A method for predicting the cyclic dynamic response of sand

Abstract

The present invention discloses a method for predicting the cyclic dynamic response of sand. First, a consolidation test is performed to obtain a curve showing the relationship between the critical porosity and confining pressure of the soil. A static triaxial apparatus is used to measure the stress-strain relationship of the sand under static load to obtain a static loading stress-strain curve. A dynamic triaxial apparatus is used to measure the stress-strain relationship of the sand under dynamic load to obtain a dynamic stress-strain curve. Then, a plastic hardening modulus field is constructed in stress space as an evaluation model. The three curves are fitted in sequence to obtain soil parameters in the model. Based on the evaluation model, the soil parameters are used to predict the cyclic dynamic response of the sand. The method provided by the present invention simultaneously evaluates the stress-strain characteristics of the sand under drained and undrained conditions. It can not only accurately describe the strength evolution law of the sand under drained conditions, but also accurately calculate the effective stress variation law of the sand under undrained conditions by accurately capturing the changes in pore pressure.

Description

Technical Field

[0001] The present invention relates to the technical field of rock and soil mechanics, and in particular to a method for predicting the cyclic dynamic response of sand. Background Art

[0002] The existing assessment methods for dynamic loading of sand mainly focus on the stress-strain response under undrained conditions, especially the vibration liquefaction phenomenon of sand under seismic loads. However, the existing assessment methods for the dynamic response of sand under drained conditions lack consideration of the stress-strain characteristics under undrained conditions. For long-term, low-frequency dynamic loads caused by wind, waves, and currents acting on marine engineering structures, the sand around the structure is basically in a completely drained state. At this time, the change in soil stiffness is affected by the evolution of the internal particle arrangement. For short-term, high-frequency cyclic loads, the sand is in an undrained state, and the stress-strain characteristics of the soil are affected by the pore pressure, which is different from the stiffness evolution in the drained state. The existing technology either focuses on the dynamic loading response of sand under undrained conditions or on the drained state, and lacks a comprehensive consideration of the two soil states. Summary of the Invention

[0003] In view of this, an embodiment of the present invention provides a method for predicting the cyclic dynamic response of sand, which constructs an evaluation model to simultaneously reflect the stress-strain characteristics of sand under drained and undrained conditions, thereby achieving better prediction accuracy for the cyclic dynamic response of sand.

[0004] In order to achieve the above object, the present invention provides the following technical solutions:

[0005] An embodiment of the present invention provides a method for predicting the cyclic dynamic response of sand, comprising:

[0006] The relationship curve between soil critical state void ratio and confining pressure is obtained through consolidation test;

[0007] The stress-strain relationship of sand under static load was measured by using a static triaxial apparatus to obtain the static loading stress-strain curve.

[0008] The dynamic triaxial apparatus is used to measure the stress-strain relationship of sand under dynamic load and obtain the dynamic stress-strain curve.

[0009] The plastic hardening modulus field is constructed in stress space as an evaluation model;

[0010] Fitting the soil critical state porosity and confining pressure curve, the static loading stress-strain curve, and the dynamic stress-strain curve in sequence to obtain soil parameters in the model;

[0011] Based on the evaluation model, the soil parameters are used to predict the cyclic dynamic response of sand.

[0012] Optionally, the construction of the plastic hardening modulus field in the stress space as an evaluation model includes: constructing an analytical surface equation reflecting the maximum stress level experienced by the soil, calculating the loading index, establishing a state-dependent shear expansion equation to calculate the soil shear expansion ratio, constructing a stress-strain increment relationship, calculating the plastic modulus and calculating the pore pressure increment.

[0013] Optionally, construct the analytical surface equation Y using the following formula:

[0014]

[0015] Where: are the coordinates of the conjugate points of the stress point (p, s) in the plastic modulus field, the symbol “:” represents the double dot product between tensors, z is the parameter that determines the strain direction of the soil, w is the control parameter of the soil hardening rate, M is the critical state stress ratio, and its values ​​in compression and tension are M c 、M e , the calculation formula is:

[0016]

[0017] Where: φ is the critical friction angle.

