Rock stress wave attenuation characteristic analysis method based on split hopkinson pressure bar device

Abstract

The application discloses a rock stress wave attenuation characteristic analysis method based on a split Hopkinson pressure bar device. The method comprises the following steps: step one, test device preparation; step two, test design and execution; step three, data acquisition; step four, data analysis; step five, attenuation characteristic analysis; step six, constitutive parameter calibration; and step seven, method verification. According to the SHPB test data analysis of rock bar samples with different lengths and the same length, the attenuation characteristics of stress waves in rocks are obtained. Based on the Maxwell model and the standard linear solid (SLS) model, a quantitative mapping relationship between the time domain and the frequency domain of the rock stress wave attenuation characteristics is established, a time-frequency determination method of the Maxwell model constitutive parameters based on the SHPB test data is proposed, and a single-variable step-by-step solution method of the standard linear solid (SLS) model constitutive parameters is proposed.

Description

Technical Field

[0001] This invention belongs to the field of rock mechanics technology and relates to a method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device. Background Technology

[0002] The mechanism of stress wave propagation attenuation in rock masses is a core scientific problem in rock mechanics, and it has important engineering value for deep resource extraction, underground space development, and geological disaster prevention and control. Natural rock masses are composed of various minerals and have formed multi-scale discontinuous structural surfaces, such as microscopic defects and macroscopic joints, through complex geological tectonic processes. These structural surfaces significantly affect the propagation characteristics of stress waves. Viscoelastic constitutive models provide a quantitative characterization method for the propagation attenuation of stress waves in rock masses. To achieve accurate identification of model parameters, researchers have developed various dynamic loading test methods, which can be divided into pendulum tests, light gas gun tests, lead-breaking tests, and split Hopkinson bar tests, depending on the loading method. Among them, the pendulum impact test based on rock rod samples with a large aspect ratio has become the main means of analyzing the traditional rock mass wave propagation attenuation characteristics due to its clear principle and simple and controllable stress wave propagation path.

[0003] Traditional pendulum testing is simple in principle and can analyze the propagation process of one-dimensional stress waves in rock rods relatively well. However, it has certain limitations: First, insufficient loading rate: due to the impact speed of the pendulum, the frequency of the generated stress wave is low, making it difficult to consider the high-frequency attenuation characteristics of the stress wave and accurately analyze the dynamic mechanical response of the rock under high strain rate. Second, sample size limitation: this method requires long rod samples with an aspect ratio greater than 10 and a length greater than 1m. The sample preparation process is difficult, and the mechanical properties of the sample often have large non-uniformity, which restricts the applicability and accuracy of the method. Summary of the Invention

[0004] This invention provides a method for analyzing the attenuation characteristics of rock stress waves based on the Hopkinson pressure bar device, in order to solve the problems mentioned in the background art.

[0005] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device includes:

[0007] Step 1: Preparation of the experimental setup;

[0008] Step two: Experiment design and execution;

[0009] Step 3, Data Collection;

[0010] Step four, data processing;

[0011] Step 5: Attenuation characteristic analysis;

[0012] Step 6: Constitutive parameter calibration;

[0013] Step 7: Method Verification;

[0014] In step one, the split Hopkinson pressure bar (SHPB) device is used for testing. The SHPB device consists of an impact bar, an incident bar, a transmission bar, and a velocimeter. Strain gauges are attached to the incident bar and the transmission bar to record the strain signals of the incident wave, reflected wave, and transmitted wave.

[0015] In step two, rock samples of different lengths are designed to meet the needs of stress wave attenuation characteristic analysis, so as to observe the attenuation of stress waves in samples of different lengths. By adjusting the speed of the impact rod, incident stress waves of different intensities are generated to ensure the diversity of experimental data.

[0016] In step three, a velocimeter is used to measure the speed of the impact rod to ensure the accuracy of the test conditions. The strain gauge data on the incident rod and the transmission rod are recorded to obtain the waveform information of the incident wave, reflected wave and transmitted wave.

[0017] In step four, the recorded waveform data is processed to extract the strain amplitudes of the incident wave, reflected wave, and transmitted wave. Through normalization, the influence of input waveform differences on the results is reduced, thereby improving the accuracy of data analysis.

[0018] In step five, the time-domain attenuation coefficient and frequency-domain attenuation coefficient are determined based on the SHPB test. For rock samples of different lengths, the relationship between the transmitted wave amplitude and the sample length is analyzed to obtain the frequency-domain and time-domain attenuation coefficients. For rock samples of the same length, neglecting the superposition of multiple reflected waves within the rock mass, the Fourier amplitude attenuation from the incident wave signal to the transmitted wave signal is analyzed to calculate the lower limit of the frequency-domain attenuation coefficient. The transmitted wave signal is superimposed with the reflected wave from within the rock, and the Fourier amplitude attenuation from the incident wave signal to the superimposed signal is analyzed to calculate the upper limit of the frequency-domain attenuation coefficient.

[0019] In step six, the parameters of the Maxwell model and the Standard Linear Solid (SLS) model are calibrated using the time-domain and frequency-domain attenuation coefficients obtained in step five. This quantifies the attenuation characteristics of stress waves in rocks, providing a more comprehensive description of the attenuation properties and facilitating numerical simulation analysis and engineering applications. Based on the time-domain and frequency-domain attenuation coefficients of rock samples of different lengths obtained in step five, the constitutive parameters of the Maxwell model are calibrated, and a mapping relationship between the time-domain and frequency-domain attenuation coefficients is proposed. Based on the frequency-domain attenuation coefficients of rock samples of the same length obtained in step five, the constitutive parameters of the Standard Linear Solid (SLS) model are calibrated, and a univariate stepwise solution method for the constitutive parameters of the SLS model is proposed, simplifying the solution process.

