A method and system for calculating the Rayleigh wave dispersion relation of a nonlinear metasurface

Abstract

The present invention relates to the field of physical acoustics technology, and in particular to a method and system for calculating the dispersion relationship of nonlinear hypersurface Rayleigh waves, the method comprising the following steps: decomposing the displacement field of a saturated soil to obtain a displacement vector and a relative displacement, and establishing a wave control equation for the saturated soil; combining the displacement vector, the relative displacement and the wave control equation to obtain an expression of a surface wave solution; constructing a potential function, obtaining a displacement expression and a relative displacement expression based on the potential function, and constructing a stress expression for the saturated soil; constructing a dynamic equilibrium equation, and obtaining a boundary formula based on the dynamic equilibrium equation combined with the Euler formula; obtaining the boundary conditions of the saturated soil, and obtaining a nonlinear hypersurface Rayleigh wave dispersion curve equation for the saturated soil in combination with the boundary formula. The present invention constructs a dispersion curve equation for nonlinear hypersurface Rayleigh waves, providing technical support for studying and analyzing the propagation characteristics of Rayleigh waves.

Description

Technical Field

[0001] The present invention relates to the field of physical acoustics technology, and in particular to a method and system for calculating the dispersion relation of Rayleigh waves on a nonlinear metasurface. Background Art

[0002] Traditional Rayleigh wave research is mostly based on linear theory, which assumes that the medium's response to the wave is linear, the wave amplitude is small, and will not cause significant changes in the medium's properties; but in practical applications, many media exhibit significant nonlinear characteristics, especially under high-amplitude or high-frequency excitation. This nonlinear characteristic can cause changes in the propagation characteristics of Rayleigh waves.

[0003] In recent years, with the rapid development of nanotechnology and micro-nano processing technology, the design and manufacture of metasurfaces with special nonlinear properties have begun. Nonlinear metasurfaces can achieve precise control of sound wave propagation characteristics at a microscopic scale through sophisticated structural design and material selection. Nonlinear metasurfaces provide a new platform and means for the study of Rayleigh waves.

[0004] However, the calculation of Rayleigh wave dispersion relations on nonlinear metasurfaces is a complex and challenging problem; the traditional linear dispersion relation calculation method is no longer applicable in nonlinear situations; in addition, the complex structure and material properties of nonlinear metasurfaces also make the calculation of dispersion relations more difficult.

[0005] Therefore, the present invention proposes a method and system for calculating the dispersion relationship of Rayleigh waves on a nonlinear metasurface, which can accurately and efficiently calculate the dispersion relationship of Rayleigh waves on a nonlinear metasurface; can fully consider the complex structure and material properties of the nonlinear metasurface, and the nonlinear propagation characteristics of Rayleigh waves, thereby realizing accurate analysis of the dispersion relationship of Rayleigh waves on the nonlinear metasurface, and further accurately predicting the band gap width of the nonlinear metasurface in saturated soil, isolating the propagation of Rayleigh waves within the band gap width; not only provides strong support for the design and optimization of nonlinear metasurfaces, but also provides a new perspective and method for the study of the propagation characteristics of Rayleigh waves in nonlinear media. Summary of the invention

[0006] In view of the defects in the prior art, the present invention provides a method and system for calculating the dispersion relationship of Rayleigh waves of a nonlinear metasurface.

[0007] In order to achieve the above-mentioned purpose, in a first aspect, the present invention provides a method for calculating the dispersion relationship of nonlinear hypersurface Rayleigh waves, the method comprising the following steps: decomposing the displacement field of a saturated soil to obtain a displacement vector and a relative displacement, and establishing a wave control equation for the saturated soil; combining the displacement vector, the relative displacement and the wave control equation to obtain an expression of a surface wave solution; constructing a potential function, obtaining a displacement expression and a relative displacement expression based on the potential function, and constructing a stress expression for the saturated soil; constructing a dynamic equilibrium equation, obtaining a boundary formula based on the dynamic equilibrium equation combined with the Euler formula; obtaining the boundary conditions of the saturated soil, and obtaining the nonlinear hypersurface Rayleigh wave dispersion curve equation for the saturated soil in combination with the boundary formula. The present invention provides a method for calculating the dispersion relationship of Rayleigh waves on a nonlinear metasurface. The method decomposes the displacement field of a saturated soil, constructs a wave control equation, and derives an expression for a surface wave solution. The potential function is introduced to simplify the process of solving the displacement and relative displacement, realize the construction of a stress expression, and improve the accuracy and efficiency of the calculation. In addition, the construction of the dynamic equilibrium equation is combined with the Euler formula to make the processing of boundary conditions more rigorous. Finally, combined with the boundary conditions of the saturated soil, the dispersion curve equation of the Rayleigh wave on the nonlinear metasurface is obtained, which provides a theoretical reference for the design and optimization of the nonlinear metasurface and opens up a new way for the study of the propagation characteristics of Rayleigh waves in complex media.

