Method for measuring radius of plastic zone in nanoindentation process of material with asymmetric strength

Abstract

The invention relates to the technical field of length scale measurement in the modern equipment manufacturing industry, and discloses a method for measuring the radius of a plastic zone in the nano-indentation process of a strength asymmetric material, which comprises the following steps: acquiring stress-strain data of the material to obtain an initial yield function and a hardening function; compiling a user material subprogram according to the initial yield function and the hardening function of the material; performing a nanoindentation experiment on the sample to obtain an experiment load displacement curve of the sample; accessing the user material subprogram into finite element software, and constructing a finite element model to perform analog simulation of a nanoindentation experiment to obtain a simulated load displacement curve of the indenter; comparing the displacement load curve with the experimental load displacement curve; and determining the distance between the initial equivalent yield stress position and the tip of the indenter in the x and y directions through an analog simulation result, and calculating the radius of the plastic zone under different indentation depths. The method can achieve the measurement of the radius of the plastic zone at any indentation depth, and is high in measurement result accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of length scale measurement in the modern equipment manufacturing industry, and particularly relates to a method for measuring the radius of the plastic zone during the nanoindentation of a strength-asymmetric material. Background Art

[0002] With the rapid development of the modern equipment manufacturing industry and the continuous emergence of new materials, the application of small-scale structures and micro-nano materials is becoming more and more extensive in many fields. As a new testing technology, nanoindentation testing has very strong advantages in micro-scale measurement, such as hardness, strength, residual stress, friction coefficient, etc. Since the micro-indentation test of metal materials is an elastoplastic coupling deformation process, with the help of contact mechanics theory, the yield strength, plastic hardening parameters, etc. of the material can be further deduced from the indentation load-displacement curve. During the process of analyzing the micro-indentation mechanical properties of materials or obtaining the plastic parameters of materials by using micro-indentation testing, the radius of the plastic zone under the indenter is a crucial parameter. Since the deformation process under the indenter cannot be observed in nanoindentation experiments, the measurement of the radius of the plastic zone during the indentation process has always been a difficult problem in this field.

[0003] At present, existing theories all regard the contact radius or a fixed multiple of the contact radius during the micro-indentation process as the radius of the plastic zone. However, the premise of regarding the contact radius or its multiple as the radius of the plastic zone is based on a large number of assumptions, such as: isotropic assumption, linear elastic assumption, ideal plasticity or power hardening assumption, tensile-compressive strength symmetry assumption, etc. These assumptions will inevitably bring serious errors to the corresponding analysis results, especially for a large number of materials with asymmetric tensile-compressive strengths. Obviously, this inherent defect of being difficult to measure the deformation in real time during the nanoindentation process limits the further application of this technology in related engineering fields. Therefore, giving an accurate nanoindentation deformation region according to the macroscopic mechanical properties of different materials can more efficiently and with high fidelity realize the application of nanoindentation testing technology in related fields. Summary of the Invention

[0004] In order to solve the technical problem of inaccurate estimation of the radius of the plastic zone during the indentation of strength-asymmetric materials in the prior art, the present invention proposes a method for measuring the radius of the plastic zone during the nanoindentation of strength-asymmetric materials to realize the visualization and continuous expression of the radius of the plastic zone.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is: a method for measuring the radius of the plastic zone during the nanoindentation of a strength-asymmetric material, comprising the following steps: Step 1: Obtain the stress-strain data of the localized deformation of the material, fit the initial yield function of the material in combination with the yield criterion of the material, and fit the hardening function of the material in combination with the stress-strain data of the material; Step 2: Based on the initial yield function and hardening function of the material, taking the equivalent plastic strain as an internal variable, determine the continuous expression of the strength asymmetry parameter, and write a user material subroutine; Step 3: Conduct nanoindentation experiments on the specimen to obtain the experimental load-displacement curve; connect the user material subroutine to the finite element software, construct a finite element model to simulate the nanoindentation experiment, and obtain the simulated load-displacement curve; Step 4: Compare the simulated load-displacement curve and the experimental load-displacement curve at the reference point to determine whether the error between the two is less than the threshold; Step 5: From the simulation results, extract the equivalent stress curves of the specimen in the x and y directions output by the finite element model at different indentation depths, and determine the distances between the positions of the initial equivalent yield stress in the x and y directions and the position of the indenter tip , and calculate the plastic zone radius at different indentation depths; Step 6: Based on the plastic zone radii at different indentation depths, fit the relationship between the indentation depth and the plastic zone radius to achieve the extraction of the plastic zone radius at any indentation depth.

