Asphalt pavement material dynamic and static modulus conversion model based on tension-compression difference

Abstract

The invention discloses an asphalt pavement material dynamic and static modulus conversion model based on tension-compression difference. According to the model, static modulus-loading rate and dynamic modulus-loading frequency correlation models of an asphalt mixture and a semi-flexible pavement material are respectively constructed by testing tension and compression dynamic moduli and static moduli of the asphalt mixture and the semi-flexible pavement material under different loading frequencies or different loading rates; and constructing a dynamic modulus-equivalent loading rate correlation model, and fitting the dynamic modulus-equivalent loading rate correlation model with the static modulus-loading rate correlation model to obtain the dynamic modulus-equivalent loading rate correlation model. The conversion model is clear in structure, wide in data source and easy to operate, a theoretical model and experimental data are effectively combined, the correlation relation between the loading rate and the tension and compression dynamic and static moduli is comprehensively reflected, the wide application range and accuracy rate are achieved, and important guiding significance is achieved for asphalt pavement structure design.

Description

Technical Field

[0001] The invention relates to a dynamic-static modulus conversion model, in particular to a dynamic-static modulus conversion model of an asphalt pavement material based on tension-compression difference, and belongs to the technical field of road engineering. Background Art

[0002] At present, the mechanical analysis of pavement structures at home and abroad adopts the traditional linear elastic theory, and the surface material adopts the dynamic compressive rebound modulus during design and verification, and does not fully consider the tension and compression difference characteristics of the material, resulting in large errors in the calculation results. my country has been studying the static modulus for a long time and has formed mature theories and test methods; studies have shown that the dynamic modulus of pavement materials is about 2 to 3 times the static modulus, and the mechanical properties of the pavement under actual dynamic loads are quite different from those of the static model. Therefore, it is necessary to establish the conversion relationship between the dynamic and static moduli of asphalt pavement materials to verify the accuracy of the dynamic modulus test results while realizing the rapid conversion of dynamic and static moduli.

[0003] Some people have taken into account the tension and compression differences of materials when calculating pavement structures, and conducted a mechanical analysis of typical asphalt pavement structures. The results show that the difference between the mechanical responses of structural calculations using compressive modulus and tensile and compressive dual modulus as material stiffness parameters can reach more than 50%; some people have tested the compressive dynamic modulus of seven different asphalt mixtures and found that the dynamic modulus of asphalt mixtures is affected by multiple factors such as temperature and loading frequency, and has significant velocity dependence.

[0004] Although some scholars have conducted partial research on the relationship between the dynamic and static moduli of asphalt pavement materials under standard test methods, the test results of the dynamic and static moduli are affected by many factors, and the one-sided establishment of the relationship curve is unreasonable.

[0005] Therefore, how to consider the influence of various internal and external factors on the dynamic and static modulus of asphalt pavement materials and realize the conversion of dynamic and static modulus of asphalt pavement materials under complex conditions is a key issue that technical personnel in this field need to solve urgently. Summary of the invention

[0006] In view of the problems existing in the prior art, the purpose of the present invention is to provide a dynamic and static modulus conversion model of asphalt pavement materials based on tension and compression differences. The model is based on the tension and compression difference characteristics of the material, and constructs the correlation model of static modulus-loading rate and dynamic modulus-loading frequency through the experimental test data of the material. Then, based on the equivalence principle, a correlation model of dynamic modulus-equivalent loading rate is constructed, and then it is fitted with the static modulus-loading rate correlation model to obtain; the conversion model has a clear structure, a wide data source, is easy to operate, and effectively combines the theoretical model with the experimental data, comprehensively reflects the correlation relationship between the loading rate and the tension and compression dynamic and static modulus, has a wide range of application and accuracy, and has important guiding significance for the design of asphalt pavement structures.

