Rock stress state equivalent calculation method based on principal stress space coordinate system rotation

Abstract

The invention discloses a principal stress space coordinate system rotation-based rock stress state equivalent calculation method, which comprises the following steps of: 1, establishing an original coordinate system, and establishing a pi axis in the original coordinate system; step 2, making a pi plane circle; 3, the original coordinate system in the step 1 is rotated around the pi axis by an angle beta, a new coordinate system is obtained, the coordinate components of the coordinate point P in the new coordinate system are (Px ', Py' and Pz '), and Px', Py 'and Pz' are three principal stress values and Pz'gt in a bidirectional equal stress state; px '= Py', and belongs to a bidirectional equal stress state; px ', Py' and Pz 'are used for judging the damage mode of the rock. By adopting the calculation method, the coordinates of the principal stress state of the rock are transformed based on the rotation matrix RT of the principal stress space coordinate system rotating around the pi axis, the equivalence of principal stress state value attributes is realized, and the accuracy of stress state equivalence is ensured.

Description

Technical Field

[0001] The invention belongs to the technical field of rock monitoring, and in particular relates to an equivalent calculation method for rock stress state based on the rotation of a principal stress space coordinate system. Background Art

[0002] For rock mass engineering, the ground stress state of the rock mass at the engineering site is extremely complex. At the same time, during the engineering excavation process, the stress state change process of the surrounding rock is also extremely complex, including the direction and magnitude of the principal stress are constantly changing. Therefore, in the process of engineering support parameter design or prevention and control measures formulation, it is necessary to obtain the stress state of the rock and the strength index of the rock under different stress states, and then design the support parameters and safety reinforcement measures to ensure the safe construction of the project.

[0003] The stress tensor [σ] can describe the stress state of rock. In the three-dimensional space coordinate system, the stress tensor [σ] has 9 components, 6 of which are independent components. Therefore, 6 independent components are needed to describe the stress state of rock. At the same time, these 6 independent components also change with different coordinate systems, which has poor engineering practicality. In order to solve this problem, the more mature method is the theory of elastic mechanics. Through rigorous mathematical derivation, this theory transforms the stress tensor of 6 independent components into 3 independent components in the principal stress coordinate system, which not only simplifies the description method of stress state, but also reveals the physical meaning of the three stress invariants, promoting the application of stress tensors in engineering. However, when the principal stress tensor is used in engineering applications, it is necessary to take into account both the value attributes (i.e., stress state attributes that are independent of the coordinate system direction, such as hydrostatic pressure, energy density, failure strength, stress value, etc.) and the direction attributes (i.e., stress state attributes that are related to the coordinate system direction) of the stress state. This makes the calculation process more cumbersome for actual engineering projects that pay more attention to the value attributes of the stress state (large deformation, unstable collapse, rock burst energy release, structural pressure failure, etc., which are mainly affected by the value attributes). Summary of the invention

[0004] The purpose of the present invention is to provide an equivalent calculation method for rock stress state based on the rotation of the principal stress space coordinate system, and a rotation matrix R based on the rotation of the principal stress space coordinate system around the π axis. T , the coordinates of the principal stress state of rock are transformed to achieve the equivalence of the principal stress state value attributes, which ensures the accuracy of the stress state equivalence and improves the practicability of the principal stress tensor in engineering.

[0005] The present invention adopts the following technical solution: an equivalent calculation method for rock stress state based on the rotation of the principal stress space coordinate system, comprising the following steps:

[0006] Step 1: Establish an original coordinate system, and establish the π axis in the original coordinate system:

[0007] Define the straight line x=y=z in the original coordinate system as the π axis;

[0008] Step 2: Make a π plane circle:

[0009] Mark a coordinate point P of the complex stress state in the original coordinate system in step 1. The three coordinate components of the coordinate point P are (P x , P y , P z );

[0010] A π plane circle is obtained by taking the vertical distance from the coordinate point P to the π axis as the radius and taking the foot of the perpendicular as the center. The π plane circle is perpendicular to the π axis, and the π axis passes through the center of the π plane circle. The stress state values ​​of the coordinate points on the circumference of the π plane circle are equivalent.

[0011] Step 3: Rotate the original coordinate system in step 1 by an angle of β around the π axis to obtain a new coordinate system. The coordinate components of the coordinate point P in the new coordinate system are (P x′ , P y′ , P z′ ), P x′ , P y′ , P z′ are the three principal stress values ​​of the bidirectional isostatic state, P z′ >P x′ =P y′ , belongs to the bidirectional iso-stress state; P x′ , P y′ , P z′ Used to determine the failure mode of rock.

