A method for solving and imaging plastic zone of surrounding rock considering shear stress direction

Abstract

The application provides a surrounding rock plastic zone solving and imaging method considering the shear stress direction: by analyzing the process that the three-direction principal stresses of any point around a roadway are deflected around three main shafts of a space rectangular coordinate system, stress expressions of three pairs of corresponding shear stresses of the roadway in the vertical, horizontal and axial directions are obtained under the deflection conditions around the respective main shafts, and then the distribution characteristics of the plastic zone of the roadway surrounding rock considering the shear stress direction are calculated, the distribution characteristics of the plastic zone are imaged in the range of 0-360 degrees by using mathematical software, and a plastic zone shape diagram is obtained. The method can further clarify the mechanism of the "asymmetric" deformation and damage of the roadway surrounding rock, so as to provide more accurate data for a supporting scheme.

Description

TECHNICAL FIELD

[0001] The present application relates to a kind of plastic zone solving and imaging method of surrounding rock considering shear stress direction, belong to coal mine engineering technical field. BACKGROUND

[0002] The natural stress in stratum not disturbed by engineering is called original rock stress, when roadway is excavated, the original geostress balance state is damaged, stress starts to redistribute.In this process, the principal stress of any point around roadway changes in size and direction, the change of stress size makes roadway produce pressure-relief zone and pressure-increase zone in different ranges, the change of stress direction makes stress transiently deflect, i.e.deviates from geodetic coordinate system by certain angle in different degrees.In addition, stress deflection phenomenon also exists in working face advancing process, with the advancing of working face, the stress deflection degree of different positions away from working face also has difference.

[0003] After stress deflection phenomenon occurs, different direction shear stress is generated, the change of shear stress direction changes the size and shape of plastic zone of roadway surrounding rock.In the current research of solving the distribution characteristics of plastic zone of roadway surrounding rock, most of them do not consider the influence of shear stress direction on plastic zone shape and size, or only study the influence of one shear stress direction on plastic zone, so that the calculation result has error.From the aspect of field application, it cannot provide accurate data for support scheme, thereby causing support failure, roof fall, rib spalling and other accidents, which is not conducive to the stability control of roadway surrounding rock. SUMMARY

[0004] In view of the above problems, the present application provides a kind of plastic zone solving and imaging method of surrounding rock considering shear stress direction: by analyzing the process that three principal stresses of any point around roadway deflect around three principal axes of space right-angle coordinate system respectively, the stress expression of three pairs of corresponding shear stresses of roadway in vertical, horizontal and axial directions under the deflection of respective principal axes is obtained, and then the distribution characteristics of plastic zone of roadway surrounding rock considering shear stress direction are calculated, the plastic zone distribution characteristics are imaged in 0°-360° range by using mathematical software, and plastic zone shape diagram is obtained.This method can further clarify the mechanism of "asymmetric" deformation and failure of roadway surrounding rock, thereby providing more accurate data for support scheme.

[0005] To achieve the above object, the present application is carried out according to the following steps.

[0006] (1) The cohesion, internal friction angle and Poisson's ratio parameters of roadway surrounding rock are determined by rock mechanics test.

[0007] (2) The cross section size of roadway and three principal stresses of roadway surrounding rock are determined by field measurement.

[0008] (3) Analysis of the stress expressions of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three principal stresses are deflected by an angle a around the X-axis of the space rectangular coordinate system.

[0009] (4) Analysis of the stress expressions of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three principal stresses are deflected by an angle β around the Y-axis of the space rectangular coordinate system.

[0010] (5) Analysis of the stress expressions of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three principal stresses are deflected by an angle γ around the Z-axis of the space rectangular coordinate system.

[0011] (6) According to the roadway rock mechanics parameters obtained in step (1), the roadway cross-section parameters obtained in step (2), the three principal stresses and the stress magnitudes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three principal stresses are deflected by an angle a around the X-axis of the space rectangular coordinate system obtained in step (4), the implicit equation expression of the plastic zone boundary of the roadway surrounding rock is obtained by using the theory of elastic-plastic mechanics.

[0012] (7) According to the roadway rock mechanics parameters obtained in step (1), the roadway cross-section parameters obtained in step (2), the three principal stresses and the stress magnitudes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three principal stresses are deflected by an angle β around the Y-axis of the space rectangular coordinate system obtained in step (4), the implicit equation expression of the plastic zone boundary of the roadway surrounding rock is obtained by using the theory of elastic-plastic mechanics.

[0013] (8) According to the roadway rock mechanics parameters obtained in step (1), the roadway cross-section parameters obtained in step (2), the three principal stresses and the stress magnitudes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three principal stresses are deflected by an angle γ around the Z-axis of the space rectangular coordinate system obtained in step (4), the implicit equation expression of the plastic zone boundary of the roadway surrounding rock is obtained by using the theory of elastic-plastic mechanics.

