Damage prediction method for surrounding rock under vibration load condition

Abstract

The invention discloses a damage prediction method for surrounding rock under a vibration load condition, and provides a research method for roadway surrounding rock damage evolution according to structural characteristics of mine roadway surrounding rock and in combination with a thermodynamic theory. According to the measured vibration characteristics of the heading machine, the change rule of the roadway floor damage degree D along with time and the change rule of the damage evolution rate along with time are analyzed and obtained; the method specifically comprises the following steps: analyzing thermodynamics of a damaged two-phase saturated pore medium; establishing a constitutive equation of the damaged two-phase saturated pore medium; analyzing the relationship between the mechanical property parameters of the two-phase saturated pore medium and the damage; deriving a wave equation of the damaged two-phase saturated pore medium; and finally, the superiority and the accuracy of the method are verified by examples.

Description

Technical Field

[0001] The present invention belongs to the field of mine tunnel surrounding rocks, and particularly relates to a method for predicting damage of surrounding rocks under vibration load conditions. Background Art

[0002] For mine tunnels with greater burial depths, the surrounding rock has a high degree of saturation and can generally be regarded as a two-phase saturated porous medium consisting of a solid phase containing pores and a liquid phase filling the pores. In the tunneling tunnel, strong vibrations are generated during the operation of the roadheader. The vibration load can easily induce damage such as pores and cracks to be generated within the surrounding rock, and promote the evolution of existing damage, aggravating the expansion and penetration of pores and cracks, seriously affecting the stability of the surrounding rock, causing significant deformation of the surrounding rock, and even causing accidents such as water inrush in the surrounding rock. Currently, related research mainly focuses on monitoring the surrounding rock using methods such as displacement monitoring, strain monitoring, microseismic monitoring, and microseismic and electrical coupling monitoring. If the damage formation and evolution laws can be predicted based on the characteristics of the vibration load (such as amplitude, frequency, etc.) and the surrounding rock characteristics in the early stages of the tunneling operation, it will help guide the tunneling operation, provide a basis for surrounding rock support, and effectively promote surrounding rock stability. Summary of the Invention

[0003] The purpose of this invention is to provide a method for predicting damage of surrounding rocks under vibration load conditions. According to the structural characteristics of mine tunnel surrounding rocks and combined with thermodynamic theory, a research method for the evolution of tunnel surrounding rock damage is proposed.

[0004] The technical solution of the present invention is a method for predicting surrounding rock damage under vibration load conditions, characterized in that, based on the measured vibration characteristics of the roadheader, the variation pattern of the damage degree D of the tunnel floor and the variation pattern of the damage evolution rate over time are analyzed and obtained;

[0005] The specific steps include

[0006] Step 1: Thermodynamic analysis of damaged two-phase saturated porous media

[0007] Step 1.1: Consider the saturated porous medium composed of solid and liquid phases as a thermodynamic system. According to the first law of thermodynamics,

[0008]

[0009] Where, are the rates of change of kinetic energy and internal energy of the studied thermodynamic system with time, J / s; is the power of the external force acting on the system, J / s; is the rate of change of the heat absorption of the system with time, J / s;

[0010] Step 1.1.1. Take a volume element from a two-phase saturated porous medium, whose total volume is V and total surface area is A. The volumes of the solid and liquid phases in the volume element are V and A respectively. s With V f , the surface areas are A s With A f , the direction cosine of the outer normal of the volume element surface is taken as n i In the volume element, the total power of the surface force and body force of the solid and liquid phases is

[0011]

[0012] Where, and are the powers acting on the solid and liquid phase loads, W; σ sij is the solid surface stress, Pa; u si is the solid phase displacement, m; is the solid phase velocity, m / s; f si is the body force per unit volume of solid phase, N / m 3 ; i and j both represent directions; σ fij is the stress on the liquid surface, Pa; according to Biot theory, σ fij =-βpδ ij , δ ij is the Kronecker symbol, β is the porosity, p is the pore fluid pressure, Pa; u fi is the liquid phase displacement, m; u· fi is the liquid phase velocity, m / s; f fi is the body force per unit volume of liquid phase, N / m 3 ;

[0013] According to Gauss's theorem, the surface integral in formula (2) is converted into volume integral, and we get

[0014]

[0015] Step 1.1.2: The rate of increase of heat energy of solid and liquid phases in the volume element is

[0016]

[0017] Where, and are the rates of change of solid and liquid phase thermal energy increments with time, W; q si is the heat flux vector of the solid phase, W / m 2 ρ s is the solid mass density, kg / m 3 ; γ s is the heat generation rate per unit mass of solid phase, W / kg; q fiis the liquid phase heat flux density vector, W / m 2 ρ f is the liquid mass density, kg / m 3 ; γ f is the heat generation rate per unit mass of liquid phase, W / kg;

[0018] According to Gauss's theorem, the surface integral in equation (4) is converted into volume integral, and we get

[0019]

[0020] Step 1.1.3: The rate of change of the internal energy of the solid and liquid phases in the volume element is

[0021]

[0022] Where, E s With E f are the internal energies per unit mass of solid and liquid phases, J / kg;

[0023] Step 1.1.4: According to Biot theory, the kinetic energy per unit volume of two-phase saturated porous medium is:

[0024]

[0025] Where, ρ 11 and ρ 22 is the mass coefficient, kg / m 3 ρ 12 is the mass coupling coefficient of the solid and liquid phases, kg / m 3 ,ρ1=(1-β)ρ s ,ρ2=βρ f , ρ 12 =-ρ a ,ρ a is the additional apparent mass, kg / m 3 , ρ 11 =ρ1+ρ a , ρ 22 =ρ2+ρ a ;

[0026] Substituting the mass coefficient and the mass coupling coefficient of the solid-liquid phase into formula (7), we can get

[0027]

[0028] After taking the derivative of Equation (8) with respect to time and integrating it with respect to volume, the increase rate of the kinetic energy of the solid and liquid phases in the volume element is obtained as

[0029]

[0030] Step 1.2: According to Biot theory, the total power dissipated by the relative motion of solid and liquid phases in the volume element is

[0031]

[0032] Where b is the Biot dissipation coefficient, kg / (m 3 ·s), b=ηβ 2 / κ; κ is permeability, m 2 ;η is the dynamic viscosity of pore fluid, kg / (m·s);

[0033] The dissipation caused by the relative motion of the solid and liquid phases is converted into heat; Substituting equations (3), (5), (6), (9) and (10) into equation (1), we can get

[0034]

[0035] According to Biot theory, the equations of motion for the solid and liquid phases are:

[0036]

[0037] Substituting equations (12) and (13) into equation (11), we can simplify to obtain

[0038]

[0039] Step 1.3: Under small strain conditions in continuum theory, the velocity gradient of the solid-liquid phase can be decomposed into a symmetric strain rate tensor and an antisymmetric spin tensor.

[0040]

[0041] The curl tensor is antisymmetric about indices i and j, and the stress tensor is symmetric about indices i and j, so

[0042]

[0043] Multiply both ends of the equal signs in formula (15) and formula (16) by σ sij and σ fij , and combined with formula (17), we get

[0044]

[0045]

[0046] Substituting equations (18) and (19) into equation (14), we can get

[0047]

[0048] Step 1.4: Calculate the total entropy of the two-phase saturated porous medium within the volume element

[0049]

[0050] Where S s 、S f are the entropies of the solid and liquid phases, J / K; η s ,η f are the entropy densities of solid and liquid phases per unit mass, J / (kg·K); M s 、M f are the masses of the solid phase and liquid phase, kg;

[0051] Step 1.4.1: According to the entropy increase principle of the second law of thermodynamics, we can get

[0052]

[0053] Where Q is the heat absorbed by the thermodynamic system, J; θ is the absolute temperature of the thermodynamic system, K; the equal sign in equation (22) corresponds to a reversible process, and the greater than sign corresponds to an irreversible process.

