Dynamic response calculation method for transverse isotropic unsaturated half space

Abstract

The invention discloses a dynamic response calculation method for a transverse isotropic unsaturated half space. In order to better simulate the deposition process of an unsaturated soil foundation and the influence of confining pressure on the unsaturated soil foundation, the method is based on the heterogeneous characteristic that a functionally graded material has continuous or quasi-continuous change in the space direction, and it is assumed that the elastic coefficient and density of a transverse isotropic unsaturated soil body are exponential functions changing along the depth; and the compressibility, the viscosity and the inertia coupling effect of the soil particles, the pore water and the pore gas are considered. Through Hankel integral transformation and the Vieta theorem, a soil dynamic control equation is derived in an axisymmetric cylindrical coordinate system, and a general solution of half-space dynamic response calculation is given in a transform domain by using a Gauss-Kronrod algorithm and corresponding inverse transformation. The method provides a theoretical basis for interaction of soil and a structure.

Description

Technical Field

[0001] The invention relates to the technical field of rock and soil dynamics and soil-structure interaction, and in particular to a dynamic response calculation method for a transversely isotropic unsaturated half space. Background Art

[0002] In the field of civil engineering, soil-structure interaction is an important part of engineering design. Due to the sedimentation process, diagenesis and its multiphase characteristics, soils show complex heterogeneity and anisotropy. Especially under unsaturated conditions, the mechanical properties of soils are closely related to the interaction between pore water and pore gas. Current research shows that the mechanical properties of heterogeneous soils have a significant impact on the dynamic response, so it is particularly important to establish a dynamic model that adapts to heterogeneous, transversely isotropic unsaturated soils.

[0003] Functionally graded materials have been widely used in aerospace, automotive and other fields because their material properties gradually change along the spatial direction. In recent years, scholars have tried to introduce the concept of functionally graded materials into soil mechanics analysis for soil-structure interaction problems, simulating the heterogeneity of soil by introducing the gradient change of material parameters. However, there are still relatively few studies on the dynamic response of transversely isotropic unsaturated media, and existing methods often have certain limitations when dealing with soil anisotropy, unsaturated characteristics and multiphase coupling problems.

[0004] Therefore, a dynamic response calculation method for transversely isotropic unsaturated half-space is proposed, which has important theoretical significance and practical application value. Summary of the invention

[0005] The purpose of the present invention is to provide a method for calculating the dynamic response of a transversely isotropic unsaturated half-space in order to solve the above-mentioned problem.

[0006] To achieve the above object, the technical solution adopted by the present invention is:

[0007] According to one aspect of the present invention, there is provided a method for calculating the dynamic response of a transversely isotropic unsaturated half-space, comprising the following steps:

[0008] S1: Consider the following assumptions: the soil is a transversely isotropic, linear elastic, heterogeneous three-phase unsaturated porous medium, each phase is compressible; the upper surface of the soil is a free boundary, there is no normal or shear stress, and the soil surface is permeable to water and air;

[0009] S2: The concept of exponential gradient material and transversely isotropic constitutive model are introduced to construct the motion equation of heterogeneous transversely isotropic unsaturated soil foundation in cylindrical coordinate system;

[0010] S3: Considering the capillary pressure and inertial coupling effects between the liquid and gas phases, the seepage continuity equation for transversely isotropic unsaturated soil is derived;

[0011] S4: combining step S2 and step S3, establishing the dynamic control equation of the heterogeneous transversely isotropic unsaturated soil foundation, and solving the control equation using Laplace-Hankel transformation;

[0012] S5: Vertical transient loads acting on the surface of a non-homogeneous, transversely isotropic, unsaturated half-space , the transient load can be a point step load, a circular uniformly distributed step load and a rectangular pulse load, and the load form is processed using Laplace-Hankel transformation;

[0013] S6: The matrix algorithm, wave radiation conditions, Vieta theorem and Newton iteration method are used to solve the heterogeneous transversely isotropic unsaturated soil foundation. The Gauss-Kronod numerical algorithm is used to realize the Hankel inverse transform, and the improved Durbin method is used to realize the Laplace inverse transform.

