Method for eliminating error of quasi-static compression mechanics experimental system

Abstract

The present application belongs to the field of mechanics, and particularly relates to a method for eliminating error of a quasi-static compression mechanics experiment system. By using a constant pressure or a constant loading speed of a pressure head on a testing machine to perform quasi-static uniaxial compression on a test piece at room temperature, and taking two test pieces of different heights, the relationship between the system stiffness, the force and deformation of the system pressure head can be obtained, so as to obtain two corresponding force-time curves and displacement-time curves. By combining the two equations and eliminating the unknown Young's modulus E, the stiffness coefficient of the test system can be obtained, and the displacement-time curve of the pressure head can be obtained. Subsequently, the Young's modulus E can be inversely solved. The present application corrects the error of the mechanics experiment by defining the system stiffness coefficient, so as to realize a higher precision of the Young's modulus E and improve the accuracy of the quasi-static mechanics experiment.

Description

Technical Field

[0001] This invention belongs to the field of mechanics, and specifically relates to a method for eliminating systematic errors in quasi-static compression mechanics experiments. Background Technology

[0002] Quasi-static compression tests are conducted to assess the compressive mechanical properties of sandwich structures. These tests reveal the structure's compressive stress-strain characteristics, deformation process, and failure modes. This has significant engineering value for describing the mechanical properties of materials and obtaining stress-strain relationships at low strain rates.

[0003] There are many methods for testing stress and strain, with quasi-static compression testing being widely recognized as the most common and effective method. However, during quasi-static compression, the stiffness of the testing system has a significant impact on the experimental results, leading to low accuracy in the displacement data obtained by the MTS testing machine. Consequently, the obtained strain values ​​are not very accurate. In the elastic stage, due to the small strain values, these experimental errors severely affect the accuracy of the yield strain of Young's modulus, and consequently the accuracy of important mechanical parameters such as sound velocity. Therefore, it is generally not recommended to use the displacement values ​​given by the testing system, as this involves the issue of experimental accuracy. However, there has been no in-depth research on this aspect both domestically and internationally. Summary of the Invention

[0004] The purpose of this invention is to provide a method for eliminating systematic errors in quasi-static compression mechanics experiments.

[0005] The technical solution to achieve the objective of this invention is: a method for eliminating systematic errors in quasi-static compression mechanics experiments, comprising the following steps:

[0006] Step (1): Prepare n specimens with different heights, the same material, and the same diameter, with initial heights of H1, H2, ... H2 respectively. n The initial cross-sectional area of ​​each specimen is A0. n specimens are subjected to quasi-static compression mechanics experiments on an MTS universal testing machine, and their respective indenter displacement time history curves Δl are recorded. i (t) and the time history curve of the indenter force F i (t), where i takes values ​​from 1 to n;

[0007] Step (2): Define the system error stiffness coefficient of the MTS universal testing machine as κ, and define the force F on the system indenter. i With deformation δL i1 The relationship between them;

[0008] Step (3): Solve for the true displacement time history curve ΔL i (t), the true displacement time history curve ΔL i (t) and deformation δL i1the displacement-time curve of the system pressure head △l i (t);

[0009] Step (4): the system pressure head force F defined by step (2) i and the relationship between the deformation δL i1 , the n group function relationship about elastic modulus E and system error stiffness coefficient κ is derived, and the elastic modulus E is solved by n group function relationship.

[0010] Step (5): after solving the error stiffness coefficient κ, according to the relationship between the system pressure head force F defined by step (2) i and the deformation δL i1 , the real elastic modulus E of the required material is solved.

[0011] Further, the zero order stiffness correction method is adopted, and the specific method is as follows:

[0012] Step (1): prepare two test pieces with initial height H1 and H2, wherein H1 ≠ H2, set the initial length of the test piece as L0, the initial cross-sectional area as A0, place the test piece on the MTS universal material testing machine, set the loading speed to 1kN / s and load for 15s, record the force-time curve F1(t) and F2(t) of the two test pieces, and record the displacement-time curve △l1(t) and △l2(t) of the test system;

[0013] Step (2): define the system error stiffness coefficient of the MTS universal material testing machine as κ, and define the relationship between the system pressure head force F i and the deformation δL i :

[0014] F i = κ·δL i ;

[0015] Step (3): solve the real displacement-time curve:

[0016]

[0017] In the formula, △L1(t) and △L2(t) represent the real displacement-time curves of test piece 1 and test piece 2 respectively, and under ideal conditions, the stiffness of the test system is infinite relative to the Young's modulus of the material, at this time the formula can be simplified as:

[0018]

[0019] Step (4): solve the system error stiffness coefficient κ:

[0020] For super strong materials, the change of cross-sectional area is ignored in the elastic stage, that is:

[0021]

[0022] The above formula eliminates the unknown Young's modulus E, and the stiffness coefficient of the test system can be obtained:

[0023]

[0024] Step (5): Solve the real elastic modulus E of the required material: the stiffness coefficient is substituted back, and the Young's modulus can be given:

[0025]

[0026] Further, the first-order stiffness correction method is adopted, which is specifically: based on the formula F i = κ·δL i , considering the weight of the material testing machine pressure head and the hydraulic characteristics, the system pressure head force F i and the deformation δL i satisfy:

[0027] F i = κ·(δL i -δL0)

[0028] That is, there is a critical deformation δL0;

[0029] At this time, there is

[0030]

[0031] If the critical deformation is not considered, the above formula is simplified as:

[0032]

[0033] In the elastic stage, the change of the cross-sectional area is ignored, that is:

[0034]

[0035] Using the first-order stiffness correction method, at least three groups of uniaxial compression tests with different heights but the same cross-sectional area need to be carried out, and at this time, the above formula can be expressed as:

[0036]

[0037] According to the above formula, the intercept can be obtained:

[0038]

[0039] The expression of the Young's modulus is:

[0040]

[0041] And the stiffness coefficient expression is:

[0042]

[0043] In the formula

[0044]

[0045] Substitute the above four formulas into any one of the following equations to obtain the corrected engineering stress-strain curve of the test piece:

[0046]

[0047] Further, the second-order stiffness correction method is adopted, specifically:

[0048] Assume that the force F of the system pressure head i and the deformation δL i satisfy the quadratic function relationship:

[0049]

[0050] That is, the critical deformation δL i0 and the nonlinear relationship between the system stiffness and the deformation are considered simultaneously; in the formula,

[0051]

[0052] respectively represent the dimensionless deformation and the dimensionless critical deformation;

[0053] At this time, that is,

[0054] △l(t) = △L + δL i = △L + L i [F(t), κ1, κ2, δL0]

[0055] That is,

[0056] △L = △l(t) - L i [F(t), κ1, κ2, δL0]

[0057] In the formula, the △l(t) and F(t) curves can be given simultaneously by testing, and are known quantities; thus there are still four unknown quantities, and at least four independent equations are required to give the analytical solution; that is, at least four groups of test pieces with different heights but the same diameter are tested on the same testing machine, so as to obtain:

[0058]

[0059] The critical deformation of the system and the two stiffness coefficients are given, so that the Young's modulus of the test piece and the corrected engineering stress and engineering strain curve of the test piece can be obtained.

[0060] Compared with the prior art, the present application has the following advantages:

[0061] (1) The present application can obtain the error stiffness coefficient K of the system, inversely solve the Young's modulus E, and thus obtain the Young's modulus E with higher accuracy, so as to improve the accuracy of the quasi-static mechanical experiment to a certain extent.

[0062] (2) The present application can not only improve the accuracy of the Young's modulus in the quasi-static experiment, but also has a significant influence on important parameters such as yield stress and yield strain in the quasi-static mechanical property experiment. In actual experiments, the accuracy of the selection of yield stress and yield strain is largely dependent on the accuracy of the Young's modulus. This method can obtain more accurate yield stress and yield strain.

[0063] (3) The present application is beneficial to reducing the experimental cost of materials in the field of mechanical technology. Compared with a large number of experiments for fitting, the present application can greatly reduce the number of test pieces by using theoretical analysis and experimental methods, and improve the practicability of the machine. Many old machines can also perform relatively high-precision analysis experiments after calculating the system stiffness, which is more in line with the needs of the mechanical laboratory and will be conducive to the development of social economy. BRIEF DESCRIPTION OF DRAWINGS

[0064] Figure 1 is the stress-strain diagram of the T12A steel test piece compression curve obtained by the extensometer method.