[0054] Take the time derivative of both sides of formula (21) and substitute it and formula (5) into formula (22) to obtain

[0055]

[0056] Where θ s ,θ f are the absolute temperatures of the solid and liquid phases, respectively, in K;

[0057] From formula (23), we can see

[0058]

[0059] Substitute equation (20) into equation (24) and eliminate γ s , γ f ,have to

[0060]

[0061] Step 1.4.2: According to thermodynamic theory, the relationship between the Helmholtz free energy per unit mass of the solid and liquid phases and their internal energy in a two-phase saturated porous medium is:

[0062] H s =E s -θ s η s (26)

[0063] H f =E f -θf η f (27)

[0064] The time derivatives of Equations (26) and (27) are respectively used to obtain the rate of change of the Helmholtz free energy per unit mass of the solid-liquid two-phase, and then substituted into Equation (25), we get

[0065]

[0066] For an isothermal process, Equation (28) is simplified to

[0067]

[0068] Formula (29) can be written as

[0069]

[0070] Where ψ is the Helmholtz free energy per unit volume of two-phase saturated porous medium, J / m 3 , ψ s and ψ f are the Helmholtz free energy per unit volume of solid and liquid phases, J / m 3 ,

[0071] Step 2: Establish the constitutive equation of the damaged two-phase saturated porous medium

[0072] Step 2.1: The Helmholtz free energy ψ of a two-phase saturated porous medium is its solid phase deformation ε sij 、Liquid phase change ε fij , and the function of damage degree D

[0073] ψ=ψ(ε sij ,ε fij ,D) (31)

[0074] Step 2.1.1: When the solid phase is uniform and isotropic, the rate of change of Helmholtz free energy with time is

[0075]

[0076] Substituting formula (32) into formula (30), we get

[0077]

[0078] The first two terms in Equation (32) are non-dissipative terms, and the third term represents the energy dissipation caused by damage evolution. The total dissipated energy of the system is non-negative. If inequality (30) is to hold for any thermodynamic process, the non-dissipative term must be zero, so

[0079]

[0080] Step 2.1.2, the damage strain energy release rate is the rate of change of the energy consumed by the medium due to damage with the damage degree. The damage strain energy release rate Y and the damage degree D are dual variables, so

[0081]

[0082] Substituting equations (34), (35) and (36) into equation (33), the damage dissipation power of the two-phase saturated porous medium satisfies

[0083]

[0084] Damage is an irreversible process, so D·≥0, so the damage strain energy release rate Y is non-negative;

[0085] Dissipation potential ψ * It is a convex function of the damage strain energy release rate Y. According to the orthogonal flow law of internal variables, we know that

[0086]

[0087] Step 2.2: Liquid phase stress tensor component σ fij and the strain tensor component ε fij They can be expressed by the liquid phase stress scalar s and the liquid phase volume strain scalar ε: fij =sδ ij , 3ε fij =εδ ij , s = -βp;

[0088] Step 2.2.1: If the two-phase saturated porous medium is elastically deformed, |ε sij |<<1,|ε|<<1,0≤D≤1,the Helmholtz free energy per unit volume ψ(ε sij ,ε,D) is

[0089]

[0090] Where ψ0 is the Helmholtz free energy of the initial state of the two-phase saturated porous medium, A (n) 、C (n) and G (n) is the scalar coefficient, and is the second-order tensor coefficient, is the fourth-order tensor coefficient,

[0091] Step 2.2.2: Substitute equation (39) into equation (34) and equation (35) respectively to obtain the stresses of the solid and liquid phases:

[0092]

[0093] Step 2.2.3: Substitute equation (39) into equation (36) to obtain the damage strain energy release rate:

[0094]

[0095] Step 2.3: During the loading process, the solid phase undergoes elastic deformation and the solid and liquid phases move relative to each other. After unloading, the stress, strain, and damage strain energy release rate of the solid and liquid phases are all zero, that is, σ sij =0,ε sij = 0, s = 0, ε = 0, Y = 0; As mentioned above, the damage formation process is irreversible, so the damage degree after unloading is not zero, that is, D ≠ 0; Therefore, the coefficients in Equations (39), (40), (41) and (42) are

[0096]

[0097] Step 2.3.1. Substitute equation (43) into equations (39), (40), (41), and (42) to obtain

[0098]

[0099] Step 2.3.2: Substitute D = 0 into Equation (45) and Equation (46) respectively to obtain the stresses of the solid and liquid phases under lossless conditions:

[0100]

[0101] In formula (48) and formula (49), and G (0) are the elastic tensor of the solid phase, the fluid-solid coupling coefficient, and the elastic modulus of the liquid phase under lossless conditions;

[0102] Step 2.3.3: Biot theory does not consider the evolution and influence of damage in porous media, and gives the constitutive equation of two-phase saturated porous media under damage-free conditions as follows:

[0103] σ sij =2Nε sij +(Ae+Qε)δ ij (50)

[0104] s=Qe+Rε (51)

[0105] Where N and A are parameters related to the Lamé constant of the solid phase, and Q and R are non-negative elastic parameters in the Biot model, which characterize the elasticity of the pore fluid and its elastic interaction with the solid phase. 2 M, Q = β (α - β) M, R = β 2 M, μ and λ are the solid-phase Lamé constants, Pa; α is the Biot coefficient, α=1-K b / K s ;M is Biot modulus, Pa, M=[(α-β) / K s +β / K f ] -1 , K s , K b and K f are the bulk modulus of solid particles, the bulk modulus of solid skeleton and the bulk modulus of liquid phase under non-destructive conditions, Pa; e is the solid phase volume strain, e = u si,i ;

[0106] From the elasticity theory, we know that the fourth-order elastic tensor of an isotropic medium is

[0107]

[0108] Step 2.3.4, equation (48), equation (49) and equation (50), equation (51) correspond to each other, and equation (48), equation (49) and equation (50), equation (51) are combined and solved to obtain

[0109]

[0110] The parameters under the damage-free condition can be multiplied by the coefficient related to the damage to obtain the corresponding parameters under the damage condition. Therefore, the relevant parameters under the damage condition are

[0111]

[0112] Where, and ζ (n) All are dimensionless non-negative coefficients;

[0113] Step 2.3.5: The pressure acting on the solid phase reduces the porosity and causes the liquid phase to be expelled from the pores, so the solid phase volume strain e and the liquid phase volume strain ε have opposite signs. According to Biot theory, A-λ, Q, and R are all positive, so and G (n) are all positive; for a uniform isotropic elastic medium, is a symmetric positive definite tensor; As mentioned above, Y in formula (36) is a non-negative physical quantity, since and G (n) are all positive. From formula (47), we can see that is a negative definite tensor

[0114]

[0115] Where, τ (n) is a dimensionless non-negative coefficient;

[0116] Step 2.4: Substitute equations (55), (56) and (57) into equations (44) to (47) to obtain

[0117]

[0118] σ sij =2N(D)ε sij +A(D)eδ ij +Q(D)εδ ij (59)

[0119] s=R(D)ε+Q(D)e (60)

[0120]

[0121] Where A(D) = AM A (D), N(D)=NM N (D), Q(D)=QM Q (D), R(D)=RM R (D), M A (D), M N (D), M Q (D) and M R (D) is the damage effect function, which represents the impact of damage on parameters such as N, A, Q and R.

[0122] Equations (58), (59), (60), and (61) are the strain energy expression, solid phase stress expression, liquid phase stress expression, and damage strain energy release rate expression of the two-phase saturated porous medium under damage conditions, respectively.