[0012] Furthermore, in step three:

[0013]

[0014]

[0015] Expanding formula (29), we get:

[0016]

[0017] The coordinate transformation calculation formula is obtained from formula (30):

[0018]

[0019] In formula (31), γ x , γ y , γ z The calculation is as follows:

[0020]

[0021] Furthermore, for equations (31) and (32), in the new coordinate system, set P x′ , P y′ , P z′ If any two coordinate components in are equal, solve for the angle β and substitute β into equations (31) to (32), we can get the three coordinate component values ​​P of the coordinate point. x′ , P y′ , P z′ .

[0022] Furthermore, in step 3, P is set x′ =P y′ , from equation (31) and equation (32), we get:

[0023] cos(β-γ x )=cos(β-γ y ) (33);

[0024] Solving equation (33) yields two solutions for β:

[0025]

[0026] Substitute β1 and β2 in equation (34) into the expression P in equation (31) z′ In the above equation, we can find the P corresponding to β1. z′1 and P corresponding to β2 z′2 , we get:

[0027]

[0028] If P z′1 ≥P z′2 When , substitute the β1 value into formula (31) and obtain:

[0029]

[0030] If P z′1 <P z′2 When , substitute the β2 value into the first two expressions in formula (31), and we get:

[0031]

[0032] Obtain the coordinate point (P x′ , P y′ , P z′ ).

[0033] The present invention also discloses a rock failure mode determination method, which is characterized by comprising the following steps based on the above-mentioned rock stress state equivalent calculation method based on the rotation of the principal stress space coordinate system:

[0034] Step A: Determine the conversion pressure σ of the rock sample 3,t :

[0035] The minimum confining pressure for rock to undergo ductile deformation failure is defined as the rock transformation pressure σ 3,t ;

[0036] Step B: The two minor principal stress values ​​P in the bidirectional iso-stress state in step 3 are x′ or P y′ and rock conversion pressure σ 3,t Compare, when P x′ =P y′ ≥σ 3,t When P x′ =P y′ <σ 3,t When the rock is subjected to brittle fracture failure mode.

[0037] The present invention also discloses a method for evaluating the quality of rock true triaxial compression test data, which comprises the following steps:

[0038] The test data of the true triaxial compression test of rock includes n groups of tests. The three stress values ​​of rock failure obtained in each group (taking the i-th group as an example) are (σ 1,i , σ 2,i , σ 3,i );

[0039] Based on any one of claims 1-4, the three stress values ​​are (σ 1,i , σ 2,i , σ 3,i ) is equivalent to a bidirectional isostatic state (σ 1,i′ , σ 3,i′ , σ 3,i′ ), the equivalent stress state is calculated according to the confining pressure σ 3,i′ Arranged from small to large: σ 3,1′ ≤σ 3,2′ ≤…≤σ 3,i′ ≤…≤σ 3,n′ ;

[0040] According to the order from small to large, the confining pressure difference Δσ between two adjacent 3,i′ as follows:

[0041]

[0042] When a certain confining pressure difference value is greater than twice the average value of other confining pressure difference values, it is determined that the test data between two adjacent confining pressure values ​​of this confining pressure difference value is missing; when a certain confining pressure difference value is less than 50% of the average value of other confining pressure difference values, it is determined that the two adjacent confining pressure values ​​of this confining pressure difference value are redundant.

[0043] The present invention also discloses a method for determining whether two stress states are equivalent. P and Q are two complex stress states. The coordinate values ​​of the two complex stress states in the original coordinate system are: P = (P x , P y , P z ) and Q=(Q x , Q y , Q z );

[0044] Step S1, determine whether the centers of the π plane circles where the two complex stress states P and Q are located are the same point:

[0045] when When , the centers of the π plane circles of the two stress states P and Q do not coincide, and the two stress states are not equivalent, and the determination ends; otherwise, execute step 2;

[0046] Step 2 S2: Determine whether the radius r of the π plane circles where the two complex stress states P and Q are located are equal:

[0047] The radius r of P's rotation on its π plane circle P for: When Q rotates on its π plane circle, its rotation radius r Q for:

[0048] When P =r Q When , the two complex stress state points are on the same π plane circle, and the two complex stress states P and Q are equivalent.

[0049] When P ≠r Q When , the two complex stress state points are not on the same π plane circle, and the two complex stress states P and Q are not equivalent.

