In Science, What Is Shear Modulus?

Modulus of rigidity, the material constant, is the ratio of shear stress to strain. Also known as shear modulus or rigid modulus. One of the mechanical properties of materials. It is the ratio of the shear stress to the shear strain within the limit of the elastic deformation ratio under the action of shear stress. It characterizes the material's ability to resist shear strain. A large modulus indicates that the material is rigid. The reciprocal of the shear modulus is called the shear compliance. It is a measure of the shear strain that occurs under the unit shear force.

Chinese name: Shear modulus
Foreign name: shear modulus of elasticity

Definition: Ratio of shear stress to strain
Alias: Shear Modulus or Rigid Modulus

Calculation of Shear Modulus Internal Force

The stiffness parameter , the shear modulus G of the concrete used may be equal to 0.425E, where E is the elastic modulus of the concrete. There is a relationship between the shear modulus G, the elastic modulus E, and the Poisson's ratio : G = E / (2 (1 + )).

Shear modulus material test

With the widespread application of fiber-reinforced composite products, and product design using computers, especially the aerospace sector, military products, calculations

Shear modulus More and more accurate, therefore, the requirements for material properties are more comprehensive, such as the requirement to measure the interlayer shear modulus G13, G23 and other properties of composite laminates. According to our long-term practical experience and theoretical analysis, GB / T1456 three-point outreach beam bending method can be used to test the G13, G23, etc. of composite laminates. The characteristic of the three-point outreach beam bending method is that the bending elastic modulus of the beam material can be independently calculated by using the displacement (deflection) of the outstretched end of the beam. From the deflection in the beam and the displacement (deflection) of the overhanging end, the interlayer shear modulus of the beam material can be calculated at one time, and it is not necessary to solve simultaneous equations like the literature, which has significant advantages. ^[1]

Shear modulus dam building stone

Crushed rock (rockfill) is the main dam building material for rockfill dams. In order to better understand the equivalent dynamic shear modulus and equivalent damping bit of rockfill

Instrument for measuring shear modulus It provides a basis for the selection of material parameters in the rockfill dam seismic response analysis. The author uses a newly developed high-precision large-scale hydraulic servo triaxial instrument [1] to perform equivalent to more than ten simulated rockfill materials for several rockfill dam projects. Tests of dynamic shear modulus and equivalent damping ratio, necessary parameter conversion or homogenization according to a unified empirical formula, an estimation formula of the maximum equivalent dynamic shear modulus of rockfill materials is given, and it is compared with that at home and abroad. In-depth comparison of on-site elastic wave tests of 8 rockfill dams, comprehensive analysis of the dependence of the equivalent dynamic shear modulus, equivalent damping ratio and dynamic shear strain amplitude of various rockfill materials, and the statistical results of the test are given. The range of the normalized equivalent dynamic shear modulus and dynamic shear strain amplitude, and the equivalent damping ratio and dynamic shear strain amplitude are shown. ^[2]

Shear modulus test

The test materials used in this paper are artificial rockfill materials. The sample gradation curve based on the actual engineering design grading requirements and the triaxial instrument sample diameter simulation is shown in Figure 1. Among them, the three main rockfill materials of the Gongboxia rockfill dam use the same gradation curve. Table 1 lists test conditions such as lithology, average particle size, non-uniformity coefficient, initial porosity, and confining pressure of each sample. With the exception of Pubugou and Guanmenshan rockfill materials, the tests for other rockfill materials were carried out under isotropic consolidation conditions, and undrained conditions were adopted during vibration. The sample is prepared by the lamination method, and the test vibration frequency is 0.1Hz.

The nonlinear properties of soils usually adopt an equivalent linear model, that is, the soil is regarded as a viscoelastic body, and the equivalent dynamic elastic modulus Eeq (or dynamic shear modulus Geq) and equivalent damping ratio h are used to reflect the soil The non-linearity and hysteresis of the dynamic stress-strain relationship are expressed as a function of the dynamic strain amplitude. It should be pointed out that during the test, the load vibration of each stage is 12 to 15 times, and the measured stress-strain hysteresis curves at different loading cycles have some differences. The equivalent dynamic elastic modulus and damping ratio calculated from this are not exactly the same. Therefore, in the analysis and arrangement of the test results, the axial strain, the equivalent dynamic elastic modulus, and the damping ratio are all given as the average values from the 3rd to the 10th.