[0008] Optionally, the displacement field of the saturated soil is decomposed to obtain a displacement vector and a relative displacement, and a wave control equation of the saturated soil is established, including: decomposing the displacement field of the saturated soil to obtain a displacement vector of a solid skeleton and a relative displacement from a fluid to the solid skeleton based on the Helmholtz theorem; and constructing the wave control equation of the saturated soil according to the displacement vector and the relative displacement. The present invention decomposes the displacement field of the saturated soil into a displacement vector of a solid skeleton and a relative displacement from a fluid to the solid skeleton by applying the Helmholtz theorem, thereby simplifying the description of the displacement field, providing a basis for the construction of the wave control equation, and being able to more accurately reflect the interaction between the solid skeleton and the fluid in the saturated soil, as well as the influence of such interaction on the propagation characteristics of Rayleigh waves, thereby improving the accuracy of the dispersion relationship calculation and providing a strong theoretical support for the design and performance analysis of nonlinear metasurfaces.

[0009] Optionally, the expression of the surface wave solution obtained by combining the displacement vector, the relative displacement and the wave control equation includes: combining the displacement vector, the relative displacement and the wave control equation to obtain the uncoupled motion control equation of the longitudinal wave and the motion control equation of the transverse wave; obtaining the first characteristic equation according to the uncoupled motion control equation, and obtaining the second characteristic equation according to the motion control equation; obtaining the body wave velocity equation according to the first characteristic equation and the second characteristic equation, and then obtaining the expression of the surface wave solution. The present invention derives the uncoupled motion control equation of the longitudinal wave and the motion control equation of the transverse wave by combining the displacement vector, the relative displacement and the wave control equation, and further obtains two characteristic equations, which provide a data basis for solving the surface wave solution. By solving the characteristic equation, the body wave velocity equation is accurately obtained, and then the expression of the surface wave solution is derived; the accuracy of the calculation is improved, and it provides strong support for the study of the dispersion relationship of Rayleigh waves on nonlinear metasurfaces.

[0010] Optionally, the construction of a potential function, obtaining a displacement expression and a relative displacement expression based on the potential function, and constructing a stress expression for the saturated soil include: constructing a potential function for a harmonic surface wave; combining the potential function with the displacement vector to obtain a displacement expression for a solid skeleton; combining the potential function with the relative displacement to obtain a relative displacement expression for a fluid; and constructing a total stress expression and a pore pressure expression for the saturated soil. The present invention improves the accuracy of calculations by constructing a potential function for a harmonic surface wave, combining it with a displacement vector and a relative displacement, and deriving a displacement expression for a solid skeleton and a relative displacement expression for a fluid, respectively; and further constructing a total stress expression and a pore pressure expression for a saturated soil, providing a powerful mathematical tool for in-depth analysis of the propagation characteristics and stress distribution of Rayleigh waves in saturated soil, which has important theoretical value and practical guiding significance.

[0011] Optionally, constructing a potential function of a harmonic surface wave comprises:

[0012] ;

[0013] ;

[0014] ;

[0015] ;

[0016] in, is the scalar potential of the solid, , and is the amplitude corresponding to each potential function, is the base of natural logarithm, , and is the wave number, express direction, is an imaginary unit, is the circular frequency, For time, is the wave number of the surface wave, express direction, is the scalar potential of the fluid, , and is the ratio of the potential function to the amplitude, and is the vector potential The present invention can comprehensively describe the propagation characteristics of waves in saturated soil by constructing a harmonic surface wave potential function including solid scalar potential, fluid scalar potential and vector potential components; key parameters such as amplitude, wave number and circular frequency are incorporated into the potential function, which provides a data basis for accurate solution and helps to deeply analyze the dispersion relationship of nonlinear metasurface Rayleigh waves.

[0017] Optionally, the construction of the dynamic balance equation, based on the dynamic balance equation combined with Euler's formula to obtain a boundary formula, includes: coupling the nonlinear resonator to the free surface of the saturated soil, establishing a dynamic balance equation of saturated soil-nonlinear spring-oscillator; based on the dynamic balance equation combined with Euler's formula, using the first harmonic balance method to obtain a boundary formula. The present invention couples the nonlinear resonator to the free surface of the saturated soil and constructs the corresponding dynamic balance equation, which can more accurately simulate the complex boundary conditions in actual engineering; the boundary formula obtained by combining the Euler formula and using the first harmonic balance method can more deeply reveal the physical mechanism of the interaction between the saturated soil and the nonlinear resonator.

[0018] Optionally, coupling the nonlinear resonator to the free surface of the saturated soil to establish a dynamic equilibrium equation of saturated soil-nonlinear spring-oscillator includes:

[0019] ;

[0020] in, is the resonator mass, represents the relative motion between the resonator mass and the vertical displacement of the saturated soil surface, is the linear stiffness of the spring, is the nonlinear stiffness of the spring, The invention couples the nonlinear resonator to the saturated soil surface, establishes a dynamic equilibrium equation, and accurately simulates the dynamic interaction between the resonator and the soil. The nonlinear spring stiffness is introduced to be closer to the actual physical situation, which helps to deeply analyze the resonance effect and soil response. The calculation accuracy is improved, and a theoretical basis and calculation basis are provided for the calculation of Rayleigh wave dispersion.