[0050] The beneficial effects of the present invention are: 1. The rotation matrix R based on the principal stress space coordinate system rotating around the π axis is adopted. T, transform the coordinates of the principal stress state of the rock to achieve the equivalence of the value attributes of the principal stress state, and it is supported by rock mechanics theory and spatial analytical geometry, avoiding the dependence on engineering experience and effectively improving the accuracy of stress state equivalence. 2. Computer programs or spreadsheet tools can be used for batch processing, which is highly efficient for large-scale projects with a large amount of data. It can quickly evaluate the quality of true triaxial compression test data of rock, and quickly and accurately make up for the quality defects of test data through conventional triaxial compression tests of rock. 3. Only the value attributes of the stress state tensor are equivalent, and the directional attributes of the stress state tensor are not considered. This is consistent with the surrounding rock collapse damage, rock burst energy release, and surrounding rock pressure that are the focus of the project site. It has good application prospects and improves the practicality of the principal stress tensor in engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 is the π axis in the original coordinate system;

[0052] Figure 2 is the π plane circle of the π axis view;

[0053] Figure 3 is the π plane circle of the coordinate system view;

[0054] Figure 4 Equivalent schematic diagram of the stress state when the coordinate system is rotated in the π-axis view;

[0055] Figure 5 The axis of the coordinate system view intersects the π plane circle;

[0056] Figure 6 It is a schematic diagram of the axis intersecting the π plane circle of the coordinate system rotation of the coordinate system view;

[0057] Figure 7 For osα z′z Calculation diagram;

[0058] Figure 8 is cosα z′y Calculation diagram;

[0059] Fig. 9 is cosα y′z Calculation diagram. DETAILED DESCRIPTION

[0060] The present invention is described in detail below with reference to the accompanying drawings and specific embodiments.

[0061] In actual engineering, rocks are generally subjected to complex stress states, such as the stress state of surrounding rocks during excavation support or the stress state of true triaxial compression test of rocks. At this time, the three principal stresses on rocks are generally not equal. The Mohr-Coulomb strength theory, rockburst criterion based on energy release, and rock ductile / brittle failure mode, which are widely used in engineering, are more based on the stress state under conventional triaxial compression test conditions. At this time, the confining principal stress σ2=σ3, and the rock is in a bidirectional iso-stress state.

[0062] Therefore, by equating the complex stress state to a biaxial isostress state, it is convenient to apply the relevant technologies based on conventional triaxial compression tests on rocks to actual projects subjected to complex stress states.

[0063] The invention provides an equivalent calculation method for rock stress state based on the rotation of principal stress space coordinate system, starting from the principal stress tensor, establishing the principal stress space coordinate system, abandoning the secondary importance of the principal stress tensor direction attribute to the actual engineering application, grasping the principal stress tensor value attribute that is the focus of the actual engineering application, and forming an equivalent calculation method for rock stress state value attribute based on the rotation of principal stress space coordinate system, which effectively improves the practicability of the principal stress tensor in engineering.

[0064] The present invention is based on the equivalent calculation method of rock stress state by rotating the principal stress space coordinate system, which is applicable to isotropic rock media. The present invention involves formula calculation, which can be processed in batches by computer programs or electronic spreadsheet tools, and is applicable to large-scale projects with extremely large amounts of data.

[0065] In the present invention, "coordinate system" specifically refers to the principal stress space coordinate system OXYZ, which defaults to the original coordinate system OXYZ before rotation; "coordinate system rotation" specifically refers to the coordinate system rotating around the π axis by an angle of β, β∈(0°,360°), where the positive direction of β is determined according to the left-hand screw rule, that is, the direction in which the left thumb points to the rotation direction vector and the four bent fingers point to the positive direction of β. "Original coordinate system" specifically refers to the principal stress space coordinate system before rotation, represented by "OXYZ"; "new coordinate system" specifically refers to the principal stress space coordinate system after rotation, represented by "OX′Y′Z′". "Stress state equivalence" specifically refers to the equivalence of the value attributes of the stress state; "bidirectional isostress state" specifically refers to the stress state in which the minor principal stresses are equal, that is, σ1>σ2=σ3, excluding the bidirectional isostress state in which the major principal stresses are equal, that is, σ1=σ2>σ3; "complex stress state" specifically refers to any other stress state that does not include the bidirectional isostress state.

[0066] The equivalent calculation method of rock stress state based on the rotation of the principal stress space coordinate system of the present invention is as follows:

[0067] 1. Establish the π axis and π plane circle in the original coordinate system:

[0068] 1.1 Establish the π axis:

[0069] Establish a coordinate system, the coordinate points (x, y, z) in the coordinate system represent the principal stress state of the rock (σ x , σ y , σ z ), that is, σ x , σ y , σ z Represents the three coordinate components of the principal stress values ​​in three directions of rock in the coordinate system.