[0014] (9) The distribution characteristics of the plastic zone are imaged by traversing from 0° to 360° by using mathematical software, and the distribution pattern of the plastic zone of the roadway surrounding rock considering the deflection of the three principal stresses is obtained.

[0015] Further, in the step (3), the original direction of the three-direction principal stress is set as: the original direction of the maximum principal stress coincides with the vertical direction of the roadway, that is, the Z-axis direction of the space rectangular coordinate system; the direction of the intermediate principal stress coincides with the axial direction of the roadway, that is, the Y-axis direction of the space rectangular coordinate system; and the direction of the minimum principal stress coincides with the horizontal direction of the roadway, that is, the X-axis direction of the space rectangular coordinate system. It should be noted that the present application only provides a method for determining the direction of the three-direction principal stress, and in practice, the method includes but is not limited to the setting, because the distribution characteristics of the ground stress, the influence degree of the excavation and the influence degree of the working face mining are different, the directions of the principal stresses are different.

[0016] The stress expression of the three-direction principal stress after the three-direction principal stress is deflected by an angle of α around the X-axis of the space rectangular coordinate system is:

[0017]

[0018] In the formula, σ' z --the vertical stress of the roadway surrounding rock, MPa; σ' x --the horizontal stress of the roadway, MPa; σ' v --the axial stress of the roadway, MPa; τ' xy --the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ' yz --the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ' xz --the shear stress of the roadway surrounding rock in the vertical and horizontal directions, MPa; σ1--the maximum principal stress of the roadway surrounding rock, MPa; σ2--the intermediate principal stress of the roadway surrounding rock, MPa; σ3--the minimum principal stress of the roadway surrounding rock, MPa; α--the angle of the three-direction stress deflection around the X-axis of the space rectangular coordinate system.

[0019] Further, in the step (4), the original direction of the three-direction principal stress is set as: the original direction of the maximum principal stress coincides with the vertical direction of the roadway, that is, the Z-axis direction of the space rectangular coordinate system; the direction of the intermediate principal stress coincides with the axial direction of the roadway, that is, the Y-axis direction of the space rectangular coordinate system; and the direction of the minimum principal stress coincides with the horizontal direction of the roadway, that is, the X-axis direction of the space rectangular coordinate system. It should be noted that the present application only provides a method for determining the direction of the three-direction principal stress, and in practice, the method includes but is not limited to the setting, because the distribution characteristics of the ground stress, the influence degree of the excavation and the influence degree of the working face mining are different, the directions of the principal stresses are different.

[0020] The stress expression of the three-direction principal stress after the three-direction principal stress is deflected by an angle of β around the Y-axis of the space rectangular coordinate system is:

[0021]

[0022] In the formula, σ'' z --the vertical stress of the roadway surrounding rock, MPa; σ''x —the vertical stress of the roadway, MPa; σ′′ v —the axial stress of the roadway, MPa; τ′′ xy —the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ′′ yz —the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ′′ xz —the shear stress of the roadway surrounding rock in the vertical and horizontal directions, MPa; σ1—the maximum principal stress of the roadway surrounding rock, MPa; σ2—the intermediate principal stress of the roadway surrounding rock, MPa; σ3—the minimum principal stress of the roadway surrounding rock, MPa; β—the angle of the three-dimensional stress deviating from the Y axis of the space rectangular coordinate system.

[0023] Further, in the step (5), the original direction of the three-dimensional principal stress is set as: the maximum principal stress original direction coincides with the vertical direction of the roadway, that is, the Z axis direction of the space rectangular coordinate system; the intermediate principal stress coincides with the axial direction of the roadway, that is, the Y axis direction of the space rectangular coordinate system; and the minimum principal stress coincides with the horizontal direction of the roadway, that is, the X axis direction of the space rectangular coordinate system. It should be noted that the present application only provides a method for determining the direction of the three-dimensional principal stress, and in practice, the method includes but is not limited to this setting. Due to the distribution characteristics of the ground stress, the influence degree of the excavation, and the influence degree of the working face mining, the principal stress directions are different.

[0024] The stress expression after the three-dimensional principal stress deviates by an angle γ from the Z axis of the space rectangular coordinate system is:

[0025]

[0026] In the formula, σ′′′ z —the vertical stress of the roadway surrounding rock, MPa; σ′′′ x —the horizontal stress of the roadway, MPa; σ′′′ v —the axial stress of the roadway, MPa; τ′′′ xy —the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ′′′ yz —the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ′′′ xz —the shear stress of the roadway surrounding rock in the vertical and horizontal directions, MPa; σ1—the maximum principal stress of the roadway surrounding rock, MPa; σ2—the intermediate principal stress of the roadway surrounding rock, MPa; σ3—the minimum principal stress of the roadway surrounding rock, MPa; γ—the angle of the three-dimensional stress deviating from the Z axis of the space rectangular coordinate system.