[0007] In order to achieve the above technical purpose, the present invention provides a dynamic and static modulus conversion model of asphalt pavement materials based on tension and compression difference, comprising: step S1: according to the test requirements of "Test Procedures for Asphalt and Mixtures in Highway Engineering" (JTG E20-2011), determine the mix ratio of asphalt mixture and semi-flexible pavement materials composed of raw materials including asphalt mixture, and obtain test pieces required for testing the two;

[0008] Step S2: After the test piece obtained in step S1 is adjusted to a constant temperature, the dynamic and static moduli of tension and compression of the asphalt mixture and the semi-flexible pavement material at different loading frequencies or loading rates are tested respectively;

[0009] Step S3: According to the test results of step S2, the correlation models of the tension and compression static modulus-loading rate of the asphalt mixture and the semi-flexible pavement material are established respectively;

[0010] Step S4: According to the test results of step S2, the correlation models of the tension and compression dynamic modulus-loading frequency of the asphalt mixture and the semi-flexible pavement material are established respectively;

[0011] Step S5: According to the correlation model obtained in step S3 and step S4, a dynamic modulus-equivalent loading rate model is established, and then fitted with the static modulus-loading rate correlation model to obtain a dynamic-static modulus conversion model.

[0012] As a preferred solution, the process of determining the mix ratio in step S1 includes:

[0013] Step S1-1, using the Marshall design method to determine the optimal asphalt-to-stone ratio of the asphalt mixture;

[0014] Step S1-2, using the results of the Schellenberg leakage test and the Kentenberg scattering test to determine the optimal asphalt dosage of the base asphalt mixture in the semi-flexible pavement material, and using an orthogonal design experiment to determine the mix ratio of cement mortar.

[0015] As a preferred solution, the test specimens required for testing in step S1 include uniaxial compression test specimens and direct tension test specimens.

[0016] As a preferred solution, the uniaxial compression test specimen is prepared by using a rotary compactor.

[0017] As a preferred solution, the direct tensile test specimen is formed by a hydraulic rutting specimen forming machine, cut into beam-type specimens, and then poured into cement mortar to obtain the specimen.

[0018] As a preferred solution, the test process of the static modulus in step S2 is: performing step-by-step loading and unloading tests on the asphalt mixture and the semi-flexible pavement material, respectively, to obtain the static modulus under each level of load.

[0019] As a preferred solution, the testing process of the dynamic modulus in step S2 is: applying an offset sinusoidal wave or a half-positive vector wave axial compressive stress to the asphalt mixture and the semi-flexible pavement material at a certain temperature and loading frequency to test the recoverable axial strain, and then obtaining the dynamic modulus under different conditions.

[0020] As a preferred solution, the process of establishing the correlation model between the tensile and compressive static modulus and the loading rate is as follows: according to the test results of the tensile and compressive static modulus of asphalt mixture and semi-flexible pavement materials and their corresponding loading rates, a relationship model between the static modulus and the loading rate is established, and the calculation process is as follows:

[0021] Formula 1:

[0022] Formula 2:

[0023] In formula 1-2: E c is the uniaxial compression static modulus, the dimension is MPa; E t is the direct tensile static modulus, with the dimension of MPa; a c , b c 、a t and b t is the fitting parameter, dimensionless; v is the loading rate, dimension is MPa / s.

[0024] As a preferred solution, the process of establishing the correlation model between the tensile and compressive dynamic modulus and the loading frequency is as follows: according to the test results of the tensile and compressive dynamic modulus of asphalt mixture and semi-flexible pavement materials and their corresponding loading frequencies, a relationship model between the dynamic modulus and the loading frequency is established, and the calculation process is as follows:

[0025] Formula 3:

[0026] Formula 4:

[0027] In formula 3-4: E* cd is the uniaxial compression dynamic modulus, the dimension is MPa; E* td is the direct tensile dynamic modulus, with the dimension of MPa; a cd , b cd 、a td and b td is the fitting parameter, dimensionless; f is the loading frequency, dimension is Hz.

[0028] As a preferred solution, the dynamic modulus-equivalent loading rate model is established by obtaining the loading time according to the loading frequency through the equivalent principle, and then calculating the equivalent loading rate of the dynamic modulus in combination with the peak stress applied during the test and the contact area of ​​the specimen.