[0070] The equation of a straight line in a coordinate system: x = y = z, is defined as the π axis, also called the isoclinic axis or hydrostatic pressure axis, such as Figure 1 As shown. The geometric characteristics of the π axis: it passes through the origin O of the original coordinate system and has equal angles with the three coordinate axes X, Y, and Z, so it is named the isoclinic axis. Therefore, when the coordinate system rotates around the π axis, the π axis of the original coordinate system remains coincident with the π axis of the new coordinate system, that is, the π axis remains unchanged during the rotation process.

[0071] The physical meaning of the π-axis: At any point on the π-axis, the three coordinate components are equal, indicating that the principal stress values ​​in the three directions of the rock are equal, similar to the hydrostatic pressure state, hence the hydrostatic pressure axis.

[0072] 1.2 Make a π plane circle:

[0073] When the rock is subjected to a certain principal stress state, the three principal stress values ​​are σ1, σ2, and σ3 respectively. The principal stress state has three coordinate points in the coordinate system, namely: N, N′, N″, and the coordinate components of these three coordinate points are: N=(σ1, σ2, σ3), N′=(σ2, σ3, σ1), N″=(σ3, σ1, σ2). The coordinate values ​​of the three coordinate points N, N′, and N″ in the coordinate system of the principal stress state are different, but the coordinate components have rotation symmetry. According to the requirements of the patent of the present invention applicable to isotropic rock media, these three coordinate points correspond to the same stress state, that is, the stress states of these three coordinate points are equivalent, and the equivalence at this time also includes the equivalence of value attributes and the equivalence of direction attributes.

[0074] An arc connects the three coordinate points N, N′, and N″ in sequence to form a circle. The π axis is perpendicular to the plane where the circle is located and passes through the center of the circle. The circle is defined as a π plane circle. They are all arcs with a central angle of 120°. Figure 2 and Figure 3 The above process can be used to produce a π plane circle of rock subjected to any principal stress state.

[0075] 2. Rotation of the original coordinate system:

[0076] 2.1 The stress state of the original coordinate system rotation is equivalent to:

[0077] According to the geometric characteristics of the π axis, there are two different stress states P and Q on the same π plane circle. These two stress states correspond to three coordinate points in the coordinate system. For the convenience of analysis, one of the coordinate points is taken. The coordinate components of the coordinate points are: P = (P x , P y , P z ) and Q=(Q x , Q y , Q z ).

[0078] By rotating the original coordinate system, the coordinate components of the coordinate point P in the new coordinate system are (P x′ , P y′ , P z′ ). When the rotation angle meets certain requirements, the coordinate component (Q x , Q y , Q z ) and (P x′ , P y′ , P z′ ) are equal, that is, P x′ =Q x , P y′ =Q y ,P=Q z At this time, the two stress states are equivalent, but because the coordinate systems of the two stress states are different, the directional properties are not equivalent, only the value properties are equivalent.

[0079] 2.2 Transformation matrix R of the original coordinate system rotation T :

[0080] The method of determining the equivalence of stress states involves the following coordinate system rotation:

[0081] The three coordinate components (X', Y', Z') of the new coordinate system are expressed using the three coordinate components (X, Y, Z) of the original coordinate system as follows:

[0082]

[0083] Formula (1) is written in matrix form as follows:

[0084]

[0085] Define the transformation matrix R for the coordinate system rotation T :

[0086]

[0087] R TIt is an orthogonal matrix. Each element of the matrix represents the direction cosine of the angle between the coordinate axis of the new coordinate system (the first subscript) and the coordinate axis of the original coordinate system (the second subscript). For example: cosα y′z Indicates the angle α between the Y′ axis of the new coordinate system and the Z axis of the original coordinate system y′z The direction cosines of .

[0088] 3. R T Calculation method:

[0089] 3.1 Axis intersecting π plane circle:

[0090] First, the coordinate component of the π axis in the original coordinate system is selected as The coordinate point O′ of the original coordinate system is the vector from the origin O to the coordinate point O′. It is the rotation direction vector of the original coordinate system around the π axis. The direction cosines of the angles with the three coordinate axes of the coordinate system are Then, a π plane circle is made with O′ as the center, intersecting the coordinate axis, with the intersection points being A, C, and E. The π plane circle is defined as the axis-intersecting π plane circle, such as Figure 5 shown.

[0091] According to the geometric characteristics of the π axis and the rotation symmetry of the coordinate system about the π axis, the π plane circle of the original coordinate system is also the π plane circle of the new coordinate system, and it intersects with the coordinate axis of the new coordinate system at B, D, and F. Figure 6 shown.

