Method for correcting error in measuring elastic modulus by instrumented spherical indentation technique

Abstract

The present invention discloses a method for error correction of elastic modulus, which comprises the following steps: obtaining the yield strain ε corresponding to the material of the elastic modulus to be measured y and the hardening index n; calculating the calculated value of the elastic modulus based on the Oliver–Pharr method; calculating the elastic modulus error value based on the maximum relative indentation depth; and correcting the calculated value of the elastic modulus by using the elastic modulus error value to obtain the corrected value of the elastic modulus. The advantages of the present invention are as follows: by correcting the elastic modulus calculated by the Oliver–Pharr method, a more accurate elastic modulus can be obtained, reducing the defect that the calculation of the elastic modulus generates errors due to the increase of the maximum indentation depth, making the corrected elastic modulus more accurate than before correction, and providing a reliable parameter testing method for measuring the elastic modulus in the engineering field.

Description

Technical Field

[0001] The present invention relates to the field of error correction, and particularly to a method for correcting the error of measuring the elastic modulus by the instrumented spherical indentation technique. Background Art

[0002] The elastic modulus is one of the most important mechanical parameters of engineering materials. Accurately measuring the elastic modulus is very important. The instrumented indentation technique is one of the efficient methods for non-destructive measurement of the elastic modulus. Among them, the most classical method for instrumented spherical indentation to measure the elastic modulus was proposed by Oliver and Pharr, namely the Oliver–Pharr method, which can be expressed as:

[0003]

[0004] In formula (1), E r is the reduced modulus, E i is the elastic modulus of the indenter, E is the elastic modulus of the specimen, v i is the Poisson's ratio of the indenter, v is the Poisson's ratio of the specimen, A c is the projected contact area of the indenter, a c is the contact radius.

[0005] As Figure 1 shown is the schematic diagram of instrumented spherical indentation for measuring the elastic modulus; from the geometric relationship of the spherical indenter, it can be known that:

[0006]

[0007] In formula (3), R is the radius of the spherical indenter, h c is the contact depth. In the Oliver–Pharr method, there is:

[0008]

[0009] In formula (4), h m is the maximum indentation depth, F m is the maximum indentation load, and S is the initial unloading stiffness.

[0010] Through experimental simulation, a phenomenon can be observed. The simulation set parameters include: the radius of the spherical indenter is 1.25 mm, the radius and thickness of the specimen are 35 mm and 50 mm respectively, the Poisson's ratio of the specimen is fixed at 0.3, and the elastic modulus is 100 GPa. Assume that the constitutive model of the specimen satisfies the Hollomon equation, that is:

[0011]

[0012] Among them, ε yis the yield strain, and its values are 0.002, 0.004 and 0.007 respectively; n is the hardening index, and its value range is 0, 0.1, 0.2 and 0.3. The friction coefficient between the specimen and the spherical indenter is 0.45. The indentation depths are 0.01R, 0.02R, 0.03R, 0.04R, 0.05R, 0.06R, 0.08R, 0.1R, 0.2R and 0.3R respectively.

[0013] After the simulation is run, the elastic modulus measured by the Oliver–Pharr method is compared with the true elastic modulus of the material, and the following is obtained: Figure 2 Schematic diagram of the elastic modulus error following the maximum relative indentation depth shown; Figure 2 It can be seen that as the indentation depth increases, the error actually increases. Therefore, when the Oliver–Pharr method is used to calculate the elastic modulus, errors will actually occur. When the instrumented spherical indentation technology is used, as the indentation depth increases, the error of the elastic modulus measured by the Oliver–Pharr method will become larger and larger. When the indentation depth is 0.3R, the measurement error can exceed 25%. Therefore, it is necessary to modify the elastic modulus measured by the Oliver–Pharr method to obtain a more accurate elastic modulus. Summary of the invention

[0014] The object of the present invention is to overcome the shortcomings of the prior art and provide an error correction method for elastic modulus, which is used to solve the defect that the elastic modulus measured by the Oliver–Pharr method in the prior art has errors, so that the calculated elastic modulus is more accurate than the elastic modulus measured by the Oliver–Pharr method in the prior art.

[0015] In order to achieve the above object, the technical solution adopted by the present invention is: an error correction method of elastic modulus, comprising the following steps:

[0016] Get the yield strain ε corresponding to the elastic modulus material to be measured y and hardening index n;

[0017] The calculated value of elastic modulus was obtained based on the Oliver–Pharr method;

[0018] Based on the maximum relative indentation depth, hardening index n, yield strain ε y , calculate the elastic modulus error value corresponding to the maximum relative indentation depth;

[0019] The elastic modulus is corrected using the calculated elastic modulus error value to obtain a corrected value of the elastic modulus.