[0029] As a preferred solution, the specific calculation process of the dynamic modulus-equivalent loading rate model is:

[0030] Formula 5:

[0031] Formula 6:

[0032] Formula 7:

[0033] Formula 8:

[0034] In formulas 5 to 8, t is the loading time, with the dimension of s; f is the loading frequency, with the dimension of Hz; P max is the peak stress value, the dimension is kN; s is the contact surface area of ​​the specimen, the dimension is mm 2 ; a' cd , b' cd 、a' td and b' td is the fitting parameter, dimensionless.

[0035] As a preferred solution, the specific expression of the dynamic-static modulus conversion model is:

[0036] Formula 9: E=A+B|E * |;

[0037] In formula 9, E is the static modulus, A and B are fitting parameters, |E * | is the dynamic modulus.

[0038] Compared with the prior art, the beneficial technical effects of the technical solution of the present invention are:

[0039] 1) The dynamic-static modulus conversion model provided by the present invention is based on the tension-compression difference characteristics of the material. The static modulus-loading rate and dynamic modulus-loading frequency correlation models are constructed through the experimental test data of the material. Then, based on the equivalence principle, the dynamic modulus-equivalent loading rate correlation model is constructed, and then it is fitted with the static modulus-loading rate correlation model. The conversion model has a clear structure, a wide data source, is easy to operate, and effectively combines the theoretical model with the experimental data, comprehensively reflects the correlation relationship between the loading rate and the tension-compression dynamic and static modulus, has a wide range of application and accuracy, and has important guiding significance for the design of asphalt pavement structures.

[0040] 2) In the technical solution provided by the present invention, a power function is selected to construct static modulus-loading rate, dynamic modulus-loading frequency and dynamic modulus-equivalent loading rate correlation models, which can accurately reflect the nonlinear relationship between the independent variable and the dependent variable, and then a static modulus-loading rate correlation model is constructed by linear fitting. The fitting results are highly consistent with the experimental results, the fitting coefficient is ≥0.99, and the regression performance is excellent. It can accurately realize the rapid conversion between dynamic and static moduli, and effectively solve the data distortion problem caused by the single dependent variable and the narrow range when converting dynamic and static moduli in the prior art. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings in the following description are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying creative work.

[0042] Figure 1 This is a graph showing the change of the static modulus of direct tension with the loading rate in Example 1 of the present invention;

[0043] Figure 2 This is a graph showing the variation of the uniaxial compression static modulus with the loading rate in Example 1 of the present invention;

[0044] Figure 3 This is a graph showing the variation of the direct tensile dynamic modulus with the loading frequency in Example 1 of the present invention;

[0045] Figure 4 This is a graph showing the variation of the uniaxial compression dynamic modulus with the loading frequency in Example 1 of the present invention;

[0046] Figure 5 This is a graph showing the variation of the uniaxial compression dynamic modulus of SFP-16 with the loading rate in Example 1 of the present invention;

[0047] Figure 6It is a fitting diagram of the dynamic and static modulus conversion model of the asphalt pavement material in Example 1 of the present invention;

[0048] Figure 7 It is an independent data set of asphalt pavement material at loading rates of 0.2, 0.3, and 0.4 MPa / s in Example 1 of the present invention;

[0049] Figure 8 This is a relative error diagram between the measured values ​​and the values ​​calculated by the conversion model at loading rates of 0.2, 0.3, and 0.4 MPa / s in Example 1 of the present invention. DETAILED DESCRIPTION

[0050] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only 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 ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0051] Example 1

[0052] This embodiment provides a dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference, and the specific process is as follows:

[0053] Step S1: According to the test requirements of the "Test Procedure for Asphalt and Mixtures for Highway Engineering" (JTG E20-2011), the mineral materials screened layer by layer are dried, and asphalt is added and stirred evenly to prepare a dense-graded asphalt mixture (AC-16) and a large-pore asphalt mixture. The uniaxial compression test of the asphalt mixture uses a rotary compactor to prepare a cylindrical specimen with a height of 100±2mm and a diameter of 100±2mm. The direct tensile standard specimen is formed by a hydraulic rutting specimen forming machine into a rutting plate with a length of 300mm×width of 300mm×height of 50mm. The dense-graded asphalt mixture is cut into 5 pieces with a length of 250mm×width of 50mm×height of 50mm after forming. The cylindrical specimens of large-pore asphalt mixture and rutting plates were placed horizontally at room temperature for 12 hours and then demolded. They were then placed in a customized test mold. The bottom of the test mold was tightly sealed with plastic wrap and tape, and then the cement mortar prepared in advance was poured in. After the cement mortar vibrated and seeped down, the excess slurry on the surface was scraped off. The specimens and the test molds were placed in an environment with a humidity of not less than 90% and a temperature of 20℃±3℃ for 28 days of curing. Finally, the semi-flexible pavement material (SFP-16) was formed, and its rutting plates were cut into beam specimens of the same size as AC-16.

[0054] Step S2: Place the prepared specimen in a constant temperature box at 20°C for 4-5 hours, then place it in the direct tension and uniaxial compression fixture of the multifunctional material testing system (MTS-Landmark), and adjust the position of the fixture so that the test head and the specimen are initially in contact, the top tensile rod is fully connected to the multifunctional material testing system, and the test process is completed in the constant temperature box; run the built-in program of the multifunctional material testing system and set the applied load, according to the "Highway Engineering Asphalt and Mixture Test Procedure" (JTG According to the relevant requirements of E20-2011, the specimens of AC-16 and SFP-16 were tested for static modulus of direct tension and uniaxial compression at loading speeds of 0.01MPa / s, 0.02MPa / s, 0.05MPa / s, 0.1MPa / s, and 0.5MPa / s. The test results are shown in Table 1. For the direct tension and uniaxial compression dynamic modulus tests of AC-16 and SFP-16, six different loading frequencies (0.1Hz, 0.5Hz, 1Hz, 5Hz, 10Hz, and 25Hz) were selected according to the specification, and sinusoidal axial tension and compression loads were applied to the specimens. The test results are shown in Table 2.

[0055] Table 1 Static modulus test results (MPa)

[0056]

[0057] Table 2 Dynamic modulus test results (MPa)

[0058]

[0059] Step S3: According to the test results obtained in step S2, the tensile and compressive static modulus-loading rate correlation models of SFP-16 and AC-16 are established at 20°C, respectively, and the power function is used to describe the above models; according to the data in Table 1, the direct tensile static modulus-loading rate and uniaxial compression static modulus-loading rate correlation models of SFP-16 and AC-16 are established, and the results are as follows: Figure 1 and Figure 2 As shown, the specific calculation formula is:

[0060] Formula 1:

[0061] Formula 2:

[0062] In formula 1-2: E c is the uniaxial compression static modulus, the dimension is MPa; E t is the direct tensile static modulus, with the dimension of MPa; a c , b c 、a t and b tis the fitting parameter, dimensionless; v is the loading rate, dimension is MPa / s.

[0063] Step S4: According to the test results obtained in step S2, the tensile and compressive dynamic modulus-loading frequency correlation models of SFP-16 and AC-16 are established at 20°C; the direct tensile dynamic modulus-loading frequency and uniaxial compression dynamic modulus-loading frequency correlation models of SFP-16 and AC-16 are established according to the data in Table 2. The results are as follows: Figure 3 and Figure 4 As shown, the specific calculation formula is:

[0064] Formula 3:

[0065] Formula 4:

[0066] In formula 3-4: E* cd is the uniaxial compression dynamic modulus, the dimension is MPa; E* td is the direct tensile dynamic modulus, with the dimension of MPa; a cd , b cd 、a td and b td is the fitting parameter, dimensionless; f is the loading frequency, dimension is Hz.

