CN109101684B - Analytic algorithm of Rayleigh wave fundamental order modal dispersion curve in regular layered semi-infinite body - Google Patents
Analytic algorithm of Rayleigh wave fundamental order modal dispersion curve in regular layered semi-infinite body Download PDFInfo
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Abstract
The invention relates to an analytical algorithm of Rayleigh fundamental order modal dispersion curve in a regular layered semi-infinite body, wherein the regular layered semi-infinite body comprises a uniform bottom layer with infinite lower layer depth and at least one uniform layered layer covering the bottom layer, the shear wave velocity of each layer increases progressively along with the layer depth, and the analytical algorithm comprises the following steps: s1, calculating a Rayleigh wave vertical displacement vibration mode function in a uniform semi-infinite body according to the Poisson ratio of the surface layer of the regular layered semi-infinite body; s2, preliminarily calculating the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body according to the displacement mode function, and correcting the relative energy in each layer by using the difference of the material mechanical parameters of the relative surface layers of each layer; and S3, taking the relative energy in each layer as a weight function, calculating the weight function and the shear wave velocity or the average value of the weighted Rayleigh wave velocity and the total relative energy of each layer to obtain an analytical expression of the Rayleigh wave fundamental mode phase velocity in the regular layered semi-infinite body.
Description
Technical Field
The method can be used for calculating the Rayleigh wave fundamental order modal dispersion curve in the regular layered semi-infinite body geotechnical medium with the layered shear wave velocity increasing with the layered depth, and can be widely applied to analysis of Rayleigh wave dispersion data obtained by surface wave testing in geotechnical engineering or engineering geophysics.
Background
Rock-soil media formed by alluvial products in different ages are distributed in a layered mode, under the action of self weight and confining pressure, the layer shear wave velocity is increased progressively along with the layer depth, and for the regular layered rock-soil media, surface wave field particle vibration is generally dominated by a fundamental mode Rayleigh wave. The ground medium surface is excited, a fundamental order modal dispersion curve can be extracted through a multi-channel surface wave test, the fundamental order modal dispersion curve contains abundant soil structure and soil physical mechanical parameter information, and the soil structure and soil physical mechanical parameter information of a detection area can be obtained through forward modeling or inversion analysis on the fundamental order modal dispersion curve, so that fundamental order modal dispersion curve calculation is very important for surface wave test data analysis.
At present, the calculation of the rayleigh wave modal dispersion curve in the layered medium in the free state is generally performed by matrix determinant calculation, such as a transfer matrix, a total stiffness matrix and a coefficient matrix method. The root of the matrix determinant, namely the wave number, is generally obtained by calculating by using a root search method, but the root search algorithm has the problems of non-convergence of a search range and root omission, so that the matrix method is unstable and the calculation efficiency is low. The layered medium is dispersed into a thin layer with small thickness relative to the wavelength, a matrix determinant root search method can be converted into an algebraic matrix decomposition method, and although the shortage of the root search method can be avoided, the method has large calculation amount. The common dispersion curve calculation method also comprises a half-wave method, and a 1/3 wavelength method and an equivalent half-space method developed on the basis of the half-wave method. The half-wave method is an empirical approximate calculation method, which utilizes the characteristic that Rayleigh wave energy is concentrated in 1/2 wavelength depth, and assumes that Rayleigh wave phase velocity is weighted average of shear wave velocity and layer thickness of each layer of medium in the energy concentration depth. The method only considers the influence of the energy concentration depth inner layer structure and the layer shear wave speed on the phase velocity calculation. Due to the discontinuity of the layered shear wave velocity, the dispersion curve calculated by the half-wave method is also discontinuous.
Disclosure of Invention
The invention aims to solve the technical problem that aiming at the defects of the prior art, an analytic algorithm of a Rayleigh wave fundamental order modal dispersion curve in a regular layered semi-infinite body is provided, the calculation complexity of the dispersion curve is reduced, the calculation efficiency is improved, and the calculation precision is ensured due to the consideration of the influence of a layered structure and the mechanical parameters of materials of each layer on the calculation of the dispersion curve.