[0018] Optionally, the external normal direction y of the analytical surface at the conjugate stress point is the plastic loading direction, which is used to calculate the loading index L, and its expression is as follows:

[0019]

[0020]

[0021] Where: K p is the plastic modulus corresponding to the current stress state. The value of L represents the magnitude of the upcoming plastic deformation. The positive and negative signs of L define the loading and unloading criteria of the soil. Its mathematical expression is as follows:

[0022]

[0023] Optionally, establish a state-dependent dilatancy equation, including:

[0024] The dilatancy equation is used to calculate the dilatancy ratio d of the soil:

[0025]

[0026] Where: d0 is the material constant that controls the volume strain and is a positive number; are the plastic volume strain increment and the plastic deviatoric strain increment n, respectively. d is the model parameter that reflects the degree of dependence of volume strain on state parameters; ρ is the distance from the starting point of loading to the stress point, is the distance from the stress conjugate point to the loading starting point; ψ is the current porosity ratio e and the critical porosity ratio e c The difference state parameter, that is

[0027] ψ=ee c (7)

[0028] in

[0029]

[0030] Where: p at represents standard atmospheric pressure, which serves as a reference value for normalizing the average normal stress p; e Γ is the limit porosity, λ is the parameter that controls the critical porosity, and ξ is the parameter that reflects the influence of stress on the critical porosity.

[0031] Optionally, construct a stress-strain increment relationship, including:

[0032] The elastic constitutive relation is described by the hypoelastic equation, where the shear modulus G is calculated using the following empirical formula:

[0033]

[0034] Where: G0 is the material constant, and the bulk modulus K is obtained by converting Poisson's ratio ν and shear modulus G. The relationship is as follows:

[0035]

[0036] The relationship between elastic strain increment and stress increment is expressed as:

[0037]

[0038] The dilatancy ratio is taken as the vector element of the plastic deformation direction m, that is, m = [d, t] T , where t is a scalar value reflecting the compression or tension state of the soil. When the soil is in compression, t = +1; when the soil is in tension, t = -1. The plastic strain increment of the soil is calculated by the following formula:

[0039]

[0040] Where: <> is the Macauley symbol, when L>0 <l>=L, L≤0 <l>=0,

[0041] The stress-strain increment relationship of soil is expressed as:

[0042]

[0043] The plastic modulus is expressed as:

[0044]

[0045] Where: s in is the deviatoric stress at the beginning of each loading event; C(ζ) is the hardening metric function, which reflects the influence of the evolution of soil particle arrangement on the stiffness change during the stress history; n p It is a model parameter that controls the degree of dependence of the plastic modulus on the state parameter.

[0046] Optionally, the hardening metric function is used to evaluate the impact of sand volume change and shear band accumulation on soil strength under drained and undrained conditions, and its expression is as follows:

[0047]

[0048] in

[0049]

[0050]

[0051] Where: v To measure the cumulative plastic volume strain length of particle arrangement failure, only the negative plastic volume strain is accumulated, ζ v is always negative, and ζ v The accumulation of will reduce the value of C(ζ); q is the length of the cumulative plastic shear strain measured by particle arrangement strengthening, ζ q Always positive, regardless of loading or unloading, ζ q Keep increasing, q The accumulation of will increase C(ζ); ω is the material constant; (1-e) is the density control term, reflecting the influence of porosity on soil stiffness; γ1 is the parameter that controls the size of the plastic modulus, γ2 is the parameter that reflects the influence of cumulative shear strain, and χ is the parameter that reflects the influence of cumulative volume strain. It represents the difference between the current stress state and the historical maximum stress state, reflecting the influence of the maximum stress history on the evolution of soil strength.

[0052] Optional, pore pressure increment du w The following formula is used for calculation:

[0053]

[0054] Optionally, the critical porosity ratio and confining pressure relationship curve is fitted using formula (8) to calibrate the critical state parameter of the soil (e Γ ,λ,ξ).

[0055] Optionally, the static loading stress-strain curve is fitted using the stress-strain increment relationship expressed by formula (13) to obtain the soil static strength parameters (G0, ν, M, d0, n d 、n p , ω, χ), the process includes:

[0056] Assume that the parameters d0, n d 、n p , ω, χ initial values ​​are calculated, and the shear stress peak values ​​obtained by comparing the test and calculation are adjusted. p Make the value of equal, and then adjust d0 to make it equal by comparing the peak value of volume strain obtained by test and calculation, and then adjust n d The values ​​of ω and χ are adjusted to make the calculated stress-strain curve consistent with the test.

