CN104077444B - Analysis method of indentation data - Google Patents

Abstract

The invention relates to a method for analyzing indentation data. The core of the method is numerical optimization based on the Levenberg-Marquardt algorithm. The optimized objective function is the difference between the indentation data obtained by finite element simulation and the experimental indentation data. The parameters are the mechanical parameters of the material to be tested. Since a variety of loading-unloading schemes and material constitutive models can be defined in the finite element model, the data analysis method has strong versatility.

Description

A Method for Analyzing Indentation Data

technical field

The invention relates to an indentation data analysis method.

Background technique

As the simplest and most feasible method, the indentation method may be the earliest mechanical property testing method used by humans. According to the difficulty of indentation or the size of the residual indentation area, various hardness indexes have been defined to evaluate the ability of materials to resist permanent deformation. The head is pressed into the material to be tested, and the softness and hardness of the material are evaluated according to the residual deformation of the pressed-in area. Depending on the indenter, load, and load retention time, there are various types of indentation hardness, such as Brinell hardness, Rockwell hardness, Vickers hardness, and microhardness.

The hardness test is simple and easy, and the data repeatability is good. However, hardness is only a comprehensive index to measure the ability of materials to resist plastic deformation, and there is no clear conversion relationship between it and constitutive parameters such as elastic modulus and yield strength of materials, which makes it difficult to test hardness in the constitutive relationship of materials. Immediate application in characterization. With the development of precision measurement and control technology, in the 1970s and 1980s, a test method born out of microhardness testing but with higher precision and continuous recording of indentation force and indentation depth, that is, instrumented indentation ( instrumented indentation) method or depth-sensing indentation (depth-sensing indentation) method. Due to the high spatial resolution (the size of the indenter can be as small as micro/nano scale), the instrumented indentation method is especially suitable for the mechanical performance testing of small-scale materials, thin films and coating material systems. After nearly 20 to 30 years of rapid development, this method has been widely used in various aspects of mechanical measurement and material deformation mechanism research, greatly expanding the application range of the indentation method.

Since the indentation process involves complex three-dimensional deformation, the indentation test cannot directly obtain constitutive parameters such as elastic modulus and yield strength from the experimental curve like uniaxial tension or uniaxial compression. For indentation testing, due to the lack of simple correspondence between constitutive parameters and test data, it is usually necessary to apply appropriate models and methods to analyze the experimental data in order to obtain the required constitutive parameters from the test data. This process is called It is the reverse analysis of indentation data.

The current commercial instrumented press-in equipment usually uses expensive capacitive distance measuring sensors and electromagnetic actuation devices. To achieve high-frequency loading, complex measurement and control techniques are often also used. This leads to the high cost of instrumented press-in equipment, which seriously restricts the application of this equipment with simple structure, diverse uses and high spatial resolution in industrial inspection. The current press-in equipment is still mainly used in laboratory research. In addition, the constitutive parameters to be measured are different, and the indentation test schemes and indentation data back analysis methods used are often also different. Due to the complexity of the indentation problem, there is no general indentation data back-analysis method at present. People often need to select or develop a suitable data analysis method according to the characteristics of the material to be tested, which undoubtedly limits instrumented indentation. Application of the method in industrial measurement.

Contents of the invention

The purpose of the present invention is to propose a method for analyzing indentation data, which can solve the problem that traditional analysis methods are not universal.

In order to achieve the above object, the technical scheme adopted in the present invention is as follows:

A method for analyzing indentation data, comprising the steps of:

Step 1, establishing a finite element model of the indentation test process, wherein the mechanical parameters of the material to be tested are defined as variables;

Step 2. Input the given initial mechanical parameters into the finite element model, and calculate the finite element model to obtain simulated indentation data;

Step 3, calling the simulated indentation data and reading the experimental indentation data, and using formula 1 to calculate the difference between the simulated indentation data and the experimental indentation data;

(Formula 1)

Among them, F(P ^k ) is the return value of the objective function; P ⁰ is the initial value of the mechanical parameters, P ^k is the mechanical parameters after k times of correction; N is the number of data contained in the experimental indentation data; f ^exp (t _i ) is the experimental indentation test result when the loading time is equal to t _i ; f ^cal (P ^k , t _i ) is the simulated indentation test result when the loading time is equal to t _i ;

Step 4, using the Levenberg-Marquardt algorithm to optimize the objective function, when it is judged that F(P ^k ) is smaller than the preset threshold, output P ^k as the final optimization result.