[0006] In the above Step 1, through mechanical experiments on the strength asymmetric material under different stress states, obtain the stress-strain curve of the maximum deformation region after the localized deformation of the strength asymmetric material as the stress-strain data of the localized deformation of the material.

[0007] In the above Step 1, the adopted yield criterion is a yield criterion considering the tensile-compressive strength asymmetry and anisotropy of the material.

[0008] In the above Step 1, when fitting the hardening function of the material, an exponential equation, a linear equation or a polynomial equation is adopted.

[0009] In the above Step 2, the strength asymmetry parameter is expressed as a function of the equivalent strain in the hardening stage to achieve continuous evolution, and it is obtained by fitting or calculating the equivalent strain curves of the tensile stress and the compressive stress.

[0010] In the above Step 3, the specific method for constructing the finite element model is as follows: (1) Connect the user material subroutine to the finite element software; (2) Conduct geometric modeling in the finite element software. When modeling, add a reference point above the indenter and bind the indenter to the reference point. When meshing, divide the indenter and the specimen into several parts, and refine the mesh within the influence range of the indenter contact.

[0011] In step 4, the following steps are further included: If the error between the two is greater than the set threshold, re-determine the initial yield function and hardening function of the material, or optimize the stress solution method of the finite element model, reconstruct the finite element model to simulate the nanoindentation experiment to obtain the simulated load-displacement curve, and re-compare the two curves until the comparison error is less than the threshold.

[0012] In step 5, the calculation formula for the plastic zone radius is: ; where R represents the plastic zone radius.

[0013] In step 6, the relationship between the indentation depth and the plastic zone radius is fitted by a linear equation.

[0014] Compared with the prior art, the present invention has the following beneficial effects: The present invention proposes a method for measuring the plastic zone radius during the nanoindentation of a strength-asymmetric material. By organically combining classical plastic mechanics theory, numerical simulation technology, and micro-indentation testing means, it solves the technical problem that the plastic zone deformation of the material under the indenter during the indentation process cannot be measured; the calculation of the plastic radius of the material can be widely applied to applications such as material strength calculation and microstructural evolution analysis to obtain relevant parameters that cannot be measured by post-characterization methods during the material analysis process; in summary, the present invention can improve the accuracy of non-destructive testing using the indentation method and provide new ideas for the development of next-generation non-destructive testing equipment. Description of the Drawings

[0015] The following further details the specific embodiments of the present invention with reference to the accompanying drawings, where: Figure 1 is a flow chart of a method for measuring the plastic zone radius during the nanoindentation of a strength-asymmetric material provided by an embodiment of the present invention; Figure 2 is a schematic diagram of the nanoindentation model established in ABAQUS and the creation of path-x and path-y; Figure 3 is a comparison of the experimental load-displacement curve and the simulated load-displacement curve; Figure 4 is a curve showing the relationship between the distance and the equivalent stress at different depths of extracting path-x; Figure 5 is a curve showing the relationship between the distance and the equivalent stress at different depths of extracting path-y; Figure 6 is a curve showing the relationship between the indentation depth and the plastic zone radius obtained by fitting. Specific Embodiments

[0016] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be described clearly and completely below. Apparently, the described embodiments are only a part of the embodiments of the present invention, rather than all of them. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts belong to the scope of protection of the present invention.