[0123] Step 2.5: As mentioned above, damage is an energy dissipation process, using the dissipation potential ψ * Characterize the dissipation characteristics of the damage process and construct the dissipation potential in the form of a power function of the damage driving force Y

[0124]

[0125] Where a and m are parameters related to the evolution of material damage;

[0126] Substituting equations (61) and (62) into equation (38), we obtain the damage evolution equation for two-phase saturated porous media:

[0127]

[0128] Step 3: Analysis of mechanical properties parameters of damaged two-phase saturated porous media

[0129] Step 3.1, Kachanov studied the elastic medium with micro-hole damage, and the damage effect functions of Young's modulus and Poisson's ratio are

[0130]

[0131] Where v is the Poisson's ratio of the elastic medium under lossless conditions;

[0132] Step 3.1.1: Based on the relationship between the Lamé coefficient of elastic media and Young's modulus and Poisson's ratio, and combined with Equations (64) and (65), the damage effect function of the solid phase Lamé coefficient in two-phase saturated porous media is obtained as follows:

[0133]

[0134] Step 3.1.2: Based on the relationship between the bulk modulus of the elastic medium, Young's modulus, and Poisson's ratio, the bulk modulus of the solid phase in the two-phase saturated porous medium under damage conditions is obtained as follows:

[0135]

[0136] Step 3.2: As mentioned above, N is the shear modulus of the solid phase in the two-phase saturated porous medium, and its damage effect function is given by Equation (67):

[0137] M N (D)=M μ (D) (69)

[0138] Step 3.2.1: Under damage conditions, the porosity of the two-phase saturated porous medium is

[0139] β(D)=β+(1-β)[1-(1-D) 2 ] (70)

[0140] Step 3.2.2: According to the definition of each parameter in Biot theory, the Biot coefficient α(D) and Biot modulus M(D) under damage condition are respectively

[0141]

[0142] Step 3.2.3: According to the definition of each parameter in Biot theory, the coefficient A(D) under damage condition is obtained by combining equations (66), (68) and equations (70) to (72):

[0143] A(D)=λM λ (D)+[α(D)-β(D)] 2 M(D) (73)

[0144] According to the definition of each parameter in Biot theory, the coefficients Q(D) and R(D) under damage conditions are obtained by combining equations (68), (70), (71) and (72).

[0145] Q(D)=β(D)[α(D)-β(D)]M(D) (74)

[0146] R(D)=[β(D)] 2 M(D) (75)

[0147] According to the definition of relevant parameters of Biot theory, the dissipation coefficient of two-phase porous medium under damage condition can be deduced as follows:

[0148]

[0149] Step 3.2.4, the Kozeny-Carman equation gives the relationship between permeability and porosity, from which the permeability of the two-phase saturated porous medium under damage conditions can be obtained as

[0150]

[0151] Where M β (D) is the damage effect function of the porosity β of the two-phase saturated porous medium.

[0152] Step 4: Establish the wave equation for the damaged two-phase saturated porous medium

[0153] Step 4.1: As shown in formula (7), the mass coefficient ρ defined in the Biot model is 11 , ρ 22 and ρ 12 are related to the density and porosity of the solid and liquid phases, respectively. As shown in Equation (70), damage affects porosity, so the mass coefficient ρ in the Biot model is 11 , ρ 22 and ρ 12 All are functions of the damage degree D;

[0154] Taking the motion of the selected volume element along the x direction as an example, the Lagrange equation gives the resultant forces of the solid and liquid phases along the x direction in the damaged two-phase saturated porous medium:

[0155]

[0156] Substituting Equations (7) and (10) into Equations (78) and (79), we can obtain the motion equations of the solid and liquid phases in the volume element along the x direction:

[0157]

[0158] Step 4.2: Similarly, the motion equations of the solid and liquid phases along the y and z directions can be obtained. Combining the stress relationship between the solid and liquid phases shown in Equations (59) and (60), as well as the geometric equations representing the displacement and strain of the solid and liquid phases, and substituting them into the motion equations of the solid and liquid phases along the x, y, and z directions, the wave equation of the two-phase saturated porous medium under damage conditions can be obtained as follows:

[0159]

[0160] Where u s 、u f are the displacement vectors of the solid and liquid phases, respectively; is the differential operator.

[0161] Step 5: Example Verification

[0162] Step 5.1: When studying the dynamic response characteristics of a two-phase saturated porous medium under damage conditions, it is necessary to solve the damage evolution equation shown in Equation (63) and the wave equations shown in Equations (82) and (83) simultaneously. The above three equations are all nonlinear partial differential equations and it is difficult to obtain analytical solutions. The generalized partial differential equation system in the COMSOL Multiphysics PDE module is in the form of

[0163]

[0164] Where, e a is the mass coefficient, d a is the damping coefficient, Г is the conserved flux, and f is the source term;

[0165] Step 5.2: When solving the wave equation, due to the inertial coupling coefficient ρ 12 The influence on the dynamic response of two-phase saturated porous media is very small. Usually, the inertial coupling coefficient ρ between the solid and liquid phases is 12 Ignore, that is, set its value to ρ 12 =0; under the damage condition, ρ in equations (82) and (83) is also 12 (D) Ignore, and rewrite the equations shown in equations (82), (83) and (63) as

[0166]

[0167] Step 5.3: From Equations (85) to (89), we can see that the mass matrix, damping matrix, dependent variable, conserved flux, and source term that satisfy the solution rules of the generalized partial differential equations in the Comsol Multiphysics PDE module are:

[0168]

[0169] Compared with the existing technology, the damage prediction method of surrounding rock under vibration load conditions provided by the present invention has the following beneficial effects:

[0170] 1. Based on the elasticity theory of porous media and the continuous damage theory, this paper establishes a wave equation for a damaged two-phase saturated porous medium, thereby predicting the degree of damage to the surrounding rock under load. The method studied in this paper fully considers the two-phase structural characteristics of the tunnel surrounding rock and is more consistent with actual working conditions. Existing research methods generally simplify the tunnel surrounding rock into an elastic medium and ignore the porous structural characteristics of the surrounding rock.

[0171] 2. In the early stage of excavation operations, the present invention can predict the formation and evolution of damage based on the characteristics of the vibration load (such as amplitude, frequency, etc.) and the characteristics of the surrounding rock. This will help guide the excavation operation and provide a basis for surrounding rock support, thereby effectively promoting surrounding rock stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0172] Figure 1 is the law of change of damage degree over time;

[0173] Figure 2 is the variation of damage evolution rate over time;

[0174] Figure 3 It is a schematic diagram of the vibration acceleration measurement system. DETAILED DESCRIPTION

[0175] The damage prediction method of surrounding rock under vibration load conditions of the present invention specifically comprises the following steps:

[0176] 1. Thermodynamic analysis of damaged two-phase saturated porous media

[0177] Considering the saturated porous medium composed of solid and liquid phases as a thermodynamic system, the first law of thermodynamics shows that

[0178]

[0179] Where, are the rates of change of kinetic energy and internal energy of the studied thermodynamic system with time, J / s; is the power of the external force acting on the system, J / s; is the rate of change of the heat absorption of the system with time, J / s.

[0180] Take a volume element from a two-phase saturated porous medium, whose total volume is V and total surface area is A. The volumes of the solid and liquid phases in the volume element are V s With V f , the surface areas are As With A f , the direction cosine of the outer normal of the volume element surface is taken as n i In the volume element, the total power of the surface force and body force of the solid and liquid phases is

[0181]

[0182] Where, and are the powers acting on the solid and liquid phase loads, W; σ sij is the solid surface stress, Pa; u si is the solid phase displacement, m; is the solid phase velocity, m / s; f si is the body force per unit volume of solid phase, N / m 3 ; i and j both represent directions; σ fij is the stress on the liquid surface, Pa; according to Biot theory, σ fij =-βpδ ij , δ ij is the Kronecker symbol, β is the porosity, p is the pore fluid pressure, Pa; u fi is the liquid phase displacement, m; is the liquid phase velocity, m / s; f fi is the body force per unit volume of liquid phase, N / m 3 .