Preferably, said step 4 specifically includes the following sub-steps:

(a) Use P ⁰ as the initial parameter of the Levenberg-Marquardt algorithm;

(b) Calculate the sensitivity matrix or Jacobian matrix using the finite difference method

(c) Solving the equation (A ^T A+λI)g ^k =-A ^T F(P ^k ) to obtain the correction value g ^k , where I is the identity matrix, and λ is a non-negative scalar parameter;

(d) Calculate P ^k+1 =P ^k +g ^k , and judge whether the objective function is smaller than the preset threshold, if yes, output P ^k as the final optimization result, if not, repeat step a to step d.

Preferably, the finite element model is an axisymmetric finite element model.

The present invention has following beneficial effects:

The indentation data reverse analysis method is applicable to any loading scheme and material type, avoiding the problem that the indentation test is not suitable for measuring complex materials.

Description of drawings

Fig. 1 is the algorithm flowchart of the indentation data analysis method of the preferred embodiment of the present invention;

Fig. 2 is the experimental indentation data curve;

Fig. 3 is the finite element model that the present invention establishes;

Fig. 4 is the stress-strain curve obtained by the present invention.

detailed description

In the following, the present invention will be further described in conjunction with the drawings and specific embodiments.

As shown in Figure 1, a method for analyzing indentation data comprises the following steps:

Step 1. Establish a finite element model of the indentation test process, wherein the mechanical parameters of the material to be tested are defined as variables. The finite element model includes a material model selected according to the type of material to be tested.

Considering that when analyzing the indentation force-indentation depth data, the pyramidal indenter such as Berkovich can be replaced by an equivalent conical indenter, so this embodiment uses an axisymmetric finite element model. A spherical indenter is defined as a rigid 1/4 arc, and a conical indenter is defined as a rigid straight line. For the Berkovich indenter, the equivalent inclination angle of the straight line is 65.03°, and for the Vicker indenter, the equivalent inclination angle of the straight line is 70.32°. The indenter is coupled to a control point and its motion is driven by a force or displacement applied to the control point. The sample is defined as a rectangular area whose length and width are both greater than 10 times the size of the indenter. The finite element grid is subdivided in the area close to the indenter, and the unit size in the subdivided area is not greater than 1/20 of the indenter-sample contact radius. Contact is defined between the indenter and the upper surface of the sample. In the finite element model, the mechanical parameters to be obtained, such as Young's modulus, yield strength, hardening index, dynamic strengthening parameters, etc., are defined as convenient replacement parameters. In addition, the boundary conditions or loads of the finite element model are equivalent to the actual test loading process. If the indentation depth is controlled by the indentation force during the test, the equivalent indentation force history is defined in the finite element model. Depth control is used as the driving control quantity, and the equivalent displacement boundary conditions are defined in the finite element model. Finally, the finite element model is saved as a data file that can be invoked by the finite element solver.

Step 2. Input the given initial mechanical parameters into the finite element model, and calculate the finite element model to obtain simulated indentation data.

In the specific implementation, the initial mechanical parameters are artificially given according to the type of the material to be tested, and the initial mechanical parameters can be a number greater than 0. It can also be estimated according to the material to be tested, within the range of the empirical value of the material to be tested The given initial mechanical parameters do not affect the calculation results, but only the calculation speed. Pass the given initial mechanical parameters into the finite element model in step 1, replace the variables in the finite element model and then perform calculations, and the output is the time-indentation force curve or indentation depth-indentation force curve (that is, the simulation indentation data). The flow of this step is defined as an indentation test simulation function.