Analysis of Shear Modulus Results

2.1 Determination of the maximum equivalent dynamic elastic modulus (Eeq) max

The confidence of the minimum axial strain measured in the test is on the order of 10 ^-5 . Although there are some data less than 10 ^-5 in the test data, the dispersion is large.

Figure II Figure 2 shows the measured results of a set of equivalent dynamic spring modes and axial strain. Previous studies have shown that sand, gravel, and soft rock have linear elastic properties when the axial strain is less than 10 ^-5 under static or dynamic load conditions. Therefore, as shown in FIG. 2, the maximum equivalent dynamic elastic modulus (E _eq ) _max is calculated based on the assumption of linear elasticity of the rockfill material in the range of _a = 10 ^-6 to 10 ^-5 . This method is different from the methods proposed in some current geotechnical test codes. The code recommends to use the inverse of the intercept on the vertical axis of the relationship between 1 / E _eq and axial strain _{a to} obtain the maximum equivalent dynamic elastic modulus. In fact, this method is based on the assumption of a hyperbolic model. For rockfill materials, 1 / E _eq _a does not necessarily satisfy a linear relationship, and it contains more uncertainty or arbitrariness when extending experimental data.

2.2 Relationship between maximum equivalent dynamic shear modulus (G _eq ) _max and average effective stress _m

The measured maximum equivalent dynamic elastic modulus (E _eq ) max and the average effective stress _m can be approximated in a straight line in logarithmic coordinates, expressed as

(E _eq ) _max = k ⁿ m (1)

In the formula: k is the equivalent elastic modulus coefficient, n is the modulus index, and the unit of E _eq and _m is kPa.

In order to facilitate comparison, the maximum equivalent dynamic elastic modulus (E _eq ) _{max is} converted into the maximum equivalent dynamic shear modulus (G _eq ) _max , and F (e) is introduced to eliminate the effect of the porosity ratio, so the maximum equivalent dynamic shear Shear modulus can be expressed as

(G _eq ) _max = AF (e) ⁿ m (2)

Where: A is the equivalent shear modulus coefficient; e is the porosity ratio; F (e) = (2.17-e) ² / (1 + e) is the porosity ratio function; (G _eq ) _max is the maximum equivalent motion Shear modulus, (G _eq ) _max = (E _eq ) _max / 2 (1 + ), where Poisson's ratio is taken according to the test conditions, that is, 0.5 in the undrained state. Shear strain and axial strain _a 's relationship is

= _a (1 + ) (3)

Table II

Table 2 lists the equivalent elastic modulus coefficient k, equivalent shear modulus coefficient A, modulus index n, and porosity function F (e) of the 13 types of rockfills. As can be seen from Table 2, although the Lithology and weathering degree, initial porosity ratio and gradation (including average particle size, non-uniformity coefficient) all have large differences, but the variation range of the modulus index n is about 0.4 to 0.6. The range of equivalent shear modulus coefficient A is large, ranging from 2000 to 10,000. Figure 3 Summary

Figure three The test results of 13 rockfill materials completed in this paper are presented. In order to compare with the on-site elastic wave test results, regression analysis is performed on all test data to give the average and upper and lower envelope lines. It can be seen that the average modulus index is 0.5 and the average equivalent dynamic shear modulus coefficient is 7645.

2.3 On-site elastic wave test and indoor triaxial test

Results comparison. In the late 1970s and early 1980s, the Japan Electric Power Central Research Institute conducted elastic wave tests on 5 different rockfill dams in Japan and compared the test results with the indoor large-scale triaxial tests. The institute has conducted on-site elastic wave tests and indoor large-scale triaxial tests on two rockfill dams in Sanbao and Qisu. The author carried out on-site elastic wave test on Guanmenshan face rockfill dam in China and compared and analyzed it with the literature [5]. This article will again cite these results, and compare the average maximum equivalent dynamic shear modulus of the 13 rockfill materials measured in laboratory tests and their upper and lower envelopes to the shear wave velocity for comparison by the following formula

(4)

In the formula: g is the acceleration of gravity, 9.81m / s ² ; _t is the rockfill density, t / m ³ ; the unit of the maximum equivalent dynamic shear modulus (G _eq ) _max should be converted to t / m ² ; The unit of shear wave speed v _s is m / s.

