Mechanical Behaviors and Constitutive Model of Polymide under Quasi-Static and Dynamic Compressive Loading
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摘要: 为研究聚酰亚胺在动静态压缩载荷作用下的力学特性，采用材料试验机（material testing system, MTS）和分离式霍普金森压杆（split Hopkinson pressure bar, SHPB）进行压缩实验，得到了材料在不同应变率下的应力-应变曲线。通过分析回收试样的形貌，得到了聚酰亚胺在裂纹形式、尺寸变形等方面的特性。聚酰亚胺屈服强度的动态增强因子与应变率的关系具有明显的双线性特性，可采用多元线性方程或Cowper-Symonds模型描述。针对聚酰亚胺的动态力学响应特性，阐述了从低应变率到高应变率范围内的压缩变形机理。采用考虑
$\,\beta $ 转变的唯象本构模型，描述了聚酰亚胺在大应变率范围内的弹塑性大变形响应，包括初始黏弹性、屈服、应变软化和应变硬化在内的力学行为。通过贝叶斯方法拟合模型参数，拟合结果与实验结果在各应变率下都具有较好的一致性。-
关键词:
- 聚酰亚胺 /
- 应变率效应 /
- $\alpha $转变 /
- $\,\beta $转变 /
- 本构模型
Abstract: To study the mechanical properties of polyimide, quasi-static and dynamic compression experiments were carried out using the materials testing system (material testing system, MTS) and the split Hopkinson pressure bar (SHPB). The stress-strain curves of the material under different strain rates were obtained. The morphology of the recovered specimens was analyzed, and the characteristics of the polyimide in terms of crack form and size deformation were obtained. A bilinear relationship between the dynamic increase factor of the polyimide and the strain rate was obtained. The bilinear characteristics were described by piecewise fitting model and Cowper-Symonds model. Based on the characteristics of the dynamic mechanical response of the polyimide, its compression deformation mechanism from low to high strain-rates was explained. A phenomenological constitutive model in consideration of the contributions of the$\,\beta $ -transition was modified to describe the large elastic-plastic deformation response of the material, in which initial viscoelasticity, yield, strain softening and strain hardening were all included. Then the constitutive model’s parameters were fitted by Bayesian approach. The Bayesian fitting results at different strain rates were in good agreement with the experimental data.-
Key words:
- polyimide /
- strain rate effect /
- $\alpha $ transition /
- $\,\beta $ transition /
- constitutive model
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图 4 准静态实验中不同样品的应力-应变曲线
Figure 4. Stress-strain curves of different samples in the quasi-static experiment
图 5
$\varnothing $ 10.00 mm×4.00 mm试样动态压缩回收后的形貌Figure 5. Morphology of the recovered samples with the initial size of
$\varnothing $ 10.00 mm×4.00 mm
图 6
$\varnothing $ 10.00 mm×4.00 mm试样的应变率随时间的变化曲线Figure 6. Strain rate of the
$\varnothing $ 10.00 mm×4.00 mm specimen versus time
图 7
$\varnothing $ 10.00 mm×4.00 mm试样的应力-应变曲线Figure 7. Stress-strain curves of the specimens with the size of
$\varnothing $ 10.00 mm×4.00 mm
图 9
$\varnothing$ 6.00 mm×2.00 mm试样的应力-应变曲线Figure 9. Stress-strain curves of the specimens with the size of
$\varnothing$ 6.00 mm×2.00 mm
图 10 多模型拟合实验动态增强因子随应变率的变化
Figure 10. Two models’ fitting results of the dynamic increase factor versus strain rate
图 11 贝叶斯方法预测本构模型材料参数流程图
Figure 11. Flow chart of predicting material parameters in the constitutive model by Bayesian approach
图 12 准静态加载下模型拟合结果与实验数据的对比
Figure 12. Comparison between the model fitting results and the experimental data under quasi-static loading
图 13 动态加载下模型拟合结果与实验数据的对比
Figure 13. Comparison between the model fitting results and the experimental data under dynamic loading
表 1 准静态压缩实验后回收试样的形貌
Table 1. Morphology of the recovered samples after quasi-static experiment
No. Strain rate/s−1 Specimens after quasi-static compression Size/(mm×mm×mm) Crack 1 0.001 
6.02×5.83×2.78 No 2 0.001 
6.15×5.72×2.74 No 3 0.001 
6.18×5.80×2.75 No 4 0.010 
6.13×5.83×2.71 No 5 0.010 
6.25×5.78×2.72 No 6 0.010 
6.46×5.70×2.72 Yes 7 0.100 
6.64×5.86×2.58 Yes 8 0.100 
6.76×5.69×2.59 Yes 9 0.100 
6.54×5.89×2.59 Yes 
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表 2 动态压缩后
$\varnothing $ 6.00 mm×2.00 mm试样的形貌Table 2. Morphology of the recovered
$\varnothing $ 6.00 mm×2.00 mm samples after dynamic compressionNo. Load pressure/MPa Strain rate/s−1 Size/(mm×mm) Specimens after dynamic compression Crack 1 0.05 3388 $\varnothing $6.13×1.91 
No 2 0.08 5195 $\varnothing $6.75×1.61 
No 3 0.10 6137 $\varnothing $7.08×1.48 
No 4 0.13 7627 $\varnothing $7.72×1.29 
No 5 0.17 8496 $\varnothing $8.48×1.10 
No 6 0.20 10371 $\varnothing $8.99×1.00 
No
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表 3 两种模型的拟合方程与判定系数
Table 3. Two models’ fitting equations and their determination coefficients
Model Equations R2 Cowper-Symonds ${ D_{ {\text{IF} } } } = {(1 + \dot \varepsilon /3\,396)^{1/3.22} }$ 0.99067 Poly-linear fitting ${ D_{ {\text{IF} } } } = \left\{ \begin{gathered} 1.065{ {\dot \varepsilon }^{0.009\,3} }\;\;{\text{ } }\dot \varepsilon \leqslant 0.100{\text{ } }{\text{s}^{ - 1} } \\ 0.275{ {\dot \varepsilon }^{0.185\,5} }\;\;{\text{ } }\dot \varepsilon \geqslant 3\,388{\text{ } }{\text{s}^{ - 1} } \\ \end{gathered} \right.$ 0.99206
0.98288
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表 4 α和β转变相关的材料参数
Table 4. Material parameters related to the α and β transformations
$ {K_\alpha } $ $ {\alpha _1} $ $ {\alpha _2} $ $ {\alpha _3} $ $ {\alpha _4} $ $ {\alpha _5} $ $ {K_\beta } $ ${\,\beta _1}$ ${\,\beta _2}$ ${\,\beta _3}$ ${\,\beta _4}$ ${\,\beta _5}$ 51.069 4.142 −0.306 15.464 1.464 0.011 0.004 20.047 0.837 81.889 −1.519 1.192
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表 5 预测屈服强度与实验屈服强度之间的误差
Table 5. Error between the predicted yield stresses and the experimental results
Strain rate/s−1 $\text{ }{\sigma }_{\text{y, exp} }$/MPa $\text{ }{\sigma }_{\text{y, pred} }$/MPa Error/% 0.001 128.55 126.96 −1.23 0.010 130.89 130.29 −0.45 0.100 134.16 133.62 −0.40 3388 155.84 161.92 3.90 5195 172.32 171.81 −0.29 6137 177.28 176.71 −0.32 7627 185.43 181.14 −2.31 8496 190.08 191.45 0.72 10371 197.88 217.53 9.93
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