[0183] According to Gauss's theorem, the surface integral in formula (2) is converted into volume integral, and we get

[0184]

[0185] The thermal energy increment of the solid and liquid phases comes from the heat transferred through their surfaces and the heat generated by their heat sources. The thermal energy increment rate of the solid and liquid phases in the volume element is

[0186]

[0187] Where, and are the rates of change of solid and liquid phase thermal energy increments with time, W; q si is the heat flux vector of the solid phase, W / m 2 ρ s is the solid mass density, kg / m 3 ; γ s is the heat generation rate per unit mass of solid phase, W / kg; q fi is the liquid phase heat flux density vector, W / m 2 ρ f is the liquid mass density, kg / m 3 ; γf is the heat generation rate per unit mass of liquid phase, W / kg.

[0188] According to Gauss's theorem, the surface integral in equation (4) is converted into volume integral, and we get

[0189]

[0190] The rate of change of the internal energy of the solid and liquid phases in the volume element is

[0191]

[0192] Where, E s With E f are the internal energies per unit mass of solid and liquid phases, J / kg, respectively.

[0193] The kinetic energy per unit volume of two-phase saturated porous medium can be obtained from Biot theory:

[0194]

[0195] Where, ρ 11 and ρ 22 is the mass coefficient, kg / m 3 ρ 12 is the mass coupling coefficient of the solid and liquid phases, kg / m 3 ,ρ1=(1-β)ρ s ,ρ2=βρ f , ρ 12 =-ρ a , ρ a is the additional apparent mass, kg / m 3 , ρ 11 =ρ1+ρ a , ρ 22 =ρ2+ρ a .

[0196] Substituting the mass coefficient and the mass coupling coefficient of the solid-liquid phase into formula (7), we can get

[0197]

[0198] After taking the derivative of Equation (8) with respect to time and integrating it with respect to volume, the increase rate of the kinetic energy of the solid and liquid phases in the volume element is obtained as

[0199]

[0200] According to Biot theory, the total dissipated power generated by the relative motion of solid and liquid phases in the volume element is

[0201]

[0202] Where b is the Biot dissipation coefficient, kg / (m 3 ·s), b=ηβ 2 / κ; κ is permeability, m 2 ; η is the dynamic viscosity of the pore fluid, kg / (m·s).

[0203] The dissipation caused by the relative motion of the solid and liquid phases is converted into heat; Substituting equations (3), (5), (6), (9) and (10) into equation (1), we can get

[0204]

[0205] According to Biot theory, the equations of motion for the solid and liquid phases are:

[0206]

[0207] Substituting equations (12) and (13) into equation (11), we can simplify to obtain

[0208]

[0209] Under small strain conditions in continuum theory, the velocity gradient of the solid-liquid phase can be decomposed into a symmetric strain rate tensor and an antisymmetric spin tensor.

[0210]

[0211] The curl tensor is antisymmetric about indices i and j, and the stress tensor is symmetric about indices i and j, so

[0212]

[0213] Multiply both ends of the equal signs in formula (15) and formula (16) by σ sij and σ fij , and combined with formula (17), we get

[0214]

[0215] Substituting equations (18) and (19) into equation (14), we can get

[0216]

[0217] According to the principles of thermodynamics, the total entropy of the two-phase saturated porous medium in the volume element is

[0218]

[0219] Where S s 、S f are the entropies of the solid and liquid phases, J / K; η s ,η fare the entropy densities of solid and liquid phases per unit mass, J / (kg·K); M s 、M f are the masses of the solid phase and liquid phase, kg respectively.

[0220] According to the entropy increase principle of the second law of thermodynamics, we can get

[0221]

[0222] Where Q is the heat absorbed by the thermodynamic system, J; θ is the absolute temperature of the thermodynamic system, K; the equal sign in equation (22) corresponds to a reversible process, and the greater than sign corresponds to an irreversible process.

[0223] Derivative both sides of equation (21) with respect to time, and substitute it and equation (5) into equation (22) to obtain

[0224]

[0225] Where θ s ,θ f are the absolute temperatures of the solid and liquid phases, respectively, in K.

[0226] From formula (23), we can see

[0227]

[0228] Substitute equation (20) into equation (24) and eliminate γ s , γ f ,have to

[0229]

[0230] The relationship between the Helmholtz free energy per unit mass of the solid and liquid phases and their internal energy in a two-phase saturated porous medium is given by thermodynamic theory:

[0231] H s =E s -θ s η s (26)

[0232] H f =E f -θ f η f (27)

[0233] The time derivatives of Equations (26) and (27) are respectively used to obtain the rate of change of the Helmholtz free energy per unit mass of the solid-liquid two-phase, and then substituted into Equation (25), we get

[0234]

[0235] For an isothermal process, Equation (28) is simplified to

[0236]

[0237] Formula (29) can be written as

[0238]

[0239] Where ψ is the Helmholtz free energy per unit volume of two-phase saturated porous medium, J / m 3 , ψ s and ψ f are the Helmholtz free energy per unit volume of solid and liquid phases, J / m 3 ,

[0240] 2. Establish the constitutive equation of damaged two-phase saturated porous medium

[0241] The Helmholtz free energy ψ of a two-phase saturated porous medium is the solid phase deformation ε sij 、Liquid phase change ε fij , and the function of damage degree D

[0242] ψ=ψ(ε sij ,ε fij ,D) (31)

[0243] When the solid phase is homogeneous and isotropic, the rate of change of Helmholtz free energy with time is

[0244]

[0245] Substituting formula (32) into formula (30), we get

[0246]

[0247] The first two terms in Equation (32) are non-dissipative terms, and the third term represents the energy dissipation caused by damage evolution. The total dissipated energy of the system is non-negative. If the inequality Equation (30) is valid for any thermodynamic process, the non-dissipative term must be zero, so

[0248]

[0249] The damage strain energy release rate is the rate of change of the energy consumed by the medium due to damage with the degree of damage. The damage strain energy release rate Y and the degree of damage D are dual variables to each other, so

[0250]

[0251] Substituting equations (34), (35) and (36) into equation (33), the damage dissipation power of the two-phase saturated porous medium satisfies

[0252]

[0253] Damage is an irreversible process, so D·≥0, so the damage strain energy release rate Y is non-negative.

[0254] Dissipation potential ψ * It is a convex function of the damage strain energy release rate Y. According to the orthogonal flow law of internal variables, we know that

[0255]

[0256] The stress tensor component σ of the liquid phase fij and the strain tensor component ε fij They can be expressed by the liquid phase stress scalar s and the liquid phase volume strain scalar ε: fij =sδ ij , 3ε fij =εδ ij , s=-βp.

[0257] If the two-phase saturated porous medium deforms elastically, |ε sij |<<1,|ε|<<1,0≤D≤1,the Helmholtz free energy per unit volume ψ(ε sij ,ε,D) is

[0258]

[0259] Where ψ0 is the Helmholtz free energy of the initial state of the two-phase saturated porous medium, A (n) 、C (n) and G (n) is the scalar coefficient, and is the second-order tensor coefficient, is the fourth-order tensor coefficient,

[0260] Substituting Equation (39) into Equation (34) and Equation (35), we can obtain the stresses of the solid and liquid phases as follows:

[0261]

[0262] Substituting Equation (39) into Equation (36), we can obtain the damage strain energy release rate:

[0263]

[0264] During loading, the solid phase deforms elastically and the solid and liquid phases move relative to each other; after unloading, the stress, strain and damage strain energy release rate of the solid and liquid phases are all zero (i.e., σ sij =0,ε sij = 0, s = 0, ε = 0, Y = 0). As mentioned above, the damage formation process is irreversible, so the damage degree after unloading is not zero (i.e., D ≠ 0); therefore, the coefficients in Equations (39), (40), (41), and (42) are

[0265]

[0266] Substituting equation (43) into equations (39), (40), (41) and (42), we get

[0267]

[0268] Substituting D = 0 into equations (45) and (46), we can obtain the stresses of the solid and liquid phases under lossless conditions:

[0269]

[0270] In formula (48) and formula (49), and G (0) They are the elastic tensor of the solid phase, the fluid-solid coupling coefficient, and the elastic modulus of the liquid phase under lossless conditions.