[0027] Further, in the step (8), the stress calculation formula of an arbitrary point on the roadway periphery in the three-dimensional space under the polar coordinates is:

[0028]

[0029] Where: σ θ —Hoop stress at any point, MPa; σ r —Radial stress at any point, MPa; σ v —Axial stress of the tunnel, MPa; τ rθ —Radial and circumferential tangential stress at any point, MPa; τ rv —Radial and axial tangential stress at any point, MPa; τ rθ —Axial and circumferential tangential stress at any point, MPa; σ1—maximum principal stress of the surrounding rock of the tunnel, MPa; σ x —Horizontal stress at the side of vertical tunnel, MPa; σ y —Stress parallel to the tunnel axis, MPa; σ z —vertical stress of the tunnel surrounding rock, MPa; v—Poisson's ratio of the tunnel surrounding rock; R—radius of the circular tunnel, m; r, θ—polar coordinates of any point.

[0030] The stress calculation formula of any point around the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses obtained by deflecting the three-dimensional stress around the X, Y and Z axes of the spatial rectangular coordinate system by angles of α, β and γ are respectively brought into the stress calculation formula of any point around the roadway in polar coordinates in three-dimensional space. The stress calculation formula of any point around the roadway considering the three-dimensional stress deflection can be obtained.

[0031] The stress calculation formula for any point around the roadway in polar coordinates in three-dimensional space after the triaxial stress is deflected by an angle α around the X-axis of the spatial rectangular coordinate system is:

[0032]

[0033] Where: σ′ θ —Hoop stress at any point, MPa; σ′ r —Radial stress at any point, MPa; σ′ v —Axial stress of the tunnel, MPa; τ′ rθ — radial and circumferential tangential stress at any point, MPa; τ′ rv —Radial and axial tangential stress at any point, MPa; τ′ rθ —Axial and circumferential tangential stresses at any point, MPa; σ1—maximum principal stress of the tunnel surrounding rock, MPa; σ2—intermediate principal stress of the tunnel surrounding rock, MPa; σ3—minimum principal stress of the tunnel surrounding rock, MPa; v—Poisson's ratio of the tunnel surrounding rock; R—radius of the circular tunnel, m; r, θ—polar coordinates of any point; α—angle of deflection of the three-dimensional stress around the X-axis of the spatial rectangular coordinate system.

[0034] The stress calculation formula for any point around the roadway in polar coordinates in three-dimensional space after the triaxial stress is deflected by an angle β around the Y-axis of the spatial rectangular coordinate system is:

[0035]

[0036] Where: σ′′ θ —Hoop stress at any point, MPa; σ′′ r —Radial stress at any point, MPa; σ′′ v —Axial stress of the tunnel, MPa; τ′′ rθ —Radial and circumferential tangential stress at any point, MPa; τ′′ rv —Radial and axial tangential stress at any point, MPa; τ′′ rθ —Axial and circumferential tangential stresses at any point, MPa; σ1—maximum principal stress of the tunnel surrounding rock, MPa; σ2—intermediate principal stress of the tunnel surrounding rock, MPa; σ3—minimum principal stress of the tunnel surrounding rock, MPa; v—Poisson's ratio of the tunnel surrounding rock; R—radius of circular tunnel, m; r, θ—polar coordinates of any point; β—angle of deflection of the three-dimensional stress around the Y-axis of the spatial rectangular coordinate system.

[0037] The stress calculation formula for any point around the roadway in polar coordinates in three-dimensional space after the three-dimensional stress is deflected by an angle γ around the Z axis of the spatial rectangular coordinate system is:

[0038]

[0039] Where: σ′′′ θ —Hoop stress at any point, MPa; σ′′′ r —Radial stress at any point, MPa; σ′′′ v —Axial stress of the tunnel, MPa; τ′′′ rθ —Radial and circumferential tangential stress at any point, MPa; τ′′′ rv —Radial and axial tangential stress at any point, MPa; τ′′′ rθ —Axial and circumferential tangential stresses at any point, MPa; σ1—maximum principal stress of the tunnel surrounding rock, MPa; σ2—intermediate principal stress of the tunnel surrounding rock, MPa; σ3—minimum principal stress of the tunnel surrounding rock, MPa; v—Poisson's ratio of the tunnel surrounding rock; R—radius of circular tunnel, m; r, θ—polar coordinates of any point; γ—angle of deflection of the three-dimensional stress around the Z-axis of the spatial rectangular coordinate system.

[0040] Furthermore, in step (6), when solving the implicit equation expression of the plastic zone boundary of the tunnel surrounding rock, the MC criterion under the three-dimensional stress state is adopted. The MC criterion under the three-dimensional stress state is:

[0041]

[0042] Where: φ—internal friction angle of surrounding rock, °; c—internal cohesion of surrounding rock; σ m—mean stress, J2—deviatoric second invariant, J3—deviatoric third invariant; θ σ —stress Lode angle.

