Construction of constitutive model for GH4169 alloy under high temperature and high strain rate
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摘要: 利用万能试验机（UTM5305）和霍普金森动态试验装置（ALT
1000 ）对GH4169合金分别进行准静态压缩试验和动态冲击试验，获得常温下应变率为0.001、0.003、0.1 s−1的准静态试验数据；以及温度为25、600、750、900 ℃和应变率为1500 、2500 、3500 、4500 s−1的动态试验数据，构建了Johnson-Cook（JC）本构模型及其修正模型。研究显示，材料的塑性硬化、热软化和速率敏感性都得到了体现，尤其在温度升至900 ℃时，软化效应尤为突出；原始JC本构方程的相关系数（r）为0.9147 ，决定系数（R2）为0.7422 ，平均相对误差（AARE）为14.53%，修正后的JC本构方程相关系数（r）提高至0.9444 ，决定系数（R2）提高至0.8867 ，平均相对误差（AARE）下降至10.77%，相较原始JC本构模型在预测精度和可靠性方面有显著提升，能够可靠正确描述材料的应力-应变行为。Abstract: The quasi-static compression test and dynamic impact test of GH4169 alloy were carried out by universal testing machine (UTM5305) and Hopkinson dynamic testing device (ALT 1000), respectively. The quasi-static test data of strain rates of 0.001, 0.003 and 0.1 s–1 at room temperature were obtained. The Johnson-Cook (JC) constitutive model and its modified model were constructed from the dynamic test data at temperatures of 25, 600, 750, 900 ℃ and strain rates of1500 ,2500 ,3500 ,4500 s–1. The result shows that the plastic hardening, thermal softening and rate sensitivity of the material happen, especially when the temperature rises to 900 degrees Celsius, the softening effect is particularly prominent. The correlation coefficient (r) of the original JC constitutive equation is0.9147 , the coefficient of determination (R2) is0.7422 , and the average relative error (AARE) is 14.53%. The revised JC constitutive equation correlation coefficient (r) is increased to0.9444 , and the coefficient of determination (R2) is increased to0.8867 . The average relative error (AARE) is reduced to 10.77%, which significantly improves the prediction accuracy and reliability compared with the original JC constitutive model, and can be used to predicate the stress-strain behavior of materials.-
Key words:
- GH4169 alloy /
- modified constitutive model /
- high temperature /
- high strain rate /
- prediction accuracy
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图 1 准静态条件下高温合金GH4169的真实应力-真应变曲线
Figure 1. True stress-true strain curve of superalloy GH4169 under quasi-static conditions
图 2 高温合金GH4169 25、600、750、900 ℃的真应力-应变曲线
(a) T=25 ℃; (b) T =600 ℃; (c) T =750 ℃; (d) T =900 ℃
Figure 2. True stress-strain curves for high temperature alloy GH4169 at different temperatures
图 3 高温合金GH4169
1500 、2500 、3500 、4500 s−1的真应力-应变曲线(a) $ \dot{\varepsilon}=1\ 500\text{ }\text{s}^{-1} $; (b) $ \dot{\varepsilon}=2\ 500\text{ }\text{s}^{-1} $; (c) $ \dot{\varepsilon}=3\ 500\text{ }\text{s}^{-1} $ ; (d) $ \dot{\varepsilon}=4\ 500\text{ }\text{s}^{-1} $
Figure 3. True stress-strain curves for high-temperature alloy GH4169 under different strain rates
图 5 $ \ln (\sigma - A) $和$ \ln \varepsilon $的关系
Figure 5. Relationship between $ \ln (\sigma - A) $ and $ \ln \varepsilon $
图 6 J-C本构方程参数$ C $和应变率$ \dot \varepsilon $拟合关系
Figure 6. Fitting diagram of the relationship between J-C constitutive equation parameters $ C $ and strain rate $ \dot \varepsilon $