[0014] Further, according to the assumption in step S1, step 2 is specifically as follows:

[0015] S21. The constitutive equation of transversely isotropic unsaturated soil considering the heterogeneity of the foundation is defined as follows:

[0016] (1a)

[0017] (1b)

[0018] (1c)

[0019] (1d)

[0020] In the formula, is the elastic coefficient of the transversely isotropic medium, χ is the effective stress coefficient, and is the compression coefficient of the soil skeleton in the horizontal and vertical directions, and denote the pore water pressure and pore gas pressure, respectively. and are stress and strain components respectively, and their relationship with displacement components is:

[0021] (2)

[0022] S22. Assuming that the elastic coefficient is an exponential function along the depth z, then:

[0023] (3)

[0024] In the formula, represents the material constant at depth z = 0, , and can be any real number, Indicates homogeneous foundation.

[0025] S23. Considering the viscous and inertial coupling between the soil skeleton, liquid phase and gas phase, the motion equation of the unsaturated porous medium in the cylindrical coordinate system is obtained as follows:

[0026] (4a)

[0027] (4b)

[0028] (4c)

[0029] In the formula, , Represent the absolute mass density of soil skeleton, liquid phase and gas phase respectively; n is the porosity of soil, S r Indicates saturation.

[0030] Substituting equations (1) to (3) into the motion equation of three-dimensional axisymmetric unsaturated porous media, the motion equation of axisymmetric heterogeneous transversely isotropic unsaturated soil in cylindrical coordinate system can be obtained as follows:

[0031] (5a)

[0032] (5b)

[0033] Furthermore, in step 3, considering the capillary pressure and inertial coupling effect between the liquid phase and the gas phase, the seepage continuity equation of the heterogeneous transversely isotropic unsaturated soil can be obtained as follows:

[0034] (6a)

[0035] (6b)

[0036] In the formula, , , , , , , , , , , m and k represent the van Genuchten parameters respectively.

[0037] Further, according to step S4, the simultaneous equations (5) and (6), after dimensionless transformation and Hankel transformation, the dynamic control equation of the axisymmetric heterogeneous transversely isotropic unsaturated soil foundation can be obtained as follows:

[0038] (7a)

[0039] (7b)

[0040] (7c)

[0041] (7d)

[0042] Wherein, the superscripts "0" and "1" represent the Hankel transform order, ξ is the independent variable in the Hankel transform, and s is the independent variable in the Laplace transform; , , , , , , , , , , , ; , , , , , , , , , , , , , , , , , , , , , , .

[0043] Further, according to step S5, in the Laplace-Hankel transform domain, the expressions of the point step load, the circular uniformly distributed step load and the rectangular pulse load are respectively:

[0044] (8a)

[0045] (8b)

[0046] (8c)

[0047] In the formula, a represents the load action radius, ξ is the independent variable in Hankel transformation, and s is the independent variable in Laplace transformation. represents the Bessel function of the first kind of order 1.

[0048] Furthermore, the half-space surface is considered to be permeable to water and air, and the boundary conditions are:

[0049] (9)

[0050] (10)

[0051] Furthermore, the step 6 includes the following specific steps:

[0052] S61: Combine equation (7) into a matrix form. Since soil displacement, pore water pressure and pore air pressure are all non-zero solutions, the coefficient matrix is ​​zero, then:

[0053] (11)

[0054] In the formula, the coefficient a i The specific expression for (i=0,1,2,3,4,5,6,7,8) is:

[0055] ;

[0056] ;

[0057] ;

[0058] ;

[0059] ;

[0060]

[0061] ;

[0062] ;

[0063] When the heterogeneity coefficient When , the coefficients a7 = a5 = a3 = a1 = 0 in the above expression, then equation (11) can be completely degenerated into a homogeneous transversely isotropic unsaturated soil foundation.