Described from the perspective of software execution, the process is as follows: when the indentation test simulation function is executed, first use the data search and replace function to analyze the finite element model defined in step 1, and replace the finite element model in step 1 with the input mechanical parameters. The corresponding mechanical parameters in the element model, and save the replaced finite element model as a new finite element model. Then call the finite element solver to solve the newly generated finite element model according to the command format of the finite element solver. After the calculation is completed, save the calculation result in the form of a file, that is, time-indentation force data or depth-indentation force data.

Step 3. Call the simulated indentation data and read the experimental indentation data. The experimental indentation data is obtained from the indentation testing equipment for the tested material, which can be time-indentation force curve or indentation depth- Indentation force curve, using formula 1 to calculate the difference between simulated indentation data and experimental indentation data;

(Formula 1)

Among them, F(P ^k ) is the return value of the objective function; P ⁰ is the initial value of the mechanical parameter (that is, the initial value given artificially), P ^k is the mechanical parameter after k times of correction; N is the experimental indentation data The number of data contained on the curve of ; f ^exp (t _i ) is the experimental indentation test result when the loading time is equal to t _i (that is, the experimental indentation data); f ^cal (P ^k ,t _i ) is the loading time is equal to t i The simulated indentation test results at time _i (ie simulated indentation data). If t _i is not equal to a certain moment output in step 2, use the interpolation method to obtain the calculated indentation test result at t _i moment.

Step 4: Optimizing the objective function using the Levenberg-Marquardt algorithm, and outputting P ^k as the final optimization result when it is judged that F(P ^k ) is less than a preset threshold (such as 10 ⁻³ ).

Preferably, said step 4 specifically includes the following sub-steps:

(a) Use P ⁰ as the initial parameter of the Levenberg-Marquardt algorithm;

(b) Calculate the sensitivity matrix or Jacobian matrix using the finite difference method Wherein F(P ^k ) is the objective function defined in step 3;

(c) Solve the equation (A ^T A+λI)g ^k =-A ^T F(P ^k ) to obtain the correction value g ^k , where I is the identity matrix, T represents the transpose of the matrix, and λ is a non-negative scalar parameter , if the iterative step reduces the objective function, then λ should be increased, otherwise it should be decreased;

(d) Calculate P ^k+1 =P ^k +g ^k , and judge whether the objective function is smaller than the preset threshold, if yes, output P ^k as the final optimization result, if not, repeat step a to step d.

In order to improve the optimization accuracy, 3-5 groups of different given initial mechanical parameters can be selected, according to step 4, the optimized mechanical parameters are obtained respectively, and then the obtained multiple groups of optimized mechanical parameters are counted to obtain each mechanical parameter respectively. The mean value and standard deviation of the parameters, if the standard deviation is greater than 10% of the mean value (or the ratio set by oneself), then it is considered that the mechanical parameters are not unique, and it is necessary to improve the test plan and repeat steps 1-4 for the improved test plan Evaluate.

In order to facilitate the understanding of this embodiment, the following description will be made in conjunction with a specific case.

The Q350 steel was tested by the indentation test equipment, and three groups of typical measured indentation curves (ie, experimental indentation data) were obtained, as shown in Figure 2.

Using the indentation data analysis method of this embodiment, the data shown in FIG. 2 is analyzed to calculate the Young's modulus, yield strength and strain hardening exponent of the tested material.

(1) Establish the finite element model of the indentation test process.

The diameter of the indenter is 1 mm, and the indenter-sample contact area is locally densified, and the smallest unit size is 0.01 mm. Assume that the material follows the Holloman model, that is, its stress-strain relationship is:

where E is Young's modulus, _σ0 is the yield strength, and n is the strain hardening exponent. In the finite element model file, define these three mechanical quantities as replaceable parameters, namely _E, _S0, _n. Linear loading-unloading, loading time 10s, unloading time 10s is consistent with the time used in the test. Displacement-controlled loading, output press-fit force-time results.