[0065] Figure 2 is the stress-strain diagram of the T12A steel test piece compression curve obtained by the DIC method.

[0066] Figure 3 is the stress-strain curve diagram directly obtained by the T12A steel test piece compression curve testing machine.

[0067] Figure 4 is the stress-strain diagram of the T12A steel test piece compression curve obtained by the present application. DETAILED DESCRIPTION

[0068] The present application will be further described in detail below with reference to the accompanying drawings.

[0069] S1, prepare and prepare n test pieces of the same material, the same diameter and different heights, the initial heights of which are H1, H2, H3……Hn respectively, and the heights of which are all different, the initial cross-sectional areas of which are all A0, place the n test pieces on the MTS universal material testing machine to perform repeated experiments n times, and record the respective pressure head displacement time history curves Δl1, Δl2, Δl3……Δln. n i ​(t) and the time history curve of the pressure head stress F i (t), wherein i is 1 to n.

[0070] S2, define the error stiffness coefficient of the MTS universal material testing machine itself as K, define the time history curve of the system pressure head stress F i and the deformation delta L i .

[0071] S3, due to the influence of the system stiffness error, the time history curve of the pressure head displacement and the actual displacement time history curve of the test piece have great error, so the true displacement curve delta L i (t) is solved. i (t) and delta L i1 . i (t) is the time history curve of the displacement of the system pressure head.

[0072] S4, the n sets of function relationship formulas about the elastic modulus E and the system error stiffness coefficient K can be derived from the relationship formula in S2, that is, the n sets of relationship formulas can be used to eliminate the elastic modulus E to inversely solve the system error stiffness coefficient K inherent in the system.

[0073] S5, after the error stiffness coefficient K is solved, the true elastic modulus E of the required material can be solved in the elastic stage by using the relationship between the system pressure head stress F i and the deformation delta L i .

[0074] In this embodiment, the function expressions of the time history curves of the pressure head displacement and the pressure head stress obtained by the test pieces of different heights are solved to explore the system stiffness of the machine itself and the higher precision Young's modulus.

[0075] In this embodiment, under the condition of room temperature, the pressure head on the testing machine is used to perform quasi-static uniaxial compression on the test piece at a constant loading speed, but the current used material strength is very large, in order to protect the loading equipment and ensure that the pressure heads below and above the test piece do not yield or break during the loading process, high-strength and high-hardness pad blocks need to be added to the upper and lower ends of the test piece.

[0076] Specifically, the specific experimental process of this embodiment is as follows:

[0077] The present application includes various experimental analysis methods: zero-order stiffness correction method, first-order stiffness correction method, second-order stiffness correction method and higher order, etc., and the zero-order stiffness correction method is used for analysis experiment in this embodiment.

[0078] (1) Experimental material: zero-order stiffness correction method: prepare two test pieces with initial heights of H1 and H2, and H1≠H2, and record the cross-sectional area A0 of the material before the experiment.

[0079] (2) Experimental equipment: MTS 810 universal material testing machine.

[0080] (3) Experimental method: First, the relevant theory needs to be clear: the initial length of the specimen is L0, and the initial cross-sectional area is A0. According to the pressure-time curve F(t) and the displacement-time curve s(t) given by the testing machine, it needs to be explained that the zero time of the pressure-time curve and the displacement-time curve corresponds to the time when the specimen starts to compress and produces force and deformation. The engineering stress-time curve can be given:

[0081]

[0082] In theory, the true stress (true stress for short) and the true strain (true strain for short) of the specimen are calculated as follows:

[0083]

[0084] Since the cross-sectional area and the length of the uniform compression interval of the specimen are constantly changing, except for zero time, generally

[0085]

[0086] For uniaxial compression test, we have:

[0087]

[0088] Thus we can get:

[0089]

[0090] According to the axial strain ε v (t) and the hoop strain ε h (t) in the elastic stage, the Poisson's ratio of the material can be calculated as:

[0091]

[0092] The negative sign in the above formula indicates that the axial strain and the hoop strain are in different directions.

[0093] In uniaxial compression test, the strains in two perpendicular directions on the horizontal hoop plane are equal:

[0094] ε h1 (t) = ε h2 (t) (7)

[0095] For metal materials, the bulk strain θ during compression is not greater than 0, i.e.