[0064] S62: Using Vieta's theorem, the expressions for the roots and coefficients of equation (11) can be expressed as:

[0065] (12a)

[0066] (12b)

[0067] (12c)

[0068] (12d) (12e)

[0069] (12f) (12g)

[0070] The equation is transformed into a real number equation, and a genetic algorithm is first used to perform a global search to obtain the initial value of the iteration, and then the Newton iteration method is used to solve it.

[0071] S63: Step S62 solves are the eight roots of equation (11). According to the wave radiation condition, the roots with positive real parts are the solutions that satisfy the problem. Let , , and are the four roots whose real parts are greater than zero. Then the solution of equation (11) that satisfies the radiation condition is:

[0072] (13)

[0073] In the formula, , (i=1, 2, 3, 4) are the unknown coefficients determined by the boundary conditions.

[0074] Substitute equation (13) into equation (7), and let , , , we can get:

[0075] (14)

[0076] in:

[0077] ;

[0078] ;

[0079] ;

[0080] .

[0081] Combining equations (7), (13) and (14), we can obtain the equation containing only the unknown number R i The expressions for displacement, pore water pressure and air pressure are:

[0082] (15a)

[0083] (15b)

[0084] (15c)

[0085] (15d)

[0086] The stress component expression expressed in displacement under axisymmetry can be written as:

[0087] (16a)

[0088] (16b)

[0089] Perform 0th-order and 1st-order Hankel transforms on equations (16a) and (16b), respectively, and apply Laplace transform and dimensionless transform, and combine with equation (15) to obtain:

[0090] (17a)

[0091] (17b)

[0092] Perform the corresponding Hankel-Laplace joint transformation on the boundary condition (3.1), and substitute (15) and (17) into it, that is:

[0093] (18a)

[0094] (18b)

[0095] (18c)

[0096] (18d)

[0097] By using formula (18), we can solve R i After the value of is obtained, it is substituted into equation (15) and equation (17), and the Hankel-Laplace integral inverse transformation is performed, and then the integral general solution expression of solid phase displacement, pore water pressure, pore gas pressure and stress in the time domain can be obtained as follows:

[0098] (19)

[0099] (20)

[0100] (twenty one)

[0101] (twenty two)

[0102] S64: The numerical implementation of the inverse Laplace transform uses the improved Durbin method, which is currently recognized to have high accuracy and reliability. Its specific mathematical expression can be expressed as:

[0103] (twenty three)

[0104] In the formula, the parameters selected for calculation are: aT=5, T=20 and NUSM=200.

[0105] Compared with the prior art, the present invention has the following beneficial effects:

[0106] The present invention is based on the concept of functional gradient materials and is a method for calculating the dynamic response of a transversely isotropic unsaturated half-space. The foundation is considered to be a material whose mechanical properties change exponentially with the coordinate system and is a multiphase soil composed of solid, liquid and gas phases, which is undoubtedly closer to the actual situation, and the invention can be completely degraded to a homogeneous foundation. Therefore, the calculation method disclosed in the present invention has a wide range of applications, and the calculation results are more in line with actual engineering. It can provide a theoretical basis for the study of the dynamic characteristics of soil under complex geological conditions, as well as the interaction between soil and structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0107] Figure 1 It is a flow chart of the core steps corresponding to the method of the present invention;

[0108] Figure 2 This is the transient load loading type described in the present invention. DETAILED DESCRIPTION

[0109] In order to make the technical means, creative features, objectives and effects achieved by the present invention easy to understand, the present invention is further explained below in conjunction with specific implementation methods.

[0110] The present invention provides a method for calculating the dynamic response of a transversely isotropic unsaturated half-space. By considering the exponential gradient change of material parameters and comprehensively processing the coupling effect of soil particles, pore water, and pore gas, the response behavior of heterogeneous soil under dynamic loads can be accurately simulated. This method can not only improve the accuracy of dynamic response analysis, but also provide a theoretical basis and technical support for soil mechanics design under complex engineering conditions.

