Beijing ,China. Keywords: Constitutive equation; Parameter Extraction; Iteration algorithm

Transcription

1 pplied Mechanics and Materials Online: ISSN: , Vol. 281, pp doi: / Trans Tech Publications, Switzerland new method based on interation algorithm to extract parameters of constitutive equations Yongzhi HU 1, a, Liwen GUN 2, b, Xinjun LIU 3, c and Yuhui ZHNG 4, d 1, 2, 3.4 Department of Precision Instruments and Mechanology,Tsinghua University, eijing ,China a huadiancun@163.com, b guanlw@mail.tsinghua.edu.cn, c xinjunliu@mail.tsinghua.edu.cn d yezaiyufengzi@163.com Keywords: Constitutive equation; Parameter Extraction; Iteration algorithm bstract. These methods that extract parameters of constitutive equations can be divided into three groups: direct search-based strategies, gradient-based methods and evolutionary algorithms. y analyzing these strategies, a new method based on iteration algorithm was proposed. To obtain parameters of JC and Z model for Ti-6l-4V, the between prediction data and SHP experiment data was set as objective function, then initial value was calculated using iteration algorithm. The effect of convergence rate and precision at various steps and experiment data was invested. The main advantage of the method are as follows: fast calculation; compatible with SHP data and orthogonal cutting data; compatible with the decoupling and coupling constitutive equations. Finally, it has shown that the algorithm is stable, and acceptable results can be obtained. Introduction Constitutive equation is a stress function with temperature, strain and strain rate happened in material deformation. Several models used for metal cutting, are respectively Johnson-Cook model [1], Power Law model [2], Zerilli-rmstrong model [3]. To ensure the accuracy of calculation, a proper constitutive equation is needed. The parameters of the equation are obtained by using material experiments. The more strain, strain rate and temperature data similar between the material experiments and metal cutting, the more accurate we get the parameters. Two methods are used to get the experiment data, split Hopkinson pressure bar [4.5], also named SHP, and orthogonal cutting [6]. The advantages of SHP test are as follows: (a) convenient to perform; (b) a great deal of data can be obtained from these experiment curves; (c) experiments can be carried out under specified strain rate and temperature; while the disadvantages of SHP are: (a) moderate strain rate and strain; (b) difficulty in heating specimens in short time [7]. The advantages of orthogonal cutting experiment are: (a) high strain and strain rate, (b) high heating rate, and also the disadvantages are: (a) the temperature is calculated instead of test, (b) experiments can t be carried out under specified strain rate and temperature. These methods used for the material model parameters identification can be divided mainly into three groups: (a) direct search-based strategies, (b) gradient-based methods, (c) evolutionary algorithms. Lee and Lin investigated the high temperature deformation behavior of Ti-6l-4V using SHP, and regression analysis is used to determine the constitutive equation parameters. Meyer and Kleponis analysis the SHP test using the least square method and extracted parameters of JC and Z models. Two step method are used by Chaparro to identify the parameters, the first step is getting the initial value using genetic algorithm, and the second step is parameters optimization using gradient-based algorithm. Sasso studied the parameters of 1018 steel using the least squares method. Direct search-based strategies are usually used to get the parameters for decoupling constitutive equation by SHP test, because the SHP data can be divided into groups of different temperature or strain rate, and the stress-strain curves can provide more data. Gradient-based methods are usually used for orthogonal cutting experiment [8~13]. y changing the cutting speed or feed in the machining, different cutting forces and the thickness of chips are recorded. Compare the simulation results with the experiment, then the parameters of the equation ll rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (# , Pennsylvania State University, University Park, US-15/09/16,19:56:47)

2 506 Mechanical Engineering, Materials and Energy II can be optimized by using iterative algorithms. The methods apply not only turning but also milling [14]. The main question for these methods is that many factors affect the simulation (such as friction and finite element mesh), so it is hard to get a good simulation by adjusting the parameters of constitutive equation. Evolutionary algorithms are also applied to optimize the parameters of the equation. The advantage of the algorithms is global optimization results would be obtained, while the disadvantage is that a great deal of experiment data is needed to guarantee the precision of the parameters. 1 Constitutive model and initial value 1.1 JC model. From the tests over a wide range of strain rates, static tensile tests, dynamic Hopkinson bar tensile tests and Hopkinson bar tests at elevated temperatures, Johnson and Cook summarized a constitutive equation which is known as JC equation.. (1) Where is the equivalent flow stress, the equivalent plastic strain, the equivalent plastic strain rate, the reference equivalent plastic strain rate, T is the workpiece temperature, T m and T r are, respectively the material melting and room temperature.,, C, n and m are constitutive constants fitted to the data obtained material experiments. 1.2 SHP experiment data and constitutive parameters. Fig.1 and Fig.2 are respectively are SHP curves at different temperature and different strain rate, which are obtained from experiments studied by Lee (1996) and Seo (2004). The temperature of Lee s is range of , with strain rate 2000s -1. The temperature of Seo s data is range of , with strain rate 1400 s -1. oth of these experiment data is used to optimize the constitutive parameters. Fig.1 SHP data where strain rate is 1400 s -1 Fig.2 SHP data where strain rate is 2000s -1 ccording to different experiments, varies JC parameters are obtained by different researchers as shown in Table 1. Table1 JC constitutive parameters of Ti-6l-4V obtained by different research Number Researcher Umbrello [15] Shivpuri Lee&Lin Li&He Ozel&Zeren [13] Meyer & Kleponis [10]