[0021] Optionally, the step of obtaining the boundary conditions of the saturated soil and obtaining the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil in combination with the boundary formula includes: constructing a homogeneous equation group based on the boundary conditions of the saturated soil in combination with the boundary formula; and obtaining the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil based on the homogeneous equation group. The present invention constructs a homogeneous equation group based on the boundary conditions of the saturated soil in combination with the boundary formula, which provides key data support for solving the nonlinear hypersurface Rayleigh wave dispersion characteristics of the saturated soil; by solving the homogeneous equation group, the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil is obtained, which is of great significance for revealing the laws of soil wave propagation, evaluating structural stability, and designing effective wave control measures.

[0022] Optionally, the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil is obtained based on the homogeneous equation group, including:

[0023] ;

[0024] in, is the circular frequency, is the effective frequency, for The wave vector of the wave, is the compression modulus of saturated soil, , is the ratio of the potential function to the amplitude, is the wave number of the surface wave, for The wave vector of the wave, and is the simplification coefficient, is the Lame constant of the soil skeleton, , and is the wave number, for The wave vector of the wave, is the resonator mass, The present invention comprehensively considers the compression modulus, wave number and wave vector of saturated soil, constructs the nonlinear metasurface Rayleigh wave dispersion curve equation, and accurately describes the propagation characteristics and dispersion law of Rayleigh waves in saturated soil.

[0025] In the second aspect, the present invention provides a nonlinear metasurface Rayleigh wave dispersion relationship calculation system, the system executes the nonlinear metasurface Rayleigh wave dispersion relationship calculation method provided by the present invention, the system includes an input device, an output device, a processor and a memory, and its gain lies in: the hardware facilities integrated by the present invention have excellent performance, the input device, the output device, the processor and the memory are interconnected, the information transmission between the various components is smooth, and an efficient information processing system is constructed through the interaction of multiple hardware facilities. The nonlinear metasurface Rayleigh wave dispersion relationship calculation system provided by the present invention integrates high-performance hardware facilities, can quickly execute the dispersion relationship calculation method, not only improves the calculation efficiency, but also ensures the accuracy and stability of data processing, and provides strong hardware system support for the research and application of nonlinear metasurface Rayleigh wave dispersion relations. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 A flow chart of a method for calculating the dispersion relation of Rayleigh waves on a nonlinear metasurface according to an embodiment of the present invention;

[0027] Figure 2 Schematic diagram of a theoretical calculation model of a nonlinear hypersurface in a saturated soil according to an embodiment of the present invention;

[0028] Figure 3 It is a partially enlarged schematic diagram of a nonlinear resonator according to an embodiment of the present invention;

[0029] Figure 4 It is a dispersion curve diagram of the saturated soil hardening model according to an embodiment of the present invention;

[0030] Figure 5 It is a dispersion curve diagram of the saturated soil softening model according to an embodiment of the present invention;

[0031] Figure 6 This is a framework diagram of a nonlinear metasurface Rayleigh wave dispersion relationship calculation system according to an embodiment of the present invention. DETAILED DESCRIPTION

[0032] The specific embodiments of the present invention will be described in detail below. It should be noted that the embodiments described herein are only for illustration and are not intended to limit the present invention. In the following description, a large number of specific details are set forth in order to provide a thorough understanding of the present invention. However, it is obvious to those of ordinary skill in the art that these specific details do not need to be adopted to implement the present invention. In other examples, in order to avoid confusing the present invention, known circuits, software or methods are not specifically described.

[0033] Throughout the specification, references to "one embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in conjunction with the embodiment or example is included in at least one embodiment of the present invention. Therefore, the phrases "in one embodiment," "in an embodiment," "an example," or "an example" appearing in various places throughout the specification do not necessarily all refer to the same embodiment or example. In addition, particular features, structures, or characteristics may be combined in one or more embodiments or examples in any suitable combination and / or subcombination. In addition, it should be understood by those of ordinary skill in the art that the figures provided herein are for illustrative purposes and that the figures are not necessarily drawn to scale.

[0034] See also Figure 1 An embodiment of the present invention provides a method for calculating the dispersion relation of Rayleigh waves on a nonlinear metasurface, the method comprising the following steps:

[0035] S1. Decomposing the displacement field of the saturated soil to obtain the displacement vector and the relative displacement, and establishing the wave control equation of the saturated soil.

[0036] Among them, S1 specifically includes the following steps:

[0037] S11. Based on the Helmholtz theorem, the displacement field of the saturated soil is decomposed to obtain the displacement vector of the solid skeleton and the relative displacement from the fluid to the solid skeleton.

[0038] In this embodiment, the Helmholtz decomposition of the saturated soil displacement field is obtained based on the Helmholtz theorem, and the displacement vector expression of the solid skeleton and the relative displacement expression from the fluid to the solid skeleton are obtained, which satisfy the following relationship:

[0039]

[0040] in, is the displacement vector of the solid skeleton, is the gradient operator, is the scalar potential of the solid, is the vector potential of the solid, is the relative displacement from the fluid to the solid skeleton, is the scalar potential of the fluid, is the vector potential of the fluid.

[0041] S12. Constructing a wave control equation of the saturated soil according to the displacement vector and the relative displacement.

[0042] In this embodiment, under the 2D plane strain state, according to the Biot theory, the wave control equation in the saturated soil can be expressed as The formula is expressed to satisfy the following relationship:

[0043]

[0044]

[0045] in, and is the Lame constant of the soil skeleton, is the compression modulus of saturated soil, is the coupling modulus of saturated soil, is the gradient operator, is the displacement vector of the solid skeleton, is the relative displacement from the fluid to the solid skeleton, is the total mass density, Indicates time Find the second-order partial derivative, is the mass density of the fluid, is the relationship parameter between fluid density and pore geometry, is the viscous coupling parameter between the solid and the fluid.