[0111] To achieve the above object, the technical solution of the present invention is as follows:

[0112] according to Figure 1-Figure 2 As shown, a method for calculating the dynamic response of a transversely isotropic unsaturated half-space includes the following steps:

[0113] S1: Consider the following assumptions: the soil is a transversely isotropic, linear elastic, heterogeneous three-phase unsaturated porous medium, each phase is compressible; the upper surface of the soil is a free boundary, there is no normal or shear stress, and the soil surface is permeable to water and air;

[0114] S2: The concept of exponential gradient material and transversely isotropic constitutive model are introduced to construct the motion equation of heterogeneous transversely isotropic unsaturated soil foundation in cylindrical coordinate system;

[0115] S3: Considering the capillary pressure and inertial coupling effects between the liquid and gas phases, the seepage continuity equation for transversely isotropic unsaturated soil is derived;

[0116] S4: combining step S2 and step S3, establishing the dynamic control equation of the heterogeneous transversely isotropic unsaturated soil foundation, and solving the control equation using Laplace-Hankel transformation;

[0117] S5: Vertical transient loads acting on the surface of a non-homogeneous, transversely isotropic, unsaturated half-space , the transient load can be a point step load, a circular uniformly distributed step load and a rectangular pulse load, and the load form is processed using Laplace-Hankel transformation;

[0118] S6: The matrix algorithm, wave radiation conditions, Vieta theorem and Newton iteration method are used to solve the heterogeneous transversely isotropic unsaturated soil foundation. The Gauss-Kronod numerical algorithm is used to realize the Hankel inverse transform, and the improved Durbin method is used to realize the Laplace inverse transform.

[0119] Preferably, according to the assumption in step S1, step 2 is specifically:

[0120] S21. The constitutive equation of transversely isotropic unsaturated soil considering the heterogeneity of the foundation is defined as follows:

[0121] (1a)

[0122] (1b)

[0123] (1c)

[0124] (1d)

[0125] In the formula, is the elastic coefficient of the transversely isotropic medium, χ is the effective stress coefficient, and is the compression coefficient of the soil skeleton in the horizontal and vertical directions, and denote the pore water pressure and pore gas pressure, respectively. and are stress and strain components respectively, and their relationship with displacement components is:

[0126] (2)

[0127] S22. Assuming that the elastic coefficient is an exponential function along the depth z, then:

[0128] (3)

[0129] In the formula, represents the material constant at depth z = 0, , and can be any real number, Indicates homogeneous foundation.

[0130] S23. Considering the viscous and inertial coupling between the soil skeleton, liquid phase and gas phase, the motion equation of the unsaturated porous medium in the cylindrical coordinate system is obtained as follows:

[0131] (4a)

[0132] (4b)

[0133] (4c)

[0134] In the formula, , Represent the absolute mass density of soil skeleton, liquid phase and gas phase respectively; n is the porosity of soil, S r Indicates saturation.

[0135] Substituting equations (1) to (3) into the motion equation of three-dimensional axisymmetric unsaturated porous media, the motion equation of axisymmetric heterogeneous transversely isotropic unsaturated soil in cylindrical coordinate system can be obtained as follows:

[0136] (5a)

[0137] (5b)

[0138] Preferably, in step 3, the capillary pressure and inertial coupling effect between the liquid phase and the gas phase are considered, and the seepage continuity equation of the heterogeneous transversely isotropic unsaturated soil can be obtained as follows:

[0139] (6a)

[0140] (6b)

[0141] In the formula, , , , , , , , , , , m and k represent the van Genuchten parameters respectively.

[0142] Preferably, according to step S4, the dynamic control equation of the axisymmetric heterogeneous transversely isotropic unsaturated soil foundation can be obtained by combining equations (5) and (6) and performing dimensionless transformation and Hankel transformation:

[0143] (7a)

[0144] (7b)

[0145] (7c)

[0146] (7d)

[0147] Wherein, the superscripts "0" and "1" represent the Hankel transform order, ξ is the independent variable in the Hankel transform, and s is the independent variable in the Laplace transform; , , , , , , , , , , , ; , , , , , , , , , , , , , , , , , , , , , , .