[0017] As Figure 1 shown, the present invention provides a method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material. This method first determines the constitutive equation suitable for the material through experiments, writes a user material (UMAT) subroutine, establishes an ABAQUS finite element model for simulation calculation, visualizes the change in the radius of the plastic zone during its nanoindentation process, extracts data, and completes the quantification of the radius of the plastic zone. Below, taking the CoCrFeNiMn high-entropy alloy as the material to be tested as an example, the method for measuring the radius of the plastic zone will be described in detail through specific embodiments.

[0018] Embodiment A method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material in this embodiment specifically includes the following steps: Step 1: Obtain the stress-strain data of the localized deformation of the material, fit the initial yield function of the material in combination with the yield criterion of the material, and fit the hardening function of the material in combination with the stress-strain data of the material.

[0019] Specifically, in Step 1, through mechanical experiments on the material under different stress states, the stress-strain curve of the maximum deformation region after the localized deformation of the strength-asymmetric material is obtained as the stress-strain data of the localized deformation of the material. Then, using the experimental data and in combination with the yield criterion, the yield function of the material can be constructed. At the same time, the hardening function of the material is obtained by fitting the stress-strain data obtained through experiments.

[0020] Specifically, in the said Step 1, the yield criterion adopted is a yield criterion considering the asymmetry of the tensile and compressive strengths and the anisotropy of the material.

[0021] In the said Step 1, through mechanical experiments on the CoCrFeNiMn high-entropy alloy under different stress states, the stress-strain curve of the maximum deformation region after the localized deformation of the strength-asymmetric material is obtained. Specifically, in this embodiment, the mechanical experiment is realized through a universal material testing machine combined with a digital image correlation measurement system (Digital Image Correlation, DIC).

[0022] First, uniaxial compression and uniaxial tension experiments were carried out. At the same time, a high-speed camera was set up during the experiment to collect photos. The digital image correlation (DIC) technology was used to obtain the stress-strain curve of the experiment. The devices for uniaxial compression and uniaxial tension experiments both adopted an Instron 5969 universal material testing machine equipped with a 50 kN sensor. For the uniaxial compression specimen, a cylindrical compression specimen was selected, with a specification of Φ6 mm×9 mm and a height-diameter ratio of 1.5, and displacement control was adopted. Flat compression heads were selected at both the upper and lower ends of the testing machine. Before starting the loading, a preload of 100 N was applied to ensure that the specimen and the compression head of the testing machine were in surface-to-surface contact. A strain rate of 0.002 s -1 was selected, and the experiment was repeated three times; for the uniaxial tension experiment, a dog-bone tensile specimen of the sheet was used, and the specific specification of its gauge section was 20 mm in length, 5 mm in width, and 2 mm in thickness. Similarly, a strain rate of 0.002 s -1 was selected, and the experiment was repeated three times. By using software to process the photos collected by the high-speed camera, the stress-strain curve could be obtained, and the initial yield strength, Young's modulus, and Poisson's ratio of the material in tension and compression were obtained. These parameters were recorded to facilitate setting the material parameters in subsequent simulation.

[0023] Specifically, in this embodiment, after obtaining the stress-strain data of the localized deformation of the material to be tested under different stress states through experiments, a yield function can be constructed or the parameters of the existing yield function can be calibrated. The form of the yield function F is as follows: ; (1) where f is a function with undetermined form, and its specific function form can be determined according to the material category. I 1 represents the first invariant of the principal stress tensor. J 2 represents the second invariant of the deviatoric stress tensor. J 3 represents the third invariant of the deviatoric stress tensor. a 1, a 2… a n are material constants, and n is an integer greater than or equal to 1, representing the number of material constants.