[0043] σ m , J2, J3 and θ σ are respectively:

[0044]

[0045]

[0046]

[0047]

[0048] The stress expression of any point in the polar coordinate system is substituted into the M-C criterion under the three-dimensional stress state to obtain the implicit equation expression of the plastic zone boundary of the surrounding rock with the three principal stresses deviated by an angle α around the X axis of the space rectangular coordinate system:

[0049]

[0050] Further, in the step (7), the M-C criterion under the three-dimensional stress state is used to solve the implicit equation expression of the plastic zone boundary of the surrounding rock. The M-C criterion under the three-dimensional stress state is:

[0051]

[0052] In the formula: φ—friction angle of surrounding rock, °; c—cohesion of surrounding rock; σ m —mean stress, J2—deviatoric second invariant, J3—deviatoric third invariant; θ σ —stress Lode angle.

[0053] σ m , J2, J3 and θ σ are respectively:

[0054]

[0055]

[0056]

[0057]

[0058] The stress expression of any point in the polar coordinate system is substituted into the M-C criterion under the three-dimensional stress state to obtain the implicit equation expression of the plastic zone boundary of the surrounding rock with the three principal stresses deviated by an angle β around the Y axis of the space rectangular coordinate system:

[0059]

[0060] Further, in the step (8), the M-C criterion under three-dimensional stress state is used to solve the implicit equation expression of the plastic zone boundary of the surrounding rock.

[0061]

[0062] In the formula: φ—friction angle of surrounding rock, °; c—cohesion of surrounding rock; σ m —average stress, J2—second invariant of deviatoric stress, J3—third invariant of deviatoric stress; θ σ —stress Lode angle.

[0063] σ m , J2, J3 and θ σ The expressions are respectively:

[0064]

[0065]

[0066]

[0067]

[0068] The stress expression of any point in the polar coordinate system is substituted into the M-C criterion under three-dimensional stress state to obtain the implicit equation expression of the plastic zone boundary of the surrounding rock with the three principal stresses deviated by an angle γ around the Z axis of the rectangular coordinate system.

[0069]

[0070] Further, in the step (9), the mathematical software is used to traverse and image the distribution characteristics of the plastic zone in the range of 0°-360° to obtain the distribution pattern of the plastic zone of the surrounding rock of the roadway considering the deviation of the three principal stresses.

[0071] Firstly, the size of the plastic zone of the roadway in the range of 0°-360° is solved, and secondly, the distribution characteristics of the plastic zone are traversed and imaged in the range of 0°-360°, including: checking whether the point list has at least two points, otherwise the traversal point list cannot be drawn; starting from the first point, connecting to the next point in turn; for each pair of adjacent points, calculating the curve path between them, calling the line drawing function; closing, connecting the last point to the first point; smoothing, outputting the final curve.

[0072] The distribution of the plastic zone of the roadway surrounding rock when the three-way stress deflects by an angle of α around the X-axis of the space rectangular coordinate system, by an angle of β around the Y-axis of the space rectangular coordinate system and by an angle of γ around the Z-axis of the space rectangular coordinate system is obtained respectively.

[0073] The application is a method for solving and imaging the plastic zone of surrounding rock considering the direction of shear stress. BRIEF DESCRIPTION OF DRAWINGS

[0074] Figure 1 The application provides a method for solving and imaging the plastic zone of surrounding rock considering the direction of shear stress.

[0075] Figure 2 The application provides a distribution diagram of the plastic zone of roadway surrounding rock when the three-way stress deflects by an angle of 20° around the X-axis of the space rectangular coordinate system.

[0076] Figure 3 The application provides a distribution diagram of the plastic zone of roadway surrounding rock when the three-way stress deflects by an angle of 40° around the Y-axis of the space rectangular coordinate system.

[0077] Figure 4 The application provides a distribution diagram of the plastic zone of roadway surrounding rock when the three-way stress deflects by an angle of 60° around the Z-axis of the space rectangular coordinate system.

[0078] Reference signs:

[0079] Figure 2 In the figure, 1 is the boundary of the roadway, and 2 is the boundary of the plastic zone of roadway surrounding rock when the three-way stress deflects by an angle of 20° around the X-axis of the space rectangular coordinate system.

[0080] Figure 3 In the figure, 1 is the boundary of the roadway, and 2 is the boundary of the plastic zone of roadway surrounding rock when the three-way stress deflects by an angle of 40° around the Y-axis of the space rectangular coordinate system.

[0081] Figure 4 In the figure, 1 is the boundary of the roadway, and 2 is the boundary of the plastic zone of roadway surrounding rock when the three-way stress deflects by an angle of 60° around the Z-axis of the space rectangular coordinate system. DETAILED DESCRIPTION

[0082] The application will be further described below in combination with the description of the drawings, and it should be understood that the preferred embodiments described herein are only used for describing and explaining the application, and are not used for limiting the application.

[0083] The specific embodiments of the present application are described below to facilitate the understanding of the present application for those skilled in the art, but it should be clear that the present application is not limited to the scope of the specific embodiments, and for those skilled in the art, all the inventions utilizing the concept of the present application are within the scope of the present application as long as various changes are obvious within the spirit and scope of the present application defined and determined by the appended claims.