[0020] In step seven above, the effectiveness of the analysis method is comprehensively verified through various means such as theoretical analysis, experimental verification, and numerical simulation.

[0021] As a further technical solution of the present invention, in step three, the split Hopkinson bar test apparatus (SHPB) can be used to achieve high strain rate loading of short rock bars. Its basic principle is that an impact bar (i.e., a bullet) strikes an incident bar at a certain velocity, generating an incident stress wave. This stress wave is transmitted to the specimen through the elastic incident bar, simultaneously generating reflected and transmitted waves. The velocity of the bullet is measured by a velocimeter, and the strain signals of the incident, reflected, and transmitted waves are obtained based on strain gauges attached to the incident and transmitted bars. The average strain rate of the specimen can be obtained based on one-dimensional stress wave theory. Mean strain ε and mean stress σ:

[0022]

[0023] In the formula, I, R, and T represent the incident, reflected, and transmitted waves recorded in the test, respectively; C is the elastic longitudinal wave velocity; l0 is the original length of the sample; E is the elastic modulus of the pressure bar; A is the cross-sectional area of ​​the pressure bar; and A0 is the cross-sectional area of ​​the sample.

[0024] In step four, it is usually difficult to ensure that the incident waves are completely identical in actual operation. However, the incident wave waveforms can be made as close as possible by adjusting the bullet velocity. Then, the transmitted wave is normalized by dividing the peak value of the incident wave by equation (4) to reduce the impact of the input waveform difference on the result.

[0025]

[0026] Where ε T For the strain record of the transmission rod, ε I This is a record of the strain of the incident rod.

[0027] As a further technical solution of the present invention, in step five, the method for analyzing the time-domain attenuation characteristics of rock samples of different lengths is to measure the normalized transmission peak value ε′ of rock samples of different lengths. Ti Then, a linear fit is performed in the logarithmic domain to calculate the time-domain attenuation coefficient β. The linear fit expression for calculating the time-domain attenuation coefficient based on rock samples of different lengths is shown below:

[0028] ln(ε′ Ti )=-βl i +ln(ε0) (5)

[0029] In the formula, ε′ Ti l represents the peak value of the normalized transmitted wave in the time domain. i denoted as the length of the rock sample.

[0030] As a further technical solution of the present invention, in step five, for rock samples of different lengths, longer samples will experience more significant attenuation compared to shorter samples. Assuming the incident wave remains constant, the formula for calculating the frequency domain attenuation coefficient based on rock samples of different lengths is as follows:

[0031]

[0032] Where α(ω) is the frequency domain attenuation coefficient, and ω is the angular frequency. and These are the Fourier amplitudes of the strain of the specimen with length l2 and the specimen with length l1, respectively (l2>l1).

[0033] In step five, the propagation of stress waves in the Hopkinson bar can be approximated as the propagation of a one-dimensional plane stress wave. Unlike a one-dimensional rock bar, when the stress wave crosses an interface with different wave impedances, reflection and transmission phenomena occur. The formulas for calculating the reflection coefficient F and the transmission coefficient T are as follows:

[0034]

[0035] In the formula, n is the dimensionless wave impedance ratio, ρ is the density, C is the elastic wave velocity, A is the cross-sectional area, and 1 and 2 below represent medium 1 and 2.

[0036] As a further technical solution of the present invention, in step five, to simplify the complexity of the experiment, an attenuation characteristic analysis method based on a single-length rock rod is proposed. In the Hopkinson bar test, for a single-length rock rod, the propagation of the incident wave to the transmitted wave undergoes three main processes: interface transmission, attenuation of the rock medium, and superposition of reflected waves from the interface. To simplify the analysis, the superposition of reflected waves at interface 1 and interface 2 is ignored first. Let the transmission coefficients at interface 1 and interface 2 be T1 and T2, respectively. Then, the lower limit estimate of the frequency domain attenuation coefficient based on a rock sample of the same length can be calculated by the following formula:

[0037]

[0038] Where α(ω) is the frequency domain attenuation coefficient. and Let F1 and F2 be the Fourier amplitudes of the transmitted wave and the incident wave, respectively, and l be the sample length. Further considering the effect of multiple reflected waves superimposed on the transmitted wave, let the reflection coefficients at interface 1 and interface 2 be F1 and F2, respectively. Then, the upper limit estimate of the frequency domain attenuation coefficient based on a rock sample of the same length can be calculated using the following formula:

[0039]

[0040] E=F2F1exp[-α(ω)×2l] (10)

[0041] Where α(ω) is the frequency domain attenuation coefficient. E reflects the superposition effect of the reflected waves and its attenuation, l is the sample length, and n represents the number of reflections at interfaces 1 and 2. Compared with equation (8), equation (9) has an additional term, which is a series term generated by the superposition of multiple reflected waves. In practical applications, since the number of reflections is difficult to determine accurately, the specific number of terms in the series is also difficult to give accurately. However, for cases where the stress equilibrium time is short, it can be assumed that the stress wave has undergone multiple reflections in a short time. It can be approximated by the superposition of infinitely many reflected waves. At this time, equation (9) can be simplified to:

[0042]

[0043] Based on the experimental results, the upper and lower limits of the attenuation coefficient can be estimated using equations (8) and (11), respectively. When the propagation time of the stress wave in the sample is greater than the wavelength, the transmitted wave signal does not undergo the superposition process of the reflected wave, and the lower limit estimate is more accurate in this case. Conversely, the upper limit estimate is more accurate. There is a time difference between the superimposed reflected wave and the transmitted wave, which can be estimated using the stress equilibrium time. For cases where the stress equilibrium time is small, the effect of the time difference is usually negligible.