[0271] Biot theory does not consider the evolution and influence of damage in porous media, and gives the constitutive equation of two-phase saturated porous media under damage-free conditions as follows:

[0272] σ sij =2Nε sij +(Ae+Qε)δ ij (50)

[0273] s=Qe+Rε (51)

[0274] Where N and A are parameters related to the Lamé constant of the solid phase, and Q and R are non-negative elastic parameters in the Biot model, which characterize the elasticity of the pore fluid and its elastic interaction with the solid phase. 2 M, Q = β (α - β) M, R = β 2 M, μ and λ are the solid-phase Lamé constants, Pa; α is the Biot coefficient, α=1-K b / K s ;M is Biot modulus, Pa, M=[(α-β) / K s +β / K f ] -1 , K s , K b and Kf are the bulk modulus of solid particles, the bulk modulus of solid skeleton and the bulk modulus of liquid phase under non-destructive conditions, Pa; e is the solid phase volume strain, e = u si,i .

[0275] From the elasticity theory, we know that the fourth-order elastic tensor of an isotropic medium is

[0276]

[0277] Formula (48), Formula (49) and Formula (50), Formula (51) correspond to each other respectively. Formula (48), Formula (49) and Formula (50), Formula (51) are combined and solved to obtain

[0278]

[0279] The parameters under the damage-free condition are multiplied by the coefficient related to the damage to obtain the corresponding parameters under the damage condition. Therefore, the relevant parameters under the damage condition are

[0280]

[0281] Where, and ζ (n) are dimensionless non-negative coefficients.

[0282] The pressure acting on the solid phase reduces the porosity and causes the liquid phase to be expelled from the pores, so the solid phase volume strain e and the liquid phase volume strain ε have opposite signs; according to Biot theory, A-λ, Q and R are all positive, so and G (n) are all positive. For a uniform isotropic elastic medium, is a symmetric positive definite tensor; As mentioned above, Y in formula (36) is a non-negative physical quantity, since and G (n) are all positive. From formula (47), we can see that is a negative definite tensor

[0283]

[0284] Where, τ (n) is a dimensionless non-negative coefficient.

[0285] Substituting equations (55), (56) and (57) into equations (44) to (47), we can get

[0286]

[0287] σ sij =2N(D)ε sij +A(D)eδ ij +Q(D)εδ ij(59)

[0288] s=R(D)ε+Q(D)e (60)

[0289]

[0290] Where A(D) = AM A (D), N(D)=NM N (D), Q(D)=QM Q (D), R(D)=RM R (D), M A (D), M N (D), M Q (D) and M R (D) is the damage effect function, which represents the impact of damage on parameters such as N, A, Q and R.

[0291] Equations (58), (59), (60), and (61) are the strain energy expressions, solid-phase stress expressions, liquid-phase stress expressions, and damage strain energy release rate expressions of two-phase saturated porous media under damage conditions, respectively.

[0292] As mentioned above, damage is an energy dissipation process, using the dissipation potential ψ * Characterize the dissipation characteristics of the damage process and construct the dissipation potential in the form of a power function of the damage driving force Y

[0293]

[0294] Where a and m are parameters related to the evolution of material damage.

[0295] Substituting equations (61) and (62) into equation (38), we obtain the damage evolution equation for two-phase saturated porous media:

[0296]

[0297] 3. Mechanical properties parameter analysis of damaged two-phase saturated porous media

[0298] Kachanov studied the elastic medium with micro-circular hole damage, and the damage effect functions of Young's modulus and Poisson's ratio are

[0299]

[0300] Where v is the Poisson's ratio of the elastic medium under lossless conditions.

[0301] According to the relationship between the Lamé coefficient of elastic media and Young's modulus and Poisson's ratio, and combined with Equations (64) and (65), the damage effect function of the solid phase Lamé coefficient in two-phase saturated porous media is obtained as follows:

[0302]

[0303] According to the relationship between the bulk modulus of elastic media, Young's modulus and Poisson's ratio, the bulk modulus of the solid phase in the two-phase saturated porous medium under damage conditions is obtained as follows:

[0304]

[0305] As mentioned above, N is the shear modulus of the solid phase in the two-phase saturated porous medium, and its damage effect function is given by Equation (67):

[0306] M N (D)=M μ (D) (69)

[0307] Under damage conditions, the porosity of the two-phase saturated porous medium is

[0308] β(D)=β+(1-β)[1-(1-D) 2 ] (70)

[0309] According to the definition of each parameter in Biot theory, the Biot coefficient α(D) and Biot modulus M(D) under damage conditions are respectively

[0310]

[0311] According to the definition of each parameter in Biot theory, the coefficient A(D) under damage condition is obtained by combining equations (66), (68) and equations (70) to (72):

[0312] A(D)=λM λ (D)+[α(D)-β(D)] 2 M(D) (73)

[0313] According to the definition of each parameter in Biot theory, the coefficients Q(D) and R(D) under damage conditions are obtained by combining equations (68), (70), (71) and (72).

[0314] Q(D)=β(D)[α(D)-β(D)]M(D) (74)

[0315] R(D)=[β(D)] 2 M(D) (75)

[0316] According to the definition of relevant parameters of Biot theory, the dissipation coefficient of two-phase porous medium under damage condition can be deduced as follows:

[0317]

[0318] The Kozeny-Carman equation gives the relationship between permeability and porosity, from which the permeability of the two-phase saturated porous medium under damage conditions can be obtained as

[0319]

[0320] Where M β (D) is the damage effect function of the porosity β of the two-phase saturated porous medium.

[0321] 4. Establish the wave equation for damaged two-phase saturated porous media

[0322] From the above analysis, we can see that the stresses of the solid phase and the liquid phase under damage conditions are shown in Equations (59) and (60), respectively. As shown in Equation (70), damage affects the porosity, so the quality coefficient ρ defined in the Biot model is 11 , ρ 22 and ρ 12 They are all functions of the damage degree D.

[0323] Taking the motion of the selected volume element along the x direction as an example, the Lagrange equation gives the resultant forces of the solid and liquid phases along the x direction in the damaged two-phase saturated porous medium:

[0324]

[0325] Substituting Equations (7) and (10) into Equations (78) and (79), we can obtain the motion equations of the solid and liquid phases in the volume element along the x direction:

[0326]

[0327] Similarly, the motion equations of the solid and liquid phases along the y and z directions can be obtained. Combining the stress relationship between the solid and liquid phases shown in Equations (59) and (60), as well as the geometric equations that characterize the displacement and strain of the solid and liquid phases, and substituting them into the motion equations of the solid and liquid phases along the x, y, and z directions, the wave equation of the two-phase saturated porous medium under damage conditions can be obtained as follows:

[0328]

[0329] Where u s 、u f are the displacement vectors of the solid and liquid phases, respectively; is the differential operator.

[0330] 5. Case analysis

[0331] When studying the dynamic response characteristics of a two-phase saturated porous medium under damage conditions, it is necessary to solve the damage evolution equation shown in Equation (63) and the wave equations shown in Equations (82) and (83) simultaneously. The above three equations are all nonlinear partial differential equations and it is difficult to obtain analytical solutions. The generalized partial differential equation system in the COMSOL Multiphysics PDE module is in the form of

[0332]

[0333] Where, e a is the mass coefficient, d a is the damping coefficient, Г is the conserved flux, and f is the source term.