[0020] The calculation method of the elastic modulus E includes:

[0021] An instrumented spherical indentation test is performed on the material to be tested, and the parameters during the test are recorded. The calculated value E of the elastic modulus is calculated using the following formula:

[0022]

[0023] In the formula, E r is the reduced modulus, E i is the elastic modulus of the indenter, E is the elastic modulus of the specimen under indentation, and v i is the Poisson's ratio of the indenter, v is the Poisson's ratio of the specimen, and A c is the projected contact area of the indenter, and a c is the contact radius; R is the radius of the spherical indenter, and h c is the contact depth. In the Oliver–Pharr method, there is:

[0024]

[0025] h m is the maximum indentation depth, F m is the maximum indentation load, and S is the initial unloading stiffness.

[0026] The elastic modulus error value is obtained according to the maximum relative indentation depth during the indentation test, and the error value is positively correlated with the maximum relative indentation depth.

[0027] The elastic modulus error value is related to the yield strain ε y , the hardening index n, and the maximum relative indentation depth.

[0028] The elastic modulus error value is calculated using the following formula:

[0029]

[0030] In the formula, a, b, and c are coefficients related to the yield strain ε y and the hardening index n, and h m / R is the maximum relative indentation depth.

[0031] a, b, and c are calculated using the following formula:

[0032]

[0033] The material to be tested is a linear elastic power hardening material with a yield strain of 0.002 to 0.007.

[0034] The maximum indentation depth during the measurement of the elastic modulus using the Oliver–Pharr method is 0 to 0.3R.

[0035] The yield strain ε y and the hardening index n of the material to be tested are calibrated through experiments.

[0036] The advantages of the present invention are as follows: By correcting the elastic modulus calculated by the Oliver–Pharr method, a more accurate elastic modulus can be obtained, reducing the defect of error in calculating the elastic modulus caused by the increase in the maximum indentation depth, making the corrected elastic modulus more accurate than before correction, and providing a reliable parameter testing method for measurement in the engineering field. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] The following briefly describes the content expressed in each drawing of the specification of the present invention and the marks in the drawings:

[0038] Figure 1 It is a schematic diagram of instrumented spherical indentation for measuring elastic modulus in the prior art;

[0039] Figure 2 It is a schematic diagram of the elastic modulus error of the present invention following the maximum indentation depth;

[0040] Figure 3 It is a schematic diagram of error comparison in the experiment of the present invention;

[0041] Figure 4 For the present invention, with different hardening indices n (taking values of 0, 0.1, 0.2, 0.3) of the linear elastic power hardening material and different materials E OP Error comparison diagram before and after correction; DETAILED DESCRIPTION OF THE EMBODIMENTS

[0042] The following further details the specific embodiments of the present invention by describing the optimal embodiments with reference to the accompanying drawings.

[0043] This application mainly focuses on instrumented spherical indentation technology. To eliminate the error in measuring the elastic modulus by the Oliver–Pharr method caused by the increase in indentation depth, the present invention proposes a depth-related correction method. For linear elastic power hardening materials with a yield strain of 0.002 to 0.007 and a maximum indentation depth not greater than 0.3R, the following technical solutions are proposed:

[0044] A method for correcting the error in measuring the elastic modulus by instrumented spherical indentation technology, the method comprising the following steps:

[0045] The first step is to obtain the yield strain ε y and the hardening index n of the material by other methods. If the above parameters of the material are known, this step can be omitted.

[0046] The second step is to record the relative maximum indentation depth h m / R during the instrumented spherical indentation using the Oliver–Pharr method, measure the load-depth curve of each test point of the test sample, and record the maximum relative indentation depth hm / R

[0047] In the third step, use the Oliver–Pharr method, namely Equations (2), (3), and (4), to calculate the elastic modulus of each test point of the test specimen.

[0048] In the fourth step, according to the maximum relative indentation depth h m / R, the material yield strain ε y , and the hardening index n, estimate the measurement error of the elastic modulus of the test specimen, and its calculation method is

[0049]

[0050] In Equation (1), a, b, and c are coefficients related to the yield strain ε y , and the hardening index n, and can be expressed as:

[0051]

[0052] Equations (1) and (7) are the core of the present invention.

[0053] In the fifth step, subtract the relative error estimated by Equations (1) and (7) from the elastic modulus measured by the Oliver–Pharr method to obtain the corrected and accurate elastic modulus.