[0044] As a further technical solution of the present invention, in step six, to more comprehensively quantify the attenuation characteristics of rock stress waves, the Maxwell constitutive model and the standard linear solid (SLS) model are calibrated using experimental data. The Hopkinson bar test has a high impact velocity and rich high-frequency components in the stress waves, allowing the Maxwell model to quantify the attenuation characteristics of these high-frequency components. Based on the frequency domain attenuation coefficients obtained from rock samples of different lengths in step 5, the constitutive parameters of the Maxwell model are calibrated using the following formula:

[0045]

[0046] Among them, C M The propagation velocity of the stress wave in the Maxwell constitutive model is equal to the rock wave velocity C. s Therefore, the variable to be solved has only one viscosity coefficient: η M (θ M =η M / E M The viscosity coefficient can be determined by fitting the experimental results using a global optimization method.

[0047] The time-domain peak value of the SHPB experiment is mainly determined by the high-frequency components, while for High-frequency waves include:

[0048]

[0049] At this point, the frequency domain attenuation coefficient α remains stable and equal to the time domain attenuation coefficient β, thus obtaining the mapping relationship between the frequency domain attenuation coefficient and the time domain attenuation coefficient. Based on the time domain attenuation coefficients obtained from rock samples of different lengths in step 5, the constitutive parameters of the Maxwell model are calibrated using the following formula:

[0050]

[0051] Since the constitutive parameters obtained from frequency domain and time domain analysis are consistent, the accuracy of the constitutive model can be evaluated by cross-validating the attenuation characteristics in the time and frequency domains.

[0052] As a further technical solution of the present invention, in step six, since the stress in the Maxwell model approaches complete relaxation under low-frequency loading, its analysis of low-frequency attenuation characteristics is not accurate enough. The Standard Linear Solid (SLS) model combines the advantages of the Maxwell model and the Kelvin-Voigt model, and can more comprehensively describe the attenuation characteristics of stress waves in various frequency bands. Therefore, the Standard Linear Solid (SLS) model is further used to analyze the attenuation characteristics of stress waves in various frequency bands. The simplest Standard Linear Solid (SLS) model includes a two-element unit formed by parallel connection of Maxwell elements and spring elements, and its frequency domain attenuation coefficient is determined by the following formula:

[0053]

[0054] Consistent with Maxwell's constitutive model, when the stress wave frequency is high, the attenuation coefficient of the SLS model remains stable and equal to the time-domain attenuation coefficient:

[0055]

[0056] Where E0 is the elastic modulus of the parallel spring, which mainly controls the propagation behavior of low-frequency stress waves. M As the relaxation modulus, since the propagation of high-frequency waves maintains a constant phase velocity, it can be considered equal to the elastic wave velocity of the rock. In this case, E M It equals the elastic modulus of the rock. θ M Let θ be the relaxation time. M From the viscosity constant η M Divided by relaxation modulus E M E M and θ M It primarily controls the propagation behavior of high-frequency stress waves.

[0057] As a further technical solution of the present invention, in step six, the constitutive parameters of the Standard Linear Solid (SLS) model are calibrated based on the test results of rock samples of the same length from step 5. To simplify the calibration process of the constitutive parameters, a single-variable stepwise solution method for the constitutive parameters of the Standard Linear Solid (SLS) is proposed:

[0058] (1) Based on the high-frequency attenuation coefficient obtained from the rock test of the same length in step 5, the constitutive parameters of the Maxwell model are calibrated using equation (14), and the time-domain attenuation coefficient β is determined.

[0059] (2) Based on the time-domain attenuation coefficient β, the constraint conditions for the constitutive parameters of the SLS model are given using equation (16). At this point, the parameters E0 and η of the standard linear solid (SLS) constitutive model are... M The variable to be solved will be subject to the constraint of equation (16), and the variable to be solved will become one E0.

[0060] (3) Based on the low-frequency attenuation coefficient obtained from the rock test of the same length in step 5, the variable E0 to be solved is determined by global optimization analysis using equation (15).

[0061] Compared with the prior art, the beneficial effects of the present invention are:

[0062] This invention proposes a method for obtaining the attenuation characteristics of stress waves in rock by analyzing SHPB test data of rock rod specimens of different lengths and single lengths. Based on the Maxwell model and the Standard Linear Solids (SLS) model, a quantitative mapping relationship between the time-domain and frequency-domain attenuation characteristics of rock stress waves is established. A time-frequency determination method for the constitutive parameters of the Maxwell model and a single-variable stepwise solution method for the constitutive parameters of the SLS model based on SHPB test data are proposed to more comprehensively quantify the attenuation characteristics of stress waves in rock and facilitate numerical simulation analysis and engineering applications. This method overcomes the limitations of traditional pendulum tests in terms of loading rate and specimen size through high strain rate loading and short rock rod test design, reducing the complexity and cost of the test. It provides a more efficient and accurate experimental and theoretical framework for the analysis of rock mass propagation attenuation characteristics, and provides a scientific basis for stress wave propagation prediction in fields such as deep resource extraction, underground space utilization, and extreme disaster prevention. Attached Figure Description

[0063] Figure 1 This is a flowchart of the method of the present invention;

[0064] Figure 2 Diagram of a split Hopkinson bar test setup;

[0065] Figure 3 This is a schematic diagram of stress wave propagation in the SHPB test.