[0092] The geometric dimensions of the circle of the axis-intersecting-π plane in the coordinate system are determined, as follows:

[0093] Rotation direction vector Length:

[0094]

[0095] The distance from any point on the axis-pi plane circle to the origin O of the coordinate system is equal, and its value is:

[0096]

[0097] For RtΔAOO′, the Pythagorean theorem is used to calculate the radius r of the circle where the axis intersects the π plane:

[0098]

[0099] The length of the arc on the circle where the axis intersects the π plane: The central angles corresponding to these six arc lengths are all 120°, as follows:

[0100] ∠AO′C=∠BO′D=∠EO′C=∠DO′F=∠EO′A=∠FO′B=120° (7);

[0101] According to the geometric characteristics of the π axis and the rotation symmetry of the coordinate system about the π axis, when the coordinate system rotates, the angles between each coordinate axis of the new coordinate system and the three coordinate axes of the original coordinate system are also rotationally symmetric and equal, such as Figure 6 The details are as follows:

[0102] cosα z′z =cosα x′x = cosα z′z (8);

[0103] cosα z′y =cosα y′x = cosα x′z (9);

[0104] cosα y′z =cosα z′x = cosα x′y (10);

[0105] 3.2 Calculation of direction cosine cosα z′z 、cosα x′x 、cosα z′z The method is as follows:

[0106] With cosα z′z Take the calculation as an example, Figure 7 shown.

[0107] Connect points A and B. The midpoint of ΔO′AB is G, and O, G and O′, G are connected. Since ΔO′AB and ΔOAB are both isosceles triangles, and are the heights of ΔO′AB and ΔOAB respectively. RtΔO′AG and RtΔOAG share a right angle side therefore:

[0108]

[0109] In the formula, β is the rotation angle of the original coordinate system. According to formula (5) In formula (6) Substituting into formula (11), we get:

[0110]

[0111] According to the half-angle formula of trigonometric functions:

[0112]

[0113] Substituting formula (13) into formula (12), we get:

[0114]

[0115] According to formula (8), we can get:

[0116]

[0117] 3.3 Calculation of direction cosine cosα z′y 、cosα y′x 、cosα x′z The method is as follows:

[0118] With cosα z′y Take the calculation as an example, Figure 8 shown.

[0119] According to formula (7): Connect points B and C. The midpoint of the triangle is H, and O, H and O′, H are connected. Since ΔO′BC and ΔOBC are both isosceles triangles, and They are the heights of ΔO′BC and ΔOBC respectively. RtΔO′BH and RtΔOBH share a right angle side therefore:

[0120]

[0121] According to formula (5) In formula (6) Substituting into formula (16), we get:

[0122]

[0123] According to formula (13), we can get:

[0124]

[0125] According to the trigonometric function and difference product formula:

[0126] cos(θ1-θ2)=cosθ1·cosθ2+sinθ1·sinθ2 (19);

[0127] Substituting formula (19) into formula (18) yields:

[0128]

[0129] According to formula (9), we can get:

[0130]

[0131] 3.4 Calculation of direction cosine cosα z′x 、cosα y′z 、cosα x′y The method is as follows:

[0132] With cosα y′z Take the calculation as an example, Fig. 9 shown.

[0133] According to formula (7): Connect points A and D. The midpoint of K is connected between O, K and O′, K. Since ΔO′AD and ΔOAD are both isosceles triangles, and They are the heights of ΔO′AD and ΔOAD respectively. RtΔO′AK and RtΔOAK share a right angle side therefore:

[0134]

[0135] According to formula (5) In formula (6) Bring it into formula (22),

[0136]

[0137] According to formula (13), we can get:

[0138]

[0139] According to the trigonometric function and difference product formula:

[0140] cos(θ1+θ2)=cosθ1·cosθ2-sinθ1·sinθ2 (25);

[0141] Substituting formula (25) into formula (24), we get:

[0142]

[0143] According to formula (10), we can get:

[0144]

[0145] 3.5R T matrix:

[0146] According to equations (15), (21), and (27), we get the rotation matrix R T ,as follows:

[0147]

[0148] 4. Equivalent calculation method of stress state:

[0149] The rock is subjected to a certain stress state, one of the coordinate points of the stress state in the original coordinate system is P, and the coordinate components of P are (P x , P y , P z ). After the original coordinate system is rotated, the coordinate components of the coordinate point P in the new coordinate system are (P x′ , P y′ , P z′ ), which can be calculated according to formula (2) and substituted into formula (28), we get:

[0150]

[0151]

[0152] Expanding formula (29), we get:

[0153]

