Method for measuring radius of plastic zone during nanoindentation of strength asymmetric material

Abstract

The present 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 the plastic zone during the nanoindentation of a strength-asymmetric material, including: obtaining the stress-strain data of the material to obtain the initial yield function and hardening function; writing a user material subroutine according to the initial yield function and hardening function of the material; performing a nanoindentation experiment on the specimen to obtain the experimental load-displacement curve of the specimen; connecting the user material subroutine to the finite element software, constructing a finite element model to simulate the nanoindentation experiment, and obtaining the simulated load-displacement curve of the indenter; comparing the displacement-load curve with the experimental load-displacement curve; determining the distances between the positions of the initial equivalent yield stress in the x and y directions and the tip of the indenter through the simulation results, and calculating the radius of the plastic zone at different indentation depths. The present invention can achieve the measurement of the radius of the plastic zone at any indentation depth, and the measurement result has high 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 strength-asymmetric materials. Background Art

[0002] With the rapid development of the modern equipment manufacturing industry and the continuous emergence of new materials, the application of small-size structures and micro-nano materials is becoming more and more extensive in many fields. As an emerging 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 derived 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] Currently, 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 the 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 achieve 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 strength-asymmetric materials, including the following steps:

[0006] 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;

[0007] 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.

[0008] Step 3: Conduct a nanoindentation 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 nanoindentation experiment, and obtain the simulated load-displacement curve.

[0009] 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.

[0010] 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, 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.

[0011] Step 6: According to 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.

[0012] 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 material's localized deformation.

[0013] In the above Step 1, the adopted yield criterion is the yield criterion considering the tensile-compressive strength asymmetry and anisotropy of the material.

[0014] 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.

[0015] 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.

[0016] In the above Step 3, the specific method for constructing the finite element model is as follows:

[0017] (1) Connect the user material subroutine to the finite element software;

[0018] (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.

[0019] 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.

[0020] In step 5, the calculation formula for the radius of the plastic zone is:

[0021] ;

[0022] where R represents the radius of the plastic zone.

[0023] In step 6, the relationship between the indentation depth and the radius of the plastic zone is fitted by a linear equation.

[0024] The present invention has the following beneficial effects compared with the prior art: The present invention proposes a method for measuring the radius of the plastic zone 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 the strength calculation of materials and the analysis of microstructural evolution 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 the next-generation non-destructive testing equipment. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The following further details the specific embodiments of the present invention in conjunction with the drawings, where:

[0026] Figure 1 is a schematic flow chart of a method for measuring the radius of the plastic zone during the nanoindentation of a strength-asymmetric material provided by an embodiment of the present invention;

[0027] Figure 2 is a schematic diagram of the nanoindentation model established in ABAQUS and the creation of path-x and path-y;

[0028] Figure 3 is the comparison between the experimental load-displacement curve and the simulated load-displacement curve;

[0029] Figure 4 is the curve of the relationship between the distance and the equivalent stress at different depths of extracting path-x;

[0030] Figure 5 is the curve of the relationship between the distance and the equivalent stress at different depths of extracting path-y;

[0031] Figure 6 The curve showing the relationship between the indentation depth and the radius of the plastic zone obtained by fitting. Specific implementation manners

[0032] 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 some but not all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the scope of protection of the present invention.

[0033] 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 material to be measured as the CoCrFeNiMn high-entropy alloy as an example, the method for measuring the radius of the plastic zone will be described in detail through specific embodiments.

[0034] Embodiment

[0035] 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:

[0036] 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.

[0037] 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.

[0038] Specifically, the yield criterion adopted in Step 1 is a yield criterion that takes into account the tensile-compressive strength asymmetry and anisotropy of the material.

[0039] In step 1, by performing mechanical experiments on CoCrFeNiMn high-entropy alloy under different stress states, the stress-strain curve of the maximum deformation region after localized deformation of the strength-asymmetric material is obtained. Specifically, in this embodiment, the mechanical experiment is realized by a universal material testing machine combined with a digital image correlation measurement system (Digital Image Correlation, DIC).