[0148] Preferably, according to step S5, in the Laplace-Hankel transform domain, the expressions of the point step load, the circular uniformly distributed step load and the rectangular pulse load are respectively:

[0149] (8a)

[0150] (8b)

[0151] (8c)

[0152] In the formula, a represents the load action radius, ξ is the independent variable in Hankel transformation, and s is the independent variable in Laplace transformation. represents the Bessel function of the first kind of order 1.

[0153] Preferably, the half-space surface is considered to be permeable to water and air, and the boundary conditions are:

[0154] (9)

[0155] (10)

[0156] Preferably, step 6 comprises the following specific steps:

[0157] S61: Combine equation (7) into a matrix form. Since soil displacement, pore water pressure and pore air pressure are all non-zero solutions, the coefficient matrix is ​​zero, then:

[0158] (11)

[0159] In the formula, the coefficient a i The specific expression for (i=0,1,2,3,4,5,6,7,8) is:

[0160] ;

[0161] ;

[0162] ;

[0163] ;

[0164] ;

[0165]

[0166] ;

[0167] ;

[0168] When the heterogeneity coefficient When , the coefficients a7 = a5 = a3 = a1 = 0 in the above expression, then equation (11) can be completely degenerated into a homogeneous transversely isotropic unsaturated soil foundation.

[0169] S62: Using Vieta's theorem, the expressions for the roots and coefficients of equation (11) can be expressed as:

[0170] (12a)

[0171] (12b)

[0172] (12c)

[0173] (12d) (12e)

[0174] (12f) (12g)

[0175] It should be noted that equation (12) is a complex equation. It is very difficult to directly search for the roots of the equation system in the complex domain. Therefore, the equation is converted into a real equation, and a genetic algorithm is first used to perform a global search to obtain the initial value of the iteration, and then the Newton iteration method is used to solve it.

[0176] S63: Step S62 solves are the eight roots of equation (11). According to the wave radiation condition, the roots with positive real parts are the solutions that satisfy the problem. Let , , and are the four roots whose real parts are greater than zero. Then the solution of equation (11) that satisfies the radiation condition is:

[0177] (13)

[0178] In the formula, , (i=1, 2, 3, 4) are the unknown coefficients determined by the boundary conditions.

[0179] Substitute equation (13) into equation (7), and let , , , we can get:

[0180] (14)

[0181] in:

[0182] ;

[0183] ;

[0184] ;

[0185] .

[0186] Combining equations (7), (13) and (14), we can obtain the equation containing only the unknown number R i The expressions for displacement, pore water pressure and air pressure are:

[0187] (15a)

[0188] (15b)

[0189] (15c)

[0190] (15d)

[0191] The stress component expression expressed in displacement under axisymmetry can be written as:

[0192] (16a)

[0193] (16b)

[0194] Perform 0th-order and 1st-order Hankel transforms on equations (16a) and (16b), respectively, and apply Laplace transform and dimensionless transform, and combine with equation (15) to obtain:

[0195] (17a)

[0196] (17b)

[0197] Perform the corresponding Hankel-Laplace joint transformation on the boundary condition (3.1), and substitute (15) and (17) into it, that is:

[0198] (18a)

[0199] (18b)

[0200] (18c)

[0201] (18d)

[0202] By using formula (18), we can solve R i After the value of is obtained, it is substituted into equation (15) and equation (17), and the Hankel-Laplace integral inverse transformation is performed, and then the integral general solution expression of solid phase displacement, pore water pressure, pore gas pressure and stress in the time domain can be obtained as follows:

[0203] (19)

[0204] (20)

[0205] (twenty one)

[0206] (twenty two)

[0207] S64: Since displacement and pore pressure contain Bessel functions, they belong to generalized oscillation integrals and are difficult to be solved directly using analytical methods. Therefore, numerical methods are needed for approximate calculations. Therefore, the Gauss-Kronod numerical algorithm is used to implement the Hankel inverse transform. However, for the numerical implementation of the Laplace inverse transform, the improved Durbin method, which is currently recognized to have high accuracy and reliability, is used. Its specific mathematical expression can be expressed as:

[0208] (twenty three)

[0209] In the formula, the parameters selected for calculation are: aT=5, T=20 and NUSM=200.

