FexCy micro-mechanical properties based on response surface methodology and molecular dynamics
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摘要: 探究了不同碳钢材料在不同温度下的力学性能响应状况，采用分子动力学建立仿真模型，通过三因素多水平正交试验和响应面法建立回归模型，研究C含量、空位比以及温度对杨氏模量E和屈服强度Q的影响。在相同的C含量和空位比条件下，利用Matlab的随机函数建立了50个模型，每种试验条件进行50次仿真。统计杨氏模量和屈服极限的中位数作为反应材料力学性能的关键参数，建立响应面回归模型。通过随机选取10组仿真试验验证，成功地探究了不同因素对碳钢力学性能的影响规律，得到了材料力学参数的可靠数学模型，并对材料成分组成进行了优化设计。Abstract: This research is to investigate the mechanical properties of different carbon steel materials at different temperatures. The simulation model was established by using molecular dynamics, and the regression model was established by the three-factor multi-level orthogonal test and response surface method to study the effects of C content, vacancy ratio and temperature on Young’s modulus and yield strength. Under the same C content and vacancy ratio conditions, 50 models were established by using the random function of Matlab, and each test condition was simulated 50 times. The median of Young’s modulus and yield limit were used as key parameters of the mechanical properties of the reaction materials, and the response surface regression model was established. By randomly selecting 10 groups of simulation experiments, this study successfully explored the influence of different factors on the mechanical properties of carbon steels, obtaining a reliable mathematical model of materials mechanical parameters, and optimizing the composition of materials.
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Key words:
- molecular dynamics /
- response surface analysis /
- Young’s modulus /
- yield strength
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图 3 杨氏模量响应曲面及等高线图
(a)(d)温度、含碳量与杨氏模量的关系；(b)(e)温度、空位比与杨氏模量的关系；(c)(f)含碳量、空位比与杨氏模量的关系
Figure 3. Young's modulus response surfaces and contour maps
图 4 屈服强度响应曲面及等高线
(a)(d)温度、含碳量和屈服强度的关系；(b)(e)温度、空位比与屈服强度的关系；(c)(f)含碳量、空位比与屈服强度的关系
Figure 4. Yield strength response surfaces and contours
表 1 因素水平设计
Table 1. Design of factors and levels
水平等级 影响因素 温度T/K 含C量R/% 空位比V/% −2 300 1 1 −1 500 1.3 2 0 700 1.6 3 1 900 1.9 4 2 2.2 5 
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表 2 响应面试验设计
Table 2. Experimental design of response surface method
编号 温度T/K 含C量R/% 空位比V/% 编号 温度T/K 含C量C/% 空位比V/% 1 500 1 3 11 300 1.6 2 2 900 1 2 12 700 1.3 2 3 900 1.6 3 13 900 2.2 5 4 500 1 3 14 900 1.6 1 5 500 1.6 5 15 300 1 5 6 500 1.6 5 16 900 1.6 3 7 700 2.2 1 17 900 1 5 8 300 2.2 2 18 300 2.2 4 9 900 2.2 3 19 700 1.3 2 10 700 2.2 1 20 300 1.6 2
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表 3 分子动力学性能平均值
Table 3. Average molecular dynamics properties
编号 E/GPa Q/GPa 编号 E/GPa Q/GPa 1 152.616 10.9617 11 205.276 12.8755 2 42.6706 9.9332 12 90.412 10.0607 3 56.319 9.4425 13 88.4735 9.7228 4 152.616 10.9617 14 36.4433 9.7911 5 168.574 10.8488 15 205.762 12.2664 6 168.574 10.8488 16 56.319 9.4425 7 86.3013 10.0222 17 81.2357 9.5083 8 213.773 12.8354 18 220.891 12.4089 9 58.739 9.4343 19 90.412 10.0607 10 86.3013 10.0222 20 205.276 12.8755
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表 4 杨氏模量和屈服强度多种模型综合分析结果