[0040] During the experiment, uniaxial compression and uniaxial tension are first carried out. At the same time, a high-speed camera is set up during the experiment to collect photos, and the stress-strain curve of the experiment is obtained by using digital image correlation (Digital Image Correlation, abbreviated as DIC) technology. The devices for uniaxial compression and uniaxial tension experiments both adopt an Instron5969 universal material testing machine equipped with a 50kN sensor. For the uniaxial compression specimen, a cylindrical compression specimen is selected, with a specification of Φ6mm×9mm and a height-diameter ratio of 1.5, and displacement control is adopted. Flat compression heads are selected at both the upper and lower ends of the testing machine. Before starting the loading, a preload of 100N is applied to ensure that the specimen and the compression head of the testing machine are in surface-to-surface contact, and a strain rate of 0.002 s -1 is selected, and the experiment is repeated three times; for the uniaxial tension experiment, a dog-bone tensile specimen of a sheet is used, and the specific specification of its gauge section is 20mm in length, 5mm in width, and 2mm in thickness. Similarly, a strain rate of 0.002 s -1 is selected, and the experiment is repeated three times. By using software to process the photos collected by the high-speed camera, the stress-strain curve can be obtained, and the initial yield strength, Young's modulus, and Poisson's ratio of the material in tension and compression can be obtained, and these parameters are recorded for setting the material parameters in subsequent simulation.

[0041] Specifically, in this embodiment, after obtaining the stress-strain data of localized deformation of the material to be tested under different stress states through experiments, a yield function can be constructed or the parameters of an existing yield function can be calibrated. The form of the yield function F is:

[0042] ; (1)

[0043] 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, n is an integer greater than or equal to 1, representing the number of material constants.

[0044] In this embodiment, after processing the data obtained from the experiment, the initial yield stresses during specimen compression and tension are obtained respectively. Considering the strength asymmetry characteristic of the material to be measured, the CoCrFeNiMn high-entropy alloy, the CB-2004 yield criterion is selected, which can be expressed as:

[0045] ; (2)

[0046] where is the equivalent stress. Specifically, the material constants of the yield function are calibrated through experimental data, and the calibration formula is as follows:

[0047] ; (3)

[0048] where represent the initial yield strengths of uniaxial tension and uniaxial compression respectively. 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.

[0049] 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 in the form of an exponential equation as follows:

[0050] ; (4)

[0051] where represents the equivalent stress, represents the equivalent strain, represents the hardening function parameter. After fitting the above hardening function with the stress-strain data obtained from the experiment, 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.

[0052] Table 1 Parameters of the material

[0053]

[0054] Step 2: Based on 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.

[0055] In this embodiment, a phenomenological constitutive model of the material is constructed by combining the flow rule and the consistency condition, and a user material subroutine is written using the stress integration algorithm. 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.

[0056] In step 2, the strength asymmetry parameter is expressed as a function of the equivalent strain in the hardening stage to achieve the continuous evolution of the strength asymmetry parameter, which is obtained by fitting or calculating the equivalent strain curve of the tensile stress and the equivalent strain curve of the compressive stress.

[0057] In this embodiment, since the constructed constitutive model is a symmetric two-dimensional model, ten state variables are required to store the variables in the calculation process, including four stress components, four strain components, the equivalent plastic strain, and the equivalent stress.

[0058] Step 3: Conduct a nanoindentation 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 nanoindentation experiment, and obtain the simulated load-displacement curve.

[0059] In this embodiment, the nanoindentation 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 with different grit sizes and then polished to a mirror surface to obtain better experimental results and reduce the influence of the uneven surface of the specimen on the experimental results.

[0060] In step 3, the specific method for constructing the finite element model is as follows:

[0061] (1) Connect the user material subroutine to the finite element software;

[0062] (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.

[0063] Specifically, in this embodiment, the ABAQUS finite element software is used to simulate the nanoindentation experiment of the material, and a nanoindentation model is established. The indentation 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, loads, and their constraints of the model are symmetric about the central axis, two-dimensional axisymmetric elements are selected. During assembly, a reference point is added above the indenter. When setting the material of the model, user-defined material parameters are assigned to the specimen, and the number of state variables added is 10. 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 is bonded to the previously added reference point.

[0064] In addition, a boundary condition restricting movement in the y-direction is set at the bottom of the specimen, and a boundary condition allowing movement in the x-direction is set on the right side of the specimen. When meshing, the indenter and the specimen are first divided into different parts, and finer meshes are drawn within the influence range of the indenter contact to improve the accuracy of the simulation results; a larger mesh size is selected at a distance more than 20 times the maximum indentation depth to facilitate efficient calculation. As Figure 2 shown, it is a schematic diagram of dividing different parts to draw meshes, and it is also a schematic diagram of the indentation model.

