The numerical stability of the constitutive calculation on viscoplastic materials
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摘要: 提出本构方程计算方法的稳定性问题，针对黏塑性本构计算的显式精确算法的稳定性进行分析，发现该算法并非无条件稳定，使用小扰动方法给出了其计算稳定的必要条件，稳定性条件对数值计算中的时间步长提出限制要求。通过有限元算例验证了分析的正确性，计算结果也表明理论推导得到的稳定性公式能够准确预测满足计算稳定性条件要求的最大时间步长与各参数之间关系。Abstract: At first, we analyzed the numerical stability of the explicit exact algorithm developed for the viscoplastic material, and then found that the explicit exact algorithm is not absolutely stable, deduced a necessary criterion that the time step should be kept below a certain value to guarantee the constitutive calculation stability. A series of numerical examples were presented to validate the reliability of the present stability analysis on the explicit exact algorithm. The results of the numerical examples show that the effective stress is unstable while the stability criterion for the constitutive calculation is not satisfied, but a complex deformation process including the elastic load, the plastic load, the elastic unload, the reverse elastic load and the reverse plastic load is accurately described while the stability criterion is satisfied. Further numerical results indicate that the stability criterion can accurately predict the relationships between the maximum time step and each parameter.
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图 2 步长dt =0.3×10-7 s时显式精确算法计算结果
Figure 2. Computational results of explicit precise algorithm with time step dt =0.3×10-7 s
图 3 步长dt =1.0×10-7 s时显式精确算法计算结果
Figure 3. Computational results of explicit precise algorithm with time step dt =1.0×10-7 s
图 4 步长dt =1.5×10-7 s时显式精确算法计算结果
Figure 4. Computational results of explicit precise algorithm with time step dt =0.3×10-7 s
图 5 步长dt =1.5×10-7 s时半径回归法计算结果
Figure 5. Computational results of radial return algorithm with time step dt =1.0×10-7 s
图 6 最大时间步长与应变率敏感因子关系
Figure 6. Relationship between the maximum time step and the strain-rate sensitivity coefficient
图 7 最大时间步长与准静态条件下屈服应力关系
Figure 7. Relationship between the maximum time step and the yield strength under quasi-static condition
图 8 最大时间步长与剪切模量关系
Figure 8. Relationship between the maximum time step and the shear modulus
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