[0046] Specifically, the Lame constant of the soil skeleton satisfies the following relationship:

[0047]

[0048]

[0049] in, and is the Lame constant of the soil skeleton, is the Young's modulus of saturated soil, is the Poisson's ratio of saturated soil.

[0050] Specifically, the compression modulus and coupling modulus of saturated soil satisfy the following relationship:

[0051]

[0052]

[0053] in, is the compression modulus of saturated soil, is the bulk modulus of the soil, is the bulk modulus of soil particles, is the coupling modulus of saturated soil, is the porosity of the saturated half space, is the bulk modulus of the liquid.

[0054] Specifically, the displacement vector of the solid skeleton satisfies the following relationship:

[0055]

[0056] in, is the displacement vector of the solid skeleton, for Displacement in direction, for Displacement in direction.

[0057] Specifically, the relative displacement from the fluid to the solid skeleton satisfies the following relationship:

[0058]

[0059] in, is the relative displacement from the fluid to the solid skeleton, for The relative displacement in the direction, for Relative displacement in direction.

[0060] Specifically, the total mass density satisfies the following relationship:

[0061]

[0062] in, is the total mass density, is the porosity of the saturated half space, is the mass density of the fluid, is the soil skeleton density in the saturated half space.

[0063] Specifically, the relationship parameters between fluid density and pore geometry satisfy the following relationship:

[0064]

[0065] in, is the relationship parameter between fluid density and pore geometry, is the curvature, is the mass density of the fluid, is the porosity of the saturated half space.

[0066] Specifically, the viscous coupling parameters between the solid and the fluid satisfy the following relationship:

[0067]

[0068] in, is the viscous coupling parameter between solid and fluid, is the fluid viscosity, is the permeability.

[0069] S2. Combining the displacement vector, the relative displacement and the wave control equation, an expression for the surface wave solution is obtained.

[0070] Among them, S2 specifically includes the following steps:

[0071] S21. Combining the displacement vector, the relative displacement and the wave control equation, the uncoupled motion control equation of the longitudinal wave and the motion control equation of the transverse wave are obtained.

[0072] In this embodiment, the displacement vector and the relative displacement are substituted into the wave control equation to obtain the uncoupled motion control equation of the longitudinal wave (P wave) and the motion control equation of the shear wave (S wave).

[0073] Specifically, the uncoupled motion control equations of longitudinal waves (P waves) satisfy the following relationship:

[0074]

[0075]

[0076] in, and is the Lame constant of the soil skeleton, is the compression modulus of saturated soil, is the coupling modulus of saturated soil, is the gradient operator, is the scalar potential of the solid, is the scalar potential of the fluid, is the total mass density, Indicates time Find the second-order partial derivative, is the mass density of the fluid, is the relationship parameter between fluid density and pore geometry, is the viscous coupling parameter between solid and fluid, Indicates time Find the partial derivative.

[0077] Specifically, the motion control equation of the shear wave (S wave) satisfies the following relationship:

[0078]

[0079]

[0080] in, is the Lame constant of the soil skeleton, is the gradient operator, is the vector potential of the solid, is the total mass density, Indicates time Find the second-order partial derivative, is the mass density of the fluid, is the vector potential of the fluid, is the relationship parameter between fluid density and pore geometry, is the viscous coupling parameter between solid and fluid, Indicates time Find the partial derivative.

[0081] S22. Obtain a first characteristic equation according to the uncoupled motion control equation, and obtain a second characteristic equation according to the motion control equation.

[0082] In this embodiment, it is assumed that the solution of the uncoupled motion control equation satisfies the following relationship:

[0083]

[0084] in, is the scalar potential of the solid, is the scalar potential of the fluid, represents transpose, is the complex amplitude of the solid, is the complex amplitude of the fluid, is the base of natural logarithm, is an imaginary unit, is the circular frequency, For time, for The wave vector of the wave, is the position vector.

[0085] Specifically, the solution of the uncoupled motion control equation is substituted into the uncoupled motion control equation, and the characteristic equation of the P wave obtained is used as the first characteristic equation, which satisfies the following relationship:

[0086]

[0087] in, is the total mass density, is the circular frequency, and is the Lame constant of the soil skeleton, is the compression modulus of saturated soil, is the coupling modulus of saturated soil, for The wave vector of the wave, is the mass density of the fluid, is the complex amplitude of the solid, is the equivalent mass density of the fluid and solid skeleton in the saturated soil during wave propagation, is the complex amplitude of the fluid.

[0088] Specifically, the equivalent mass density of the fluid and solid skeleton in the saturated soil during wave propagation satisfies the following relationship:

[0089]

[0090] in, is the equivalent mass density of the fluid and solid skeleton in the saturated soil during wave propagation, is the relationship parameter between fluid density and pore geometry, is an imaginary unit, is the viscous coupling parameter between solid and fluid, is the circular frequency.