[0210] It can be seen from the above technical solutions that the dynamic response calculation method of the transversely isotropic unsaturated half-space based on the concept of functional gradient materials of the present invention, the foundation is considered to be a material whose mechanical properties change exponentially with the coordinate system and is a multiphase soil composed of solid, liquid and gas phases, which is undoubtedly closer to the actual situation, and the invention can be completely degraded to a homogeneous foundation. Therefore, the calculation method disclosed in the present invention has a wide range of applications, and the calculation results are more in line with actual engineering. It can provide a theoretical basis for the study of the dynamic characteristics of soil under complex geological conditions, as well as the interaction between soil and structures.

[0211] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

[0212] In addition, it should be understood that although the present specification is described according to implementation modes, not every implementation mode contains only one independent technical solution. This description of the specification is only for the sake of clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment may also be appropriately combined to form other implementation modes that can be understood by those skilled in the art.

Claims

1. A method for calculating the dynamic response of a transversely isotropic unsaturated half-space, characterized in that: The following steps are involved: S1: Consider the following assumptions: the soil is a transversely isotropic, linear elastic, heterogeneous three-phase unsaturated porous medium, each phase is compressible; the upper surface of the soil is a free boundary, there is no normal or shear stress, and the soil surface is permeable to water and air; S2: The concept of exponential gradient material and transversely isotropic constitutive model are introduced to construct the motion equation of heterogeneous transversely isotropic unsaturated soil foundation in cylindrical coordinate system; S3: Considering the capillary pressure and inertial coupling effects between the liquid and gas phases, the seepage continuity equation for transversely isotropic unsaturated soil is derived; S4: combining step S2 and step S3, establishing the dynamic control equation of the heterogeneous transversely isotropic unsaturated soil foundation, and solving the control equation using Laplace-Hankel transformation; S5: Vertical transient loads acting on the surface of a non-homogeneous, transversely isotropic, unsaturated half-space , the transient load can be a point step load, a circular uniformly distributed step load and a rectangular pulse load, and the load form is processed using Laplace-Hankel transformation; S6: The matrix algorithm, wave radiation conditions, Vieta theorem and Newton iteration method are used to solve the heterogeneous transversely isotropic unsaturated soil foundation. The Gauss-Kronod numerical algorithm is used to realize the Hankel inverse transform, and the improved Durbin method is used to realize the Laplace inverse transform.

2. The method for calculating the dynamic response of a transversely isotropic unsaturated half-space according to claim 1, characterized in that: According to the assumption in step S1, step 2 is specifically as follows: S21. The constitutive equation of transversely isotropic unsaturated soil considering the heterogeneity of the foundation is defined as follows: （1a） （1b） （1c） （1d） In the formula, is the elastic coefficient of the transversely isotropic medium, χ is the effective stress coefficient, and is the compression coefficient of the soil skeleton in the horizontal and vertical directions, and denote the pore water pressure and pore gas pressure, respectively. and are stress and strain components respectively, and their relationship with displacement components is: （2） S22. Assuming that the elastic coefficient is an exponential function along the depth z, then: （3） In the formula, represents the material constant at depth z = 0, , and can be any real number, Indicates homogeneous foundation; S23. Considering the viscous and inertial coupling between the soil skeleton, liquid phase and gas phase, the motion equation of the unsaturated porous medium in the cylindrical coordinate system is obtained as follows: （4a） （4b） （4c） In the formula, , Represent the absolute mass density of soil skeleton, liquid phase and gas phase respectively; n is the porosity of soil, S r Indicates saturation; Substituting equations (1) to (3) into the motion equation of three-dimensional axisymmetric unsaturated porous media, the motion equation of axisymmetric heterogeneous transversely isotropic unsaturated soil in cylindrical coordinate system can be obtained as follows: （5a） （5b）。 3. The method for calculating the dynamic response of a transversely isotropic unsaturated half-space according to claim 1, characterized in that: In step 3, considering the capillary pressure and inertial coupling effect between the liquid phase and the gas phase, the seepage continuity equation of the heterogeneous transversely isotropic unsaturated soil can be obtained as follows: （6a） （6b） In the formula, , , , , , , , , , , m and k represent the van Genuchten parameters respectively.