[0054] Among them, the Oliver–Pharr method formulas mentioned in the third step include:

[0055]

[0056] In Equation (1), E r is the reduced modulus, E i is the indenter elastic modulus, E is the specimen indentation elastic modulus, v i is the indenter Poisson's ratio, v is the specimen Poisson's ratio, A c is the indenter projected contact area, a c is the contact radius. From the geometric relationship of the spherical indenter, it can be known that:

[0057]

[0058] In Equation (3), R is the spherical indenter radius, h c is the contact depth. In the Oliver–Pharr method, there is:

[0059]

[0060] In Equation (4), h m is the maximum indentation depth, F m is the maximum indentation load, and S is the initial unloading stiffness.

[0061] Using the instrumented spherical indentation technique, within the range of the maximum indentation depth from 0 to 0.3R, for linear elastic power hardening materials with a yield strain of 0.002 to 0.007, the present invention can effectively eliminate the measurement error of the elastic modulus introduced by the increase in the maximum indentation depth in Oliver–Pharr method.

[0062] To verify the effectiveness of the method of this application, this application is verified through experiments. The verification examples include:

[0063] Select two typical metals (AA 7075 and AA 2014) as experimental materials, and their yield strain ε y and hardening index n have been measured through other tests. The specific parameters are shown in Table 1. Among them, E ref is the conventional reference true value of the elastic modulus, and the error magnitude can be calculated by the difference between the calculated value and the reference value.

[0064] First step, the plastic parameters of the materials are known, as shown in Table 1. Therefore, there is no need to adopt other testing means to obtain the plastic parameters of the two materials.

[0065] Table 1 Elastoplastic parameters of AA 7075 and AA 2014 measured by other tests

[0066]

[0067] Second step, adopt the Oliver–Pharr method to carry out indentation. Fix the indentation depths at 0.01R, 0.05R, 0.1R, 0.2R, and 0.3R respectively. For the polished AA 7075 and AA 2014, at least 5 test points are indented at each maximum indentation depth. The indenter radius R is 500 μm, and the material is carbide, and its elastic modulus and Poisson's ratio are 537 GPa and 0.23 respectively. Measure the load-depth curve of each indentation point. Record the maximum relative indentation depth h m / R.

[0068] Third step, for each test point, use the Oliver-Pharr method, that is, Equation (1), Equation (2), Equation (4), to calculate the elastic modulus of AA2014 and AA 7075 at each test point.

[0069] Fourth step, for each test point of the two materials, substitute the maximum relative indentation depth h m / R, the material yield strain ε y , and the hardening index n into Equation (1) and Equation (7) respectively, and calculate the relative error of the elastic modulus at each test point.

[0070] In the fifth step, for each test point, subtract the relative error (error value) estimated in the fourth step from the elastic modulus (calculated value) calculated by the Oliver–Pharr method, and the relatively accurate elastic modulus can be obtained.

[0071] Finally, the comparison chart of the elastic modulus calculated before correction and the elastic modulus calculated after correction is as Figure 3 . From Figure 3 (a) and Figure 3 (b), it can be seen that before correction, the measured elastic modulus value increases with the increase of the indentation depth, and its maximum relative error exceeds 20%. From Figure 3 (d), it can be seen that as the relative indentation depth increases, the error of the elastic modulus after correction is basically within 5%. Therefore, the present invention can effectively correct the measurement error of the elastic modulus caused by the increase of the indentation depth in the Oliver–Pharr method.

[0072] This application can correct errors, making the obtained elastic modulus more accurate and reliable. The theoretical basis of this application includes:

[0073] In this solution, the finite element simulation software ABAQUS is used to construct a two-dimensional axisymmetric finite element simulation model. The boundary condition of this model is that the bottom surface of the specimen is fixed. The indenter only moves along the indentation direction, and its radius is 1.25 mm. To meet the assumption of the semi-infinite space of the specimen, the radius and thickness of the specimen are 35 mm and 50 mm respectively, which are approximately 93 times and 133 times the maximum indentation depth respectively. The Poisson's ratio of the specimen is fixed at 0.3, and the elastic modulus is 100 GPa. It is assumed that the constitutive model of the specimen satisfies the Hollomon equation, that is:

[0074]

[0075] Among them, ε y is the yield strain, and the values are 0.002, 0.004, and 0.007 respectively; n is the hardening index, and the value range is 0, 0.1, 0.2, and 0.3. The friction coefficient between the specimen and the spherical indenter is 0.45. The indentation depths are 0.01R, 0.02R, 0.03R, 0.04R, 0.05R, 0.06R, 0.08R, 0.1R, 0.2R, and 0.3R respectively.