[0084] The present application provides a method for solving and imaging the plastic zone of surrounding rock considering the direction of shear stress: by analyzing the process of the three principal stresses of any point around the roadway deviating around the three principal axes of the space rectangular coordinate system, the expressions of the roadway in the vertical, horizontal and axial directions and the three direction shear stresses under the deviation around the respective principal axes are obtained, and then the distribution characteristics of the plastic zone of the roadway considering the direction of shear stress are calculated, and the plastic zone distribution characteristics are imaged in the range of 0°-360° by using mathematical software, and the plastic zone shape map is obtained. In order to make the above-mentioned purposes, characteristics and advantages of the present application more obvious and easy to understand, the present application will be further described in detail below in combination with the drawings and specific embodiments.

[0085] (1) The cohesion c=2.75 MPa, the internal friction angle φ=32°, and the Poisson ratio v=0.2 of the roadway surrounding rock are determined by rock mechanics test; R is the radius of the circular roadway, m;

[0086] (2) The cross-sectional size of the roadway is determined by field measurement: length=5.6 m, width=3.8 m, and equivalent circle radius=3.38; the three principal stresses of the roadway surrounding rock are measured as: σ1=25.38 MPa, σ2=17.77 MPa, and σ1=13.40 MPa.

[0087] (3) The stress expression after the three principal stresses deviate around the X axis of the space rectangular coordinate system by 20° is:

[0088]

[0089] In the formula: σ' z —vertical stress of the roadway surrounding rock, MPa; σ' x —horizontal stress of the roadway, MPa; σ' v —axial stress of the roadway, MPa; τ' xy —shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ' yz —shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ' xz —shear stress of the roadway surrounding rock in the vertical and horizontal directions, MPa; σ1—maximum principal stress of the roadway surrounding rock, MPa; σ2—intermediate principal stress of the roadway surrounding rock, MPa; σ3—minimum principal stress of the roadway surrounding rock, MPa.

[0090] Substitute σ1=25.38 MPa, σ2=17.77 MPa, σ1=13.40 MPa into the above expression, and the stress sizes of the roadway in the vertical, horizontal, axial directions and the corresponding three shear stresses after the three-dimensional principal stress deflects 20° around the X axis are respectively: σ' x =13.40 MPa, σ' y =18.66 MPa, σ' z =24.49 MPa, τ' xy =0 MPa, τ' yz =2.44 MPa, τ' xz =0 MPa

[0091] (4) The stress expression of the three-dimensional principal stress after deflecting 40° around the Y axis of the space rectangular coordinate system is:

[0092]

[0093] In the formula: σ'' z is the vertical stress of the roadway surrounding rock, MPa; σ'' x is the horizontal stress of the roadway, MPa; σ'' v is the axial stress of the roadway, MPa; τ'' xy is the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ'' yz is the shear stress of the roadway surrounding rock in the vertical and axial directions, MPa; τ'' xz is the shear stress of the roadway surrounding rock in the vertical and horizontal directions, MPa; σ1 is the maximum principal stress of the roadway surrounding rock, MPa; σ2 is the intermediate principal stress of the roadway surrounding rock, MPa; σ3 is the minimum principal stress of the roadway surrounding rock, MPa; β is the angle of the three-dimensional stress deflecting around the Y axis of the space rectangular coordinate system.

[0094] Substitute σ1=25.38 MPa, σ2=17.77 MPa, σ1=13.40 MPa into the above expression, and the stress sizes of the roadway in the vertical, horizontal, axial directions and the corresponding three shear stresses after the three-dimensional principal stress deflects 40° around the Y axis are respectively: σ'' x =18.35 MPa, σ'' y =17.77 MPa, σ'' z =20.43 MPa, τ'' xy =0 MPa, τ'' yz =0 MPa, τ'' xz =5.90 MPa

[0095] (5) The stress expression of the three-dimensional principal stress after deflecting 60° around the Z axis of the space rectangular coordinate system is:

[0096]

[0097] wherein σ''' z —vertical stress of roadway surrounding rock, MPa; σ''' x —horizontal stress of roadway, MPa; σ''' v —axial stress of roadway, MPa; τ''' xy —shear stress of roadway surrounding rock in vertical and axial directions, MPa; τ''' yz —shear stress of roadway surrounding rock in vertical and axial directions, MPa; τ''' xz —shear stress of roadway surrounding rock in vertical and horizontal directions, MPa; σ1—maximum principal stress of roadway surrounding rock, MPa; σ2—intermediate principal stress of roadway surrounding rock, MPa; σ3—minimum principal stress of roadway surrounding rock, MPa.