[0066] Figure 4 A diagram of the Maxwell model;

[0067] Figure 5 The SLS plot is for a Maxwell-type standard linear solid.

[0068] Figure 6This is a graph showing the normalized measurement results of the transmission rod strain gauge;

[0069] Figure 7 Here is a diagram of the SHPB finite element model;

[0070] Figure 8 The incident, transmitted, and reflected wave waveforms are for operating conditions 1-3.

[0071] Figure 9 Comparison of calculated and measured values ​​of the Fourier amplitude of the strain of the transmission rod;

[0072] Figure 10 Comparison of attenuation coefficients between Maxwell constitutive model and standard linear solid model (SLS);

[0073] Figure 11(a) is a comparison of the Fourier amplitude results of the 80mm sample;

[0074] Figure 11(b) is a comparison of the Fourier amplitude results of the 140mm sample;

[0075] Figure 11(c) is a comparison of the Fourier amplitude results for the 200mm sample;

[0076] Figure 12 To estimate the change in error with respect to strain rate;

[0077] Figure 13(a) shows a comparison of the waveforms of the transmission rod of the 80mm sample;

[0078] Figure 13(b) shows a comparison of the waveforms of the transmission rod of the 140mm sample;

[0079] Figure 13(c) shows a comparison of the waveforms of the transmission rod for a 200mm sample;

[0080] Figure 14(a) is a comparison of the numerical simulation results and the measured results of the spectrum of the 80mm sample;

[0081] Figure 14(b) shows a comparison between the numerical simulation results and the measured results of the spectrum of the 140mm sample;

[0082] Figure 14(c) shows a comparison between the numerical simulation results and the measured results of the spectrum of the 200mm sample. Detailed Implementation

[0083] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0084] Please see the appendix Figure 1 - Figure 14 shows an embodiment of the present invention:

[0085] Test apparatus

[0086] To verify the effectiveness of the rock stress wave attenuation characteristic analysis method, a series of SHPB tests were conducted on granite specimens. The tests were carried out on a 74.4 mm split Hopkinson pressure bar (SHPB) apparatus in the laboratory of Ningbo University. The bar used was high-strength steel with an elastic modulus of 214 GPa and a wave velocity of 5236.8 m / s. The bullet length was 398 mm, the incident bar length was 2745 mm, and the transmission bar length was 1792 mm. The strain gauge recording point on the incident bar was 1727 mm from the specimen end face, and the strain gauge recording point on the transmission bar was 892 mm from the specimen end face. The strain gauge sensitivity was 2252E-6, and the strain signal sampling frequency was 1 MHz. A 60 mm diameter, 5 mm thick paper sheet was used as a pulse shaper in the test. To reduce the frictional effect between the compression bar and the specimen contact surface, a thin layer of Vaseline was uniformly applied to the contact surface between the specimen and the pressure bar. The results of the empty bar impact test showed that the coaxiality of the compression bar was good, and the stress wave propagated in a one-dimensional manner within the compression bar.

[0087] rock sample

[0088] The rock sample is a sesame grey granite from Rizhao, Shandong Province. The sample was cut from a single large rock face to ensure minimal dispersion. Preliminary static load tests indicate the following basic physical parameters of the rock: density 2730 kg / m³. 3 The wave velocity was 4573 m / s, the elastic modulus was 5.71 GPa, the Poisson's ratio was 0.28, and the compressive strength was 113 MPa. The rock samples were processed into four different cylindrical specimens of different lengths: 40 mm, 80 mm, 140 mm, and 200 mm, with a diameter of 69 mm for each specimen.

[0089] Test conditions

[0090] By measuring the strain waveforms at both ends of the specimen, the attenuation characteristics of stress waves in specimens of different lengths were analyzed. Specific test conditions are shown in Table 1. Conditions 1 to 3 used specimens with an aspect ratio greater than 1, with lengths of 80mm, 140mm, and 200mm, respectively. Traditional Hopkinson bar tests typically use specimens with an aspect ratio less than 1 to achieve stress equilibrium more quickly. However, this example uses specimens with a larger aspect ratio to better reflect the propagation of one-dimensional stress waves. The influence of stress equilibrium will be discussed later. Conditions 1 to 3 used similar bullet velocities to ensure consistency of the incident wave waveform. Taking conditions 1 to 3 as examples, [Table 1 is attached]. Figure 8 Typical incident and transmitted wave waveforms are shown. To ensure the integrity of the experimental data, this invention does not filter the original waveforms, thus preserving the complete frequency components of the actual waveforms. (See attached...) Figure 8It can be seen that, at similar bullet velocities, the incident wave waveforms of 80mm, 140mm, and 200mm rock rods are highly consistent, while the transmitted wave waveforms show significant attenuation characteristics. Conditions 4 to 9 used 40mm specimens and different bullet velocities to study the effects of aspect ratio and different strain rates on the test results.