The technical scheme for solving the technical problems is as follows: an analytical algorithm for Rayleigh fundamental mode dispersion curves in a regular layered semi-infinite body, the regular layered semi-infinite body comprising a uniform bottom layer with infinite layer depth, namely a uniform bottom semi-infinite body, and at least one uniform layer covering the bottom layer, wherein shear wave velocity of each layer in the regular layered semi-infinite body increases with layer depth, the analytical algorithm comprising the steps of:
s1, calculating related parameters of a Rayleigh wave vertical displacement mode function in a uniform semi-infinite body according to the Poisson ratio of the surface layer of the regular layered semi-infinite body, and further obtaining the displacement mode function;
s2, preliminarily calculating the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body according to the displacement mode function, and correcting the relative energy in each layer by using the difference of the material mechanical parameters of the relative surface layers of each layer;
and S3, taking the relative energy in each layer as a weight function, calculating the weight function and the shear wave velocity or the average value of the weighted Rayleigh wave velocity and the total relative energy of each layer to obtain an analytical expression of the Rayleigh wave fundamental mode phase velocity in the regular layered semi-infinite body.
The invention has the beneficial effects that: the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body is calculated on the basis of the Rayleigh wave vertical displacement vibration type function in the uniform semi-infinite body, so that the calculation process is greatly simplified, and the calculation efficiency is improved; meanwhile, the relative energy in each layer is corrected through the mechanical parameter difference of the relative surface layer material of each layer in the regular layered semi-infinite body, the influence of the parameters of each layer in the layered semi-infinite body on the Rayleigh fundamental mode dispersion curve is fully considered, and the calculation precision is ensured.
On the basis of the technical scheme, the invention can be further improved as follows:
further: the step S1 specifically includes:
s11, the longitudinal wave speed and the shear wave speed of the uniform semi-infinite body satisfy the relation:
wherein, c p Longitudinal wave velocity of uniform semi-infinite volume, c s Is the shear wave velocity of the uniform semi-infinite body, and v is the Poisson's ratio of the uniform semi-infinite body;
the Rayleigh wave velocity and the shear wave velocity of the uniform semi-infinite body satisfy a regression relation:
wherein, c R A uniform semi-infinite rayleigh wave velocity;
obtaining a relation between the Rayleigh wave velocity and the longitudinal wave velocity of the uniform semi-infinite body according to the relation (1) and the relation (2):
s12, calculating a relevant parameter alpha of a Rayleigh wave vertical displacement mode function in the uniform semi-infinite body according to the Poisson ratio of the surface layer of the regular layered semi-infinite body β and γ:
ratio c in equation (4) R /c s And c R /c p Calculated by equation (2) and equation (3), respectively:
wherein v is 1 The surface Poisson's ratio of the regular lamellar semi-infinite body is shown;
after parameters alpha, beta and gamma are determined, a relational expression of the Rayleigh wave vertical displacement vibration mode in the uniform semi-infinite body along with the change of depth and wavelength is obtained:
wherein, the first and the second end of the pipe are connected with each other,
the method is a Rayleigh wave vertical displacement vibration mode function in a uniform semi-infinite body, k is the wave number of Rayleigh waves, k =2 pi/lambda, lambda is the wavelength of the Rayleigh waves, omega is the angular frequency of the Rayleigh waves, z is the depth, and e is a natural constant.
The beneficial effects of the above further scheme are: the Rayleigh wave vertical displacement vibration mode function in the uniform semi-infinite body is used as a basis for calculating the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body, and the analytic expression of the Rayleigh wave vertical displacement vibration mode function in the uniform semi-infinite body is known, so that the calculation process is simple and the calculation speed is high.