[0091] In this embodiment, it is assumed that the solution of the motion control equation satisfies the following relationship:

[0092]

[0093] in, and is the vector potential The weight, is the complex amplitude of the solid, is the complex amplitude of the fluid, is the base of natural logarithm, is an imaginary unit, is the circular frequency, For time, for The wave vector of the wave, is the position vector.

[0094] Specifically, the solution of the motion control equation is substituted into the motion control equation, and the characteristic equation of the S wave obtained is used as the second characteristic equation, which satisfies the following relationship:

[0095]

[0096] in, is the total mass density, is the circular frequency, is the Lame constant of the soil skeleton, for The wave vector of the wave, is the mass density of the fluid, is the equivalent mass density of the fluid and solid skeleton in the saturated soil during wave propagation, is the complex amplitude of the solid, is the complex amplitude of the fluid.

[0097] S23. Obtain a body wave velocity equation according to the first characteristic equation and the second characteristic equation, and then obtain an expression for a surface wave solution.

[0098] In this embodiment, the first characteristic equation has two non-zero solutions, satisfying the following relationship:

[0099]

[0100] in, For saturated soil The speed of body waves, is the circular frequency, for The wave vector of the wave, and is the coefficient, For saturated soil The speed of body waves, for The wave vector of the wave;

[0101]

[0102] in, and is the coefficient, and is the Lame constant of the soil skeleton, is the compression modulus of saturated soil, is the coupling modulus of saturated soil, is the equivalent mass density of the fluid and solid skeleton in the saturated soil during wave propagation, is the mass density of the fluid, is the total mass density.

[0103] In this embodiment, the second characteristic equation has a non-zero solution that satisfies the following relationship:

[0104]

[0105] in, is the shear wave velocity in saturated soil, is the circular frequency, for The wave vector of the wave, is the Lame constant of the soil skeleton, is the total mass density, is the mass density of the fluid, is the equivalent mass density of the fluid and solid skeleton in the saturated soil during wave propagation.

[0106] Furthermore, two non-zero solutions of the first characteristic equation and one non-zero solution of the second characteristic equation are used together as the body wave velocity equation, and the expression of the surface wave solution in the saturated half space is obtained through the body wave velocity equation in the saturated soil.

[0107] S3. Construct a potential function, obtain a displacement expression and a relative displacement expression based on the potential function, and construct a stress expression for the saturated soil.

[0108] Among them, S3 specifically includes the following steps:

[0109] S31. Construct the potential function of harmonic surface waves.

[0110] In this embodiment, the four potential functions of the harmonic surface wave propagating in the x-direction and exhibiting exponential decay in the z-direction satisfy the following relationship:

[0111]

[0112]

[0113]

[0114]

[0115] in, is the scalar potential of the solid, , and is the amplitude corresponding to each potential function, is the base of natural logarithm, , and is the wave number, express direction, is an imaginary unit, is the circular frequency, For time, is the wave number of the surface wave, express direction, is the scalar potential of the fluid, , and is the ratio of the potential function to the amplitude, and is the vector potential The amount.

[0116] Specifically, is the amplitude corresponding to the potential function, is the ratio of the potential function amplitudes, satisfying the following relationship:

[0117]

[0118]

[0119]

[0120] in, , and is the ratio of the potential function to the amplitude, and is the complex amplitude of the fluid, and is the complex amplitude of the solid, is the mass density of the fluid, For saturated soil The speed of body waves, is the compression modulus of saturated soil, is the coupling modulus of saturated soil, is the equivalent mass density of the fluid and solid skeleton in the saturated soil during wave propagation, For saturated soil The speed of body waves, is the complex amplitude of the fluid, is the complex amplitude of the solid.

[0121] Specifically, , and is the wave number, satisfying the following relationship:

[0122]

[0123] in, , and is the wave number, is the wave number of the surface wave, for The wave vector of the wave, for The wave vector of the wave, for The wave vector of the wave.

[0124] S32. Combining the potential function with the displacement vector to obtain a displacement expression of the solid skeleton.

[0125] In this embodiment, the potential function is substituted into the displacement vector to obtain the displacement expression of the solid skeleton, which satisfies the following relationship:

[0126]

[0127]

[0128] in, For soil skeleton Displacement in direction, is the scalar potential of the solid, Express Find the first-order partial derivative, is the vector potential The weight, Express Find the first-order partial derivative, is an imaginary unit, is the wave number of the surface wave, , and is the amplitude corresponding to each potential function, is the base of natural logarithm, , and is the wave number, express direction, is the circular frequency, For time, express direction, For soil skeleton Displacement in direction.

[0129] S33. Combining the potential function with the relative displacement to obtain an expression for the relative displacement of the fluid.

[0130] In this embodiment, the potential function is substituted into the relative displacement to obtain the relative displacement expression of the fluid, which satisfies the following relationship:

[0131]

[0132]

[0133] in, For fluid The relative displacement in the direction, is the scalar potential of the fluid, Express Find the first-order partial derivative, is the vector potential The weight, Express Find the first-order partial derivative, is an imaginary unit, is the wave number of the surface wave, , and is the amplitude corresponding to each potential function, , and is the ratio of the potential function to the amplitude, is the base of natural logarithm, , and is the wave number, express direction, is the circular frequency, For time, express direction, For fluid Relative displacement in direction.