[0091] Table 1 Test Conditions

[0092]

[0093] Analysis of attenuation characteristics based on rock rods of different lengths

[0094] Calculation of time-domain attenuation coefficient

[0095] For the test results of rock rods of different lengths, it is relatively simple to first analyze the time-domain attenuation characteristics. The strain of the transmission rod is normalized using equation (4), and then the normalized peak value of the transmission rod is linearly fitted using equation (5) to calculate the time-domain attenuation coefficient. The results are shown in the appendix. Figure 6 As shown, the normalized peak values ​​at 80mm, 140mm, and 200mm are 0.8420, 0.7526, and 0.6423, respectively. The calculated time-domain attenuation coefficient is 2.2571, and the coefficient of determination for linear fitting is 0.9953, indicating a high goodness of fit and reliable results. However, since only three samples were used, the uncertainty of the experiment has a significant impact on the fitting results, and the calculated results may not be robust. Therefore, it is necessary to verify the results by combining the frequency domain attenuation characteristics.

[0096] Frequency domain attenuation coefficient calculation

[0097] Based on the correspondence between the time-domain attenuation coefficient and the frequency-domain attenuation coefficient of high-frequency stress waves, substituting the time-domain attenuation coefficient (2.2571) into equation (14) yields the viscosity coefficient of the Maxwell constitutive model as 4.8472e-05, and thus the corresponding frequency-domain attenuation coefficient is obtained. Substituting the frequency-domain attenuation coefficient into equation (6), and combining the transmitted wave data of 80mm and 140mm rock rods, the theoretical predicted values ​​of the Fourier amplitude of the transmitted waves of 140mm and 200mm rock rods can be calculated respectively. The results are shown in the appendix. Figure 9 As shown. (From the appendix) Figure 9 As can be seen, when the frequency exceeds 4000Hz, the transmitted wave spectrum curves corresponding to rock rods of different lengths show significant overlap, indicating that the effective component of the signal in this frequency band is relatively low and it is more susceptible to high-frequency noise interference. Therefore, 4000Hz is set as the high-frequency cutoff frequency. For the low-frequency components, see the attached... Figure 6The bandwidth of the transmitted wave was approximately 0.5 ms, while the sampling frequency of the experiment was 1 MHz, resulting in a Fourier spectrum resolution of 2000 Hz. Although zero-padding or periodic extension can improve the spectral resolution, the effective signal components do not substantially increase, and frequency bands below 2000 Hz will contain numerical errors, with the error increasing as the frequency approaches zero. Based on the above analysis, 2000 Hz-4000 Hz was defined as the effective high-frequency component, while 1000-2000 Hz was selected as the effective low-frequency band to reduce error interference. Within the effective high-frequency band, the spectral curves of the predicted and measured values ​​showed a high degree of consistency. The mean absolute percentage error (MAPE) for each frequency band was calculated, showing that the error was largest in the 3000-4000 Hz range at 9.6%, while the average error across the entire frequency band was 5.81%. Overall, the errors in each frequency band were relatively small, verifying the correspondence between the time-domain and frequency-domain attenuation characteristics and the effectiveness of using the Maxwell constitutive model for attenuation characteristic analysis. It is worth noting that, since Equation (6) ignores the attenuation of the reflected wave at the left-hand interface, the theoretical prediction of the Maxwell model is slightly higher than the measured value. However, the measured results show that the overall error is within the acceptable range for engineering, which proves the effectiveness of the Maxwell constitutive model and the correspondence between the time-frequency attenuation characteristics, and provides a reliable basis for the subsequent analysis of the attenuation characteristics based on single-size rock rod samples.

[0098] Attenuation characteristics analysis based on rock rods of the same length

[0099] High-frequency attenuation characteristic analysis

[0100] The method based on rock rod samples of different lengths requires obtaining high-precision test results from rock samples of multiple sizes, which significantly increases the complexity of the test. To overcome this limitation, the feasibility of attenuation characteristic analysis based on rock rod samples of the same length was explored. The effectiveness of the Maxwell constitutive model and the correspondence between time-frequency attenuation characteristics have been verified. Therefore, the Maxwell constitutive model can be used to first determine the time-domain attenuation coefficient and the corresponding high-frequency attenuation characteristics. Using equations (8) and (11), the lower limit and upper limit of the frequency-domain attenuation coefficient can be estimated respectively. For this invention, the longest sample length is 200 mm, the rock wave velocity is 4573 m / s, and the corresponding stress wave transit time through the sample is 4.37e-5 s, which is much smaller than the stress wave wavelength (0.5e-3 s). Therefore, the time difference of the superposition of reflected waves can be ignored, so it is more reasonable to use the upper limit value for estimation. The attenuation characteristics of the 2000-4000Hz high-frequency band were spectrally fitted using the upper limit formula (11), and the corresponding time-domain attenuation coefficient was calculated. The average time-domain attenuation coefficient of the 80mm, 140mm and 200mm rock rods was 2.1565, and the error (MAPE) of the results with rock rods of different lengths was only 4.40%. This result shows that it is feasible to use a single rock rod sample for attenuation characteristic analysis. This method can obtain more accurate results with fewer experiments.