Further: the step S2 specifically includes:
step S21, in the regular layered semi-infinite body, the unit volume energy E of the fundamental mode Rayleigh wave at the mth layer m And a fundamental mode Rayleigh wave vertical displacement mode function phi z The (ω, z) square integral is proportional:
wherein h is m Is the depth of the layer above the mth layer, h m+1 Is the depth of the layer below the mth layer, p m Is the density of the mth layer of media;
step S22, using the Rayleigh wave vertical displacement vibration mode function in the uniform semi-infinite body
Replaces the vertical displacement mode function phi of the fundamental mode Rayleigh wave in the regular layered semi-infinite body in the relational expression (6) z (ω, z), calculating the relative energy of the rayleigh waves in each layer, and then correcting the relative energy in each layer by using the difference of the material mechanical parameters of each layer relative to the surface layer:
wherein the content of the first and second substances,
to correct the coefficients, the power exponent n s ＝n ρ ＝n v ＝0.5；
The integral expression in the formula (7),
the beneficial effects of the further scheme are as follows: the relative energy of the Rayleigh waves in each layer is in direct proportion to the displacement and the layer thickness of the Rayleigh waves in each layer, and the Rayleigh wave vertical displacement vibration type function describes the relation of the displacement shape of the Rayleigh waves along with the change of the depth. The vertical displacement mode function of the fundamental mode Rayleigh wave in the regular layered semi-infinite body has high correlation with the vertical displacement mode function of the Rayleigh wave in the uniform semi-infinite body, so that the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body can be estimated through the vertical displacement mode function of the Rayleigh wave in the uniform semi-infinite body. And correcting the relative energy in each layer by utilizing the relative change of the mechanical parameters of each layer relative to the surface layer material in the regular layered semi-infinite body, so that the calculation precision of the relative energy of the fundamental mode Rayleigh wave in each layer is higher.
Further: the step S3 specifically comprises the following steps:
step S31, taking the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body as a weight function, calculating the average value of the weighted weight function and the shear wave velocity or the weighted Rayleigh wave velocity of each layer and the total relative energy, and obtaining the relation of the Rayleigh fundamental mode phase velocity in the regular layered medium along with the change of the wavelength:
wherein N is the total number of layers of the regular layered semi-infinite body,
taking the wave velocity of the m-th layer in the regular layered semi-infinite body as the surface Rayleigh wave velocity c of the surface layer R,1 That is, the amount of the oxygen present in the gas,
the Rayleigh wave velocity of the surface layer is calculated by the Poisson's ratio of the surface layer and the shear wave velocity according to the formula (2), and for the bottom layer, the wave velocity is taken as the Rayleigh wave velocity c of the bottom layer R,N That is to say that,
the Rayleigh wave velocity of the bottom layer is calculated by the Poisson's ratio of the bottom layer and the shear wave velocity according to the formula (2), and the wave velocity of each layer is taken as the shear wave velocity c of each layer for the middle layer s,j That is to say that,
2≤j≤N-1；
step S32, obtaining the change of the phase velocity c (f) with the frequency f from the change of the phase velocity c (λ) with the wavelength λ according to the relationship f = c (λ)/λ between the wavelength λ, the phase velocity c (λ) and the frequency f.
The beneficial effects of the further scheme are as follows: the relative energy of the basic-order mode Rayleigh waves in each layer of the regular layered semi-infinite body is weighted with the layer shear wave velocity or the layer Rayleigh wave velocity, and then the total relative energy is averaged to obtain the phase velocity, so that the basic-order mode Rayleigh wave dispersion curve is obtained.
Drawings
FIG. 1 is a flow chart of an analytical algorithm of the present invention;
FIG. 2 is a schematic view of a layer structure of a regular layered semi-infinite body
FIG. 3 is a graph comparing calculated values of dispersion curves with theoretical values in example 1;
FIG. 4 is a graph comparing the calculated value of the dispersion curve with the theoretical value in example 2.