[0024] In this embodiment, after processing the data obtained from the experiment, the initial yield stresses of the specimen in compression and tension were obtained respectively. Considering the strength asymmetry characteristic of the material to be tested, the CoCrFeNiMn high-entropy alloy, the CB-2004 yield criterion was selected, which can be expressed as: ; (2) where is the equivalent stress. Specifically, the material constants of the yield function were obtained through experimental data. Calibration is carried out, and the calibration formula is as follows: ; (3) where respectively represent the initial yield strength of uniaxial tension and uniaxial compression. In addition, it should be noted that when dealing with traditional materials such as low-carbon steel without obvious tensile-compressive asymmetry characteristics, the classical Mises yield criterion can be selected; for polymer materials, a hydrostatic pressure-sensitive yield function such as the DP criterion or the Mohr-Coulomb criterion can be selected.

[0025] In this embodiment, the compressive stress is defined as the equivalent stress. By fitting the compressive stress-strain curve with Matlab, the hardening function is selected as an exponential equation in the following form: ; (4) where represents the equivalent stress, represents the equivalent plastic strain, represents the hardening function parameter. By fitting the stress-strain data obtained from experiments to the above hardening function, the hardening function parameter can be obtained . In addition, for materials with a plastic stress-strain curve approximately linear, a linear equation can be selected as the hardening function; for materials with relatively complex deformation, a polynomial equation can also be used as the hardening function. Specifically, in this embodiment, the obtained parameters are shown in Table 1.

[0026] Table 1 Parameters of the material Step 2: According to the initial yield function and hardening function of the material, taking the equivalent plastic strain as the internal variable, determine the continuous expression of the strength asymmetry parameter, and write a user material (UMAT) subroutine.

[0027] In this embodiment, combining the flow rule and the consistency condition, a phenomenological constitutive model of the material is constructed, and a stress integration algorithm is used to write a user material subroutine. To avoid the error accumulation and instability of the explicit integration algorithm, a fully implicit integration algorithm is used to write the user material subroutine.

[0028] In step 2, the strength asymmetry parameter is expressed as a function of the equivalent plastic strain during the hardening stage to achieve the continuous evolution of the strength asymmetry parameter, which is obtained by fitting or calculating the tensile stress-equivalent plastic strain curve and the compressive stress-equivalent plastic strain curve.

[0029] In this embodiment, since the constructed constitutive model is a symmetric two-dimensional model, ten state variables are required to store the variables during the calculation process, including four stress components, four strain components, equivalent plastic strain, and equivalent stress.

[0030] Step 3: Conduct a nano-indentation experiment on the specimen to obtain the experimental load-displacement curve; connect the user material subroutine to the finite element software, construct a finite element model to simulate the nano-indentation experiment, and obtain the simulated load-displacement curve.

[0031] In this embodiment, the nano-indentation experiment uses a Nano-indenter G200 test system. When testing, a cylindrical specimen is used. Before the indentation test, the specimen needs to be polished with a series of SiC sandpapers of different grit sizes and then polished to a mirror surface to obtain better experimental results and reduce the influence of the uneven specimen surface on the experimental results.

[0032] In the said Step 3, the specific method for constructing the finite element model is as follows: (1) Connect the user material subroutine to the finite element software; (2) Conduct geometric modeling in the finite element software. When modeling, add a reference point above the indenter and bind the indenter to the reference point. When meshing, divide the indenter and the specimen into several parts, and refine the mesh within the influence range of the indenter contact.

[0033] Specifically, in this embodiment, the ABAQUS finite element software is used to simulate the nano-indentation experiment of the material, and a nano-indentation model is established. The indented specimen model is a cylinder with a diameter of Φ8 mm and a height of 5 mm. The Berkovich indenter is used in the experiment, so the Berkovich indenter is also used in the modeling, and its equivalent conical semi-angle is 70.32°. Since the geometric dimensions, load, and its constraints of the model are symmetric about the central axis, two-dimensional axisymmetric elements are selected. During assembly, add a reference point above the indenter. When setting the material of the model, assign the user-defined material parameters to the specimen, and add 10 state variables. The given material parameters need to correspond to those in the UMAT subroutine. The symmetry axis is set as the symmetry boundary condition; the contact between the indenter and the specimen is set as smooth and frictionless hard contact; the indenter is set as a rigid body and bound to the previously added reference point.