[0065] When submitting the job, the pre-edited UMAT subroutine is called using the interface provided by ABAQUS. In the field output during post-processing, the previously set reference point is selected to output the simulated load-displacement curve of the reference point.

[0066] Step 4: Compare the simulated 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.

[0067] The simulated load-displacement curve obtained by finite element software simulation is compared with the experimental load-displacement curve obtained from the nanoindentation 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 the UMAT subroutine writing.

[0068] 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 nanoindentation experiment to obtain the simulated load-displacement curve, and re-compare the two curves until the comparison error is less than the threshold.

[0069] Step 5: From the simulation results, extract the equivalent stress curves of the specimen output by the finite element model in the x and y directions 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.

[0070] After the simulation is completed, open the post-processing file of the completed simulation, select the contour plot representing the equivalent stress state variable, and view the stress contour plots at different increment steps to visually observe the change process of the plastic zone radius. 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 indentation depths at equal intervals. After determining the increment steps, use the path to output the equivalent stress at each increment step to obtain the data of the distance between different indentation depths and the indenter tip and the equivalent stress.

[0071] Determine the value of the initial equivalent yield stress (in this embodiment, uniaxial compression is defined as the equivalent stress state), and use this value as the boundary to determine whether the specimen has reached yield when pressed in. Obtain the distances between each position and the indenter tip and the corresponding equivalent stress data at different indentation depths, 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 indenter tip at different depths , as Figures 4 - 5 shown, respectively taking the indenter tip as the origin, the schematic diagrams of the equivalent stress curves at 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 initial equivalent yield stress straight line is the distance between the positions of the initial equivalent yield stress in the x and y directions and the indenter tip and values.

[0072] In the said Step 5, the calculation formula for the plastic zone radius is:

[0073] ; (5)

[0074] where R represents the plastic zone radius.

[0075] Step 6: According to the plastic zone radii at different indentation depths, fit the relationship between the indentation depth and the plastic zone radius to realize the extraction of the plastic zone radius of the indentation area at any indentation depth.

[0076] Specifically, in step 6 of the present invention, the relationship between the indentation depth and the radius of the plastic zone is fitted by a linear equation.

[0077] As Figure 6 shown, in this embodiment, according to the obtained different indentation depths and their corresponding plastic zone radii, a dot line graph is drawn in Origin, and the relationship equation between the depth and the plastic zone radius can be obtained through numerical fitting. That is, the relationship between the indentation depth and the plastic zone radius 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 of the indentation area.

[0078] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them; 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 described in the foregoing embodiments, or perform equivalent replacements for 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 the nanoindentation of a strength-asymmetric material, characterized in that, It 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 by combining with the yield criterion of the material, and fit the hardening function of the material by combining with the stress-strain data 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 the user material subroutine; Step 3: Conduct a nanoindentation 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 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 output by the finite element model in the x and y directions 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: According to the plastic zone radius at different indentation depths, fit the relationship between the indentation depth and the plastic zone radius to realize the extraction of the plastic zone radius at any indentation depth.

2. A method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material according to claim 1, characterized in that In the said Step 1, through mechanical experiments on the strength asymmetric material under different stress states, 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. A method for measuring the radius of the plastic zone in the nanoindentation process of an intensity asymmetric material according to claim 1, characterized in that In the said Step 1, the yield criterion adopted is the yield criterion considering the tensile-compressive strength asymmetry and anisotropy of the material.

4. A method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material according to claim 1, wherein In the said Step 1, when fitting the hardening function of the material, an exponential equation, a linear equation or a polynomial equation is adopted.

5. A method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material according to claim 1, characterized in that, In the said 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 tensile stress equivalent strain curve and the compressive stress equivalent strain curve.

6. A method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material according to claim 1, characterized in that In the said Step 3, the specific method for constructing the finite element model is: (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.

7. A method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material according to claim 1, characterized in that, In the said Step 4, it also includes the following steps: 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, re-construct 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.

8. A method for measuring the radius of the plastic zone during nanoindentation of a strength-asymmetric material according to claim 1, characterized in that In the said Step 5, the calculation formula for the plastic zone radius is: ； where, R represents the plastic zone radius.

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

Images