Detailed Description
The principles and features of this invention are described below in conjunction with the following drawings, which are set forth by way of illustration only and are not intended to limit the scope of the invention.
The present invention will be described with reference to the accompanying drawings.
As shown in fig. 1, an embodiment of the present invention provides an analytic algorithm for a rayleigh fundamental order modal dispersion curve in a regular layered semi-infinite body, where the regular layered semi-infinite body includes a uniform bottom layer with a depth of a lower layer approaching infinity, i.e., a uniform bottom semi-infinite body, and at least one uniform layer covering the bottom layer, and a shear wave velocity of each layer in the regular layered semi-infinite body increases with a layer depth, i.e., c s,1 ＜c s,2 ...＜c s,N-1 ＜c s,N Wherein c is s,1 ,c s,2 ...c s,N-1 Respectively represent the shear wave velocities of the first to N-1 th layers, c s,N Representing the shear wave velocity of the underlying layer, the analytical algorithm comprising the steps of:
s1, calculating related parameters of a Rayleigh wave vertical displacement mode function in a uniform semi-infinite body according to the Poisson ratio of the surface layer of the regular layered semi-infinite body, and further obtaining the displacement mode function;
s2, preliminarily calculating the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body according to the displacement mode function, and correcting the relative energy in each layer by using the difference of the material mechanical parameters of the relative surface layers of each layer;
and S3, taking the relative energy in each layer as a weight function, calculating the weight function and the shear wave velocity or the average value of the weighted Rayleigh wave velocity and the total relative energy of each layer to obtain an analytical expression of the Rayleigh wave fundamental mode phase velocity in the regular layered semi-infinite body.
The invention aims to solve the problems of complexity of the traditional matrix algorithm and poor precision of a half-wave empirical analysis method, provides a quick analysis algorithm for calculating a Rayleigh wave fundamental mode dispersion curve in a regular layered semi-infinite body, and provides a new analysis method for surface wave test data.
Based on the rule that the energy of the fundamental mode Rayleigh wave in the regular layered semi-infinite body is distributed along with the depth and the rule that the phase velocity of the fundamental mode Rayleigh wave changes along with the frequency, the invention provides a method for estimating the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body on the basis of a vertical displacement vibration type function of the Rayleigh wave in the uniform semi-infinite body; then correcting the relative energy in each layer through the relative difference of the mechanical parameters of each layer and the surface layer material; and then, taking the corrected relative energy in each layer as a weight function, weighting the relative energy with the shear wave velocity of each layer or the Rayleigh wave velocity of each layer, and then calculating the Rayleigh wave fundamental mode phase velocity on the average of the total relative energy.
Compared with the traditional layer transfer matrix, rigidity matrix or coefficient matrix method, the method for calculating the Rayleigh wave fundamental mode phase velocity has the advantages of simplicity and rapidness, and is not limited by the analysis frequency range, so that the problems of non-convergence of root search, root omission, high-frequency numerical precision loss, time consumption, complexity and the like of the traditional matrix algorithm are solved. Compared with approximation methods such as a half-wave empirical algorithm and the like, the method has higher calculation precision because the influence of mechanical parameters of all layer materials in the regular layered semi-infinite body on the Rayleigh wave fundamental mode phase velocity is considered.