[0057] Optionally, the dynamic loading stress-strain curve is fitted using formula (13) to obtain the cyclic parameters (γ1, γ2, z), and the process includes:

[0058] Assuming initial values ​​of parameters γ1, γ2, and z, by comparing the stress-strain curves of the volume expansion stage obtained from the test and the calculation, γ1 is adjusted to make the trends consistent, then the value of γ2 is adjusted to make the calculated cyclic loading strain accumulation rate consistent with the test, and finally the value of z is adjusted to make the shape of the hysteresis loop of the cyclic loading curve consistent with the test.

[0059] The technical solution of the present invention has the following advantages:

[0060] The present invention provides a method for predicting the cyclic dynamic response of sand. First, a consolidation test is performed to obtain a curve showing the relationship between the critical porosity and confining pressure of the soil. A static triaxial apparatus is used to measure the stress-strain relationship of the sand under static load to obtain a static loading stress-strain curve. A dynamic triaxial apparatus is used to measure the stress-strain relationship of the sand under dynamic load to obtain a dynamic stress-strain curve. Then, a plastic hardening modulus field is constructed in stress space as an evaluation model. The three curves are fitted in sequence to obtain soil parameters in the model. Based on the evaluation model, the soil parameters are used to predict the cyclic dynamic response of the sand. The method provided by the present invention simultaneously evaluates the stress-strain characteristics of the sand under drained and undrained conditions. It can not only accurately describe the strength evolution law of the sand under drained conditions, but also accurately calculate the effective stress variation law of the sand under undrained conditions by accurately capturing the changes in pore pressure. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the specific embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0062] Figure 1 Flowchart of a method for predicting cyclic dynamic response of sand in an embodiment of the present invention;

[0063] Figure 2 is a graph showing the relationship between the porosity ratio and the confining pressure of sand according to an embodiment of the present invention;

[0064] Figure 3 (a) and Figure 3 (b) is a comparison chart of the prediction of the evaluation model constructed using the embodiment of the present invention and the actual test in the static loading test of sand;

[0065] Figure 4 (a) and Figure 4 (b) is a comparison diagram of the prediction of the evaluation model constructed using an embodiment of the present invention and the actual test in the dynamic loading test of sand. DETAILED DESCRIPTION

[0066] To make the purpose, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative efforts shall fall within the scope of protection of the present invention.

[0067] This embodiment provides a method for predicting the cyclic dynamic response of sand. Figure 1 As shown in FIG, the method for predicting the cyclic dynamic response of sand includes the following steps:

[0068] Step S1: Obtain the relationship curve between the critical state porosity and confining pressure of the soil through consolidation test.

[0069] Specifically, since soil has the characteristic of shrinking in volume under pressure, the compressibility of the soil is mainly caused by the reduction of pore volume. The consolidation test is a test method for measuring the drainage, degassing, and bubble compression properties of the soil under the action of external force. The process of conducting a consolidation test in the embodiment of the present invention to obtain the curve is a relatively conventional test in this field and will not be repeated here.

[0070] Step S2: using a static triaxial apparatus to measure the stress-strain relationship of sand under static load, and obtaining a static loading stress-strain curve.

[0071] In the embodiment of the present invention, the sand density and the test confining pressure are changed to obtain stress-strain curves under different stress levels and soil states as static loading stress-strain curves.

[0072] Step S3: Using a dynamic triaxial apparatus to measure the stress-strain relationship of the sand under dynamic load, and obtaining a dynamic stress-strain curve.

[0073] Specifically, the dynamic characteristics of soil mainly refer to the deformation characteristics and strength characteristics of soil. The deformation characteristics are the dynamic stress-strain relationship. The dynamic characteristics of soil are observed indoors using a dynamic triaxial instrument. This is a more conventional method and will not be described in detail here.

[0074] Step S4: Constructing the plastic hardening modulus field in the stress space as an evaluation model.

