Parametric Analysis and Parameter Inversion of the Crystal Plasticity Constitutive Model for as-Cast TiZrNbV Refractory High Entropy Alloys
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摘要: 难熔高熵合金因其卓越的力学性能而备受关注，但其细观特征行为对其宏观力学行为的影响尚未被充分理解. 随着对材料细观力学行为研究需求的增加，晶体塑性有限元方法已成为揭示晶体材料细观机制的关键工具. 由于晶体塑性本构模型包含众多复杂参数，深入分析这些参数对于理解合金的细观力学行为至关重要. 研究中采用的晶体塑性本构模型考虑了Peierls应力，这一因素能够反映材料的短程势垒，从而更准确地模拟材料的应变率行为. 通过试验设计和极差分析，识别了影响合金力学性能的关键本构参数. 单因素分析明确了关键参数对材料力学特性的具体影响. 在参数反演方面，提出了一种基于优化设计的参数反演方法，该方法结合支持向量回归法和优化算法，能够有效地从宏观力学测试数据中反演出晶体塑性本构参数. 针对铸态TiZrNbV合金，成功反演出一组最优参数，仿真与试验的一致性验证了该方法的有效性. 研究为难熔高熵合金的力学行为预测、材料设计以及性能优化提供了有力的支撑.Abstract: Refractory high-entropy alloys (RHEAs) have attracted considerable attention due to their outstanding mechanical properties. However, the influence of their microstructural behavior on macroscopic mechanical performance remains poorly understood. With the increase of study on material micromechanical behaviors, the crystal plasticity finite element methods become essential tools for uncovering the underlying mechanisms of crystalline materials. Since crystal plasticity constitutive models involve numerous complex parameters, a thorough analysis of these parameters is critical for a deeper understanding of the micromechanical behaviors of alloys. The crystal plasticity model used in this study incorporates the Peierls stress, which accounts for the short-range potential barriers of the material, thereby enabling a more accurate simulation of its strain-rate behavior. Through experimental design and range analysis, the key constitutive parameters affecting the alloy's mechanical properties were identified. Univariate analysis was then employed to clarify the specific effects of these critical parameters on the mechanical characteristics of the material. For parameter inversion, an optimization-based approach was developed, combining the support vector regression with optimization algorithms. This method effectively inverts crystal plasticity constitutive parameters from macroscopic mechanical testing data. For the cast TiZrNbV alloy, a set of optimal parameters was successfully inverted, and the agreement between simulation results and experimental data validated the method's effectiveness. This study provides valuable insights for predicting the mechanical behaviors, guiding material design, and optimizing the performances of refractory high-entropy alloys.
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图 2 参数反演优化设计流程
Figure 2. The flow chart of the parameter inversion optimization design method
图 5 各设计变量对误差响应指标的影响权重
Figure 5. The influence weight of each design variable on the error response indexes
图 6 不同加载率下，各设计变量对各力学指标的影响程度
Figure 6. The magnitudes of effects of each design variable on each mechanical index at different loading rates
图 7 不同n0下，多晶RVE模型的准静态、动态力学响应结果
Figure 7. Quasi-static and dynamic mechanical response results of the polycrystalline RVE model with different n0 values
图 8 不同τ0下，RVE模型的准静态、动态力学响应结果
Figure 8. Quasi-static and dynamic mechanical response results of the polycrystalline RVE model with different τ0 values
图 9 不同τp0下，RVE模型的准静态、动态力学响应结果
Figure 9. Quasi-static and dynamic mechanical response results of the polycrystalline RVE model with different τp0 values
图 10 不同Hk下，RVE模型的准静态、动态力学响应结果
Figure 10. Quasi-static and dynamic mechanical response results of the polycrystalline RVE model with different Hk values
图 11 晶体塑性本构模型参数对材料力学性能的影响行为
Figure 11. Effects of crystal plasticity constitutive model parameters on the mechanical properties of materials
图 12 代理模型训练集、测试集的预测情况
Figure 12. Prediction of the surrogate model training set and test set
图 13 经50次优化求解后各材料参数优化结果的箱线图
Figure 13. Box plots of optimization results for each material parameter after 50 optimization solutions
图 14 参数反演后的晶体塑性有限元模拟结果与试验结果对比
Figure 14. Comparison of crystal plasticity model simulation results after parameter inversion with experimental results
图 15 不同加载条件下仿真结果与试验结果的力学响应特性对比
Figure 15. Comparison of mechanical response characteristics between simulation results and experimental results under different loading conditions
表 1 试验设计各因素的取值范围
Table 1. The range of values for each factor in the experimental design
design variable symbol lower limit upper limit initial hardening modulus h0/MPa 100 500 saturation stress τs/MPa 220 780 initial critical resolved shear stress τ0/MPa 200 700 rate sensitivity exponent n0 30 80 reference slip shear rate $\dot{\gamma}$0/s-1 2×10-3 5×10-3 Peierls stress τp0/MPa 300 900 reference strain rate $\dot{\gamma}$p0/s-1 5.7×107 1.51×108 kink-pair formation enthalpy Hk/eV 0.38 0.56 
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表 2 参数单因素分析的设计变量取值情况
Table 2. Design variable values for parametric univariate analysis of variance
design variable symbol level 1 level 2 level 3 level 4 level 5 rate sensitivity exponent n0 30 15 22.5 37.5 45 initial critical resolved shear stress τ0/MPa 300 150 225 375 450 Peierls stress τp0/MPa 500 250 375 625 750 kink-pair formation enthalpy Hk/eV 0.62 0.31 0.47 0.78 0.94 saturation stress τs/MPa 600 - - - - initial hardening modulus h0/MPa 200 - - - - reference slip shear rate $\dot{\gamma}$0/s-1 3×10-3 - - - - reference strain rate $\dot{\gamma}$p0/s-1 1×108 - - - -
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表 3 铸态TiZrNbV高熵合金的晶体塑性本构模型参数
Table 3. Crystal plasticity constitutive model parameters for cast TiZrNbV high-entropy alloys
design variable symbol value initial hardening modulus h0/MPa 91 saturation stress τs/MPa 430 initial critical resolved shear stress τ0/MPa 410 rate sensitivity exponent n0 34 reference slip shear rate $\dot{\gamma}$0/s-1 3.6×10-3 Peierls stress τp0/MPa 320 reference strain rate $\dot{\gamma}$p0/s-1 9.2×107 kink-pair formation enthalpy Hk/eV 0.60
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