Preferably, the step S1 specifically includes:
s11, the longitudinal wave speed and the shear wave speed of the uniform semi-infinite body satisfy the relation:
wherein, c p Longitudinal wave speed of uniform semi-infinite body, c s Is the shear wave velocity of the uniform semi-infinite body, and v is the Poisson's ratio of the uniform semi-infinite body;
the Rayleigh wave velocity and the shear wave velocity of the uniform semi-infinite body satisfy a regression relation:
wherein, c R A uniform semi-infinite rayleigh wave velocity;
obtaining a relation between the Rayleigh wave velocity and the longitudinal wave velocity of the uniform semi-infinite body according to the relation (1) and the relation (2):
step S12, calculating relevant parameters alpha, beta and gamma of a Rayleigh wave vertical displacement mode function in the uniform semi-infinite body according to the surface Poisson ratio of the regular layered semi-infinite body:
ratio c in equation (4) R /c s And c R /c p Calculated by equation (2) and equation (3), respectively:
wherein v is 1 Is the surface Poisson's ratio of the regular lamellar semi-infinite body;
after parameters alpha, beta and gamma are determined, a relational expression of the rayleigh wave vertical displacement vibration mode in the uniform semi-infinite body along with the change of depth and wavelength is obtained:
wherein the content of the first and second substances,
the method is a Rayleigh wave vertical displacement mode function in a uniform semi-infinite body, k is the wave number of Rayleigh waves, k =2 pi/lambda, lambda is the wavelength of the Rayleigh waves, omega is the angular frequency of the Rayleigh waves, z is the depth, and e is a natural constant.
The analytical algorithm provided by the invention is based on the Rayleigh wave vertical displacement vibration mode function in the uniform semi-infinite body, and the function analytical relation is known.
Preferably, the step S2 specifically includes:
step S21, in the regular layered semi-infinite body, relative energy in each layer of the fundamental mode Rayleigh waves is related to displacement and layer thickness of the Rayleigh waves in each layer, and a displacement mode function describes the relation that the displacement shape of the fundamental mode Rayleigh waves in the layered medium changes along with the depth, so that the displacement mode can be usedThe function predicts the relative energy of the fundamental mode rayleigh waves in each layer. Energy E of fundamental mode Rayleigh wave in unit volume of m layer m And a fundamental mode Rayleigh wave vertical displacement mode function phi z The (ω, z) square integral is proportional:
wherein h is m Is the depth of the layer above the mth layer, h m+1 Is the depth of the layer below the mth layer, p m Is the density of the mth layer of media;
step S22, using the Rayleigh wave vertical displacement vibration mode function in the uniform semi-infinite body
Replaces the vertical displacement mode function phi of the fundamental mode Rayleigh wave in the regular layered semi-infinite body in the relation (6) z (ω, z), calculating the relative energy in each layer, and then correcting the relative energy in each layer using the difference in the material-mechanical parameters of each layer relative to the surface layer:
wherein, the first and the second end of the pipe are connected with each other,
to correct the coefficient, the power exponent n s ＝n ρ ＝n v ＝0.5；
The integral expression in the formula (7) is shown,
preferably, the step S3 specifically includes:
step S31, taking the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body as a weight function, calculating the average value of the weighted weight function and the shear wave velocity or the weighted Rayleigh wave velocity of each layer and the total relative energy, and obtaining the relation of the Rayleigh fundamental mode phase velocity in the regular layered medium along with the change of the wavelength:
wherein N is the total number of layers of the regular layered semi-infinite body,
taking the wave velocity of the m-th layer in the regular layered semi-infinite body as the surface Rayleigh wave velocity c of the surface layer R,1 That is to say that,
the Rayleigh wave velocity of the surface layer is calculated by the Poisson's ratio of the surface layer and the shear wave velocity according to the formula (2), and for the bottom layer, the wave velocity is taken as the Rayleigh wave velocity c of the bottom layer R,N That is, the amount of the oxygen present in the gas,
the Rayleigh wave velocity of the bottom layer is calculated by the Poisson's ratio of the bottom layer and the shear wave velocity according to the formula (2), and the wave velocity of each layer is taken as the shear wave velocity c of each layer for the middle layer s,j That is to say that,
2≤j≤N-1；
step S32, obtaining the change of the phase velocity c (f) with the frequency f from the change of the phase velocity c (λ) with the wavelength λ according to the relationship f = c (λ)/λ between the wavelength λ, the phase velocity c (λ) and the frequency f.