It should be noted that the average effective stress _m = 1/3 (1 + ) (1 + K) _t ^z (6)

In the formula: the Poisson's ratio is taken as 0.35, the principal stress ratio K is taken as 1.5, and z is the depth m.

Figure 4 shows the comparison of the results from the on-site elastic wave test and the indoor triaxial test, where curve 4 is the average line equation suggested in Figure 3 of this article, and curves 5 and 6 are the upper and lower envelopes in Figure 3, respectively. Curve 7 is the result of the on-site elastic wave test of the Guanmenshan face dam.

Figure four

It can be seen that the scope given by the indoor large-scale triaxial test in this paper basically envelopes the results of field elastic wave tests of 8 rockfill dams in Japan and China. Modern rockfill dams use mechanized roller compaction construction technology. The density of the rockfill dams is relatively high and all are close. Therefore, the results of the field elastic wave tests of the 8 rockfill dams are basically consistent. In general, the results obtained by the indoor large-scale triaxial test are lower than the results of the on-site elastic wave test. This is mainly due to the stable structure of the actual engineering rockfill particles, and the rockfill materials were severely disturbed and the samples were tested during the indoor test. Due to size restrictions.

2.4 Relationship between normalized equivalent dynamic shear modulus G _eq / (G _eq ) _max and dynamic shear strain amplitude

Figure 5 gives a typical example of the dependence of the normalized equivalent dynamic shear modulus on the dynamic shear strain amplitude, that is, the two main rockfill dams of Jilintai and Hongjiadu.

Figure five

Stone test results. In general, the normalized equivalent dynamic shear modulus decreases with increasing dynamic shear strain amplitude, and the degree of attenuation is mainly affected by the confining pressure _c or the average effective stress _m . The lower the confining pressure, the faster the normalized equivalent dynamic shear modulus decays (ie, the attenuation curve is to the lower left), which is similar to the research results of sand. It can be seen from Fig. 5 that the normalized equivalent dynamic shear modulus varies with the dynamic shear strain amplitude within a certain range, and the variation range varies with different materials. The upper limit of Hongjiadu rockfill is slightly higher than that of Jilintai rockfill, and the range of normalized equivalent dynamic shear modulus with dynamic shear strain amplitude is also larger than that of Jilintai rockfill. But overall, the difference is not very significant.

In order to compare the test results of various rockfill materials, the dependence of the normalized equivalent dynamic shear modulus and dynamic shear strain amplitude of the various rockfill materials measured by the author using this method is summarized in Figure 6. Each curve represents the average value of a test rockfill material G _eq / (G _eq ) _max to . From the results in the figure, we can see that although the lithology and gradation of these rockfills are quite different, and the range of the maximum equivalent dynamic shear modulus is also large, the normalization of various rockfills, etc. The dispersion of the dependence of the effective dynamic shear modulus and the dynamic shear strain amplitude is not large. In order to facilitate the application, the test results of various rockfill materials in Figure 6 are averaged again, and the range of normalized equivalent dynamic shear modulus and dynamic shear strain dependence of general rockfill materials is suggested as shown in Figure 7. As shown.