[0067] Step S5: According to the correlation model obtained in step S3 and step S4, a dynamic modulus-equivalent loading rate model is established. In order to facilitate the demonstration of the establishment process of the model, the dynamic modulus of SFP-16 uniaxial compression at 20°C and a loading frequency of 10 Hz is taken as an example for calculation. The calculation process is as follows:

[0068] Formula 5:

[0069] Formula 6:

[0070] According to the conversion relationship between the loading rate and loading frequency, the equivalent loading rates of each loading frequency in the SFP-16 uniaxial compression dynamic modulus test are: 0.11MP / s (0.1Hz), 0.55MPa / s (0.5Hz), 1.1MPa / s (1Hz), 5.5MPa / s (5Hz), 11MPa / s (10Hz), 27.5MP / s (25Hz), and the SFP-16 uniaxial compression dynamic modulus-equivalent loading rate correlation model is established. The results are as follows: Figure 5 As shown, the specific calculation process is:

[0071] Formula 7:

[0072] Formula 8:

[0073] In formulas 5 to 8, t is the loading time, with the dimension of s; f is the loading frequency, with the dimension of Hz; P max is the peak stress value, the dimension is kN; s is the contact surface area of ​​the specimen, the dimension is mm 2 ; a ’ cd , b ’ cd 、a ’ td and b ’ td is the fitting parameter, dimensionless.

[0074] Depend on Figure 5 It can be seen that the uniaxial compression dynamic modulus of SFP-16 increases with the increase of equivalent loading rate, and the growth rate gradually decreases, showing a power function relationship, and the fitting coefficient is greater than 0.99, with a high degree of fitting. The dynamic modulus at each loading rate can be known through the fitting relationship. The calculation results are shown in Table 3:

[0075] Table 3 Static and dynamic modulus of tension and compression of pavement materials

[0076]

[0077] According to the tensile and compressive dynamic and static moduli of SFP-16 and AC-16 in Table 3, the two results are fitted to obtain the dynamic-static modulus conversion model, and the results are as follows: Figure 6 As shown, the specific calculation process is:

[0078] Formula 9: E=A+B|E * |;

[0079] In formula 9, E is the static modulus, A and B are fitting parameters, |E * | is the dynamic modulus.

[0080] To further verify the accuracy and precision of the conversion model, the dynamic and static moduli of tension and compression of SFP-16 and AC-16 were tested at loading rates of 0.2MPa / s, 0.3MPa / s, and 0.4MPa / s. Independent data sets were used to identify and evaluate them, and the test results were plotted on Figure 7 , and the static modulus is calculated according to the conversion model based on the dynamic modulus test results, so as to calculate the relative error between the two, such as Figure 8 As shown, it can be seen that the maximum relative error between the measured value and the calculated value based on the conversion model does not exceed 3.5%, indicating that the dynamic and static modulus conversion model of asphalt pavement materials based on tension and compression differences invented by this patent has high accuracy.

[0081] From the above description, it can be seen that the dynamic-static modulus conversion model provided by the present invention fully considers the correlation between the dynamic and static models and the loading rate and loading frequency, and effectively solves the problem that the prior art only conducts a simple quantitative analysis on the mechanical parameters such as the modulus of asphalt pavement materials, and rarely studies the changing laws of the mechanical parameters of materials under the influence of multiple factors, which leads to the problem of single consideration factors and incomplete scope; and, the model is suitable for the conversion of dynamic and static moduli under different loading frequencies, overcoming the difficulty that the prior art only considers the dynamic and static correspondence under the loading frequency of 10Hz, and cannot achieve a one-to-one correspondence between the dynamic modulus and the static modulus conversion under different loading frequencies; the model makes the selection of pavement material mechanical parameters more reasonable, and the pavement structure design more accurate and scientific, and provides a theoretical basis for the mechanical parameter selection and structural design of asphalt pavement materials.

Claims

1. A dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference, characterized in that: include: Step S1: According to the test requirements of "Testing Procedures for Asphalt and Mixtures in Highway Engineering" (JTG E20-2011), the mix ratio of asphalt mixture and semi-flexible pavement material composed of raw materials including asphalt mixture is determined, and test specimens required for testing the two are obtained; Step S2: After the test piece obtained in step S1 is adjusted to a constant temperature, the dynamic and static moduli of tension and compression of the asphalt mixture and the semi-flexible pavement material at different loading frequencies or loading rates are tested respectively; Step S3: According to the test results of step S2, the correlation models of the tension and compression static modulus-loading rate of the asphalt mixture and the semi-flexible pavement material are established respectively; Step S4: According to the test results of step S2, the correlation models of the tension and compression dynamic modulus-loading frequency of the asphalt mixture and the semi-flexible pavement material are established respectively; Step S5: According to the correlation model obtained in step S3 and step S4, a dynamic modulus-equivalent loading rate model is established, and then fitted with the static modulus-loading rate correlation model to obtain a dynamic-static modulus conversion model.

2. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 1 is characterized by: The process of determining the mix ratio in step S1 includes: Step S1-1, using the Marshall design method to determine the optimal asphalt-to-stone ratio of the asphalt mixture; Step S1-2, using the results of the Schellenberg leakage test and the Kentenberg scattering test to determine the optimal asphalt dosage of the base asphalt mixture in the semi-flexible pavement material, and using an orthogonal design experiment to determine the mix ratio of cement mortar.

3. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 1 is characterized by: The test specimens required for the test in step S1 include uniaxial compression test specimens and direct tensile test specimens; the uniaxial compression test specimens are prepared by a rotary compactor; the direct tensile test specimens are formed by a hydraulic rutting specimen forming machine and then cut into beam-type specimens, which are then poured with cement mortar.

4. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 1 is characterized by: The testing process of the static modulus in step S2 is: performing loading and unloading tests on the asphalt mixture and the semi-flexible pavement material step by step, respectively, to obtain the static modulus under each level of load; The testing process of the dynamic modulus in step S2 is: applying offset sinusoidal wave or half-positive vector wave axial compressive stress to the asphalt mixture and semi-flexible pavement material at a certain temperature and loading frequency to test the recoverable axial strain, and then obtain the dynamic modulus under different conditions.

5. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 1 is characterized by: The process of establishing the correlation model of the tensile and compressive static modulus and the loading rate is as follows: according to the test results of the tensile and compressive static modulus of asphalt mixture and semi-flexible pavement materials and their corresponding loading rates, a relationship model between the static modulus and the loading rate is established, and the calculation process is as follows: Formula 1: Formula 2: In formula 1-2: E c is the uniaxial compression static modulus, the dimension is MPa; E t is the direct tensile static modulus, with the dimension of MPa; a c , b c 、a t and b t is the fitting parameter, dimensionless; v is the loading rate, dimension is MPa / s.

6. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 1 is characterized by: The process of establishing the correlation model between the tensile and compressive dynamic modulus and the loading frequency is as follows: according to the test results of the tensile and compressive dynamic modulus of asphalt mixture and semi-flexible pavement materials and their corresponding loading frequencies, a relationship model between the dynamic modulus and the loading frequency is established, and the calculation process is as follows: Formula 3: Formula 4: In formula 3-4: E* cd is the uniaxial compression dynamic modulus, the dimension is MPa; E* td is the direct tensile dynamic modulus, with the dimension of MPa; a cd 、b cd 、a td and b td is the fitting parameter, dimensionless; f is the loading frequency, dimension is Hz.

7. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 1 is characterized by: The dynamic modulus-equivalent loading rate model is established by obtaining the loading time according to the loading frequency through the equivalent principle, and then calculating the equivalent loading rate of the dynamic modulus in combination with the peak stress applied during the test and the contact area of ​​the specimen.

8. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 7 is characterized by: The specific calculation process of the dynamic modulus-equivalent loading rate model is: Formula 5: Formula 6: Formula 7: Formula 8: In formulas 5 to 8, t is the loading time, with the dimension of s; f is the loading frequency, with the dimension of Hz; P max is the peak stress value, the dimension is kN; s is the contact surface area of ​​the specimen, the dimension is mm 2 ;;a ’ cd 、b ’ cd 、a ’ td and b ’ td is the fitting parameter, dimensionless.

9. The dynamic and static modulus conversion model of asphalt pavement material based on tension and compression difference according to claim 7 is characterized by: The specific expression of the dynamic-static modulus conversion model is: Equation 9: E=A+B|E * |; In formula 9, E is the static modulus, A and B are fitting parameters, |E * | is the dynamic modulus.

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