The energy of the fundamental mode Rayleigh waves in the regular layered semi-infinite body is concentrated in the depth of half wavelength, distributed in the depth of one wavelength, and rapidly attenuated along with the depth outside the wavelength, and the distribution rule has high correlation with the Rayleigh wave energy distribution rule in the uniform semi-infinite body. Thus, based on the vertical displacement vibration type function of Rayleigh waves in the uniform semi-infinite body with the Poisson ratio being the same as the surface Poisson ratio of the regular layered semi-infinite body, the Rayleigh waves in the fundamental mode in each regular layered semi-infinite body can be estimatedRelative energy in the layer. Compared with a uniform semi-infinite body, the energy of the fundamental mode Rayleigh wave in the regular layered semi-infinite body is gathered to a weaker layer, and the estimated relative energy in each layer can be corrected through the relative difference of the mechanical parameters of each layer of the regular layered semi-infinite body and the surface layer material of the regular layered semi-infinite body. The larger the relative energy of the Rayleigh wave in the layer is, the larger the influence of the mechanical parameters of the layer on the Rayleigh wave phase velocity is, and the phase velocity is calculated on the average of the total relative energy by taking the relative energy of the Rayleigh wave in the layer in the fundamental mode as a weight function and weighting the relative energy with the layer shear wave velocity or the Rayleigh wave velocity. For high-frequency Rayleigh waves, because the wavelength is smaller than the thickness of the surface layer, the energy of fundamental mode Rayleigh waves is gathered to the surface layer of the regular layered semi-infinite body, the fundamental mode phase velocity of the Rayleigh waves in the regular layered semi-infinite body tends to the Rayleigh wave velocity of the surface layer, and the wave velocity of the surface layer in the relation (8) is the Rayleigh wave velocity c of the surface layer R,1 That is to say that,
for low-frequency Rayleigh waves, the wavelength is larger than the total thickness of the upper layers of the bottom semi-infinite body, the Rayleigh wave energy of the fundamental mode is gathered to the bottom semi-infinite body of the regular layered semi-infinite body, the Rayleigh fundamental mode phase velocity in the regular layered semi-infinite body tends to the Rayleigh wave velocity of the bottom semi-infinite body, and the wave velocity of the middle and bottom layers in the relation (8) is taken as the Rayleigh wave velocity c of the bottom layer R,N That is to say that,
for the intermediate layer, the greater the relative energy of the rayleigh wave in the layer, the closer the rayleigh wave velocity and the shear wave velocity of the layer become, and therefore, the wave velocity of each intermediate layer in the relation (8) is taken as the shear wave velocity c of each intermediate layer s,j That is to say that,
2≤j≤N-1。
specifically, the analytical algorithm provided by the present invention is used to calculate the rayleigh wave fundamental mode dispersion curve in the regular layered semi-infinite body under two different layering conditions, and the dispersion curve calculated according to the analytical algorithm provided by the present invention is compared with the theoretical dispersion curve, so as to verify the analytical algorithm provided by the present invention.
Example 1:
TABLE 1 case I stratified Material mechanics parameters
Table 1 shows the mechanical parameters of the layered material in case I, and the calculation steps of the Rayleigh wave fundamental order modal dispersion curve in the regular layered semi-infinite body in case I are as follows:
as can be seen from the above table, the Poisson's ratio v of the surface layer of the regular lamellar semi-infinite body in the calculation example 1 1 =0.35; and (S12) calculating parameters alpha, beta and gamma of the Rayleigh wave vertical displacement vibration mode function in the uniform semi-infinite body according to the formula in the step (S) so as to obtain the Rayleigh wave vertical displacement vibration mode function in the uniform semi-infinite body:
the wavelength interval [0.05m,60m]Dispersing into M equally spaced intervals, M =1200, wavelength lambda corresponding to the ith discrete point i ＝λ 0 + (i-1) (60-0.05)/1200, (i =1,2.., M + 1), according to formula k i ＝2π/λ i Calculating the wave number k corresponding to the ith discrete point i 。
Combining the depth h of the upper and lower layers of each layer in the upper table m And h m+1 M =1,2,3,4, the integral of each layer is calculated:
combining the above table and the formula
Calculating a correction factor
Calculating the phase velocity c corresponding to the ith discrete point according to the formula in the step S31 i (λ i )。
According to the formula f i ＝c i (λ i )/λ i Calculating the frequency f corresponding to the ith discrete point i (λ i )。
And calculating the phase velocity and frequency corresponding to all the discrete points to obtain a dispersion curve of the phase velocity changing along with the frequency.