Figures 6, 7, 8, 9

2.5 Relationship between equivalent damping ratio h and dynamic shear strain amplitude

A large number of studies have shown that the higher the dynamic shear modulus, the lower the equivalent damping ratio. The equivalent damping ratio not only increases with the increase of the dynamic shear strain amplitude , but also depends on the confining pressure _c or the average effective stress _m With the same dynamic shear strain amplitude, the confining pressure _c increases and the equivalent damping ratio decreases. In addition, the consolidation stress ratio K also affects the equivalent damping ratio. That is, under the same confining pressure _c and dynamic shear strain amplitude, the equivalent damping ratio decreases when the consolidation stress ratio K increases. This paper summarizes the relationship between the equivalent damping ratio of various rockfill materials and the dynamic shear strain amplitude as shown in Fig. 8. Each curve in the figure represents the average value of the h ~ variation range of a test rockfill material. It can be seen that the dispersion of the equivalent damping ratio of various rockfill materials as a function of the dynamic shear strain amplitude is greater than the normalized equivalent dynamic shear modulus dispersion of the dynamic shear strain amplitude. FIG. 9 is an average treatment of the test results of various rockfill materials in FIG. 8. It is suggested that the range of value dependence of equivalent damping ratio and dynamic shear strain amplitude of general rockfill materials is suggested. Generally speaking, the equivalent damping ratio of rockfill materials is not high. When the dynamic shear strain amplitude = 10 ^-5 , the equivalent damping ratio is about 2%, and when = 10 ^-4 , the equivalent damping ratio is close to 5%. When the dynamic shear strain amplitude is greater than = 10 ^-4 , the damping ratio rises faster, which indicates that the rockfill material enters a stronger nonlinearity, and the phenomenon that the strain lags behind the stress becomes more obvious. It should be pointed out that the discrete range of the equivalent damping ratio is relatively large. This is caused by the uncertainty contained in the rockfill itself, and it is also related to the analysis and arrangement of the test data. ^[3]

Shear Modulus Report Results

(1) The estimation formula of the maximum equivalent dynamic shear modulus of more than ten types of rockfill materials based on indoor high-precision large-scale triaxial tests in this paper is basically consistent with the results of on-site elastic wave test of 8 rockfill dams at home and abroad. Although the grading, initial porosity, lithology, and weathering degree of rockfill dam materials are different, but due to the use of heavy-duty mechanized construction, the actual filling density of modern rockfill dams is high, and the dam body is sheared. The wave velocity distribution is also approximately close.

(2) When rockfill dam seismic response analysis has not been obtained before the rockfill test data is obtained, the maximum equivalent dynamic shear modulus can be roughly estimated with reference to Figures 3 and 4 of this paper, and the normalization is determined with reference to Figures 7 and 9 The relationship between equivalent dynamic shear modulus, equivalent damping ratio and dynamic shear strain amplitude. The influence of rock hardness and static shear strength on the maximum equivalent dynamic shear modulus and attenuation relationship should be considered when selecting the calculation parameters. It should be said that according to the formula proposed in this article or the range estimation given, it can meet the needs of the project.

(3) Compared with clay and sand, there are many difficulties in the testing equipment and testing technology of dam-filled rockfill materials. So far, there are few test data on the dynamic shear modulus and damping ratio of rockfill materials. The author will further accumulate data for further research.

Shear modulus related content

The material is deformed by external forces. When the external force is small, elastic deformation occurs. Elastic deformation is reversible. When unloaded, the deformation disappears and returns to its original state. In the range of elastic deformation, the linear relationship between stress and strain is maintained, that is, it obeys Hooke's law:

Elastic modulus is a physical quantity that characterizes the strength of the bonding between atoms in a crystal, so it is an insensitive parameter of the tissue structure. In engineering, the elastic modulus is a measure of the stiffness of a material.

In fact, there is no ideal elastic body. When most engineering materials are elastically deformed, elastic incompleteness such as the misalignment of the loading line and the unloading line, and the strain lagging behind the stress change may occur. Elastic incompleteness includes Bauschinger effect, elastic aftereffect, elastic hysteresis and cyclic toughness.

For amorphous, and even for some polycrystalline, viscoelastic phenomena may occur with less stress. Viscoelastic deformation is both time-dependent and recoverable elastic deformation, that is, it has two characteristics of elastic and viscous deformation. Viscoelastic deformation is one of the important mechanical properties of polymer materials.

When the applied stress exceeds the elastic limit, the material undergoes plastic deformation, that is, irreversible permanent deformation. Through plastic deformation, not only can the material obtain the desired external dimensions, but also the internal structure and properties of the material can be changed.