图 7 J-C本构方程参数$ m $和温度$ T $拟合关系
Figure 7. Fitting diagram of the relationship between parameters $ m $ and temperature $ T $ of the J-C constitutive equation
图 8 两种本构模型的真应力预测和试验值比较
黑色：25℃；红色：600℃；蓝色：750℃；紫色：900℃(a)$ \dot{\varepsilon}=1\ 500\text{ }\text{s}^{-1} $ ; (b) $ \dot{\varepsilon}=2\ 500\text{ }\text{s}^{-1} $;(c) $ \dot{\varepsilon}=3\ 500\text{ }\text{s}^{-1} $ ; (d) $ \dot{\varepsilon}=4\ 500\text{ }\text{s}^{-1} $
Figure 8. Comparison of true stress predictions and experimental values for two constitutive models
图 9 两种本构模型的相关系数(r)、决定系数（R2）和平均相对误差(AARE)
(a）原JC本构模型；(b)修正JC本构模型
Figure 9. Correlation coefficient (r), determination coefficient (R2) and average relative error (AARE) of the two constitutive models
表 1 25、600、750、900 ℃下GH4169合金材料不同应变率时的屈服强度
Table 1. Yield strength of GH4169 alloy at different strain rates at 25, 600, 750 and 900 ℃
温度/℃ 应变率/s−1 屈服强
度/MPa塑性流动
段应变失效
应变失效应
力/MPa25 1500 1221.30 0.0747 0.0761 1263.98 2500 1240.02 0.2028 0.2130 931.75 3500 1305.97 0.3390 0.3611 1020.32 4500 1404.72 0.4357 0.4475 1535.05 600 1500 900.83 0.2079 0.2106 898.18 2500 1000.46 0.2254 0.2345 1062.14 3500 1054.44 0.3434 0.3557 1058.25 4500 1085.02 0.3101 0.3541 595.29 750 1500 891.21 0.1709 0.1756 974.66 2500 910.31 0.2765 0.2987 579.53 3500 950.69 0.3693 0.3868 872.06 4500 979.47 0.4749 0.4910 820.53 900 1500 350.86 0.1605 0.1632 458.72 2500 380.28 0.2519 0.2578 708.53 3500 401.32 0.3718 0.3793 632.80 4500 420.20 0.4506 0.4560 766.36 
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表 2 不同应变率下加工硬化参数(25、600、750、900 ℃下）
Table 2. Work hardening parameters at different strain rates for temperatures of 25, 600, 750, and 900 ℃
温度/℃ k n 1500 s−12500 s−13500 s−14500 s−11500 s−12500 s−13500 s−14500 s−125 3289 1327 1098 862 0.78 0.61 0.70 0.61 600 1655 1920 1386 2536 0.67 0.70 0.71 1.20 750 2430 2131 2207 1153 0.84 0.99 1.05 0.84 900 4622 2188 1 918 1769 1.24 0.93 0.94 0.88
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表 3 GH4169合金传动J-C本构模型参数C的值
Table 3. Values of parameter C of the J-C constitutive model of GH4169 alloy transmission
$ T $/℃ 应变率$ \dot \varepsilon $/s−1 屈服强度$ \sigma $/MPa 参数$ C $值 参数$ C $平均值 25 1500 1175.10 0.0081 0.0127 2500 1305.08 0.0091 3500 1471.96 0.0135 4500 1618.27 0.0201
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表 4 GH4169合金传动J-C本构模型参数m的值
Table 4. Values of parameter m of the J-C constitutive model of GH4169 alloy transmission
应变率$ \dot \varepsilon $/s−1 温度$ T $/℃ 屈服应力$ \sigma $/MPa 参数$ m $值 参数$ m $平均值 1500 600 900.83 1.4931 1.8429 750 891.21 2.0468 900 350.86 0.8153 2500 600 1054.43 2.0961 750 950.69 2.3037 900 401.32 0.9510 3500 600 1187.94 3.0772 750 1048.92 2.8663 900 355.53 0.8192 4500 600 1085.02 2.2265 750 979.47 2.4221 900 420.20 0.9978
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表 5 GH4169合金的修正J-C本构模型参数值
Table 5. Parameters of modified J-C constitutive model for GH4169 alloy
$ A $/MPa $ B $/MPa $ n $ $ {C_1} $ $ {C_2} $ $ {m_1} $ $ {m_2} $ 1113.84 645.69 0.34674 − 0.0006 4.0301 ×10−65.16143 − 0.00442
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