[0154] After rearranging formula (30), we get the coordinate transformation calculation formula with spatial geometric meaning:

[0155]

[0156] In formula (31), γ x , γ y , γ z The calculation is as follows:

[0157]

[0158] In formula (31), P x′ , P y′ , P z′ The first term of each expression represents the rotation amount of the original coordinate system transformation, where the cosine value represents the angle of rotation of the coordinate point P on the π plane circle, and the square root value represents the rotation radius of the coordinate point P on the π plane circle; the second term of the expression represents the hydrostatic pressure value of the coordinate point P, which is constant for a certain stress state and will not change with the rotation of the coordinate system. In addition, the geometric meaning of formula (31) is: when a certain stress state coordinate point P rotates in the original coordinate system, its coordinate components in the new coordinate system are: As the center of the circle, is the radius, (γ x , γ y , γ z ) is the starting angle and β is the rotation angle. Therefore, when the coordinate system rotates, the coordinate points corresponding to the stress state are always located on the same π plane circle, that is, the stress state is equivalent.

[0159] Engineering application 1: Complex stress state is equivalent to bidirectional iso-stress state:

[0160] In actual engineering, rocks are generally subjected to complex stress states, such as the stress state of surrounding rocks during excavation support or the stress state of true triaxial compression test of rocks. At this time, the three principal stresses on rocks are generally not equal. The Mohr-Coulomb strength theory, rockburst criterion based on energy release, and rock ductile / brittle failure mode, which are widely used in engineering, are more based on the stress state under conventional triaxial compression test conditions. At this time, the confining principal stress σ2=σ3, and the rock is in a bidirectional iso-stress state.

[0161] By equating the complex stress state to a biaxial iso-stress state, it is easy to apply the relevant technology based on conventional triaxial compression test of rock to actual projects subjected to complex stress states. The specific method is as follows:

[0162] The three coordinate components of the complex stress state at the coordinate point P in the original coordinate system are (P x , P y , P z ), after the coordinate system is rotated, the coordinate components of the coordinate point P in the new coordinate system are (P x′ , P y′ , P z′ ). Since the coordinate components have rotation symmetry, according to equations (31) and (32), let any two of the three coordinate components in the new coordinate system be equal, solve for the angle β, and substitute this β into equations (31) and (32) to obtain the three coordinate component values ​​P of the coordinate point x′ , P y′ , P z′ , that is, the complex stress state is equivalent to the three principal stress values ​​of the bidirectional isostress state.

[0163] P x′ =P y′ As an example, we sort out the first two expressions in equation (31) and equation (32) to get the following:

[0164] cos(β-γ x )=cos(β-γ y ) (33)

[0165] Solving equation (33) yields:

[0166]

[0167] Substitute β1 and β2 in equation (34) into the third expression in equation (31) to obtain the P corresponding to β1. z′1 and P corresponding to β2 z′2 ,as follows:

[0168]

[0169] According to formula (35), P z′1 and P z′2 , one of them corresponds to the stress state of equal major principal stress, that is, σ1=σ2>σ3. This situation does not belong to the bidirectional equal stress state specifically referred to in the present invention, so it should be discarded. The specific method is as follows:

[0170] When P z′1 ≥P z′2 When , substitute the β1 value into the first two expressions in formula (31) to obtain:

[0171]

[0172] When P z′1 <P z′2 When , substitute the β2 value into the first two expressions in formula (31), and we get:

[0173]

[0174] Complex stress state coordinate point (P x , P y , P z ) After the equations (34) to (37) are equivalent, the coordinate point in the new coordinate system is (P x′ , P y′ , P z′ ), at this time P z′ >P x′ =P y′ , which belongs to the bidirectional isostatic state.

[0175] After the complex stress state is equivalent to the bidirectional iso-stress state, the Mohr-Coulomb strength theory can be used to perform strength safety verification. x′ or P y′ The ductile / brittle failure mode of rock can be identified by the size of rock conversion pressure, laying a foundation for the application of high ground stress energy release rockburst criterion.

[0176] Engineering application 2: Quality evaluation of rock true triaxial compression test data.

[0177] For a batch of rock true triaxial compression test data, the batch contains n groups of tests. The three stress values ​​of rock failure obtained in each group (taking the i-th group as an example) are (σ 1,i , σ 2,i , σ 3,i ).