[0076] Figure 4 (a)–(c) are the calculation results of the elastic modulus error E OP of the simulation material. It can be seen that as the relative maximum indentation depth h m / R increases, the E OP error increases rapidly, and its maximum error can exceed 25%. For all materials, the change trend of E OP with h m / R can be fitted by the following equation

[0077]

[0078] Among them, a, b, and c are constants related to the yield strain and the hardening index.

[0079] By fitting with formula (1), the corresponding values of a and the hardening index n can be obtained, as shown in Table 1. The coefficients b, c, and the hardening index n and the yield strain ε y The original fitting values are shown in Table 2.

[0080] Table 1 Coefficient a Hardening index Original fitting values

[0081] a n 33 0 25 0.1 18 0.2 14 0.3

[0082] Table 2 Coefficients b, c, hardening index n, and yield strain ε y Original fitting values

[0083] b c <![CDATA[ε y > n -5.484 0.005 0.002 0 -5.089 0.019 0.002 0.1 -3.929 0.022 0.002 0.2 -4.128 0.047 0.002 0.3 -7.71 0.026 0.004 0 -6.852 0.05 0.004 0.1 -5.018 0.05 0.004 0.2 -5.186 0.096 0.004 0.3 -12.095 0.098 0.007 0 -11.143 0.156 0.007 0.1 -7.085 0.122 0.007 0.2 -6.88 0.178 0.007 0.3

[0084] Using quadratic polynomial fitting for the data in Table 1 and Table 2 respectively can obtain

[0085]

[0086] Finally, use formula (2) and formula (3) for Figure 4 (a)–(c)E OP to correct the error, and the results are as shown in Figure 4 (d)–(f). It can be seen from the figure that after correction, the error of E OP is basically within 5%, which proves that this method can effectively reduce the error introduced by the increase in the maximum indentation depth to E OP introduced.

[0087] Obviously, the specific implementation of the present invention is not limited by the above methods. As long as various non-substantive improvements are made by adopting the method concept and technical solution of the present invention, they are all within the protection scope of the present invention.

Claims

1. A method for error correction of elastic modulus, characterized in that : The following steps Obtain the yield strain ε corresponding to the material with the elastic modulus to be measured y and the hardening index n; Calculate the calculated value of the elastic modulus based on the Oliver–Pharr method; Obtain the corresponding elastic modulus error value related to the maximum relative indentation depth during the measurement of the elastic modulus based on the Oliver–Pharr method; Use the elastic modulus error value to correct the calculated value of the elastic modulus to obtain the corrected value of the elastic modulus; The elastic modulus error value is calculated using the following formula: where a, b, c are coefficients related to the yield strain ε y and the hardening index n, h m / R is the maximum relative indentation depth; a, b, c are calculated using the following formula:

2. The error correction method for elastic modulus according to claim 1, wherein : The calculation method of the calculated value E of the elastic modulus includes: Perform an instrumented spherical indentation test on the material to be measured and record the parameters during the test. Calculate the calculated value E of the elastic modulus using the following formula: where E r is the reduced modulus, E i is the elastic modulus of the indenter, E is the elastic modulus of the specimen under indentation, v i is the Poisson's ratio of the indenter, v is the Poisson's ratio of the specimen, A c is the projected contact area of the indenter, a c is the contact radius; R is the radius of the spherical indenter, h c is the contact depth. In the Oliver–Pharr method, there is: h m is the maximum indentation depth, F m is the maximum indentation load, and S is the initial unloading stiffness.

3. The error correction method for elastic modulus according to claim 1, characterized in that : The elastic modulus error value is obtained based on the maximum relative indentation depth during the indentation test, and the error value is positively correlated with the maximum relative indentation depth.

4. The error correction method for elastic modulus according to claim 1, characterized in that : The elastic modulus error value is related to the yield strain ε y , the hardening index n, and the maximum relative indentation depth.

5. A method for correcting the error of elastic modulus according to any one of claims 1-4, characterized in that : The material whose elastic modulus is to be measured is a linear elastic power hardening material with a yield strain of 0.002 to 0.

007.

6. A method for error correction of elastic modulus according to any one of claims 1-4, characterized in that : The maximum indentation depth during the measurement of the elastic modulus using the Oliver–Pharr method is 0 to 0.3R.

7. A method for error correction of elastic modulus according to any one of claims 1-4, characterized in that : Yield strain ε of the material to be tested y And the hardening index n is calibrated experimentally.

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