Two basic ways of plastic deformation of a single crystal are slip and twinning. Slip and twinning are both shear strains, and can only begin when the component of the applied shear stress is greater than the critical shear stress tC of the crystal. However, slip is uneven shear and twins are uniform shear.

For polycrystals, each grain must have at least 5 independent slip systems to meet the constraints and coordination of each grain during the deformation process. In polycrystals, the presence of grain boundaries at room temperature hinders slippage, and it has been proven that the strength of polycrystals increases with the refinement of their grains, which can be described by the well-known Hall-Petch formula

Shear modulus spring steel

There are many types of metal spring materials, and spring steel is widely used. When selecting spring steel for calculation of spring design, the shear modulus or elastic modulus of the material is used. Almost all design materials and related textbooks at home and abroad, as well as GB / T 123.9.6-92 "Design and Calculation of Cylindrical Helical Springs", give the shear modulus of metal spring materials as fixed values. But among the cylindrical coil springs, volute springs, non-linear characteristic line coil springs, and multi-strand coil springs, if the values of the shear modulus given in the above traditional design data are taken, then the calculated spring deformation amount is actually There is a large error in the amount of deformation measured. Now take the force measuring spring used in the NYL-2000 type pressure testing machine produced by our factory as an example to try to describe as follows.

1 Designed and calculated spring elongation and measured elongation

Large and small force measuring springs (processed by Shanghai China Spring Factory) are ordinary cylindrical spiral tension springs. The spring material is 60Si2MnA, and heat treatment is 45-50HRc. Some of its design parameters are shown in Table 1.

Table I

For example, according to the design parameters in Table 1, and taking the traditional shear modulus value G = 8 × 10 ⁴ MPa, the calculated elongations of the large and small force-measuring springs under the rated load are 91.55mm and 90.85mm, respectively.

As is known to all, because of the finished springs after processing, especially the springs that are hot-wound and require heat treatment, there is inevitably a certain dimensional deviation. For example, the diameter of the spring steel wire and the middle diameter of the spring may be different from the parameters at the time of design, and even have large deviations. This leads to a certain error between the actual extension of the spring and the calculated extension. Table 2 is the comparison of the elongation calculated by the author based on the relevant dimensions of the spring measured during the inspection, and then based on the traditional material shear modulus.

Table II

It can be seen from Table 2 that the elongation at the rated load, in which the elongation calculated based on the actual measured dimensions of the spring, is larger than the calculated elongation by design (-1.76 to 20.93) mm and (0.34 ~ 22.16) mm. However, it still differs from the actual measured value by 3.21% to 4.15%. Why is the design calculation of the spring elongation so different from the actual measured value? As stated in "Spring": "Even the most precise and careful calculation of the characteristic line of a spring, the results and actual values always have a certain degree. The difference is due to the inevitable existence of certain process errors in the manufactured spring and the non-absolute uniformity of the material structure. " "Because of the dimensional error and the influence of material factors, the calculated characteristic line is different from the measured value." "Therefore, springs that have stricter requirements on the characteristic line should be tested and repeatedly modified before they can be produced in batches." It can be seen that there is indeed a certain error between the measured value of the spring deformation and its design calculation. However, why does the elongation calculated based on the actual measured spring size still have a larger error than the actual measured value? The author believes that excluding the "dimensional error" (including measurement error) and "material factor" (internal organization) of the spring (Non-absolute uniformity), the error between the actual elongation of the spring and the calculated elongation is mainly due to the change in the shear modulus of the spring material after heat treatment.

2 Shear modulus of heat treated spring steel

In order for the spring to obtain higher yield limit, elastic limit, high yield strength ratio and fatigue strength, the spring is generally subjected to heat treatment. However, the elastic modulus and the shear modulus of the heat-treated spring material have changed. Among them, the shear modulus changes greatly. For example, the elastic modulus and shear modulus of the commonly used spring steel 60Si2MnA after quenching and tempering at different temperatures are listed in Table 3.