[0178] The stress state obtained from the true triaxial compression test of rock (taking the i-th group as an example) is equivalent to the biaxial isostatic state (σ 1,i′, σ 3,i′ , σ 3,i′ ), the equivalent stress state is calculated according to the confining pressure σ 3,i′ Arranged from smallest to largest:

[0179] Confining pressure 3,i′ Arrange from small to large: (σ 3,1′ ≤σ 3,2′ ≤…≤σ 3,i′ ≤…≤σ 3,n′ )

[0180] After the above arrangement, the confining pressure difference between two adjacent 3,i′ :

[0181]

[0182] By comparing all adjacent confining pressure differences Δσ 3,i′ , combined with the test conditions and test objectives, comprehensively judge the data quality of the batch of rock true triaxial compression tests. For example: a confining pressure difference Δσ 3,j′ It is obviously much larger than other confining pressure differences, indicating that the confining pressure difference Δσ 3,j′ The two confining pressure levels σ 3,j+1′ and σ 3,j′ The test data between the two are missing, so a set of conventional triaxial compression tests of rock needs to be added between these two confining pressure levels; on the contrary, when a certain confining pressure difference Δσ 3,k′ It is obviously much smaller than other confining pressure difference values, indicating that the confining pressure difference Δσ 3,k′ The two confining pressure levels σ 3,k+1′ and σ 3,k′ The test data nearby is redundant, resulting in a certain waste of test resources.

[0183] Engineering Application 3: Determine whether two stress states are equivalent.

[0184] In the present invention, when judging whether two stress states are equivalent, since an analytical formula is used, a certain amount of calculation is required, so data batch processing can be achieved with the help of tools such as computer programs or spreadsheets. This is very efficient and has huge advantages for large-scale projects with extremely large amounts of data (hundreds, thousands or tens of thousands).

[0185] According to the stress state equivalent calculation method in step (4), combined with Figure 4 And the stress state equivalence principle of step (2) ① is used to determine whether the two complex stress states P and Q are equivalent. The method is as follows:

[0186] The three principal stress values ​​of the two complex stress states P and Q are known. The coordinate values ​​of these two stress states in the original coordinate system are: P = (P x , P y, P z ) and Q=(Q x , Q y , Q z ).

[0187] According to step (4), when determining whether the two stress states P and Q are equivalent, it is necessary to determine whether the π plane circles where the two stress states are located are the same π plane circle. This determination method is divided into two steps. The first step is to determine whether the centers of the π plane circles where the two stress states P and Q are located coincide. If the centers coincide, the second step is to determine whether the radii of the two π plane circles are equal. If the radii are equal, the stress states of the two points P and Q are equivalent. The specific process is as follows:

[0188] First, determine whether the centers of the π plane circles where the two stress states P and Q are located are the same point.

[0189] The center of the π plane circle is on the π axis. According to the physical meaning of the π axis in step (1) ① and the elastic stress tensor theory: the hydrostatic pressure value of any stress state is equal to the average value of its three principal stresses in the principal stress coordinate system, and is also equal to the average value of its three normal stresses in the non-principal stress coordinate system. Therefore, the determination method is as follows:

[0190] when When , the centers of the π plane circles of the two stress states P and Q do not coincide, the two stress states are not equivalent, and the judgment ends; otherwise, enter the second step of the judgment process below.

[0191] Then, determine whether the radius r of the π plane circle where the two stress states P and Q are located is equal. The details are as follows:

[0192] According to equation (31) in step (4), the radius r of the rotation of the coordinate point P on its π plane circle is P for: Similarly, when the coordinate point Q rotates on its π plane circle, its rotation radius r Q for: When the rotation radius of the two stress state coordinate points P and Q is equal P =r Q When the two stress states are on the same π plane circle, the two stress states P and Q are considered equivalent. On the contrary, when the rotation radius of the coordinate points of the two stress states P and Q are not equal, P ≠r Q When , although the centers of the π plane circles of the two stress states coincide, the sizes of the π plane circles are different, so they are not on the same π plane circle. At this time, the two stress states P and Q are determined to be equivalent.

Claims

1. An equivalent calculation method for rock stress state based on the rotation of the principal stress space coordinate system is characterized by: The steps include: Step 1: Establish an original coordinate system, and establish the π axis in the original coordinate system: Define the straight line x=y=z in the original coordinate system as the π axis; Step 2: Make a π plane circle: Mark a coordinate point P of the complex stress state in the original coordinate system in step 1. The three coordinate components of the coordinate point P are (P x , P y , P z ); A π plane circle is obtained by taking the vertical distance from the coordinate point P to the π axis as the radius and taking the foot of the perpendicular as the center. The π plane circle is perpendicular to the π axis, and the π axis passes through the center of the π plane circle. The stress state values ​​of the coordinate points on the circumference of the π plane circle are equivalent. Step 3: Rotate the original coordinate system in step 1 by an angle of β around the π axis to obtain a new coordinate system. The coordinate components of the coordinate point P in the new coordinate system are (P x′ , P y′ , P z′ ), P x′ , P y′ , P z′ are the three principal stress values ​​of the bidirectional isostatic state, P z′ >P x′ =P y′ , belongs to the bidirectional iso-stress state; P x′ , P y′ , P z′ Used to determine the failure mode of rock.