Table 4. Results of comprehensive analysis of Young’s modulus and yield strength models
模型 $ P $ $ {R}^{2} $ 顺序P值 校正值 预测值 E 线性 < 0.0001 0.9868 0.9799 两因素 < 0.0001 0.9975 0.9956 平方 0.0471 0.9985 0.9936 三次 1.0000 Q 线性 < 0.0001 0.9887 0.8635 两因素 0.2947 0.9053 0.7766 平方 < 0.0001 0.9970 0.9847 三次 1.0000
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表 5 杨氏模量回归模型方差分析
Table 5. Variance analysis of Young’s modulus of compressive strength regression model
平方和 自由度 均方值 F值 P值 是否显著 $ E\left(T,R,V\right) $ 78703.48 6 13117.25 1275.1 < 0.0001 显著 $ T $ 68447.57 1 68447.57 6653.63 < 0.0001 显著 $ R $ 246.39 1 246.39 23.95 0.0003 显著 $ V $ 2374.5 1 2374.5 230.82 < 0.0001 显著 $ TR $ 73.67 1 73.67 7.16 0.019 显著 $ TV $ 535.98 1 535.98 52.1 < 0.0001 显著 $ RV $ 20.23 1 20.23 1.97 0.1843 残差 133.73 13 10.29 总变异值 78837.22 19
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表 6 屈服极限回归模型方差分析
Table 6. Variance analysis of yield limit of compressive strength regression model
平方和 自由度 均方值 F值 P值 是否显著 $ Q\left(T,R,V\right) $ 29.82 9 3.31 709.77 < 0.0001 显著 $ T $ 26.93 1 26.93 5769.72 < 0.0001 显著 $ R $ 0.0588 1 0.0588 12.61 0.0053 显著 $ V $ 0.4652 1 0.4652 99.65 < 0.0001 显著 $ TR $ 0.0069 1 0.0069 1.47 0.2534 $ TV $ 0.1378 1 0.1378 29.51 0.0003 显著 $ RV $ 0.1589 1 0.1589 34.03 0.0002 显著 $ {T}^{2} $ 1.87 1 1.87 401.02 < 0.0001 显著 $ {R}^{2} $ 0.005 1 0.005 1.08 0.3234 $ {V}^{2} $ 0.1421 1 0.1421 30.43 0.0003 显著 残差 0.0467 10 0.0047 总变异值 29.87 19
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表 7 模型可信度检验分析
Table 7. Reliability test and analysis of the model
模型 标准差 相关系数 调整系数 变异系数/% 信噪比 $ E\left(T,R,V\right) $ 3.21 0.9983 0.9975 2.6 100.5393 $ Q\left(T,R,V\right) $ 0.0683 0.9984 0.9970 0.6376 72.3359
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表 8 验证不同温度T、含碳量R以及空位率V下的仿真试验结果
Table 8. Verification of the simulation results under different temperatures T, carbon contents R and vacancy rates V
编号 组合 $ E $/GPa $ Q $/GPa 温度T/K 含碳量R/% 空位率V/% 1 300 1.9 1 205.1847 13.0851 2 500 1.9 2 156.9726 11.1803 3 500 1.9 4 166.5242 11.0399 4 700 1.9 3 99.7662 10.3489 5 900 1.9 5 85.7702 9.7070 6 300 1 1 192.1814 13.2412 7 500 1.3 4 160.2058 10.9185 8 500 2.2 3 164.0080 11.1295 9 700 1.6 1 85.3685 10.0065 10 900 1 2 42.6706 9.9332
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表 9 仿真值与预测值误差
Table 9. The errors of simulation and prediction values
编号 $ E $ $ Q $ 绝对误差/GPa 相对误差/% 绝对误差/GPa 相对误差/% 1 0.99 0.48 0.09 0.66 2 3.88 2.47 0.16 1.47 3 1.01 0.60 0.25 2.30 4 9.04 9.06 0.57 5.46 5 0.39 0.45 0.04 0.43 6 2.73 1.42 0.34 2.58 7 0.39 0.24 0.16 1.50 8 0.72 0.44 0.30 2.72 9 1.27 1.48 0.29 2.90 10 0.35 0.83 0.10 1.05
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表 10 优化结果预测值与仿真值
Table 10. Optimized results of the predicted values and simulation values
T/K R/% V/% $ E $ $ Q $ 预测
/GPa仿真
/GPa相对
误差/%预测
/GPa仿真
/GPa相对
误差/%750 1.018 1.004 71.108 70.349 1.067 10.442 10.220 2.126 800 1.004 1.120 58.572 59.145 0.978 10.266 10.153 1.101 850 1.063 1.499 49.431 50.017 1.185 9.996 10.092 0.960 900 1.000 2.000 42.317 42.671 0.837 9.829 9.933 1.058
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