[0101] Low-frequency attenuation characteristic analysis

[0102] While the Maxwell constitutive model effectively characterizes high-frequency attenuation, it has certain errors in predicting the attenuation of low-frequency components. Therefore, the Standard Linear Solid State (SLS) model is used to improve the accuracy of low-frequency attenuation characteristic prediction. Based on the theoretical framework, the mean time-domain attenuation coefficient (2.1565) is taken for further calculation and analysis. At this time, the constitutive parameters of the Standard Linear Solid State (SLS) model will satisfy the constraint of equation (16). The constitutive parameter E0 of the Standard Linear Solid State (SLS) model is solved using the global optimization method and is 12.34 GPa. Its corresponding frequency domain attenuation characteristics are shown in the attached figure. Figure 10 As shown. From the appendix Figure 10As can be seen, the frequency domain attenuation coefficient increases monotonically with increasing frequency and then gradually stabilizes, with its stable value being basically consistent with the time domain attenuation coefficient (2.1565). The high-frequency attenuation characteristics of the Maxwell constitutive and the standard linear solid (SLS) constitutive are basically consistent, but there are significant differences in the low-frequency range. Based on the frequency domain attenuation coefficient, the transmitted wave spectrum is predicted using Equation (11). The comparison between the frequency domain distribution of the transmitted wave calculated by the Maxwell constitutive and the standard linear solid (SLS) constitutive and the measured results is shown in Figure 11. The results show that the prediction results of the two are basically consistent in the high-frequency range, while for the low-frequency range below 2000Hz, the estimated value of the Maxwell constitutive is too small, and the estimated value of the standard linear solid (SLS) constitutive is more accurate. Comparing the prediction errors (MAPE) across different frequency bands, the Maxwell model exhibits a high prediction error of 15.84% in the low-frequency range (1000-2000Hz), while the Standard Linear Solid (SLS) model, by introducing a low-frequency spring, reduces the maximum low-frequency error to 2.16%. However, its prediction accuracy in the high-frequency range (3000-4000Hz) decreases slightly. Nevertheless, for the entire frequency range, the SLS constitutive model's prediction error is 6.38%, lower than the Maxwell model's (8.29%), indicating overall higher accuracy. This model can significantly improve the prediction accuracy of low-frequency attenuation characteristics while maintaining high high-frequency prediction accuracy, providing a more universal theoretical tool for multi-band stress wave propagation analysis.

[0103] Numerous experimental studies have shown that the mechanical response of rock specimens is significantly dependent on their geometric dimensions. Therefore, the influence of size effect must be considered when conducting error analysis. Based on the standard linear solid (SLS) constitutive model, the prediction error of specimens of different sizes was analyzed. For 80 mm, 140 mm, and 200 mm specimens, the prediction error showed a significant increasing trend with the increase of specimen length, from 6.38% to 16.84%. This phenomenon may be due to the stress equilibrium time being extended with the increase of specimen length, resulting in an increase in error caused by the time difference. Furthermore, the superposition effect of actual reflected waves is not an infinite superposition of terms. The stress wave disperses significantly after a finite number of reflections, causing the upper limit estimate to be higher than the measured result. This deviation increases with the increase of the specimen length. However, smaller specimen size is not always better. According to the International Society for Rock Mechanics (ISRM) SHPB test specifications, the specimen diameter should not be less than 50 mm, and the aspect ratio should be controlled between 0.5 and 1. Experimental data shows that under similar strain rate conditions, the error of a 40 mm specimen is as high as 19.15%, while the error of an 80 mm specimen is only 6.38%, significantly lower than the former. This difference may be due to the significant transverse inertia effect and geometric attenuation effect in specimens with small aspect ratios, which reduces the applicability of the one-dimensional stress wave assumption, thus affecting the accuracy of the test results. Based on the above analysis, it is recommended to select specimens with an aspect ratio close to 1 (i.e., the 80 mm specimen in this invention) in the attenuation relationship analysis to obtain better test accuracy. The mechanical behavior of rock masses changes under different strain rates, therefore, further research is needed on the influence of strain rate on the prediction results. Analysis shows that some defects in the basic physical assumptions of the 40mm sample lead to a certain reference error. This section mainly discusses the influence of strain rate on the trend of error variation, focusing on the fluctuation of the error. Figure 12 The trend of the overall error of the 40mm specimen as a function of strain rate is shown. The results indicate that the error first decreases and then increases with increasing strain rate, fluctuating between 12.5% ​​and 22.5%. The error is basically stable without a significant linear trend. This demonstrates that predictions based on the average constitutive parameters of 80mm, 140mm, and 200mm specimens maintain high robustness, verifying the applicability of the constitutive model. The model's adaptability can be attributed to the effective control capability of Maxwell elements over a wide strain rate range.

[0104] Numerical simulation verification

[0105] Numerical model design

[0106] Numerical simulations were performed using LS-DYNA software to analyze conditions 1-3 and verify the accuracy and applicability of the constitutive parameters. A three-dimensional solid element model was established based on the actual dimensions of the Hopkinson bar, where both the bar and the specimen were modeled using three-dimensional solid elements. The incident and transmitted bars adopted linear elastic constitutive models, with material parameters consistent with experimental results. The rock specimen used a standard linear solid (SLS) constitutive model, selecting material number 76 (General-Viscoelastic constitutive) in LS-DYNA software, and the calculated average values ​​were used for the material parameters. Since a shaper was used in the actual experiment, to fully preserve the incident wave waveform in the numerical simulation, the incident bar strain was converted into an equivalent pressure at the bar end. The magnitude of the equivalent pressure P was the product of the incident bar strain and the elastic modulus of the input bar. The contact between the input rod and the rock adopts a frictionless contact model. To ensure simulation accuracy, a suitable mesh size needs to be determined. According to wave field propagation theory, an excessively large mesh size will lead to a high-pass filtering effect. To better preserve wave field components, the mesh size should be 1 / 8 to 1 / 16 of the minimum wavelength. Considering that the high-frequency components above 15000Hz are mainly experimental noise, this embodiment uses this frequency as the high-pass cutoff frequency. The mesh size corresponding to 1 / 16 of the minimum wavelength is 1.9cm. This embodiment selects a mesh size of 1cm, which can balance computational accuracy and efficiency. The model uses uniform mesh generation, and its finite element model is attached. Figure 7 As shown.

