IMAGE058
，
Figure 651009DEST_PATH_IMAGE059
and
Figure 142033DEST_PATH_IMAGE060
is shown inPPT，
Figure 793594DEST_PATH_IMAGE061
，
Figure 573331DEST_PATH_IMAGE062
Are all known coefficients;
KKT conditions are listed:
Figure 499699DEST_PATH_IMAGE063
（21）
Figure 845230DEST_PATH_IMAGE064
（22）
Figure 215162DEST_PATH_IMAGE065
（23）
Figure 216616DEST_PATH_IMAGE066
（24）
the above-mentioned
Figure 946675DEST_PATH_IMAGE067
And
Figure 349974DEST_PATH_IMAGE068
the expression of (c) is:
Figure 405655DEST_PATH_IMAGE069
。
3. the method for reconstructing an ellipsoid model of a rock body surface fracture as recited in claim 1, further comprising the steps of: analyzing the influence of the rock mass surface crack structure on the macroscopic and microscopic mechanical properties of the rock mass according to the ellipsoidal crack field and the microscopic solid mechanical theory; the method comprises the following steps:
solving the influence of the single ellipsoidal fracture on the elastic property of the rock mass;
solving the influence of all the ellipsoidal fractures on the elastic property of the rock mass;
in the step of solving the influence of the single ellipsoidal fracture on the elastic property of the rock mass, an equivalent elastic modulus formula of the rock mass containing the single fracture is obtained based on an equivalent method of a Mori-Tanaka model:
Figure 628826DEST_PATH_IMAGE070
（25）
wherein the content of the first and second substances,
Figure 428155DEST_PATH_IMAGE071
is the tensor of the elastic modulus of the rock matrix,
Figure 685961DEST_PATH_IMAGE072
i.e. the equivalent elastic modulus tensor to be solved
Figure 400626DEST_PATH_IMAGE073
The volume fraction of the fractures is expressed,V 1 is the volume of the fracture,Vis the volume of the matrix; s is the fourth order Eshelby tensor, whose non-zero components are:
Figure 376672DEST_PATH_IMAGE074
（26）
the remaining components of the tensor S are all 0;
in the step of solving the influence of all the ellipsoidal fractures on the elastic property of the rock body, when a 2 nd ellipsoidal fracture is added in the RVE containing the 1 st ellipsoidal fracture, the REV of the rock containing the 1 st fracture is equivalent to a uniform medium, the 2 nd fracture is taken as a new inclusion, and the calculation of the formula (25) is repeated; by analogy, assuming that the coal rock has n ellipsoidal fractures, taking the rock containing n-1 fractures as a lossless equivalent elastic medium each time, and calculating the influence of the nth ellipsoidal fracture on the fractured coal rock to obtain the equivalent elastic modulus of the rock under the influence of all n fractures.
4. An ellipsoid model reconstruction device of rock mass face crack is characterized by comprising:
the scanning module is used for carrying out CT scanning on the natural fractured rock mass to obtain the topological form and the space coordinates of the rock mass face cracks which take the dominant effect;
the dispersion module is used for integrally dispersing the fracture field of the CT scanning surface of the rock mass by adopting an Ear-Clipping algorithm and decomposing to obtain a set of a plurality of space triangular fracture surfaces;
the control equation building module is used for fitting each space triangle crack surface by adopting a surface ellipsoid, building a control equation corresponding to the space triangle crack surface and realizing ellipsoid reconstruction fitting on a crack field of the CT scanning surface of the rock mass;
the ellipsoidal fracture field construction module is used for iteratively solving a control equation of each space triangular fracture surface to obtain an ellipsoidal fracture field simplified corresponding to the natural fracture of the rock body;
the step of establishing a control equation corresponding to the spatial triangular fracture surface comprises:
establishing a target function for reconstructing a space triangular crack surface ellipsoid; wherein the constraint objective of the objective function is to minimize the number of ellipsoids and maximize the coverage;
establishing a constraint equation of triangular crack surface ellipsoid reconstruction in a two-dimensional space; wherein, the constraint condition is that all ellipsoids are positioned in the triangle and do not cover each other;
establishing a constraint equation for triangular crack surface ellipsoid reconstruction in a three-dimensional space; the constraint condition is that the cross section of the surface ellipsoid and the space triangle are projected into a coordinate plane for constraint;
in the step of establishing the objective function of the spatial triangular slit surface ellipsoid reconstruction, the mathematical expression of the objective function is as follows:
Figure 714112DEST_PATH_IMAGE075
（1）
wherein the variables in two dimensions
Figure 826425DEST_PATH_IMAGE076
Wherein, in the step (A),
Figure 223908DEST_PATH_IMAGE077
、
Figure 421671DEST_PATH_IMAGE078
is a two-dimensional space coordinate of an ellipsoid circle section;
variation in three dimensions
Figure 562803DEST_PATH_IMAGE079
Wherein, in the step (A),
Figure 60780DEST_PATH_IMAGE080
、
Figure 301268DEST_PATH_IMAGE081
、
Figure 64956DEST_PATH_IMAGE082
is a three-dimensional space coordinate of an ellipsoid circular section; the number of ellipsoids isn，KIs the coverage rate;