[0034] In addition, set the boundary condition that restricts the movement along the y direction at the bottom of the specimen, and set the boundary condition that allows the movement along the x direction on the right side of the specimen. When meshing, first divide the indenter and the specimen into different parts, and draw a finer mesh within the influence range of the indenter contact to improve the accuracy of the simulation results; select a larger mesh size at a distance more than 20 times the maximum indentation depth to facilitate efficient calculation. As Figure 2 shown, it is the schematic diagram of dividing different parts to draw the mesh, and it is also the schematic diagram of the indentation model.

[0035] When submitting the job, use the built-in interface of ABAQUS to call the pre-edited UMAT subroutine. In the field output of the post-processing, select the previously set reference point and output the simulation load-displacement curve of the reference point.

[0036] Step 4: Compare the simulation load-displacement curve of the reference point with the experimental load-displacement curve and determine whether the error between the two is less than the threshold.

[0037] Compare the simulation load-displacement curve obtained by finite element software simulation with the experimental load-displacement curve obtained from the nano-indentation experiment of the specimen. As Figure 3 shown, it can be seen that the simulation results of this embodiment are in good agreement with the experimental results, which can verify the accuracy of the finite element model establishment and UMAT subroutine writing.

[0038] In the said Step 4, the following steps are further included: If the error between the two is greater than the set threshold, re-determine the material initial yield function, hardening function, or optimize the stress solution method of the finite element model, re-construct the finite element model to conduct the simulation of the nano-indentation experiment to obtain the simulation load-displacement curve, and re-compare the two curves until the comparison error is less than the threshold.

[0039] Step 5: Through the simulation results, extract the equivalent stress curves of the specimen in the x and y directions output by the finite element model at different indentation depths, and determine the distances between the positions of the initial equivalent yield stress in the x and y directions and the position of the indenter tip , and calculate the plastic zone radius at different indentation depths.

[0040] After the simulation is completed, open the post-processing file of the completed simulation, select the contour plot representing the equivalent stress state variable, view the stress contour plots of different increment steps, and the change process of the plastic zone radius can be visually seen. As Figure 2 shown, create two path paths respectively. On the upper surface of the specimen, sequentially select the nodes from the symmetry center to the edge along the x direction and create it as path-x; similarly, select the nodes on the symmetry axis along the y direction and create it as path-y. Use the function of outputting the path path in ABAQUS, select different increment steps, first output the displacement in the y direction, and find the increment steps corresponding to several equally spaced indentation depths. After determining the increment steps, use the path to output the equivalent stress of each increment step to obtain the data of the distance between different indentation depths and the indenter tip and the equivalent stress.

[0041] Determine the value of the initial equivalent yield stress (uniaxial compression is defined as the equivalent stress state in this embodiment). Taking this value as the boundary, determine whether the specimen has reached yield during indentation. Obtain the distances between each position and the tip of the indenter at different indentation depths and the corresponding equivalent stress data, plot the corresponding curves using Origin, and then insert the value of the initial equivalent yield stress into the curve graph to obtain the distances between the positions of the initial equivalent yield stress in the x and y directions and the tip of the indenter at different depths. , such as Figures 4 - 5 shown. Respectively, taking the tip of the indenter as the origin, the schematic diagrams of the equivalent stress curves corresponding to different positions in the x and y directions at each indentation depth are obtained. The abscissa corresponding to the intersection point of the stress curve and the straight line of the initial equivalent yield stress is the distance between the positions of the initial equivalent yield stress in the x and y directions and the tip of the indenter. and values.

[0042] In the said step 5, the calculation formula for the radius of the plastic zone is: ; (5) where R represents the radius of the plastic zone.