[0107] Simulation Results and Analysis

[0108] Figure 13 shows a comparison between the numerical simulation results and the measured transmitted wave waveforms of the standard linear solid (SLS) constitutive and linear elastic constitutive models. The results show that the standard linear solid (SLS) constitutive model can accurately reproduce the time-domain waveform characteristics of the transmitted wave, and its predicted curve is highly consistent with the measured data. The maximum prediction error of the peak value of the transmitted wave under the standard linear solid (SLS) constitutive model is 4.58%, while the error of the linear elastic constitutive model gradually increases due to the failure to consider the attenuation effect of the medium, with a maximum error reaching 22.44%. This result illustrates the necessity of attenuation characteristic analysis in the simulation.

[0109] Further analysis of the time-domain attenuation coefficient revealed that the simulation result for the Standard Linear Solid (SLS) constitutive model was 2.1084, with an error of 2.23% compared to the theoretical value (2.1565) and 6.53% compared to the measured value (2.2571) from different rock rod tests. This demonstrates the simulation accuracy of the SLS constitutive model. Waveform difference analysis showed that the simulation waveform was relatively smooth, which may be due to the high-frequency filtering effect of the finite element model suppressing noise components in the measured signal. Furthermore, the wavelength of the simulation result was slightly smaller than the measured value, with the difference mainly occurring during the unloading phase. This may be because the invention did not consider the influence of macroscopic joints, which cause waveform hysteresis. Macroscopic joints have a smaller impact on the loading phase but a larger impact on the unloading phase. Since the focus in practical engineering scenarios is primarily on the characteristics of the loading stage, the impact of this deviation is relatively small. Figure 14 compares the spectral differences between the simulation and measured results. The results show that the numerical simulation results and the measured data are in high agreement within the main frequency bands. The maximum overall error for each frequency band is 9.63%, which is better than the upper limit estimate of the theoretical results (16.84%). This may be due to the simplified assumptions made by the theoretical model regarding the number of superpositions of reflected waves and the time difference effect, while the numerical simulation can more realistically reflect the wave field evolution process. It is noteworthy that the error growth rate of the standard linear solid (SLS) constitutive model is significantly lower than that of the linear elastic model, with an error increase of less than one-third that of the linear elastic model, highlighting the advantages of the viscoelastic constitutive model. The numerical simulation results further verify the applicability of the constitutive model and the effectiveness of the calculation method.

[0110] This invention analyzes SHPB test data from rock samples of different lengths and rock rod samples of the same length to obtain the attenuation characteristics of stress waves in rocks. A quantitative mapping relationship between the time-domain and frequency-domain attenuation characteristics of rock stress waves is established. A method for determining constitutive model parameters characterizing the dynamic mechanical properties of rocks is presented. The effectiveness of this method is comprehensively verified through theoretical analysis, experimental verification, and numerical simulation. The specific conclusions are as follows: A method is proposed to obtain the attenuation characteristics of stress waves in rocks by analyzing SHPB test data of rock specimens of different lengths and rock rod specimens of the same length. This method overcomes the limitations of loading rate and specimen size in traditional pendulum tests by using high strain rate loading and short rock rod test design, thus reducing the complexity and cost of the test. Based on the Maxwell model and the Standard Linear Solids (SLS) model, a quantitative mapping relationship between the time-domain and frequency-domain attenuation characteristics of rock stress waves is established. A time-frequency determination method for the constitutive parameters of the Maxwell model and a single-variable stepwise solution method for the constitutive parameters of the SLS model based on SHPB test data are proposed. A series of SHPB tests on rock media with different specimen lengths and strain rates are designed and carried out. Based on the proposed method, the constitutive parameters of the Maxwell model and the SLS model are determined. Combined with experimental data, the accuracy of the correspondence between time-frequency attenuation characteristics and the applicability of the Maxwell model and the SLS model for different frequency components are verified. The influence of aspect ratio and strain rate on the calculation accuracy of the method of this invention is revealed.

Claims

1. A method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device, characterized in that, include: Step 1: Conduct the test using a split Hopkinson pressure bar device; Step two: Experiment design and execution; To design rock samples of different lengths to meet the needs of stress wave attenuation characteristic analysis, observe the attenuation of stress waves in samples of different lengths, and generate incident stress waves of different intensities by adjusting the speed of the impact rod, thus ensuring the diversity of experimental data. Step 3, Data Collection; The velocity of the striking rod is measured using a velocimeter, and the strain gauge data on the incident rod and the transmission rod are recorded to obtain waveform information of the incident wave, reflected wave and transmitted wave. Step four: Process the recorded waveform data. Step 5: Attenuation characteristic analysis; The time-domain attenuation coefficient and frequency-domain attenuation coefficient were determined based on the SHPB test. For rock samples of different lengths, the relationship between the transmitted wave amplitude and the sample length was analyzed to obtain the frequency-domain and time-domain attenuation coefficients. For rock samples of the same length, the superposition of multiple reflected waves inside the rock mass was ignored, and the Fourier amplitude attenuation from the incident wave signal to the transmitted wave signal was analyzed to calculate the lower limit of the frequency-domain attenuation coefficient. The transmitted wave signal was superimposed with the reflected wave inside the rock mass, and the Fourier amplitude attenuation from the incident wave signal to the superimposed signal was analyzed to calculate the upper limit of the frequency-domain attenuation coefficient. Step 6: Constitutive parameter calibration; In step six, based on the time-domain and frequency-domain attenuation coefficients of rock samples of different lengths from step five, the constitutive parameters of the Maxwell model are calibrated, and the mapping relationship between the time-domain attenuation coefficient and the frequency-domain attenuation coefficient is proposed. Based on the frequency domain attenuation coefficient of rock samples of the same length from step 5, the constitutive parameters of the Standard Linear Solid (SLS) model are calibrated. A single-variable stepwise solution method for the constitutive parameters of the SLS model is proposed to simplify the solution process.