[0134] S34. Construct the total stress expression and pore pressure expression of the saturated soil.

[0135] In this embodiment, it is assumed that the saturated soil satisfies the isotropic linear elastic constitutive relation. Then, the total stress expression and pore pressure expression of the saturated soil satisfy the following relationship:

[0136]

[0137]

[0138]

[0139] in, is the total stress, and is the Lame constant of the soil skeleton, For soil skeleton Displacement in direction, Express Find the first-order partial derivative, Express Find the first-order partial derivative, For soil skeleton Displacement in direction, is the compression modulus of saturated soil, is the mass density of the fluid, is the coupling modulus of saturated soil, , is the ratio of the potential function to the amplitude, for The wave vector of the wave, , and is the wave number, , and is the amplitude corresponding to each potential function, is the base of natural logarithm, express direction, is an imaginary unit, is the circular frequency, For time, is the wave number of the surface wave, express direction, for The wave vector of the wave, is the physical quantity of shear stress between z and x directions inside the soil, for The wave vector of the wave, is the pore pressure, For fluid The relative displacement in the direction, For fluid Relative displacement in direction.

[0140] S4. Construct a dynamic balance equation, and obtain a boundary formula based on the dynamic balance equation combined with Euler's formula.

[0141] Wherein, S4 specifically includes the following steps:

[0142] S41, coupling a nonlinear resonator to the free surface of the saturated soil, and establishing a dynamic equilibrium equation of saturated soil-nonlinear spring-oscillator.

[0143] In this embodiment, the dynamic equilibrium equation of saturated soil-nonlinear spring-oscillator satisfies the following relationship:

[0144]

[0145] in, is the resonator mass, represents the relative motion between the resonator mass and the vertical displacement of the saturated soil surface, is the linear stiffness of the spring, is the nonlinear stiffness of the spring, is the vertical displacement of the saturated soil surface.

[0146] Specifically, the relative motion between the resonator mass and the vertical displacement of the saturated soil surface satisfies the following relationship:

[0147]

[0148] in, represents the relative motion between the resonator mass and the vertical displacement of the saturated soil surface, Expressed as the vertical displacement of the resonator, is the vertical displacement of the saturated soil surface, is the relative vertical displacement between the saturated soil and the resonator mass.

[0149] Furthermore, the relative vertical displacement between the saturated soil and the resonator mass satisfies the following relationship:

[0150]

[0151] in, is the relative vertical displacement between the saturated soil and the resonator mass, is the vertical displacement of the saturated soil surface, is the vertical displacement of the resonator.

[0152] In this embodiment, The displacement expression that satisfies the solid skeleton The expression of , let the first boundary condition:

[0153]

[0154] in, is the amplitude, , and is the amplitude corresponding to each potential function, , is the wave number, is an imaginary unit, is the wave number of the surface wave.

[0155] Furthermore, according to the first boundary condition, the vertical displacement expression of the saturated soil surface satisfies the following relationship:

[0156]

[0157] in, is the vertical displacement of the saturated soil surface, is the amplitude, is the base of natural logarithm, is an imaginary unit, is the circular frequency, For time, is the wave number of the surface wave, express direction.

[0158] In this embodiment, it is assumed that Adopt and In a similar form, the relative motion expression satisfies the following relationship:

[0159]

[0160] in, represents the relative motion between the resonator mass and the vertical displacement of the saturated soil surface, is the amplitude, is the base of natural logarithm, is an imaginary unit, is the circular frequency, For time, is the wave number of the surface wave, express direction.

[0161] Furthermore, the vertical displacement expression and relative motion expression of the saturated soil surface are substituted into the dynamic equilibrium equation to obtain the expanded expression of the dynamic equilibrium equation, which satisfies the following relationship:

[0162]

[0163] in, is the circular frequency, is the amplitude, is the base of natural logarithm, is an imaginary unit, For time, is the wave number of the surface wave, express direction, is the resonator mass, is the linear stiffness of the spring, is the nonlinear stiffness of the spring, is the amplitude.

[0164] S42. Based on the dynamic balance equation and Euler's formula, a boundary formula is obtained using the first harmonic balance method.

[0165] In this embodiment, the Euler relationship is obtained according to the Euler formula, which satisfies the following relationship:

[0166]

[0167] in, is the base of natural logarithm, is an imaginary unit, is the circular frequency, For time, is the wave number of the surface wave, express direction.

[0168] Furthermore, the Euler relationship is substituted into the expanded expression of the dynamic balance equation, and the first harmonic balance method is adopted to approximately ignore the high-frequency or low-frequency harmonics generated by nonlinearity, thereby obtaining a boundary formula that satisfies the following relationship:

[0169]

[0170]

[0171] in, is the resonator mass, is the effective frequency, is the circular frequency, is the amplitude, is the amplitude, is the resonant frequency, is the correlation coefficient of the strength of the nonlinear effect.

[0172] Specifically, the correlation coefficient between the resonance frequency and the nonlinear effect intensity satisfies the following relationship:

[0173]

[0174]

[0175] in, is the resonant frequency, is the linear stiffness of the spring, is the resonator mass, is the correlation coefficient of the intensity of nonlinear effect, is the nonlinear stiffness of the spring.

[0176] Furthermore, when When , the nonlinear resonator degenerates into a linear one.