[0096] θ = ε v (t) + ε h1 (t) + ε h2 (t) ≤ 0 (8)

[0097] i.e.

[0098]

[0099] In plastic flow stage, since for metal material, its plastic stage volume is incompressible, i.e. the body strain is 0, at this time

[0100] v = 0.5 (10)

[0101] In elastic stage, since the axial strain of the specimen is small, according to formula (6), therefore the hoop strain is smaller, so we can ignore the change of the cross-sectional area of the specimen in the compression process, i.e. we can consider that:

[0102]

[0103] Combined with formula (2), in the elastic stage, since the deformation is very small, we can approximately consider that the engineering stress and engineering strain are equal to the corresponding true stress and true strain:

[0104]

[0105] And in the plastic deformation stage, as described above, since the plastic incompressible assumption Poisson's ratio is approximately 0.5, at this time

[0106]

[0107] Let the system stiffness be constant K, the system pressure head force F i and deformation δL i satisfy:

[0108] F i = κ·δL i (14)

[0109] According to formula (13), the engineering stress engineering strain curve of the specimen can be converted into the true stress true strain curve.

[0110] However, in the quasi-static compression process, the stiffness of the test system has a great influence on the experimental results, so the displacement data obtained by the MTS testing machine has low precision; thus, the accuracy of the obtained strain value is not high. For the elastic stage, since the strain value is small, these test errors seriously affect the accuracy of the yield strain of Young's modulus, and further affect the accuracy of important mechanical parameters such as sound velocity; therefore, it is generally not recommended to use the displacement value given by the test system. The current relatively accurate conventional methods for giving real-time deformation of the test piece mainly include the strain gauge method and the extensometer method. Under the condition that the size of the test piece and other conditions are permitted, in general, the data given by the strain gauge method is intuitive, reliable and accurate; the extensometer method is relatively simple and accurate, and its precision is slightly worse than that of the strain gauge method. In addition to the above two basic methods, there are currently popular DIC and video extensometer methods based on graphical image shooting and data processing, which can give strain field and relatively accurate test results. The test curves given by the four methods, especially the strain curve accuracy, are all greater than the data given by the material testing machine, as shown in Figure 1 、 Figure 2 and Figure 3

[0111] By comparison Figures 1-3 , it can be found that the yield stress given by the three methods is approximately 3.57GPa, but the stress-strain curve, especially the Young's modulus, is not the same; the Young's modulus given by the DIC method is 183.22GPa, which is slightly larger than that of the extensometer method, 173.23GPa, and the values given by the two methods are relatively close, both of which are significantly greater than the value obtained by the material testing machine itself. Since the Young's modulus value of typical metal steel materials is basically 200GPa, the curve given by the material testing machine is obviously inaccurate, only 69.69GPa, indicating that when using the MTS testing machine directly, the given curve and data are not accurate and cannot be directly used.

[0112] Specifically, the test piece is placed on the MTS universal material testing machine and the center position is adjusted, and is fixed with a clamp. A loading speed of 1kN / s is set for 15s, and after loading is completed, unloading is carried out, and the experiment is ended. The force-time curves of the two test pieces are recorded as F1(t) and F2(t), respectively. At this time, the displacement-time curves of the test system are given as △l1(t) and △l2(t), respectively, and it is easy to give, in the elastic stage:

[0113] wherein △L1(t) and △L2(t) represent the real length deformation time curves of test piece 1 and test piece 2, respectively.

[0114]

[0115] ​In the formula, ΔL1(t) and ΔL2(t) represent the time history curves of the actual length deformation of specimen 1 and specimen 2, respectively. Under ideal conditions, the stiffness of the test system is infinitely large relative to the Young's modulus of the material, and the above formula can be simplified to:

[0116]

[0117] For soft materials or relatively soft metals, the error caused by the above assumptions is limited; however, for ultra-strong materials, the influence of the system's own stiffness on the strain measurement results cannot be ignored.

[0118] Neglecting the change in cross-sectional area during the elastic phase, we have:

[0119]

[0120] Eliminating the unknown Young's modulus E from the above equation yields the stiffness coefficients of the experimental system:

[0121]

[0122] Substituting into equation (17), we can obtain Young's modulus:

[0123]

[0124] However, if we consider the entire elastoplastic compression process, then according to equation (15):

[0125]

[0126] According to equations (19) and (20), based on the force and displacement curves given by the testing machine, by eliminating the system stiffness error, a relatively accurate Young's modulus and engineering stress-strain curves of the material can be given, and then the uniaxial compressive true stress-strain curves of the material can be given.