3 pplied Mechanics and Materials Vol lgorithm analyses s we can see in the Fig.3, five numbers is choose as the initial value of constitutive equation, set k as step length; then three values derive from each initial value by add subtract step; 243 sets of parameters are obtained by permutation and combination of 15 parameters. For each set of constitutive parameter, experiment data including SHP test and cutting test is used to calculate the sum s. Then the parameters with the least sum s will be choose as the new constitutive value, and if the parameter is the same as the last one, stop the loop and import the parameter. The sum s for each set of parameter are calculated as follows:. (2). (3) Where n is the num of experiment data, k,, k,c k, m k and n k are one set of constitutive parameters, (i) is true strain, is true strain rate, T(i) is true temperature, σ p (i) is stress calculated by these experiment data, is true stress, is the sum. Start Set initial model parameters( C m n) Set step k=2% = (1+k) (1+k) = (1+k) (1+k) n=n n(1+k) n(1+k) Set new model parameters Permute the parameters above, get 243 groups of parameters Calculate the sum results from each set of parameters Load experiment data Load the set of parameters which is minimal No The same as the former one Yes Output the parameters and the and iteration End Fig.3 lgorithm flowchart 3 Comparison and results 3.1 Local convergence. ccording the algorithm above, 827 experiment data are obtained from Lee and Seo s SHP curves. Using the initial values in Table 1, JC parameters are calculated, and step is 2%, the results are shown it Table 2. The algorithm is local convergence as the same with most algorithms. We can see different results are obtained for various initial values. To get global optimal solution, the average s of different results are studied and the least s were chose as the parameters.

4 508 Mechanical Engineering, Materials and Energy II Table 2 Different parameters after optimization verage Parameters Optimization Parameters Optimization Parameters Optimization Parameters Optimization Parameters Optimization Parameters Optimization The step. The step of the loop affects not only the loop speed but also the precision of the results. ig step make it converge more quickly, however small step lead to more accurate results. To get better results, the process are divide into two parts, k is big for the first part and small for the second. s we can see in the Table 2, smaller step get more accuracy results, then the step 0.1% is choose as the parameters of Ti-6l-4V. Table 3 Optimization results in big step verage Initial values Step 2% Step 4% Step 6% Table 4 Optimization results in small step verage Initial values Step 0.1% Step 0.2% Step 0.4% Precision. For the same initial value, the results are different with different iteration. s shown in Table 5, the s of parameters will be improved after optimization using Lee s data or Seo s data, while the better results will be obtained when these two sets of data are combined. Table 5 Optimization results using different experiment data verage Initial values Lee Seo Lee and Seo

5 pplied Mechanics and Materials Vol Data compatibility. What makes the algorithm outstanding is the compatibility of experiment data, both SHP data and orthogonal cutting experiment data can be used in the algorithm at same time. The unified form of experiment data can be [stress, strain, strain rate, temperature], and many point can be extracted from the SHP curves, also these data can be calculated using the cutting forces and thickness of chips in the cutting experiment. 3.5 Decoupled equation and coupled equation. The method applys not only decoupled constitutive equation, but also coupled constitutive equation. s we all know, the Z- model are coupled equation as Eq.4.. (4) Where σ is stress, is strain, is strain rate, T is absolute temperature, C 0, C 1, C 3, C 4 and C 5 are material parameters. The material parameters of Ti-6l-4V for Z- model are calculated by Meyer as shown in table 6. The parameters after optimization are shown in table 7. fter optimization, the sum s are decreased from 508 to 86, which means these s are improved. 3.6 Fast calculation. Compared with other gradient-based methods, this method can run faster. There are two reasons for rapid calculation: (a) finite element iteration algorithm for constitutive equation, simulation results are compared with the experiment, then modify the parameters, the simulation cost much time; (b) each time after comparison of these results, only one parameter will be modified in the iteration algorithm while all parameters can be modified at the same time, so it can save much time. 4 Validation of constitutive parameters ccording the parameters determined above, the JC equation can be expressed as follows:. (5) To validate the constitutive parameters, stress at different stain, strain rate and temperatures are predicted. y observing the Fig.4 and Fig.5, we can see the calculated data and experiment data fit very well, which show the accuracy of the parameters. Fig.4 Prediction where strain rate is 1400s -1 Fig.5 Prediction where strain rate is 2000s -1 5 Conclusions y optimizing the JC parameters of Ti-6l-4V using SHP data, we can draw the conclusions as follows: (1) The method proposed in the paper is an effective way to determine the parameters of constitutive models, no matter the equation are coupled or not. The process shows it can convergence quickly. (2) ecause the initial value affect the parameters, so many initial values should be choose to get the optimal solution.

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