[0177] S5. Obtain the boundary conditions of the saturated soil, and obtain the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil in combination with the boundary formula.

[0178] Among them, S5 specifically includes the following steps:

[0179] S51. Construct a homogeneous set of equations based on the boundary conditions of the saturated soil and the boundary formula.

[0180] In this embodiment, the dispersion relationship between the wave number of the surface wave and the circular frequency is determined by the boundary conditions; the boundary conditions of the saturated soil surface (z=0) satisfy the following relationship:

[0181]

[0182]

[0183]

[0184] in, is the physical quantity of shear stress between z and x directions inside the soil, express direction, For time, is the total stress, is the effective unit area of ​​the resonator, For each resonant unit, a vertical load is applied to the saturated soil surface. is the pore pressure.

[0185] Specifically, the vertical load applied by each resonant unit on the saturated soil surface satisfies the following relationship:

[0186]

[0187] in, For each resonant unit, a vertical load is applied to the saturated soil surface. is the linear stiffness of the spring, is the nonlinear stiffness of the spring, Represents the relative motion between the resonator mass and the vertical motion displacement of the saturated soil surface.

[0188] Furthermore, in order to obtain the dispersion relation between the wave number and the circular frequency of the surface wave under the nonlinear resonator, the first boundary condition, the boundary formula and the boundary condition are combined, a total of five boundary condition expressions are generated for the five amplitudes A1, A2, A3, Y, Bw, and a homogeneous equation system is generated to satisfy the following relationship:

[0189]

[0190] in, is an imaginary unit, is the wave number of the surface wave, , and is the wave number, , and is the amplitude corresponding to each potential function, for The wave vector of the wave, and is the Lame constant of the soil skeleton, is the compression modulus of saturated soil, is the coupling modulus of saturated soil, , is the ratio of the potential function to the amplitude, is the resonator mass, is the effective frequency, is the effective unit area of ​​the resonator, for The wave vector of the wave, for The wave vector of the wave, is the circular frequency, is the amplitude, is the amplitude.

[0191] S52. Based on the homogeneous equation group, obtain the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil.

[0192] In this embodiment, in order for the homogeneous equation system to have a non-zero solution, its coefficient determinant must be zero; satisfying the following relationship:

[0193]

[0194] in, is an imaginary unit, is the wave number of the surface wave, , and is the wave number, for The wave vector of the wave, and is the simplification coefficient, is the Lame constant of the soil skeleton, is the resonator mass, is the effective frequency, is the effective unit area of ​​the resonator, is the coupling modulus of saturated soil, is the compression modulus of saturated soil, , is the ratio of the potential function to the amplitude, for The wave vector of the wave, for The wave vector of the wave, is the circular frequency.

[0195] Furthermore, the determinant is expanded to obtain the nonlinear hypersurface Rayleigh wave dispersion curve equation of saturated soil, which satisfies the following relationship:

[0196]

[0197]

[0198]

[0199] in, is the circular frequency, is the effective frequency, for The wave vector of the wave, is the compression modulus of saturated soil, , is the ratio of the potential function to the amplitude, is the wave number of the surface wave, for The wave vector of the wave, and is the simplification coefficient, is the Lame constant of the soil skeleton, , and is the wave number, for The wave vector of the wave, is the resonator mass, is the effective area per unit of the resonator.

[0200] See also Figure 2 , the figure is a schematic diagram of the theoretical calculation model of nonlinear super surface in saturated soil. The calculation model of the present invention is composed of a uniform nonlinear resonator array placed on a semi-infinite isotropic saturated soil, and the saturated soil parameters are displayed at the same time.

[0201] See also Figure 3 , the figure is a local enlarged schematic diagram of the nonlinear resonator, which shows in detail the local physical analysis of the nonlinear resonator.

[0202] See also Figure 4 , the figure shows the hardening model of saturated soil The dispersion curve diagram shows in detail the dispersion curve and band gap width of the nonlinear resonator hardening model under different incident wave amplitude conditions.

[0203] See also Figure 5 , the figure shows the saturated soil softening model The dispersion curve diagram shows in detail the dispersion curve and band gap width of the nonlinear resonator softening model under different incident wave amplitude conditions.

[0204] See also Figure 6 In an optional embodiment, in order to efficiently execute the nonlinear metasurface Rayleigh wave dispersion relationship calculation method provided by the present invention, the present invention provides a nonlinear metasurface Rayleigh wave dispersion relationship calculation system, the system includes an input device, an output device, a processor and a memory, the hardware facilities are interconnected, wherein the memory is used to store a computer program, the computer program includes program instructions, and the processor is configured to call the program instructions to execute the technical steps of the specific implementation method of the present invention. The nonlinear metasurface Rayleigh wave dispersion relationship calculation system provided by the present invention has a complete structure, objective stability, and improves the overall applicability and practical application ability of the present invention.

[0205] In summary, the method and system provided by the present invention provide a nonlinear metasurface Rayleigh wave dispersion relationship calculation method, obtain the dispersion curve equation of the nonlinear metasurface Rayleigh wave, and provide new technical inspiration for the study of Rayleigh wave propagation characteristics. The method of the present invention is easy to understand, simple to calculate, with a small workload, and is convenient for engineering application, providing a theoretical basis and technical support for the further development of physical acoustic technology.