[0075] Specifically, the plastic deformation of the soil is calculated by the distance between the conjugate points of the stress points. The analytical equation Y of the evaluation model is in the following form:

[0076]

[0077] Where: are the coordinates of the conjugate points of the stress point (p, s) in the plastic modulus field, the symbol ":" represents the double dot product between tensors, z is the parameter that determines the strain direction of the soil, and w is the control parameter of the soil hardening rate; M is the critical stress ratio, and its values ​​in compression and tension are M c 、M e , which can be calculated by the critical friction angle φ:

[0078]

[0079] The external normal direction y of the analytical surface at the conjugate stress point is the plastic loading direction, which can be used to calculate the loading index L, and its expression is as follows:

[0080]

[0081] in

[0082]

[0083] Where: K p are the plastic modulus corresponding to the current stress state. The value of L represents the magnitude of the upcoming plastic deformation. At the same time, the positive and negative signs of L define the loading and unloading criteria of the soil. Its mathematical expression is as follows:

[0084]

[0085] To better predict the dilatancy characteristics of sand under drained or undrained conditions, the embodiment of the present invention uses the dilatancy equation to calculate the dilatancy ratio d of the soil:

[0086]

[0087] Where: d0 is the material constant that controls the volume strain, which is usually a positive number; ψ is the state parameter; n d is the model parameter that reflects the degree of dependence of volume strain on state parameters. ψ is the current porosity e and the critical state porosity e c The difference, that is

[0088] ψ=ee c (7)

[0089] e c The following calculation formula is used:

[0090]

[0091] Where: p at represents standard atmospheric pressure, which serves as a reference value for normalizing the average normal stress p; e Γ is the limit porosity, λ is the parameter that controls the critical porosity, and ξ is the parameter that reflects the degree of influence of stress on the critical porosity. The introduction of ψ in formula (6) is equivalent to incorporating the influence of soil density into the calculation of shear dilatancy; It takes into account the stage of cyclic loading and stress history; Represents the current stress state, taking into account the magnitude of deviatoric stress and normal stress.

[0092] In order to reflect the influence of density and average normal stress on the elastic deformation of sand, the present invention adopts the subelastic equation to describe the elastic constitutive relationship, wherein the shear modulus G is calculated by the following empirical formula:

[0093]

[0094] Where: G0 is the material constant, and the bulk modulus K can be converted from Poisson's ratio ν and shear modulus G, as shown in the following relationship:

[0095]

[0096] The relationship between elastic strain increment and stress increment is expressed as

[0097]

[0098] In the embodiment of the present invention, the dilatancy ratio is used as the vector element of the plastic deformation direction m, that is, m = [d, t] T , where t is a scalar value reflecting the compression or tension state of the soil. When the soil is in compression, t = +1, and when the soil is in tension, t = -1. The plastic strain increment of the soil can be calculated by the following formula:

[0099]

[0100] Where: <> is the Macauley symbol, that is, when L>0 <l>=L, L≤0 <l>= 0. The stress-strain increment relationship of the soil is expressed as:

[0101]

[0102] The plastic modulus is expressed as:

[0103]

[0104] Where: s in is the deviatoric stress at the beginning of each loading event; C(ζ) is the hardening metric function, which reflects the influence of the evolution of soil particle arrangement on the stiffness change during the stress history; n p is a model parameter that controls the degree of dependence of the plastic modulus on the state parameter, ρ is the distance from the loading starting point to the stress point, is the distance from the stress conjugate point to the loading starting point.

[0105] The hardening metric function is used to evaluate the effects of sand volume change and shear band accumulation on soil strength under drained and undrained conditions. Its expression is as follows:

[0106]

[0107] in:

[0108]

[0109]

[0110] Where: v To measure the cumulative plastic volume strain length of particle arrangement failure, only the negative plastic volume strain is accumulated. It can be seen from the above formula that ζ v is always negative, and ζ v The accumulation of will reduce the value of C(ζ); q is the length of the cumulative plastic shear strain measured by particle arrangement strengthening, ζ q Always positive, regardless of loading or unloading, ζ q Keep increasing, q The accumulation of will increase C(ζ); ω is the material constant; (1-e) is the density control term, reflecting the influence of porosity on soil stiffness; γ1 is the parameter that controls the size of the plastic modulus, γ2 is the parameter that reflects the influence of cumulative shear strain, and χ is the parameter that reflects the influence of cumulative volume strain. Represents the difference between the current stress state and the historical maximum stress state, reflecting the influence of the maximum stress history on the evolution of soil strength. In order to improve the calculation accuracy of the evaluation model in calculating the drainage and undrained sand, formula (15) adopts The position of the stress point within the analytical surface can be reflected in the calculation of the plastic modulus, that is, the gap between the current stress state and the ultimate stress state represented by the analytical surface is considered.