[0334] When solving the wave equation, due to the inertial coupling coefficient ρ 12 The influence on the dynamic response of two-phase saturated porous media is very small. Usually, the inertial coupling coefficient ρ between the solid and liquid phases is 12 Ignore (i.e., ρ 12 =0). Under the damage condition, ρ in equations (82) and (83) is also 12 (D) is ignored. Taking two-dimensional saturated soil as an example, the equations shown in Equations (82), (83) and (63) are rewritten as

[0335]

[0336]

[0337] From Equations (85) to (89), we can see that the mass matrix, damping matrix, dependent variable, conserved flux, and source term that satisfy the solution rules of the generalized partial differential equations in the Comsol Multiphysics PDE module are:

[0338]

[0339] u={u sx u sy u fx u fy D} T (92)

[0340]

[0341]

[0342] When using this method to analyze the evolution law of tunnel floor damage, the vibration acceleration a of the tunnel boring machine body is first measured using a vibration measurement system. c (downward is the positive direction), and on this basis, the stress on the bottom plate is obtained as

[0343]

[0344] Where m is the total mass of the TBM; g is the acceleration of gravity; A is the total ground contact area of ​​the TBM crawler; and k is the comprehensive ground stiffness of the TBM, which is related to the TBM signal and the performance of the tunnel floor.

[0345] As mentioned above, when the roadheader is working, vibration signals are collected and this method is used to analyze the evolution of tunnel floor damage. Based on the measured vibration characteristics of the roadheader, the change of the tunnel floor damage degree D over time is analyzed as follows: Figure 1 As shown in the figure, the variation of damage evolution rate with time is as follows: Figure 2 shown.

[0346] Depend on Figure 1 and Figure 2 It can be seen that the damage evolution of the tunnel floor rock layer can be divided into three stages: in the first stage, the damage degree D increases rapidly with time, and the damage evolution rate gradually decreases with time; in the second stage, the damage degree D continues to increase with time, but the damage evolution rate basically remains constant with time; in the third stage, the damage degree D increases sharply with time, and the damage evolution rate increases sharply with time.

[0347] The vibration detection system of the tunnel boring machine body includes acceleration sensor A, acceleration sensor B, acceleration sensor C, acceleration sensor D, programmable logic controller PLC (composed of CPU module and analog module), power supply, computer, etc. When measuring acceleration, the four acceleration sensors are respectively arranged at the four corners of the tunnel boring machine body. When the tunnel boring machine starts working, the acceleration sensors transmit the collected data to the computer through the programmable logic controller PLC to complete the collection of relevant vibration signals. The arithmetic mean of the acceleration signals collected by each acceleration sensor is used as the acceleration a used in formula (95) c .

[0348] The acceleration measurement system of the present invention is as follows Figure 3 As shown, the positive and negative poles of the 24V power supply are respectively connected to the L pole and M pole of the CPU module of the programmable logic controller PLC (model SIEMENS200Smart T40) to power the programmable logic controller PLC.

[0349] The positive pole (VCC) of acceleration sensor A (model PCB356A17) is connected to the positive pole of the 24V power supply, and the x pole, y pole and z pole are respectively connected to the 0+ port, 2+ port and 1+ port of analog module 1 (model EM AM06) of the programmable logic controller PLC.

[0350] The positive pole (VCC) of acceleration sensor B (model PCB356A17) is connected to the positive pole of the 24V power supply, and the x pole, y pole and z pole are respectively connected to the 0+ port, 2+ port and 1+ port of analog module 2 (model EM AM06) of the programmable logic controller PLC.

[0351] The positive pole (VCC) of the acceleration sensor C (model PCB356A17) is connected to the positive pole of the 24V power supply, and the x pole, y pole and z pole are respectively connected to the 0+ port, 2+ port and 1+ port of the analog module 3 (model EM AM06) of the programmable logic controller PLC.

[0352] The positive pole (VCC) of the acceleration sensor D (model PCB356A17) is connected to the positive pole of the 24V power supply, and the x pole, y pole and z pole are respectively connected to the 0+ port, 2+ port and 1+ port of the analog module 4 (model EM AM06) of the programmable logic controller PLC.

[0353] The computer's Ethernet port is connected to the Ethernet port of the CPU module of the programmable logic controller (PLC) (model SIEMENS 200SmartT40). The PLC collects signals from each acceleration sensor and transmits them to the computer via a network cable, thereby completing relevant calculations and predictions.

Claims

1. A method for predicting surrounding rock damage under vibration load conditions, characterized in that: Based on the measured vibration characteristics of the roadheader, the variation of the damage degree D of the roadway floor and the variation of the damage evolution rate over time during the tunneling operation were analyzed. The specific steps include: Step 1: Thermodynamic analysis of damaged two-phase saturated porous media; Step 2: Establish the constitutive equation of the damaged two-phase saturated porous medium; Step 3: Analyze the mechanical properties parameters of the damaged two-phase saturated porous medium; Step 4: Establish the wave equation of the damaged two-phase saturated porous medium; Step 5: Example verification.

2. The method for predicting surrounding rock damage under vibration load conditions according to claim 1, characterized in that: Step 1 specifically includes the following steps: Step 1.1: Consider the saturated porous medium composed of solid and liquid phases as a thermodynamic system. According to the first law of thermodynamics, Where, are the rates of change of kinetic energy and internal energy of the studied thermodynamic system with time, J / s; is the power of the external force acting on the system, J / s; is the rate of change of the heat absorption of the system with time, J / s; Step 1.1.