[0127] right Figure 3 By performing stiffness correction, the Young's modulus data for this T12A steel material can be provided, such as... Figure 4 As shown in the figure, the system stiffness coefficients given by different specimens are all close to 263.72 kN / mm, and the Young's modulus obtained after stiffness correction is also very close, with an average value of 196.49 GPa, which is much larger than the 69.69 GPa before stiffness correction. Comparing the Young's modulus result of 196.49 GPa given by the zero-order stiffness correction method and 183.22 GPa given by the DIC method, it can be seen that the two are relatively close; combined with the Young's modulus of 197.57 GPa given by the DIC method in the tensile test of this material, it can be seen that the Young's modulus given by the stiffness correction method is quite accurate; and comparing it with the result given by the extensometer method, it can be seen that the Young's modulus obtained by the latter is smaller.

[0128] The method can be adjusted according to different testing machines and different material properties. If higher accuracy is required, higher order stiffness correction method can be used, which is of great significance to guide the mechanical property test method, and the order can be extended to n order. The following is a higher order stiffness correction method:

[0129] 1. First-order stiffness correction method - two-constant linear stiffness correction method

[0130] Zero-order stiffness correction method is always an idealized stiffness correction method, and its premise is that formula (14) is established. If the weight of the material testing machine pressure head and the hydraulic characteristics are initially considered, the system pressure head stress F can be further defined as i and the deformation δL i satisfy:

[0131] F i = κ·(δL i - δL0) (21)

[0132] That is, there is a critical deformation δL0.

[0133] At this time, there is

[0134]

[0135] If the critical deformation is not considered, the above formula can be simplified to formula (15). The above formula can also be written as:

[0136]

[0137] In the elastic stage, the change of the cross-sectional area is ignored, that is:

[0138]

[0139] It is easy to see that there are two unknown quantities in the above formula, but only two independent equations, which cannot give an accurate analytical solution; therefore, using the first-order stiffness correction method, at least three groups of uniaxial compression tests with different heights but the same cross-sectional area need to be carried out, at this time, the above formula can be expressed as:

[0140]

[0141] According to the above formula, the intercept can be obtained:

[0142]

[0143] The expression of Young's modulus is:

[0144]

[0145] And the stiffness coefficient expression is:

[0146]

[0147] where

[0148]

[0149] Substitute the above four formulas into any one of the following equations to obtain the corrected engineering stress-strain curve of the test specimen:

[0150]

[0151] 2. Second-order stiffness correction method - two-constant linear stiffness correction method

[0152] The so-called second-order stiffness correction method is a zero-order stiffness correction method that more accurately assumes that the system pressure head force F i and the deformation δL i satisfy a quadratic function relationship:

[0153]

[0154] That is, the critical deformation δL i0 and the nonlinear relationship between the system stiffness and the deformation are considered simultaneously. In the formula,

[0155]

[0156] respectively represent the dimensionless deformation and the dimensionless critical deformation.

[0157] At this time, there is

[0158] △l(t) = △L + δL i = △L + L i [F(t), κ1, κ2, δL0] (33)

[0159] That is,

[0160] △L = △l(t) - L i [F(t), κ1, κ2, δL0] (34)

[0161] In the formula, the △l(t) and F(t) curves can be given simultaneously by testing, and are known quantities. Thus, there are still four unknown quantities, and at least four independent equations are needed to give the analytical solution; that is, at least four test specimens with different heights but the same diameter need to be tested on the same testing machine to obtain:

[0162]

[0163] The system critical deformation and the two stiffness coefficients can be obtained by combining the above formula and formula (31), and thus the Young's modulus of the test specimen and the corrected engineering stress and engineering strain curve of the test specimen can be obtained.