[0111] When the soil is in a drained state, the effective stress is equal to the total stress, and there is no need to calculate the pore pressure. The volume change of the soil can be directly calculated according to formulas (6) and (13). When the soil is in an undrained state, the change in pore pressure will directly affect the effective stress of the soil. Therefore, the calculation of pore pressure will determine the calculation accuracy of the evaluation model under undrained conditions. In order to comprehensively consider the effects of shear stress and elastic strain of sand on pore pressure accumulation, the increment of pore pressure in this evaluation model is du w Calculate using formula (18)

[0112]

[0113] Combined with formulas (6), (13), and (18), the evaluation model provided by the embodiment of the present invention can accurately track the change pattern of soil volume and effective stress under undrained cyclic loading conditions, and reflect the influencing factors such as stress level, compaction state, loading and unloading stage, and stress history in the calculation process of pore pressure. In formula (6), The term ensures that d is always positive in the calculation unloading stage, that is, the soil is in shear contraction state, and the effective stress and volume decrease with unloading; when the soil is in loading state, only when the stress is large, Greater than When d is negative, the soil is in a dilatant state. Formula (18) combines the dilatant ratio d with the plastic modulus K p The soil elastic bulk modulus K can comprehensively reflect the physical laws that when the soil is loaded with shear expansion, the pore pressure increases and the effective stress decreases, or when the soil is loaded with shear contraction, the pore pressure decreases and the effective stress increases.

[0114] Step S5: fitting the soil critical state porosity and confining pressure relationship curve, the static loading stress-strain curve, and the dynamic stress-strain curve in sequence to obtain soil parameters in the model.

[0115] In one embodiment, the embodiment of the present invention calibrates the critical state parameter (e) of the soil by fitting the critical void ratio and confining pressure relationship curve using formula (8). Γ , λ, ξ). Specifically, by fitting the critical porosity data under different average normal stresses through formula (8), we can obtain Figure 2 The relationship curve between the porosity e and confining pressure of sand (ep curve) is shown, and the parameter e is obtained. Γ =0.934, λ=0.019, ξ=0.7. It should be noted that the data values ​​obtained in the present invention are only applicable to the sand used in the test. In practice, the critical state parameters of the soil body vary according to the type of sand.

[0116] In one embodiment, the stress-strain increment relationship expressed by formula (13) is used to fit the static loading stress-strain curve to obtain the static strength parameters of the soil. The specific method is as follows: First, assume that the parameters d0, n d 、n p , ω, χ initial values ​​are calculated, and the shear stress peak values ​​obtained by comparing the test and calculation are adjusted. p Make the value of equal, and then adjust d0 to make it equal by comparing the peak value of volume strain obtained by test and calculation, and then adjust n d The values ​​of ω and χ are adjusted to make the calculated stress-strain curve consistent with the test.

[0117] In determining the 11 parameters (e Γ ,λ,ξ,G0,ν,M,d0,n d 、n p After ,ω,χ), it is necessary to calibrate the three cycle parameters (γ1, γ2, z) related to the cycle dynamics. The embodiment of the present invention uses formula (13) to fit the dynamic stress-strain curve to obtain the cycle parameters. The specific method is to first assume the initial values ​​of the parameters γ1, γ2, and z, and then adjust γ1 to match the trend by comparing the stress-strain curves of the volume expansion stage obtained by the test with the calculated ones. Then, adjust the value of γ2 to make the calculated cyclic loading strain accumulation rate consistent with the test. Finally, adjust the value of z to make the shape of the hysteresis loop of the cyclic loading curve consistent with the test.

[0118] Step S6: Based on the evaluation model, the soil parameters are used to predict the cyclic dynamic response of sand.