Table three

Table 3 shows that the shear modulus G of the spring material after quenching and tempering changes greatly, and increases with the increase of the tempering temperature in a certain range, and is no longer the traditional 8 × 104MPa.

3 Comparison of spring elongation calculated from shear modulus after heat treatment with actual measured value

For example, the shear modulus value after tempering at 450 ° C in Table 3 is 83160 MPa, and the hardness is about 47HRc. The calculated results are shown in Table 4 according to the measured dimensions of the force measuring spring in Table 2.

Table four

Obviously, the spring elongation calculated in Table 4 based on the value of the shear modulus after heat treatment is close to the actual measured value. The biggest error is -0.71%. This shows that when the spring size, load, etc. are the same, its elongation depends on the shear modulus of the material. Or when other conditions are not considered, the shear modulus of the material is changed only by heat treatment. For example, the shear modulus of 60Si2MnA after tempering at 450 is 83160MPa, which can make the spring The difference between the deformation amounts is about 3.95%; and the difference from the 78 × 10 ³ N / mm ² specified in GB / T1239.6-92 is 6.62%. If the spring material is chrome-vanadium steel, such as 50CrVA, the shear modulus G value when tempered at 600 (hardness is about 47.5HRc) is 86600MPa [6] G = 8 × 10 ⁴ MPa and 78 × 10 ³ N / mm ² , the difference is 8.25% and 11.03% respectively. That is, when the spring material, wire diameter, spring middle diameter, effective turns, structure, load, etc. remain unchanged, it is only because the shear modulus value of the material is changed after heat treatment, which will make the spring deformation amount earlier. In the design calculation, a congenital error of 3.95% or 6.62%, or even 8.25% or 11.03%, has occurred. This error is not caused by the size of the spring and the non-uniformity of the internal structure of the material, but by artificial mishandling or neglecting the effect of heat treatment on the shear modulus of the material. Because the shear modulus is not only inherent to the material itself, but also related to the heat treatment state, and determines the relationship between the amount of spring deformation and the load. For this reason, the author believes that when designing and calculating coil springs with high characteristics, it should be based on the service conditions of the spring, such as working temperature and load, and the influence of heat treatment on its shear modulus should be considered. That is, the value of the shear modulus of the spring material after heat treatment is taken, instead of the traditional given value. Even for coil springs that do not require high characteristic lines, the change in the shear modulus of the spring after heat treatment should not be considered. As for what value should be taken, it is mainly determined according to the working conditions and load properties of the spring. Generally, springs need to be quenched and tempered at moderate temperatures. According to GB / T1239.6-92, heat treatment 45HRc 50HRc. Just select the shear modulus within the corresponding tempering temperature and hardness requirements.

When the alloy is a single-phase solid solution, due to the presence of solute atoms, it will exhibit a solid solution strengthening effect, and some materials will also experience yielding and strain aging. When the alloy has a multi-phase structure, its deformation will be affected by the second phase Effect, showing a diffuse strengthening effect.

Ceramic crystals, due to the nature of their bonding bonds (ionic bonds, covalent bonds), coupled with the small slip system in ceramic crystals, and large dislocations b, make their plastic deformation much more difficult than metallic materials, only Single crystal ceramics dominated by ionic bonding can undergo large plastic deformation. For polymer materials, the plastic deformation is caused by viscous flow instead of slip, so it is closely related to the viscosity of the material, and it is greatly affected by temperature.

After the material is plastically deformed, the work part of the external force exists in the material in the form of stored energy, so that the free energy of the system increases and is in an unstable state. Therefore, recovery recrystallization is the spontaneous tendency of the material after cold deformation, and heating accelerates this process.

When the heating temperature is low and the time is short, recovery occurs. At this time, the main manifestation is the change of the substructure and the process of multilateralization. The first type of internal stress is largely eliminated, and the resistivity is reduced, but it has little effect on the structure and mechanical properties.

When the heating temperature is high, the recrystallization occurs when the time is long. During recrystallization, new distortion-free equiaxed crystals will replace the cold-deformed structure, and their properties will basically return to the state before the cold-deformation.

When heating is continued after recrystallization is complete, grain growth will occur.