The dotted line in fig. 3 is a calculated value of a rayleigh wave fundamental order modal dispersion curve in the case I regular layered semi-infinite body obtained by using the analytic algorithm provided by the present invention, the solid line in fig. 3 is a theoretical value of a rayleigh wave fundamental order modal dispersion curve in the case I regular layered semi-infinite body obtained according to a matrix method, and it can be seen that the calculated value of the present invention has higher accuracy by comparing the two curves.
Example 2:
table 2 case II layered material mechanics parameters
The calculation method of the Rui Li Boji order modal dispersion curve in the wavelength interval [0.05m,60m ] in case II is similar to that in case I, and is not repeated here. The dotted line in fig. 4 shows the calculated value of the rayleigh fundamental order modal dispersion curve in the case II regular layered semi-infinite body obtained by the analytic algorithm provided by the present invention, and the solid line in fig. 4 shows the theoretical value of the rayleigh fundamental order modal dispersion curve in the case II regular layered semi-infinite body obtained by the matrix method.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.
Claims (4)
1. An analytic algorithm of Rayleigh fundamental mode dispersion curves in a regular layered semi-infinite body, wherein the regular layered semi-infinite body comprises a uniform bottom layer with infinite layer depth, namely a uniform bottom semi-infinite body, and at least one uniform layer covering the bottom layer, and the shear wave velocity of each layer in the regular layered semi-infinite body increases progressively along with the layer depth, and the analytic algorithm is characterized by comprising the following steps:
s1, calculating related parameters of a Rayleigh wave vertical displacement mode function in a uniform semi-infinite body according to the Poisson ratio of the surface layer of the regular layered semi-infinite body, and further obtaining the displacement mode function;
s2, preliminarily calculating the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body according to the displacement mode function, and correcting the relative energy in each layer by using the difference of the material mechanical parameters of the relative surface layers of each layer;
and S3, taking the relative energy in each layer as a weight function, calculating the weight function and the shear wave velocity or the average value of the weighted Rayleigh wave velocity and the total relative energy of each layer to obtain an analytical expression of the Rayleigh wave fundamental mode phase velocity in the regular layered semi-infinite body.
2. The analytical algorithm for the rayleigh wave fundamental order modal dispersion curve in the regular layered semi-infinite body according to claim 1, wherein the step S1 specifically comprises:
s11, the longitudinal wave speed and the shear wave speed of the uniform semi-infinite body satisfy the relation:
wherein, c p Longitudinal wave speed of uniform semi-infinite body, c s Is the shear wave velocity of the uniform semi-infinite body, and v is the Poisson's ratio of the uniform semi-infinite body;
the Rayleigh wave velocity and the shear wave velocity of the uniform semi-infinite body satisfy a regression relation:
wherein, c R A uniform semi-infinite rayleigh wave velocity;
obtaining a relation between the Rayleigh wave velocity and the longitudinal wave velocity of the uniform semi-infinite body according to the relation (1) and the relation (2):
step S12, calculating relevant parameters alpha, beta and gamma of a Rayleigh wave vertical displacement mode function in the uniform semi-infinite body according to the surface Poisson ratio of the regular layered semi-infinite body:
ratio c in equation (4) R /c s And c R /c p Calculated by equation (2) and equation (3), respectively:
wherein v is 1 The surface Poisson's ratio of the regular lamellar semi-infinite body is shown;
after parameters alpha, beta and gamma are determined, a relational expression of the Rayleigh wave vertical displacement vibration mode in the uniform semi-infinite body along with the change of depth and wavelength is obtained:
wherein the content of the first and second substances,
the method is a Rayleigh wave vertical displacement mode function in a uniform semi-infinite body, k is the wave number of Rayleigh waves, k =2 pi/lambda, lambda is the wavelength of the Rayleigh waves, omega is the angular frequency of the Rayleigh waves, z is the depth, and e is a natural constant.