[0098] Substituting σ1=25.38 MPa, σ2=17.77 MPa, σ1=13.40 MPa into the above expressions, the stress sizes of the roadway in vertical, horizontal, axial directions and the corresponding three shear stresses after deflecting 60° around Z axis are respectively: σ''' x =16.68 MPa, σ''' y =14.49 MPa, σ''' z =25.28 MPa, τ''' xy =1.89 MPa, τ''' yz =0 MPa, τ''' xz =0 MPa

[0099] (6) According to the roadway rock mechanics parameters obtained in step (1), the roadway cross section parameters obtained in step (2), the three principal stresses and the stress sizes of the roadway in vertical, horizontal, axial directions and the corresponding three shear stresses after deflecting α angle around X axis of the space rectangular coordinate system obtained in step (4), the implicit equation expression of the boundary of the plastic zone of the roadway surrounding rock is obtained by using the theory of elastic-plastic mechanics.

[0100] After deflecting 20° around X axis of the space rectangular coordinate system, the stress calculation formula of any point on the periphery of the roadway in polar coordinates is:

[0101]

[0102] wherein σ' θ —hoop stress of any point, MPa; σ' r —radial stress of any point, MPa; σ' v —axial stress of roadway, MPa; τ' rθ —shear stress of any point in radial and hoop directions, MPa; τ'rv τ'—any point radial and axial tangential stress, MPa rθ σ1—maximum principal stress of roadway surrounding rock, MPa; σ2—intermediate principal stress of roadway surrounding rock, MPa; σ3—minimum principal stress of roadway surrounding rock, MPa; v—Poisson's ratio of roadway surrounding rock; R—radius of circular roadway, m; r, θ—polar coordinates of any point.

[0103] In solving the implicit equation expression of the plastic zone boundary of roadway surrounding rock, the M-C criterion under three-dimensional stress state is adopted. The M-C criterion under three-dimensional stress state is:

[0104]

[0105] In the formula: φ—friction angle of surrounding rock, °; c—cohesion of surrounding rock; σ m —average stress, J2—second invariant of deviatoric stress, J3—third invariant of deviatoric stress; θ σ —stress Lode angle.

[0106] σ m , J2, J3 and θ σ The expressions are respectively:

[0107]

[0108]

[0109]

[0110]

[0111] Substitute the stress expression of any point in the polar coordinate system into the M-C criterion under three-dimensional stress state, and get the implicit equation expression of the plastic zone boundary of surrounding rock with three principal stresses deviated by 20° around the X axis of the space rectangular coordinate system:

[0112]

[0113] (7) According to the roadway rock mechanics parameters obtained from step (1) test, the roadway cross section parameters obtained from step (2) field measurement, the stress size of three principal stresses and the corresponding three shear stresses of the roadway after the three principal stresses are deviated by β angle around the Y axis of the space rectangular coordinate system, and using the theory of elastic-plastic mechanics, the implicit equation expression of the plastic zone boundary of roadway surrounding rock is analyzed and solved.

[0114] After the three principal stresses are deviated by 40° around the Y axis of the space rectangular coordinate system, the stress calculation formula of any point on the roadway periphery in the polar coordinate system is:

[0115]

[0116] where: σ'' - hoop stress at any point, MPa; σ'' - radial stress at any point, MPa; τ'' - shear stress at any point, MPa; σ1 - maximum principal stress of roadway surrounding rock, MPa; σ2 - intermediate principal stress of roadway surrounding rock, MPa; σ3 - minimum principal stress of roadway surrounding rock, MPa; v - Poisson's ratio of roadway surrounding rock; R - radius of circular roadway, m; r, θ - polar coordinates of any point. θ r v rθ rv rθ where: σ'' - hoop stress at any point, MPa; σ'' - radial stress at any point, MPa; τ'' - shear stress at any point, MPa; σ1 - maximum principal stress of roadway surrounding rock, MPa; σ2 - intermediate principal stress of roadway surrounding rock, MPa; σ3 - minimum principal stress of roadway surrounding rock, MPa; v - Poisson's ratio of roadway surrounding rock; R - radius of circular roadway, m; r, θ - polar coordinates of any point.

[0117] The M-C criterion under three-dimensional stress state is used to solve the implicit equation expression of the plastic zone boundary of roadway surrounding rock. The M-C criterion under three-dimensional stress state is:

[0118]

[0119] where: φ - internal friction angle of surrounding rock, °; c - cohesion of surrounding rock; σ m - average stress, J2 - second invariant of deviatoric stress, J3 - third invariant of deviatoric stress; θ σ - stress Lode angle.

[0120] σ m , J2, J3 and θ σ are respectively:

[0121]

[0122]

[0123]

[0124]

[0125] The stress expression of any point in the polar coordinate system is substituted into the M-C criterion under three-dimensional stress state to obtain the implicit equation expression of the plastic zone boundary of surrounding rock with the three principal stresses deviated by an angle of β around the Y axis of the rectangular coordinate system in space:

[0126]

[0127] ​​​​​(8) Based on the rock mechanics parameters of the tunnel obtained from the experiment in step (1), the cross-sectional parameters and three-dimensional principal stresses of the tunnel obtained from the field measurement in step (2), and the stress magnitudes of the tunnel in the vertical, horizontal, and axial directions and the corresponding three shear stresses after the three-dimensional principal stresses are deflected by an angle γ around the Z axis of the spatial rectangular coordinate system obtained from the analysis in step (4), the implicit equation expression of the plastic zone boundary of the tunnel surrounding rock is analyzed and solved using the theory of elastic-plastic mechanics.