2. The rock stress state equivalent calculation method based on the rotation of the principal stress space coordinate system according to claim 1 is characterized in that: In step three: Expanding formula (29), we get: The coordinate transformation calculation formula is obtained from formula (30): In formula (31), γ x , γ y , γ z The calculation is as follows:

3. The rock stress state equivalent calculation method based on the rotation of the principal stress space coordinate system according to claim 2 is characterized in that: For equations (31) and (32), in the new coordinate system, set P x′ , P y′ , P z′ If any two coordinate components in are equal, solve for the angle β and substitute β into equations (31) to (32), we can get the three coordinate component values ​​P of the coordinate point. x′ , P y′ , P z′ .

4. The rock stress state equivalent calculation method based on the rotation of the principal stress space coordinate system according to claim 3 is characterized in that: In step 3, set P x′ =P y′ , from equation (31) and equation (32), we get: cos(β-γ) x )=cos(β-γ y ) (33); Solving equation (33) yields two solutions for β: Substitute β1 and β2 in equation (34) into the expression P in equation (31) z′ In the above equation, we can find the P corresponding to β1. z′1 and P corresponding to β2 z′2 , we get: If P z′1 ≥P z′2 When , substitute the β1 value into formula (31) and obtain: If P z′1 <P z′2 When , substitute the β2 value into the first two expressions in formula (31), and we get: Obtain the coordinate point (P x′ , P y′ , P z′ ).

5. A rock failure mode determination method, characterized in that: The equivalent calculation method of rock stress state based on the rotation of the principal stress space coordinate system according to any one of claims 1 to 4 comprises the following: Step A: Determine the conversion pressure σ of the rock sample 3,t : The minimum confining pressure for rock to undergo ductile deformation failure is defined as the rock transformation pressure σ 3,t ; Step B: The two minor principal stress values ​​P in the bidirectional iso-stress state in step 3 are x′ or P y′ and rock conversion pressure σ 3,t Compare, when P x′ =P y′ ≥σ 3,t When P x′ =P y′ <σ 3,t When the rock is subjected to brittle fracture failure mode.

6. A method for evaluating the quality of rock true triaxial compression test data, characterized in that: The method includes the following: The test data of the true triaxial compression test of rock includes n groups of tests. The three stress values ​​of rock failure obtained in each group (taking the i-th group as an example) are (σ 1,i , σ 2,i , σ 3,i ); Based on any one of claims 1-4, the three stress values ​​are (σ 1,i , σ 2,i , σ 3,i ) is equivalent to a bidirectional isostatic state (σ 1,i′ , σ 3,i′ , σ 3,i′ ), the equivalent stress state is calculated according to the confining pressure σ 3,i′ Arranged from small to large: σ 3,1′ ≤σ 3,2′ ≤…≤σ 3,i′ ≤…≤σ 3,n′ ; According to the order from small to large, the confining pressure difference Δσ between two adjacent 3,i′ as follows: When a certain confining pressure difference value is greater than twice the average value of other confining pressure difference values, it is determined that the test data between two adjacent confining pressure values ​​of this confining pressure difference value is missing; when a certain confining pressure difference value is less than 50% of the average value of other confining pressure difference values, it is determined that the two adjacent confining pressure values ​​of this confining pressure difference value are redundant.

7. A method for determining the equivalence of two stress states, characterized in that: P and Q are two complex stress states, and the coordinate values ​​of the two complex stress states in the original coordinate system are: P = (P x , P y , P z ) and Q=(Q x , Q y , Q z ); Step S1, determine whether the centers of the π plane circles where the two complex stress states P and Q are located are the same point: when When , the centers of the π plane circles of the two stress states P and Q do not coincide, and the two stress states are not equivalent, and the judgment ends; Otherwise, go to step 2; Step 2 S2: Determine whether the radius r of the π plane circles where the two complex stress states P and Q are located are equal: The radius r of P's rotation on its π plane circle P for: When Q rotates on its π plane circle, its rotation radius r Q for: When P =r Q When , the two complex stress state points are on the same π plane circle, and the two complex stress states P and Q are equivalent. When P ≠r Q When , the two complex stress state points are not on the same π plane circle, and the two complex stress states P and Q are not equivalent.

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