Claims

1. A method for eliminating errors of a quasi-static compression mechanics experimental system, characterized in that, Comprising the following steps: Step (1): Prepare n specimens with different heights, the same material, and the same diameter, with initial heights of H1, H2, ... H2 respectively. n The initial cross-sectional area of ​​each specimen is A0. n specimens are subjected to quasi-static compression mechanics experiments on an MTS universal testing machine, and their respective indenter displacement time history curves Δl are recorded. i (t) and the time history curve of the indenter force F i (t), where i takes values ​​from 1 to n, and n takes values ​​that are positive integers not less than 4; Step (2): define the system error stiffness coefficient of the MTS universal material testing machine as K, and define the system pressure head force F i The relationship between the deformation δL i1 ; Step (3): solving the true displacement time history curve ΔL i (t), the true displacement time history curve ΔL i (t) and the deformation δL i1 of their sum is the displacement time history curve Δl i (t) of the system pressure head; Step (4): the system pressure head force F defined by step (2) i and the relationship between the deformation δL i1 between the elastic modulus E and the system error stiffness coefficient κ, and the n sets of function relationship formulas are used to eliminate the elastic modulus E to inversely solve the system error stiffness coefficient κ. Step (5): After solving the error stiffness coefficient K, the system pressure force F is calculated according to the system pressure force F defined in step (2) i The relationship between the deformation δL i1 The real elastic modulus E of the required material is solved.

2. The method of claim 1, wherein, The zero-order stiffness correction method is adopted, and the specific method is as follows: Step (1): Prepare two test pieces with initial heights H1 and H2, wherein H1≠H2, and the initial length of the test piece is L0 and the initial cross-sectional area is A0. Place the test piece on the MTS universal material testing machine, set the loading speed to 1kN / s, load for 15s, record the force-time curve F1(t) and F2(t) of the two test pieces, and record the displacement-time curve △l1(t) and △l2(t) given by the test system; Step (2): define the system error stiffness coefficient of the MTS universal material testing machine as K, and define the system pressure head force F i The relationship between the deformation δL i and the relationship between the deformation δL F i = κ · δL i ; Step (3): Solve the real displacement-time curve: In the formula, △L1(t) and △L2(t) represent the real displacement-time curves of test piece 1 and test piece 2 respectively. Under ideal conditions, the stiffness of the test system is infinite relative to the Young's modulus of the material, and at this time the formula can be simplified as: Step (4): Solve the system error stiffness coefficient κ: For super-strong materials, the change in cross-sectional area is ignored in the elastic stage, that is: The above formula eliminates the unknown Young's modulus E, and the stiffness coefficient of the test system can be obtained: Step (5): Solve the real elastic modulus E of the required material: the stiffness coefficient is substituted back, and the Young's modulus can be given:

3. The method of claim 1, wherein, The first-order stiffness correction method is adopted, specifically, based on the formula F i = κ·δL i , considering the weight of the material testing machine pressure head and the hydraulic characteristics, the system pressure head force F i and the deformation δL i satisfy: F i = κ · (δL i - δL0) That is, there is a critical deformation amount δL0; At this time, there is If the critical deformation amount is not considered, the above formula is simplified as: In the elastic stage, the change in cross-sectional area is ignored, that is: Using the first-order stiffness correction method, at least three groups of uniaxial compression tests of test pieces with different heights but the same cross-sectional area are needed, and at this time the above formula can be expressed as: According to the above formula, the intercept can be obtained: The expression of the Young's modulus is: And the stiffness coefficient expression is: In the formula Substitute the above four formulas into any one of the following equations to obtain the corrected engineering stress-strain curve of the test piece:

4. The method of claim 1, wherein, Using the second-order stiffness correction method, the specific method is as follows: Assume that the system pressure head force F i and the deformation δL i satisfy a quadratic function relationship: i.e. considering simultaneously the critical deformation δL i0 and the non-linear relationship between the system stiffness and the deformation δL; where, Respectively represent the dimensionless deformation and the dimensionless critical deformation; At this time, there is Δl(t) = ΔL + δL i = ΔL + L i [F(t), κ1, κ2, δL0] That is ΔL = Δl(t) - L i [F(t), K1, K2, δL0] In the formula, the △l(t) and F(t) curves can be given by the test simultaneously, and are known quantities; Therefore, there are still four unknown quantities, and at least four independent equations are needed to give the analytical solution; That is, at least four groups of test pieces with different heights but the same diameter are tested on the same test machine, so that: The system critical deformation and the two stiffness coefficients are given, so that the Young's modulus of the test piece and the corrected engineering stress and engineering strain curve of the test piece can be obtained.

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