2. The method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device according to claim 1, characterized in that, In step four, the transmitted wave is normalized by dividing the peak value of the incident wave by equation (4) to reduce the impact of input waveform differences on the results. Where ε T For the strain record of the transmission rod, ε I Record the strain of the incident rod.

3. The method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device according to claim 1, characterized in that, In step five, the method for analyzing the time-domain attenuation characteristics of rock samples of different lengths is to measure the normalized transmission peak value ε' of the rock samples of different lengths. Ti Then, a linear fit is performed in the logarithmic domain to calculate the time-domain attenuation coefficient β; the linear fit expression for calculating the time-domain attenuation coefficient based on rock samples of different lengths is shown in equation (5): ln(e' Ti )=-βl i +ln(ε0) (5) In the formula, ε' Ti Indicates a length of l i The peak value of the normalized transmitted wave in the time domain corresponding to the rock sample, l i The length of the rock sample; When the incident wave is consistent, the formula for calculating the frequency domain attenuation coefficient based on rock samples of different lengths is shown in equation (6): Where α(ω) is the frequency domain attenuation coefficient, and ω is the angular frequency; and These are the Fourier amplitudes of the strain of the specimen with length l2 and the specimen with length l1, respectively (l2>l1); Let the transmission coefficients at interface 1 and interface 2 be T1 and T2, respectively. Then, the lower limit estimate of the frequency domain attenuation coefficient based on rock samples of the same length can be calculated by equation (8): Where α(ω) is the frequency domain attenuation coefficient; and , respectively, are the Fourier amplitudes of the transmitted wave and the incident wave, and l is the sample length; Let the reflection coefficients at interface 1 and interface 2 be F1 and F2, respectively. Then, the upper limit estimate of the frequency domain attenuation coefficient based on rock samples of the same length can be calculated by equation (11): E reflects the superposition effect of the reflected waves and their attenuation. Based on the experimental results, the upper and lower limits of the attenuation coefficient were estimated by Equation (8) and Equation (11), respectively.

4. The method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device according to claim 1, characterized in that, In step six, the frequency domain attenuation coefficients obtained from rock samples of different lengths in step five are used to calibrate the constitutive parameters of the Maxwell model using equation (12): Among them, C M The propagation velocity of the stress wave in the Maxwell constitutive model is equal to the rock wave velocity C. s Therefore, the variable to be solved has only one viscosity coefficient: η M (θ M =η M / E M The viscosity coefficient can be determined by fitting the experimental results using a global optimization method; ω is the angular frequency. Based on the time-domain attenuation coefficients obtained from rock samples of different lengths in step 5, the constitutive parameters of the Maxwell model are calibrated using equation (14): β is the time-domain decay coefficient; Since the constitutive parameters obtained from frequency domain and time domain analysis are consistent, the accuracy of the constitutive model can be evaluated by cross-validating the attenuation characteristics in the time and frequency domains. Consistent with Maxwell's constitutive model, when the stress wave frequency is high, the attenuation coefficient of the SLS model remains stable and equal to the time-domain attenuation coefficient β: Where E0 is the elastic modulus of the parallel spring, E M For relaxation modulus, θ M This is the relaxation time.

5. The method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device according to claim 4, characterized in that, In step six, to simplify the calibration process of constitutive parameters, a single-variable stepwise solution method for standard linear solid constitutive parameters is proposed: (1) Based on the high-frequency attenuation coefficient obtained from the rock test of the same length in step 5, the constitutive parameters of the Maxwell model are calibrated using equation (14), and the time-domain attenuation coefficient β is determined. (2) Based on the time-domain attenuation coefficient β, the constraint conditions for the constitutive parameters of the SLS model are given using equation (16); at this time, the parameters E0 and η of the standard linear solid (SLS) constitutive model are... M The variable to be solved will be subject to the constraint of equation (16), and the variable to be solved will become one E0; (3) Based on the low-frequency attenuation coefficient obtained from the rock test of the same length in step 5, the variable E0 to be solved is determined by global optimization analysis using equation (15); 6. The method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device according to claim 1, characterized in that, It also includes step seven, method verification; through theoretical analysis, experimental verification and numerical simulation, the effectiveness of the analysis method is fully verified.

7. The method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device according to claim 1, characterized in that, Step four, as described above, involves the following specific operations: The recorded waveform data is processed to extract the strain amplitudes of the incident wave, reflected wave, and transmitted wave, and then normalized.

8. The method for analyzing the attenuation characteristics of rock stress waves based on a split Hopkinson pressure bar device according to claim 1, characterized in that, The specific operations in step one are as follows: The split Hopkinson pressure bar device includes an impact bar, an incident bar, a transmission bar, and a velocimeter. Strain gauges are attached to the incident bar and the transmission bar to record the strain signals of the incident wave, reflected wave, and transmitted wave.

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