the coverage K is expressed as:
Figure 681882DEST_PATH_IMAGE083
（2）
wherein the content of the first and second substances,
Figure 831104DEST_PATH_IMAGE084
is the radius of the section of an ellipsoid circle,
Figure 242494DEST_PATH_IMAGE085
is a triangular surfaceProduct of 0<K<1；
In the step of establishing the constraint equation for reconstructing the ellipsoid of the triangular fracture surface in the two-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure 613095DEST_PATH_IMAGE002
（3）
non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle:
Figure 163362DEST_PATH_IMAGE087
（4）
wherein ABC is the vertex of the space triangle, and the coordinates of the three points ABC are expressed as follows: a (A)
Figure 104773DEST_PATH_IMAGE088
，
Figure 14961DEST_PATH_IMAGE089
）、B（
Figure 674612DEST_PATH_IMAGE090
，
Figure 446390DEST_PATH_IMAGE091
）、C（
Figure 507887DEST_PATH_IMAGE092
，
Figure 588976DEST_PATH_IMAGE093
）
Figure 470344DEST_PATH_IMAGE094
Is as followsiCenter of section of ellipsoid
Figure 295080DEST_PATH_IMAGE095
Distance to triangle side AB;
Figure 211084DEST_PATH_IMAGE096
is as followsiCenter of section of ellipsoid
Figure 463074DEST_PATH_IMAGE097
Distance to triangle side BC;
Figure 362897DEST_PATH_IMAGE098
is as followsiCenter of section of ellipsoid
Figure 663428DEST_PATH_IMAGE095
Distance to triangle side AC;
Figure 246987DEST_PATH_IMAGE099
（5）
the section of the ellipsoid of the constrained surface is inside the space triangle:
Figure 607561DEST_PATH_IMAGE100
（6）
Figure 525839DEST_PATH_IMAGE101
（7）
Figure 630061DEST_PATH_IMAGE102
（8）
wherein the content of the first and second substances,
Figure 582973DEST_PATH_IMAGE103
respectively, equation of a straight line AB, a straight line BC and a straight line AC, and the expressions are respectively:
Figure 114449DEST_PATH_IMAGE104
（9）
Figure 520022DEST_PATH_IMAGE105
（10）
Figure 427936DEST_PATH_IMAGE106
（11）
in the step of establishing a constraint equation for triangular fracture surface ellipsoid reconstruction in three-dimensional space, the constraint equation is expressed as:
non-coverage condition between the ellipsoidal sections:
Figure 156334DEST_PATH_IMAGE004
（12）
the non-coverage condition between the cross section of the ellipsoid of the surface and the boundary of the space triangle is as follows:
Figure 419955DEST_PATH_IMAGE108
（13）
wherein ABC is the vertex of the space triangle,
Figure 516087DEST_PATH_IMAGE109
is as followsiCenter of section of ellipsoid
Figure 24429DEST_PATH_IMAGE110
Distance to triangle side AB;
Figure 889616DEST_PATH_IMAGE111
is as followsiCenter of a section of a spheroid
Figure 825211DEST_PATH_IMAGE110
Distance to triangle side BC;
Figure 877481DEST_PATH_IMAGE112
is as followsiCenter of section of ellipsoid
Figure 189514DEST_PATH_IMAGE113
Distance to triangle side AC;
Figure 174787DEST_PATH_IMAGE114
（14）
wherein
Figure 218967DEST_PATH_IMAGE115
，
Figure 571582DEST_PATH_IMAGE116
，
Figure 359409DEST_PATH_IMAGE117
And
Figure 995927DEST_PATH_IMAGE118
are respectively a pointAAnd pointBThe coordinates of (a);
the section of the ellipsoid of the constrained surface is inside the space triangle:
the center of the section circle of the ellipsoid of the constraint surface is in the plane ABC:
Figure 211007DEST_PATH_IMAGE119
（15）
the section of the constraint surface ellipsoid coincides with the plane ABC:
Figure 565765DEST_PATH_IMAGE120
（16）
wherein the content of the first and second substances,
Figure 157284DEST_PATH_IMAGE121
is the normal direction vector of the section of the ellipsoid circle,
Figure 851570DEST_PATH_IMAGE122
and
Figure 299869DEST_PATH_IMAGE123
are respectively an edgeABAndBCthe direction vector of (a) is,
Figure 814027DEST_PATH_IMAGE124
the focal radii of three main shafts of the ellipsoid are respectively;
the center of the section circle of the ellipsoid of the constraint surface is positioned inside the triangle ABC:
projecting the central point M of the cross section of the ellipsoid and the triangle ABC into a coordinate plane for constraint, wherein the constraint conditions are as follows:
Figure 756706DEST_PATH_IMAGE125
（17）
Figure 305499DEST_PATH_IMAGE126
（18）
Figure 924700DEST_PATH_IMAGE127
（19）
wherein the content of the first and second substances,
Figure 191733DEST_PATH_IMAGE128
are respectively straight line
Figure 125054DEST_PATH_IMAGE129
Straight line, straight line
Figure 590670DEST_PATH_IMAGE130
Straight line, straight line
Figure 318455DEST_PATH_IMAGE131
The equation of (a) is given,
Figure 869522DEST_PATH_IMAGE132
is the projection of ABC in the coordinate plane, and M' is the projection of the center M in the coordinate plane.
5. A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the method according to any one of claims 1-3 when executing the computer program.
6. A non-transitory computer-readable storage medium having stored thereon a computer program, wherein the computer program, when executed by a processor, implements the method of any one of claims 1-3.
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CN112345340A (en) * 2020-10-16 2021-02-09 中国地质大学（武汉） Controllable angle crack surface generation device for cylindrical rock sample
CN113240805B (en) * 2021-04-27 2022-04-12 浙江大学 Method for representing rock mass crack surface by universal elliptic disc model

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