[0119] In order to verify the evaluation model constructed in the embodiment of the present invention and its simulation ability of the stress-strain response of sand under cyclic loading, the embodiment of the present invention adopts triaxial test of sand for investigation. Before simulating the cyclic loading test, three groups of monotonic loading tests of sand with different densities (e0=0.59, e0=0.69, e0=0.80) are adopted. According to the basic parameters of the calibration model in Table 1, the test is carried out with a confining pressure of 200 kPa.

[0120]

[0121] The evaluation model constructed by the embodiment of the present invention is used to calculate the Figure 3 (a) Shear stress and axial strain q-ε a The curve and Figure 3 (b) Volume strain and axial strain ε v -ε a The curves are in good agreement with the test results, indicating that the evaluation model constructed in the embodiment of the invention can reasonably reflect the monotonic loading stress-strain characteristics of sand under different densities and stress levels.

[0122] The embodiment of the present invention predicts the cyclic loading test and uses the calibrated soil parameters to predict the cyclic dynamic response of sand soil, and obtains the prediction results as follows: Figure 4 (a) and Figure 4 As shown in (b), it can be seen that the evaluation model constructed in the embodiment of the present invention has good prediction accuracy for the cyclic dynamic response of sand.

[0123] The method for predicting the cyclic dynamic response of sand provided by the embodiment of the present invention can simultaneously evaluate the stress-strain characteristics of sand under drained conditions and undrained conditions. It can not only accurately describe the strength evolution law of sand under drained conditions, but also accurately calculate the effective stress change law of sand under undrained conditions by accurately capturing the changes in pore pressure.

[0124] Although the embodiments of the present invention have been described with reference to the accompanying drawings, those skilled in the art may make various modifications and variations without departing from the spirit and scope of the present invention. Such modifications and variations are all within the scope defined by the appended claims.< / l> < / l> < / l> < / l>

Claims

1. A method for predicting the cyclic dynamic response of sand, characterized in that: include: The relationship curve between soil critical state void ratio and confining pressure is obtained through consolidation test; The stress-strain relationship of sand under static load was measured by using a static triaxial apparatus to obtain the static loading stress-strain curve. The dynamic triaxial apparatus is used to measure the stress-strain relationship of sand under dynamic load and obtain the dynamic stress-strain curve. The plastic hardening modulus field is constructed in stress space as an evaluation model, including: constructing an analytical surface equation reflecting the maximum stress level experienced by the soil, calculating the loading index, establishing a state-dependent dilatancy equation to calculate the soil dilatancy ratio, constructing a stress-strain increment relationship, calculating the plastic modulus, and calculating the pore pressure increment. The construction of the stress-strain increment relationship includes: The analytical surface equation Y is constructed by the following formula: Where: are the coordinates of the conjugate points of the stress point (p, s) in the plastic modulus field, the symbol ":" represents the double dot product between tensors, z is the parameter that determines the strain direction of the soil, w is the control parameter of the soil hardening rate, and M is the critical state stress ratio, which takes the values ​​of M in compression and tension respectively. c 、M e , the calculation formula is: Where: φ is the critical friction angle; The elastic constitutive relation is described by the hypoelastic equation, where the shear modulus G is calculated using the following empirical formula: Where: G0 is the material constant, p represents the average normal stress, p at represents standard atmospheric pressure, and the bulk modulus K is obtained by converting Poisson's ratio ν and shear modulus G, as shown in the following relationship: The relationship between elastic strain increment and stress increment is expressed as: The dilatancy ratio is taken as the vector element of the plastic deformation direction m, that is, m = [d, t] T , where t is a scalar value reflecting the compression or tension state of the soil. When the soil is in compression, t = +1; when the soil is in tension, t = -1. The plastic strain increment of the soil is calculated by the following formula: Where: < > is the Macauley symbol, when L>0 <l>=L, L≤0 <l>=0, d represents the dilatancy ratio, are the plastic volume strain increment and the plastic deviator strain increment respectively;< / l> < / l> The stress-strain increment relationship of soil is expressed as: Among them, the plastic modulus K p Expressed as: Where: s in is the deviatoric stress at the beginning of each loading event; C(ζ) is the hardening metric function, which reflects the influence of the evolution of soil particle arrangement on the stiffness change during the stress history; n p is a model parameter that controls the degree of dependence of the plastic modulus on the state parameter, ρ is the distance from the starting point of loading to the stress point, is the distance from the stress conjugate point to the loading starting point, ψ=ee c Indicates the current porosity e and the critical porosity e c The hardening metric function is used to evaluate the influence of sand volume change and shear band accumulation on soil strength under drained and undrained conditions. Its expression is as follows: in Where: v To measure the cumulative plastic volume strain length of particle arrangement failure, only the negative plastic volume strain is accumulated, ζ v is always negative, and ζ v The accumulation of will reduce the value of C(ζ); q is the length of the cumulative plastic shear strain measured by particle arrangement strengthening, ζ q Always positive, regardless of loading or unloading, ζ q Keep increasing, q The accumulation of will increase C(ζ); ω is the material constant; (1-e) is the density control term, reflecting the influence of porosity on soil stiffness; γ1 is the parameter that controls the size of the plastic modulus, γ2 is the parameter that reflects the influence of cumulative shear strain, and χ is the parameter that reflects the influence of cumulative volume strain. It represents the difference between the current stress state and the historical maximum stress state, reflecting the influence of the maximum stress history on the evolution of soil strength; The increment of pore pressure du w The following formula is used for calculation: Fitting the soil critical state porosity and confining pressure curve, the static loading stress-strain curve, and the dynamic stress-strain curve in sequence to obtain soil parameters in the model; Based on the evaluation model, the soil parameters are used to predict the cyclic dynamic response of sand.