1. Take a volume element from a two-phase saturated porous medium, whose total volume is V and total surface area is A. The volumes of the solid and liquid phases in the volume element are V and A respectively. s With V f , the surface areas are A s With A f , the direction cosine of the outer normal of the volume element surface is taken as n i In the volume element, the total power of the surface force and body force of the solid and liquid phases is Where, and are the powers acting on the solid and liquid phase loads, W; σ sij is the solid surface stress, Pa; u si is the solid phase displacement, m; is the solid phase velocity, m / s; f si is the body force per unit volume of solid phase, N / m 3 ; i and j both represent directions; σ fij is the stress on the liquid surface, Pa; according to Biot theory, σ fij =-βpδ ij , δ ij is the Kronecker symbol, β is the porosity, p is the pore fluid pressure, Pa; u fi is the liquid phase displacement, m; is the liquid phase velocity, m / s; f fi is the body force per unit volume of liquid phase, N / m 3 ; According to Gauss's theorem, the surface integral in formula (2) is converted into volume integral, and we get Step 1.1.2: The rate of increase of heat energy of solid and liquid phases in the volume element is Where, and are the rates of change of solid and liquid phase thermal energy increments with time, W; q si is the heat flux vector of the solid phase, W / m 2 ; ρ s is the solid mass density, kg / m 3 ; γ s is the heat generation rate per unit mass of solid phase, W / kg; q fi is the liquid phase heat flux density vector, W / m 2 ; ρ f is the liquid mass density, kg / m 3 ; γ f is the heat generation rate per unit mass of liquid phase, W / kg; According to Gauss's theorem, the surface integral in equation (4) is converted into volume integral, and we get Step 1.1.3: The rate of change of the internal energy of the solid and liquid phases in the volume element is Where, E s With E f are the internal energies per unit mass of solid and liquid phases, J / kg; Step 1.1.4: According to Biot theory, the kinetic energy per unit volume of two-phase saturated porous medium is: Where, ρ 11 and ρ 22 is the mass coefficient, kg / m 3 ; ρ 12 is the mass coupling coefficient of the solid and liquid phases, kg / m 3 ,ρ1=(1-β)ρ s ,ρ2=βρ f , ρ 12 =-ρ a , ρ a is the additional apparent mass, kg / m 3 , ρ 11 =ρ1+ρ a , ρ 22 =ρ2+ρ a ; Substituting the mass coefficient and the mass coupling coefficient of the solid-liquid phase into formula (7), we can get After taking the derivative of Equation (8) with respect to time and integrating it with respect to volume, the increase rate of the kinetic energy of the solid and liquid phases in the volume element is obtained as Step 1.2: According to Biot theory, the total power dissipated by the relative motion of solid and liquid phases in the volume element is Where b is the Biot dissipation coefficient, kg / (m 3 ·s), b=ηβ 2 / κ; κ is permeability, m 2 ; η is the dynamic viscosity of the pore fluid, kg / (m·s); The dissipation caused by the relative motion of solid and liquid phases is converted into heat; Substituting equations (3), (5), (6), (9) and (10) into equation (1), we can get According to Biot theory, the equations of motion for the solid and liquid phases are: Substituting equations (12) and (13) into equation (11), we can simplify to obtain Step 1.3: Under small strain conditions in continuum theory, the velocity gradient of the solid-liquid phase is decomposed into a symmetric strain rate tensor and an antisymmetric spin tensor. The curl tensor is antisymmetric about indices i and j, and the stress tensor is symmetric about indices i and j, so Multiply both ends of the equal signs in formula (15) and formula (16) by σ sij and σ fij , and combined with formula (17), we get Substituting equations (18) and (19) into equation (14), we can get Step 1.4: Calculate the total entropy of the two-phase saturated porous medium within the volume element Where S s 、S f are the entropies of the solid and liquid phases, J / K; η s ,η f are the entropy densities of solid and liquid phases per unit mass, J / (kg·K); M s 、M f are the masses of the solid phase and liquid phase, kg; Step 1.4.1: According to the entropy increase principle of the second law of thermodynamics, we can get Where Q is the heat absorbed by the thermodynamic system, J; θ is the absolute temperature of the thermodynamic system, K; the equal sign in equation (22) corresponds to a reversible process, and the greater than sign corresponds to an irreversible process; Take the time derivative of both sides of formula (21) and substitute it and formula (5) into formula (22) to obtain Where θ s ,θ f are the absolute temperatures of the solid and liquid phases, respectively, in K; From formula (23), we can see Substitute equation (20) into equation (24) and eliminate γ s , γ f ,have to Step 1.4.2: According to thermodynamic theory, the relationship between the Helmholtz free energy per unit mass of the solid and liquid phases and their internal energy in a two-phase saturated porous medium is: H s =E s -θ s or s (26) H f =E f -θ f or f (27) The time derivatives of Equations (26) and (27) are respectively used to obtain the rate of change of the Helmholtz free energy per unit mass of the solid-liquid two-phase, and then substituted into Equation (25), we get For an isothermal process, Equation (28) is simplified to Formula (29) can be written as Where ψ is the Helmholtz free energy per unit volume of two-phase saturated porous medium, J / m 3 , ψ s and ψ f are the Helmholtz free energy per unit volume of solid and liquid phases, J / m 3 , 3. The method for predicting surrounding rock damage under vibration load conditions according to claim 1, characterized in that: Step 2 specifically includes the following steps: Step 2.1: The Helmholtz free energy ψ of a two-phase saturated porous medium is its solid phase deformation ε sij 、Liquid phase change ε fij , and the function of damage degree D ψ=ψ(e sij ,he fij ,D) (31) Step 2.1.1: When the solid phase is uniform and isotropic, the rate of change of Helmholtz free energy with time is Substituting formula (32) into formula (30), we get The first two terms in Equation (32) are non-dissipative terms, and the third term represents the energy dissipation caused by damage evolution. The total dissipated energy of the system is non-negative. If inequality (30) is to hold for any thermodynamic process, the non-dissipative term must be zero, so Step 2.1.2, the damage strain energy release rate is the rate of change of the energy consumed by the medium due to damage with the damage degree. The damage strain energy release rate Y and the damage degree D are dual variables, so Substituting equations (34), (35) and (36) into equation (33), the damage dissipation power of the two-phase saturated porous medium satisfies Damage is an irreversible process, so Therefore, the damage strain energy release rate Y is non-negative; Dissipation potential ψ * It is a convex function of the damage strain energy release rate Y. According to the orthogonal flow law of internal variables, we know that Step 2.2: Liquid phase stress tensor component σ fij and the strain tensor component ε fij They can be expressed by the liquid phase stress scalar s and the liquid phase volume strain scalar ε: fij =sδ ij , 3ε fij =εδ ij , s = -βp; Step 2.2.1: If the two-phase saturated porous medium is elastically deformed, |ε sij |<<1,|ε|<<1,0≤D≤1,the Helmholtz free energy per unit volume ψ(ε sij ,ε,D) is Where ψ0 is the Helmholtz free energy of the initial state of the two-phase saturated porous medium, A (n) 、C (n) and G (n) is the scalar coefficient, and is the second-order tensor coefficient, is the fourth-order tensor coefficient, Step 2.2.2: Substitute equation (39) into equation (34) and equation (35) respectively to obtain the stresses of the solid and liquid phases: Step 2.2.3: Substitute equation (39) into equation (36) to obtain the damage strain energy release rate: Step 2.3: During the loading process, the solid phase undergoes elastic deformation and the solid and liquid phases move relative to each other. After unloading, the stress, strain, and damage strain energy release rate of the solid and liquid phases are all zero, that is, σ sij =0,ε sij = 0, s = 0, ε = 0, Y = 0; As mentioned above, the damage formation process is irreversible, so the damage degree after unloading is not zero, that is, D ≠ 0; Therefore, the coefficients in Equations (39), (40), (41) and (42) are Step 2.3.

1. Substitute equation (43) into equations (39), (40), (41), and (42) to obtain Step 2.3.2: Substitute D = 0 into Equation (45) and Equation (46) respectively to obtain the stresses of the solid and liquid phases under lossless conditions: In formula (48) and formula (49), and G (0) are the elastic tensor of the solid phase, the fluid-solid coupling coefficient, and the elastic modulus of the liquid phase under lossless conditions; Step 2.3.3: Biot theory does not consider the evolution and influence of damage in porous media, and gives the constitutive equation of two-phase saturated porous media under damage-free conditions as follows: s sij =2Nε sij +(Ae+Qε)δ ij (50) s=Qe+Rε (51) Where N and A are parameters related to the Lamé constant of the solid phase, and Q and R are non-negative elastic parameters in the Biot model, which characterize the elasticity of the pore fluid and its elastic interaction with the solid phase. 2 M, Q=β(α-β)M,R=β 2 M, μ and λ are the solid-phase Lamé constants, Pa; α is the Biot coefficient, α=1-K b / K s ; M is the Biot modulus, Pa, M = [(α-β) / K s +β / K f ] -1 , K s , K b and K f are the bulk modulus of solid particles, the bulk modulus of solid skeleton and the bulk modulus of liquid phase under non-destructive conditions, Pa; e is the solid phase volume strain, e = u si,i ; From the elasticity theory, we know that the fourth-order elastic tensor of an isotropic medium is Step 2.3.4, equation (48), equation (49) and equation (50), equation (51) correspond to each other, and equation (48), equation (49) and equation (50), equation (51) are combined and solved to obtain The parameters under the damage-free condition can be multiplied by the coefficient related to the damage to obtain the corresponding parameters under the damage condition. Therefore, the relevant parameters under the damage condition are Where, χ (n) and ζ (n) All are dimensionless non-negative coefficients; Step 2.3.5: The pressure acting on the solid phase reduces the porosity and causes the liquid phase to be expelled from the pores, so the solid phase volume strain e and the liquid phase volume strain ε have opposite signs. According to Biot theory, A-λ, Q, and R are all positive, so and G (n) are all positive; for a uniform isotropic elastic medium, is a symmetric positive definite tensor; As mentioned above, Y in formula (36) is a non-negative physical quantity, since and G (n) are all positive. From formula (47), we can see that is a negative definite tensor Where, τ (n) is a dimensionless non-negative coefficient; Step 2.4: Substitute equations (55), (56) and (57) into equations (44) to (47) to obtain s sij =2N(D)ε sij +A(D)eδ ij +Q(D)ed ij (59) s=R(D)ε+Q(D)e (60) Where A(D) = AM A (D), N(D)=NM N (D), Q(D)=QM Q (D), R(D)=RM R (D), M A (D), M N (D), M Q (D) and M R (D) is the damage effect function, which represents the impact of damage on parameters such as N, A, Q and R. Equations (58), (59), (60), and (61) are the strain energy expression, solid phase stress expression, liquid phase stress expression, and damage strain energy release rate expression of the two-phase saturated porous medium under damage conditions, respectively. Step 2.5: As mentioned above, damage is an energy dissipation process, using the dissipation potential ψ * Characterize the dissipation characteristics of the damage process and construct the dissipation potential in the form of a power function of the damage driving force Y Where a and m are parameters related to the evolution of material damage; Substituting equations (61) and (62) into equation (38), we obtain the damage evolution equation for two-phase saturated porous media:

4. The method for predicting surrounding rock damage under vibration load conditions according to claim 1, wherein: Step 3 specifically includes the following steps Step 3.1, Kachanov studied the elastic medium with micro-hole damage, and the damage effect functions of Young's modulus and Poisson's ratio are Where v is the Poisson's ratio of the elastic medium under lossless conditions; Step 3.1.1: Based on the relationship between the Lamé coefficient of elastic media and Young's modulus and Poisson's ratio, and combined with Equations (64) and (65), the damage effect function of the solid phase Lamé coefficient in two-phase saturated porous media is obtained as follows: Step 3.1.2: Based on the relationship between the bulk modulus of the elastic medium, Young's modulus, and Poisson's ratio, the bulk modulus of the solid phase in the two-phase saturated porous medium under damage conditions is obtained as follows: Step 3.2: As mentioned above, N is the shear modulus of the solid phase in the two-phase saturated porous medium, and its damage effect function is given by Equation (67): M N (D)=M μ (D) (69) Step 3.2.1: Under damage conditions, the porosity of the two-phase saturated porous medium is β(D)=β+(1-β)[1-(1-D) 2 ] (70) Step 3.2.2: According to the definition of each parameter in Biot theory, the Biot coefficient α(D) and Biot modulus M(D) under damage condition are respectively Step 3.2.3: According to the definition of each parameter in Biot theory, the coefficient A(D) under damage condition is obtained by combining equations (66), (68) and equations (70) to (72): A(D)=λM λ (D)+[α(D)-β(D)] 2 M(D) (73) According to the definition of each parameter in Biot theory, the coefficients Q(D) and R(D) under damage conditions are obtained by combining equations (68), (70), (71) and (72). Q(D)=β(D)[α(D)-β(D)]M(D) (74) R(D)=[β(D)] 2 M(D) (75) According to the definition of relevant parameters of Biot theory, the dissipation coefficient of two-phase porous medium under damage condition can be deduced as follows: Step 3.2.4, the Kozeny-Carman equation gives the relationship between permeability and porosity, from which the permeability of the two-phase saturated porous medium under damage conditions can be obtained as Where M β (D) is the damage effect function of the porosity β of the two-phase saturated porous medium.

5. The method for predicting surrounding rock damage under vibration load conditions according to claim 1, characterized in that: Step 4 specifically includes the following steps Step 4.1, as shown in formula (7), the mass coefficient ρ defined in the Biot model is 11 , ρ 22 and ρ 12 are related to the density and porosity of the solid and liquid phases, respectively. As shown in Equation (70), damage affects porosity, so the mass coefficient ρ in the Biot model is 11 , ρ 22 and ρ 12 All are functions of the damage degree D; Taking the motion of the selected volume element along the x direction as an example, the Lagrange equation gives the resultant forces of the solid and liquid phases along the x direction in the damaged two-phase saturated porous medium: Substituting Equations (7) and (10) into Equations (78) and (79), we can obtain the motion equations of the solid and liquid phases in the volume element along the x direction: Step 4.2: Similarly, the motion equations of the solid and liquid phases along the y and z directions are obtained; the stress relationship between the solid and liquid phases shown in Equations (59) and (60), as well as the geometric equations representing the displacement and strain of the solid and liquid phases, are combined and substituted into the motion equations of the solid and liquid phases along the x, y, and z directions to obtain the wave equation of the two-phase saturated porous medium under damage conditions: Where u s 、u f are the displacement vectors of the solid and liquid phases, respectively; is the differential operator.

6. The method for predicting surrounding rock damage under vibration load conditions according to claim 1, characterized in that: Step 5 specifically includes the following steps Step 5.1: When studying the dynamic response characteristics of a two-phase saturated porous medium under damage conditions, it is necessary to solve the damage evolution equation shown in Equation (63) and the wave equations shown in Equations (82) and (83) simultaneously. The above three equations are all nonlinear partial differential equations and it is difficult to obtain analytical solutions. The generalized partial differential equation system in the COMSOL Multiphysics PDE module is in the form of Where, e a is the mass coefficient, d a is the damping coefficient, Г is the conserved flux, and f is the source term; Step 5.2: When solving the wave equation, due to the inertial coupling coefficient ρ 12 The influence on the dynamic response of two-phase saturated porous media is very small. Usually, the inertial coupling coefficient ρ between the solid and liquid phases is 12 Ignore, that is, set its value to ρ 12 =0; under the damage condition, ρ in equations (82) and (83) is also 12 (D) Ignore, and rewrite the equations shown in equations (82), (83) and (63) as Step 5.3: From Equations (85) to (89), we can see that the mass matrix, damping matrix, dependent variable, conserved flux, and source term that satisfy the solution rules of the generalized partial differential equations in the Comsol Multiphysics PDE module are: u={u sx u sy u fx u fy D} T (92) 7. A system for predicting surrounding rock damage under vibration load conditions, characterized in that: A method for predicting tunnel floor damage under the vibration load of a tunnel boring machine is implemented, comprising an acceleration sensor A, an acceleration sensor B, an acceleration sensor C, an acceleration sensor D, a programmable logic controller (PLC), a power supply and a computer. The four acceleration sensors are respectively arranged at the four corner points of the tunnel boring machine body. The programmable logic controller (PLC) is composed of a CPU module and an analog module.

8. The damage prediction system for surrounding rock under vibration load conditions according to claim 7, characterized in that: The positive pole and negative pole of the power supply in the vibration detection system are respectively connected to the L pole and M pole of the CPU module of the programmable logic controller PLC to supply power to the programmable logic controller PLC; The positive poles (VCC) of the four accelerometers are connected to the positive pole of the 24V power supply, and the x-pole, y-pole, and z-pole are connected to the 0+ port, 2+ port, and 1+ port of analog modules 1, 2, 3, and 4 of the programmable logic controller (PLC), respectively. The computer Ethernet port is connected to the Ethernet port of the CPU module of the programmable logic controller PLC. The programmable logic controller PLC is used to collect signals from each acceleration sensor and transmit the signals to the computer through a network cable.

9. The damage prediction system for surrounding rock under vibration load conditions according to claim 8, characterized in that: When the tunnel boring machine starts working, the acceleration sensors in the vibration detection system transmit the collected data to the computer via the programmable logic controller (PLC) to complete the collection of relevant vibration signals. The arithmetic mean of the acceleration signals collected by each acceleration sensor is used as the acceleration of the tunnel boring machine body. After analysis, the characteristics of the total vibration load on the surrounding rock are obtained, which are applied to the established mathematical model of surrounding rock damage. Through simulation calculations, the degree of damage to the surrounding rock and its evolution process are predicted.

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