4. The method for calculating the dynamic response of a transversely isotropic unsaturated half-space according to claim 1, characterized in that: According to step S4, by combining equations (5) and (6), after dimensionless transformation and Hankel transformation, the dynamic control equation of the axisymmetric heterogeneous transversely isotropic unsaturated soil foundation can be obtained as follows: （7a） （7b） （7c） （7d） Wherein, the superscripts "0" and "1" represent the Hankel transform order, ξ is the independent variable in the Hankel transform, and s is the independent variable in the Laplace transform; , , , , , , , , , , , ; , , , , , , , , , , , , , , , , , , , , , , .

5. The method for calculating the dynamic response of a transversely isotropic unsaturated half-space according to claim 1, characterized in that: According to step S5, in the Laplace-Hankel transform domain, the expressions of the point step load, the circular uniformly distributed step load and the rectangular pulse load are respectively: （8a） （8b） （8c） In the formula, a represents the load action radius, ξ is the independent variable in Hankel transformation, and s is the independent variable in Laplace transformation. represents the Bessel function of the first kind of order 1.

6. The method for calculating the dynamic response of a transversely isotropic unsaturated half-space according to claim 1, characterized in that: The half-space surface is considered to be permeable to water and air, and the boundary conditions are: （9） （10）。 7. The method for calculating the dynamic response of a transversely isotropic unsaturated half-space according to claim 1, characterized in that: The step 6 comprises the following specific steps: S61: Combine equation (7) into a matrix form. Since soil displacement, pore water pressure and pore air pressure are all non-zero solutions, the coefficient matrix is ​​zero, then: （11） In the formula, the coefficient a i The specific expression for (i=0,1,2,3,4,5,6,7,8) is: ； ； ； ； ； ； ； When the heterogeneity coefficient When , the coefficients a7 = a5 = a3 = a1 = 0 in the above expression, at this time, equation (11) can be completely degenerated into a homogeneous transversely isotropic unsaturated soil foundation; S62: Using Vieta's theorem, the expressions for the roots and coefficients of equation (11) can be expressed as: （12a） （12b） （12c） （12d） （12e） （12f） （12g） The equation is transformed into a real number equation, and a genetic algorithm is first used to perform a global search to obtain the initial value of the iteration, and then the Newton iteration method is used to solve it; S63: Step S62 solves are the eight roots of equation (11). According to the wave radiation condition, the roots with positive real parts are the solutions that satisfy the problem. Let , , and are the four roots whose real parts are greater than zero respectively; then the solution of equation (11) that satisfies the radiation condition is: （13） In the formula, , (i=1, 2, 3, 4) are unknown coefficients determined by boundary conditions; Substitute equation (13) into equation (7), and let , , , we can get: （14） in: ； ； ； ； Combining equations (7), (13) and (14), we can obtain the equation containing only the unknown number R i The expressions for displacement, pore water pressure and air pressure are: （15a） （15b） （15c） （15d） The stress component expression expressed in displacement under axisymmetry can be written as: （16a） （16b） Perform 0th-order and 1st-order Hankel transforms on equations (16a) and (16b), respectively, and apply Laplace transform and dimensionless transform, and combine with equation (15) to obtain: （17a） （17b） Perform the corresponding Hankel-Laplace joint transformation on the boundary condition (3.1), and substitute (15) and (17) into it, that is: （18a） （18b） （18c） （18d） By using formula (18), we can solve R i After the value of is obtained, it is substituted into equation (15) and equation (17), and the Hankel-Laplace integral inverse transformation is performed, and then the integral general solution expression of solid phase displacement, pore water pressure, pore gas pressure and stress in the time domain can be obtained as follows: （19） （20） （21） （22） S64: The numerical implementation of the inverse Laplace transform uses the improved Durbin method, which is currently recognized to have high accuracy and reliability. Its specific mathematical expression can be expressed as: （23） In the formula, the parameters selected for calculation are: aT=5, T=20 and NUSM=200.

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