2. The method for predicting the cyclic dynamic response of sand according to claim 1, characterized in that: The external normal direction y of the analytical surface at the conjugate stress point is the plastic loading direction, which is used to calculate the loading index L. Its expression is as follows: Where: K p is the plastic modulus corresponding to the current stress state. The value of L represents the magnitude of the upcoming plastic deformation. The positive and negative signs of L define the loading and unloading criteria of the soil. Its mathematical expression is as follows:

3. The method for predicting the cyclic dynamic response of sand according to claim 2, characterized in that: Establish the state-dependent dilatancy equation, including: The dilatancy equation is used to calculate the dilatancy ratio d of the soil: Where: d0 is the material constant that controls the volume strain and is a positive number; are the plastic volume strain increment and plastic deviatoric strain increment, n d is the model parameter that reflects the degree of dependence of volume strain on state parameters; ρ is the distance from the starting point of loading to the stress point, is the distance from the stress conjugate point to the loading starting point; ψ is the current porosity ratio e and the critical porosity ratio e c The difference state parameter, that is ψ=ee c (7) in Where: p at represents standard atmospheric pressure, which serves as a reference value for normalizing the average normal stress p; e Γ is the limit porosity, λ is the parameter that controls the critical porosity, and ξ is the parameter that reflects the influence of stress on the critical porosity.

4. The method for predicting the cyclic dynamic response of sand according to claim 3, characterized in that: The critical state parameter e of the soil is calibrated by fitting the relationship curve between the critical porosity and confining pressure using formula (8). Γ ,λ,ξ.

5. The method for predicting the cyclic dynamic response of sand according to claim 1, characterized in that: The static loading stress-strain curve is fitted using the stress-strain increment relationship expressed by formula (13) to obtain the soil static strength parameters G0, ν, M, d0, and n d 、n p , ω, χ, the process includes: Assume that the parameters d0, n d 、n p , ω, χ initial values ​​are calculated, and the shear stress peak values ​​obtained by comparing the test and calculation are adjusted. p Make the value of equal, and then adjust d0 to make it equal by comparing the peak value of volume strain obtained by test and calculation, and then adjust n d The values ​​of ω and χ are adjusted to make the calculated stress-strain curve consistent with the test.

6. The method for predicting the cyclic dynamic response of sand according to claim 1, characterized in that: The dynamic loading stress-strain curve is fitted using formula (13) to obtain the cyclic parameters γ1, γ2, and z. The process includes: Assuming initial values ​​of parameters γ1, γ2, and z, by comparing the stress-strain curves of the volume expansion stage obtained from the test and the calculation, γ1 is adjusted to make the trends consistent, then the value of γ2 is adjusted to make the calculated cyclic loading strain accumulation rate consistent with the test, and finally the value of z is adjusted to make the shape of the hysteresis loop of the cyclic loading curve consistent with the test.

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