(2) Define the indentation test simulation function double*VirturalIndentation(double E, double S0, double n), where E, S0 and n are Young's modulus, yield strength and strain hardening exponent, respectively. Use the string search function to obtain the positions of the three variable names "_E", "_S0", and "_n" in the finite element model file, and use the string replacement function to replace these three parameters with E, S0 respectively and the initial value of n, save the replaced model file as a new model file, such as newIndent.comm. Call the system command system("waster-m512newIndent.comm") to perform finite element calculation, where waster is the name of the finite element solver, and -m512 specifies that the maximum memory is 512M. After the calculation is completed, read the pressing force-time data from the result file and return it in the form of an array.

(3) Define the optimization objective function double ObjectiveFun (double E, double S0, double n). The optimization objective function (process) first calls the indentation simulation function VirtualIndentation(E,S0,n) to obtain the corresponding simulated indentation data. Then read the experimental indentation data, calculate the difference between the simulated indentation data and the experimental indentation data according to the number of data points contained in the experimental indentation data, and return it.

(4) Optimize the objective function ObjectiveFun with the Levenberg-Marquardt algorithm, and the threshold is 10 ^-5 .

According to the characteristics of Q345 steel, three sets of mechanical parameter estimates were selected as the initial values for optimization. In order to confirm the uniqueness of the mechanical parameters, there is a large deviation between the initial values and typical values:

After optimization, the elastoplastic parameters obtained are:

It can be judged that all three mechanical parameters can be uniquely determined.

The corresponding stress-strain curves are shown in Fig. 4.

Generally speaking, this embodiment relates to a method for analyzing indentation data. The core of the method is numerical optimization based on the Levenberg-Marquardt algorithm. The difference between them, the optimized parameters are the mechanical parameters to be found for the tested material. Since various loading-unloading schemes and material constitutive models can be defined in the finite element model, the data analysis method has strong versatility.

Compared with the prior art, the advantages of this embodiment are:

(1) Mature technology and high stability: The force measurement, distance measurement and control technologies used in this embodiment are all mature technologies (the traditional indentation test equipment can be used), which has the advantages of controllable cost and good working stability. ;

(2) Wide range of applicable materials: suitable for testing various engineering materials such as metals, plastics, and ceramics;

(3) The uniqueness of the result can be judged.

For those skilled in the art, various other corresponding changes and modifications can be made according to the technical solutions and ideas described above, and all these changes and modifications should fall within the protection scope of the claims of the present invention.

Claims (2)

1. a kind of indentation data analysis method, it is characterised in that comprise the following steps：

Step 1, the FEM model for setting up impression test process, wherein, the mechanics parameter of measured material is defined as variable；

Step 2, given initial mechanics parameter is input into the FEM model, and the FEM model is counted Calculate, obtain simulating indentation data；

Step 3, call it is described simulation indentation data and read experiment indentation data, using the calculating simulation indentation data of formula one With the difference between experiment indentation data；

Wherein, F (P^k) for object function return value；P⁰For the initial value of mechanics parameter, P^kTo correct the mechanics parameter after k time； N is to test the data amount check that indentation data is included；f^exp(t_i) it is equal to t for the loading moment_iWhen experiment impression test result；f^cal (P^k, t_i) it is equal to t for the loading moment_iWhen simulation impression test result；

Step 4, object function is optimized using Levenberg-Marquardt algorithms, when determining F (P^k) less than default During threshold value, P is exported^kAs final optimum results；The step 4 specifically includes following sub-step：

A () is by P⁰As the initial parameter of Levenberg-Marquardt algorithms；

B () calculates sensitivity matrix or Jacobi an matrixes using finite difference method

(c) solving equation (A^TA+λI)g^k=-A^TF(P^k), obtain correction g^k, wherein I is unit matrix, and λ is the scalar of non-negative Parameter；

D () calculates P^k+1=P^k+g^k, and judge that object function, whether less than predetermined threshold value, if so, then exports P^kAs final excellent Change result, if it is not, then repeat step a is to step d.

2. indentation data analysis method as claimed in claim 1, it is characterised in that the FEM model is that axial symmetry is limited Meta-model.