3. The analytical algorithm for the rayleigh wave fundamental order modal dispersion curve in the regular layered semi-infinite body according to claim 2, wherein the step S2 specifically comprises:
step S21, in the regular layered semi-infinite body, the unit volume energy E of the fundamental mode Rayleigh wave at the mth layer m And a fundamental mode Rayleigh wave vertical displacement mode function phi z The (ω, z) square integral is proportional:
wherein h is m Is the depth of the layer above the mth layer, h m+1 Is the depth of the layer below the mth layer, p m Is the density of the mth layer of media;
step S22, using the vertical displacement mode function of Rayleigh wave in the uniform semi-infinite body
Replaces the vertical displacement mode function phi of the fundamental mode Rayleigh wave in the regular layered semi-infinite body in the relation (6) z (ω, z) calculating relative energies in the layers, and then correcting the relative energies in the layers using differences in material-mechanical parameters of the layers relative to the surface layer:
wherein the content of the first and second substances,
to correct the coefficient, the power exponent n s ＝n ρ ＝n v ＝0.5；
The integral expression in the formula (7),
4. the analytical algorithm for Rayleigh fundamental mode dispersion curves in the regular layered semi-infinite body according to claim 3, wherein the step S3 is specifically:
step S31, taking the relative energy of the fundamental mode Rayleigh wave in each layer of the regular layered semi-infinite body as a weight function, calculating the average value of the weighted weight function and the shear wave velocity or the weighted Rayleigh wave velocity of each layer and the total relative energy, and obtaining the relation of the Rayleigh fundamental mode phase velocity in the regular layered medium along with the change of the wavelength:
wherein N is the total number of layers of the regular layered semi-infinite body,
taking the wave velocity of the m-th layer in the regular layered semi-infinite body as the surface Rayleigh wave velocity c of the surface layer R,1 That is to say that,
the Rayleigh wave velocity of the surface layer is calculated by the Poisson's ratio of the surface layer and the shear wave velocity according to the formula (2), and for the bottom layer, the wave velocity is taken as the Rayleigh wave velocity c of the bottom layer R,N That is to say that,
the Rayleigh wave velocity of the bottom layer is calculated by the Poisson's ratio of the bottom layer and the shear wave velocity according to the formula (2), and the wave velocity of each layer is taken as the shear wave velocity c of each layer for the middle layer s,j That is to say that,
step S32, obtaining the change of the phase velocity c (f) with the frequency f from the change of the phase velocity c (λ) with the wavelength λ according to the relationship f = c (λ)/λ between the wavelength λ, the phase velocity c (λ) and the frequency f.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN104711965A (en) * | 2015-03-23 | 2015-06-17 | 中冶集团武汉勘察研究院有限公司 | Method for dividing density degrees of gravelly soil foundations through surface-wave inversion shear wave velocities |
| CN105676281A (en) * | 2016-01-22 | 2016-06-15 | 河北省电力勘测设计研究院 | Method for determining mechanical parameters of stratum via Rayleigh wave speed |
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| CN104711965A (en) * | 2015-03-23 | 2015-06-17 | 中冶集团武汉勘察研究院有限公司 | Method for dividing density degrees of gravelly soil foundations through surface-wave inversion shear wave velocities |
| CN105676281A (en) * | 2016-01-22 | 2016-06-15 | 河北省电力勘测设计研究院 | Method for determining mechanical parameters of stratum via Rayleigh wave speed |
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