[0043] Step 6: According to the radii of the plastic zones at different indentation depths, fit the relationship between the indentation depth and the radius of the plastic zone to extract the radius of the plastic zone in the indentation area at any indentation depth.

[0044] Specifically, in the said step 6, the relationship between the indentation depth and the radius of the plastic zone is fitted by a linear equation.

[0045] Such as Figure 6 shown. In this embodiment, according to the obtained different indentation depths and their corresponding radii of the plastic zones, a dot-line graph is drawn in Origin, and the relationship formula between the depth and the radius of the plastic zone can be obtained through numerical fitting. The relationship between the indentation depth and the radius of the plastic zone of the material to be measured is: R =7.05 h . Where h represents the indentation depth. Thus, this embodiment realizes the extraction of the effective plastic zone radius in the indentation area.

[0046] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that: they can still modify the technical solutions recorded in the foregoing embodiments, or perform equivalent replacements on some or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for measuring the radius of the plastic zone during nano-indentation of strength-asymmetric materials, characterized in that: The following steps are involved: Step 1: Obtain the stress-strain data of the localized deformation of the material, fit the initial yield function of the material in combination with the yield criterion of the material, and fit the hardening function of the material in combination with the stress-strain data of the material; Step 2: According to the initial yield function and hardening function of the material, take the equivalent plastic strain as the internal variable, determine the continuous expression of the strength asymmetry parameter, and write the user material subroutine; Step 3: Perform a nanoindentation experiment on the sample to obtain an experimental load-displacement curve; connect the user material subroutine to the finite element software, build a finite element model to simulate the nanoindentation experiment, and obtain a simulated load-displacement curve; Step 4: Compare the simulated load-displacement curve and the experimental load-displacement curve of the reference point to determine whether the error between the two is less than the threshold; Step 5: Through simulation results, extract the equivalent stress curves of the sample along the x-direction and y-direction output by the finite element model at different indentation depths, and determine the distance between the initial equivalent yield stress in the x-direction and y-direction and the indenter tip position. , calculate the radius of the plastic zone at different indentation depths; Step 6: According to the plastic zone radius at different indentation depths, the relationship between the indentation depth and the plastic zone radius is fitted to achieve the extraction of the plastic zone radius at any indentation depth.

2. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In the step 1, mechanical experiments are performed on the strength asymmetric material under different stress states to obtain the stress-strain curve of the maximum deformation area after the localized deformation of the strength asymmetric material as the stress-strain data of the localized deformation of the material.

3. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In step 1, the yield criterion adopted is the yield criterion that takes into account the asymmetry and anisotropy of the tensile and compressive strength of the material.

4. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In the step 1, when fitting to obtain the hardening function of the material, an exponential equation, a linear equation or a polynomial equation is used.

5. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In the step 2, the strength asymmetry parameter is expressed as a function of equivalent strain in the hardening stage to achieve continuous evolution, which is obtained by fitting or calculating the equivalent strain curve of tensile stress and the equivalent strain curve of compressive stress.

6. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In step 3, the specific method of constructing the finite element model is: (1) User material subroutine is connected to finite element software; (2) Perform geometric modeling in finite element software. When modeling, add a reference point above the indenter and bind the indenter to the reference point. When meshing, divide the indenter and the specimen into several parts, and refine the mesh within the influence range of the indenter contact.

7. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In step 4, The method comprises the following steps: if the error between the two is greater than a set threshold, the initial yield function and hardening function of the material are redetermined, or the stress solution method of the finite element model is optimized, the finite element model is reconstructed to simulate the nanoindentation experiment to obtain the simulated load-displacement curve, and the two curves are compared again until the comparison error is less than the threshold.

8. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In step 5, the calculation formula of the plastic zone radius is: ； Where R represents the radius of the plastic zone.

9. The method for measuring the radius of the plastic zone during nano-indentation of a strength-asymmetric material according to claim 1, characterized in that: In step 6, the relationship between the indentation depth and the plastic zone radius is fitted by a linear equation.

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