[0206] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein by equivalents. These modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention, and they should all be included in the scope of the claims and specification of the present invention.

Claims

1. A method for calculating the dispersion relation of Rayleigh waves on a nonlinear metasurface, characterized in that: The steps include: Decomposing the displacement field of the saturated soil to obtain a displacement vector and a relative displacement, and establishing a wave control equation of the saturated soil; Combining the displacement vector, the relative displacement and the wave control equation to obtain an expression for a surface wave solution; Constructing a potential function, obtaining a displacement expression and a relative displacement expression based on the potential function, and constructing a stress expression of the saturated soil; The constructing of the potential function, obtaining the displacement expression and the relative displacement expression based on the potential function, and constructing the stress expression of the saturated soil body, includes: Construct potential functions for harmonic surface waves; Combining the potential function with the displacement vector to obtain a displacement expression of the solid skeleton; Combining the potential function with the relative displacement to obtain a relative displacement expression of the fluid; Constructing total stress expression and pore pressure expression of the saturated soil; The potential function of constructing the harmonic surface wave comprises: in, is the scalar potential of the solid, , and is the amplitude corresponding to each potential function, is the base of natural logarithm, , and is the wave number, express direction, is an imaginary unit, is the circular frequency, For time, is the wave number of the surface wave, express direction, is the scalar potential of the fluid, , and is the ratio of the potential function to the amplitude, and is the vector potential The amount of Constructing a dynamic balance equation, and obtaining a boundary formula based on the dynamic balance equation and the Euler formula; The boundary conditions of the saturated soil are obtained, and the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil is obtained in combination with the boundary formula.

2. The method for calculating the dispersion relation of Rayleigh waves of nonlinear metasurface according to claim 1, characterized in that: The displacement field of the saturated soil is decomposed to obtain a displacement vector and a relative displacement, and a wave control equation of the saturated soil is established, including: Based on the Helmholtz theorem, the displacement field of the saturated soil is decomposed to obtain the displacement vector of the solid skeleton and the relative displacement from the fluid to the solid skeleton; A wave control equation of the saturated soil is constructed according to the displacement vector and the relative displacement.

3. The method for calculating the dispersion relation of Rayleigh waves of nonlinear metasurface according to claim 1, characterized in that: The expression for obtaining the surface wave solution by combining the displacement vector, the relative displacement and the wave control equation comprises: Combining the displacement vector, the relative displacement and the wave control equation, an uncoupled motion control equation of a longitudinal wave and a motion control equation of a transverse wave are obtained; A first characteristic equation is obtained according to the uncoupled motion control equation, and a second characteristic equation is obtained according to the motion control equation; The body wave velocity equation is obtained according to the first characteristic equation and the second characteristic equation, and then the expression of the surface wave solution is obtained.

4. The method for calculating the dispersion relation of Rayleigh waves of nonlinear metasurface according to claim 1, characterized in that: The constructing of the dynamic balance equation, based on the dynamic balance equation combined with the Euler formula to obtain the boundary formula, includes: A nonlinear resonator is coupled to the free surface of the saturated soil, and a dynamic equilibrium equation of the saturated soil-nonlinear spring-oscillator is established; Based on the dynamic balance equation combined with the Euler formula, the boundary formula is obtained using the first harmonic balance method.

5. The method for calculating the dispersion relation of Rayleigh waves of nonlinear metasurface according to claim 4, characterized in that: The nonlinear resonator is coupled to the free surface of the saturated soil to establish a dynamic equilibrium equation of saturated soil-nonlinear spring-oscillator, including: in, is the resonator mass, represents the relative motion between the resonator mass and the vertical displacement of the saturated soil surface, is the linear stiffness of the spring, is the nonlinear stiffness of the spring, is the vertical displacement of the saturated soil surface.

6. The method for calculating the dispersion relation of Rayleigh waves of nonlinear metasurface according to claim 1, characterized in that: The step of obtaining the boundary conditions of the saturated soil and obtaining the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil in combination with the boundary formula includes: According to the boundary conditions of the saturated soil, a homogeneous set of equations is constructed in combination with the boundary formula; Based on the homogeneous equation group, the nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil is obtained.

7. The method for calculating the dispersion relation of Rayleigh waves of nonlinear metasurface according to claim 6, characterized in that: The nonlinear hypersurface Rayleigh wave dispersion curve equation of the saturated soil is obtained based on the homogeneous equation group, including: in, is the circular frequency, is the effective frequency, for The wave vector of the wave, is the compression modulus of saturated soil, , is the ratio of the potential function to the amplitude, is the wave number of the surface wave, for The wave vector of the wave, and is the simplification coefficient, is the Lame constant of the soil skeleton, , , is the wave number, for The wave vector of the wave, is the resonator mass, is the effective area per unit of the resonator.

8. A nonlinear metasurface Rayleigh wave dispersion relation calculation system, characterized in that: The system includes an input device, an output device, a processor and a memory, wherein the input device, the output device, the processor and the memory are interconnected, wherein the memory is used to store a computer program, the computer program includes program instructions, and the processor is configured to call the program instructions to execute the nonlinear metasurface Rayleigh wave dispersion relationship calculation method as described in any one of claims 1 to 7.

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