[0128] After the triaxial stress is deflected 60° around the Z axis of the spatial rectangular coordinate system, the stress calculation formula at any point around the roadway in polar coordinates is:

[0129]

[0130] Where: σ′′′ θ —Hoop stress at any point, MPa; σ′′′ r —Radial stress at any point, MPa; σ′′′ v —Axial stress of the tunnel, MPa; τ′′′ rθ —Radial and circumferential tangential stress at any point, MPa; τ′′′ rv —Radial and axial tangential stress at any point, MPa; τ′′′ rθ —Axial and circumferential tangential stresses at any point, MPa; σ1—maximum principal stress of the tunnel surrounding rock, MPa; σ2—intermediate principal stress of the tunnel surrounding rock, MPa; σ3—minimum principal stress of the tunnel surrounding rock, MPa; v—Poisson's ratio of the tunnel surrounding rock; R—radius of circular tunnel, m; r, θ—polar coordinates of any point.

[0131] When solving the implicit equation expression of the plastic zone boundary of the tunnel surrounding rock, the MC criterion under three-dimensional stress state is adopted. The MC criterion under three-dimensional stress state is:

[0132]

[0133] Where: φ—internal friction angle of surrounding rock, °; c—internal cohesion of surrounding rock; σ m —mean stress, J2—second invariant of deviatoric stress, J3—third invariant of deviatoric stress; θ σ —Stress Lode angle.

[0134] σ m , J2, J3 and θ σ The expressions are:

[0135]

[0136]

[0137]

[0138]

[0139] Substituting the stress expression of any point in the polar coordinate system into the MC criterion under the three-dimensional stress state, the implicit equation expression of the plastic zone boundary of the surrounding rock with the three principal stresses deflected by an angle γ around the Z axis of the spatial rectangular coordinate system is obtained:

[0140]

[0141] (9) Mathematical software is used to perform traversal imaging of the distribution characteristics of the plastic zone in the range of 0°-360°, and the distribution morphology of the plastic zone of the tunnel surrounding rock is obtained under the condition of considering the three-dimensional principal stress deflection.

[0142] First, the size of the plastic zone in the range of 0°-360° in the roadway is solved. Secondly, the distribution characteristics of the plastic zone are traversed and imaged in the range of 0°-360°, including: checking whether the point list has at least two points, otherwise the traversal point list cannot be drawn. In this calculation case, the output data points after deflection around the X, Y, and Z axes are all 73, respectively represented by (r1, θ1), (r2, θ2)... (r 73 ,θ 73 ) numbering; starting from the first point (r1, θ1), connect to the next point in sequence; for each pair of adjacent points, calculate the curve path between them and call the line drawing function. Here, 72 curves need to be called because according to the size of the plastic zone obtained by the solution, the first point (r1, θ1) and the 73rd point (r 73 ,θ 73 ) indicates the same position, that is, 0° and 360° are the same position; close, and the last point (r 73 ,θ 73 ) is connected to the first point (r1,θ1); smooth and output the final curve.

[0143] The plastic zone image of the tunnel surrounding rock after the triaxial stress is deflected 20° around the X-axis of the spatial rectangular coordinate system is obtained as follows: Figure 2 As shown in the figure, the plastic zone image of the tunnel surrounding rock after the three-dimensional stress is deflected 40° around the Y axis of the spatial rectangular coordinate system is as follows: Figure 3 As shown in the figure, the image of the plastic zone of the roadway surrounding rock after the three-dimensional stress is deflected 60° around the Z axis of the spatial rectangular coordinate system is as follows: Figure 4 shown.

Claims

1. A method for solving and imaging plastic zone of surrounding rock considering shear stress direction, characterized in that, The application relates to a method for determining the plastic zone boundary of a roadway surrounding rock. The method comprises the following steps: determining the cohesion, internal friction angle and Poisson's ratio parameters of the roadway surrounding rock through rock mechanics experiments; determining the roadway section size and the three-direction principal stresses of the roadway surrounding rock through field measurement; analyzing the stress expressions of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle alpha around the X-axis of the space rectangular coordinate system; analyzing the stress expressions of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle beta around the Y-axis of the space rectangular coordinate system; analyzing the stress expressions of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle gamma around the Z-axis of the space rectangular coordinate system; according to the rock mechanics parameters of the roadway obtained through the experiments, the roadway section parameters obtained through the field measurement, the three-direction principal stresses and the stress sizes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle alpha around the X-axis of the space rectangular coordinate system, the plastic zone boundary implicit equation expression of the roadway surrounding rock is analyzed and solved by using the elastic-plastic mechanics theory; according to the rock mechanics parameters of the roadway obtained through the experiments, the roadway section parameters obtained through the field measurement, the three-direction principal stresses and the stress sizes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle beta around the Y-axis of the space rectangular coordinate system, the plastic zone boundary implicit equation expression of the roadway surrounding rock is analyzed and solved by using the elastic-plastic mechanics theory; according to the rock mechanics parameters of the roadway obtained through the experiments, the roadway section parameters obtained through the field measurement, the three-direction principal stresses and the stress sizes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle gamma around the Z-axis of the space rectangular coordinate system, the plastic zone boundary implicit equation expression of the roadway surrounding rock is analyzed and solved by using the elastic-plastic mechanics theory; 2. The method of claim 1, wherein, The three-direction principal stresses of the roadway cross section size and the roadway surrounding rock determined by the field measurement include: the length and width of the roadway cross section; the maximum principal stress σ1, the intermediate principal stress σ2 and the minimum principal stress σ3 of any point of the roadway surrounding rock 3。 3. The method of claim 1, wherein the method further comprises: the plastic zone distribution characteristics are imaged in the 0-360 degree range by using the mathematical software, and the distribution form of the plastic zone of the roadway surrounding rock considering the shear stress direction is obtained. The stress sizes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle alpha around the X-axis of the space rectangular coordinate system are analyzed and solved by using the elastic-plastic mechanics theory, and the plastic zone boundary implicit equation expression of the roadway surrounding rock comprises the following steps: By the average stress σ m , deviatoric stress second invariant J2, deviatoric stress third invariant J3 and stress Lode angle θ σ M-C criterion under three-dimensional stress state is represented; the stress expression of an arbitrary point on the periphery of the roadway in the polar coordinate system is obtained according to the generalized plane strain problem model; 4. The method of claim 1, wherein, the stress expression of the arbitrary point in the polar coordinate system is substituted into the M-C criterion under the three-dimensional stress state, and the plastic zone boundary implicit equation expression of the roadway surrounding rock after the three-direction principal stresses are deflected by an angle alpha around the X-axis of the space rectangular coordinate system is obtained. The stress sizes of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three-direction principal stresses are deflected by an angle beta around the Y-axis of the space rectangular coordinate system are analyzed and solved by using the elastic-plastic mechanics theory, and the plastic zone boundary implicit equation expression of the roadway surrounding rock comprises the following steps: the stress expression of an arbitrary point on the periphery of the roadway in the polar coordinate system is obtained according to the generalized plane strain problem model; the stress expression of the arbitrary point in the polar coordinate system is substituted into the M-C criterion under the three-dimensional stress state, and the plastic zone boundary implicit equation expression of the roadway surrounding rock after the three-direction principal stresses are deflected by an angle beta around the Y-axis of the space rectangular coordinate system is obtained. By the average stress σ m , deviatoric stress second invariant J2, deviatoric stress third invariant J3 and stress Lode angle θ σ M-C criterion under three-dimensional stress state is represented; The stress expression of any point in the polar coordinate system is substituted into the M-C criterion under the three-dimensional stress state, so that the implicit equation expression of the plastic zone boundary of the roadway surrounding rock after the three principal stresses are deflected by an angle β around the Y axis of the space rectangular coordinate system is obtained.

5. The method of claim 1, wherein, The stress size of the roadway in the vertical, horizontal and axial directions and the corresponding three shear stresses after the three principal stresses are deflected by an angle γ around the Z axis of the space rectangular coordinate system is analyzed and solved by using the theory of elastic-plastic mechanics, so that the implicit equation expression of the plastic zone boundary of the roadway surrounding rock is obtained, including: The stress expression of any point on the roadway periphery in the polar coordinate system is obtained according to the generalized plane strain problem model; By the average stress σ m , deviatoric stress second invariant J2, deviatoric stress third invariant J3 and stress Lode angle θ σ M-C criterion under three-dimensional stress state is represented; The stress expression of any point in the polar coordinate system is substituted into the M-C criterion under the three-dimensional stress state, so that the implicit equation expression of the plastic zone boundary of the roadway surrounding rock after the three principal stresses are deflected by an angle γ around the Z axis of the space rectangular coordinate system is obtained.

6. The method of claim 1, wherein, The plastic zone distribution characteristics are imaged in the range of 0°-360° by using the mathematical software, so that the distribution form of the plastic zone of the roadway surrounding rock considering the shear stress direction is obtained, including: The plastic zone distribution characteristics are imaged in the range of 0°-360°; The distribution form of the plastic zone of the roadway surrounding rock when the three stresses are deflected by an angle α around the X axis of the space rectangular coordinate system is obtained; the distribution form of the plastic zone of the roadway surrounding rock when the three stresses are deflected by an angle β around the Y axis of the space rectangular coordinate system is obtained; and the distribution form of the plastic zone of the roadway surrounding rock when the three stresses are deflected by an angle γ around the Z axis of the space rectangular coordinate system is obtained.

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