A Macroscopic Dynamic Constitutive Model for Ceramic Materials
doi: 10.11858/gywlxb.20190863
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摘要: 基于已有混凝土材料的相关研究，建立了陶瓷材料在动态载荷作用下的宏观本构模型。模型中状态方程采用多项式描述，强度面模型中考虑了压力相关性、Lode角效应、应变率效应、剪切损伤及拉伸软化等的影响。采用一个新的函数来描述陶瓷材料的强度面，其在较高压力下趋于一个平台值；并采用动态增强因子（DIF）考察剥除惯性效应后陶瓷材料的真实应变率效应。通过将模型预测的压力-体应变响应、准静态强度面以及应变率效应与相关实验数据进行对比，验证了该模型。单个单元测试模拟得到的结果与三轴实验数据以及侵彻实验数据高度吻合，进一步验证了此模型。为显示模型的优越性，还与JH-2模型的预测结果进行了比较。结果表明：所提出的本构模型能够很好地预测陶瓷材料在不同加载条件下的力学行为，且优于现有的模型。Abstract: A macroscopic constitutive model is presented herein for ceramic materials subjected to dynamic loadings by closely following a previous study on concrete. The equation of state is described by a polynomial equation and the strength model takes into account various effects such as pressure hardening, Lode angle, strain rate, shear damage and tensile softening. In particular, the strength surface of ceramic materials is characterized by a new function which levels out at very high pressures and strain rate effect is taken into account by dynamic increase factor (DIF) which excludes inertial effect. The present model is verified against some available experimental data for ceramic materials in terms of pressure-volumetric response, quasi-static strength surface and strain rate effect. The model is further verified against the data for triaxial test by single element simulation approach and the test data for depth of penetration in AD99.5/RHA struck by tungsten alloy penetrators. Furthermore, comparisons are also made between numerical results of the present model and the JH-2 model. It is demonstrated that the present model can be employed to describe the mechanical behavior of ceramic materials under different loading conditions with reasonable confidence and is advantageous over the existing model.
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Figure 1. Comparison between the present model predictions (Eq.(9) and Eq.(10)) and the experimental data for ceramic materials
Figure 2. Comparisons between the present model predictions (Eq.(4) and Eq.(9)) with the experimental data
Figure 3. Comparison between the present model predictions (Eq.(6)) with available experimental data for different ceramic materials at different strain rates (Unit of strain rate: s–1)
Figure 4. Comparisons between the present model predictions (Eq.(4)) and the experimental data obtained from plate impact tests
Figure 5. Comparison between the experimental data[25] and the predictions by the present model of strength varies with pressure at different strain rates of AlN
Figure 6. Comparison of stress-strain curves for BeO under triaxial compression between the present model and experimental data[22]
Figure 7. Variation of pressure, effective stress with maximum principal strain for AlN under quasi-static uniaxial tension
Figure 8. Variation of pressure and effective stress with maximum principal strain for AlN under quasi-static biaxial tension
Figure 9. Numerically predicted relationship between pressure and maximum principal strain of AlN under both quasi-static triaxial tension
Figure 10. Schematic diagrams of the geometric dimensions of the projectile-target combination under the impact velocity of 1 500 m/s (Same materials used for all target configurations, and all configuration are axisymmetric.)
Figure 11. Comparison between the numerical results and the test data for the depth of penetration in AD99.7/RHA targets by flat-nosed tungsten alloy penetrators[35]
Table 1. Values of parameters for BeO in the present model
Equation of state parameters Constitutive model parameters K1/GPa K2/GPa K3/GPa ρc/(kg·m−3) Fm Wx Wy S 181.5 1 207.9 −2 991 3 030 3 3.8 2 1.25 Constitutive model parameters ${f_{{\rm{c}}}'}$/GPa ft/GPa B G/GPa λm λs l r 1.5 0.15 1.2 125 0.3 7.5 0.8 0.3 
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Table 2. Values of parameters for AlN in the present model
Equation of state parameters Constitutive model parameters ρc/(kg·m−3) K1/GPa K2/GPa K3/GPa K4/GPa K5/GPa K6/GPa Fm Wx Wy 3 229 181.5 1 207.9 −2 991 181.9 335.6 −283 3 3.8 2 Constitutive model parameters ${f_{{\rm{c}}}'}$/GPa ft/GPa B G/GPa S λm λs l r 3 0.3 1.7 127 1.25 0.3 7.5 0.8 0.3
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Table 3. Values of various parameters for AlN ceramic (JH-2 model)
Equation of state parameters Constitutive model parameters ρc/(kg·m−3) K1/GPa K2/GPa K3/GPa a b C n m 3 229 201 260 0 1.36 1.0 0.013 0.75 0.65 Constitutive model parameters HEL/GPa pHEL/GPa σHEL/GPa μHEL T/GPa β d1 d2 9.0 5.0 6.0 0.024 2 0.32 1.0 0.02 1.85
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Table 4. Values of various parameters for AD99.7 ceramic[35] (The present model)
Equation of state parameters Constitutive model parameters ρc/(kg·m−3) K1/GPa K1/GPa K3/GPa Fm Wx Wy S 3 809 181.5 1 207.9 −2 991 3 3.8 2 1.25 Constitutive model parameters ${f_{{\rm{c}}}'}$/GPa ft/GPa B G/GPa λm λs l r 3 0.3 1.4 135 0.3 7.5 0.8 0.3
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Table 5. Values of various parameters for tungsten alloy[35] (JC model)
Constitutive model parameters ρc/(kg·m−3) G/GPa A1/GPa B1/GPa N1 C1 M1 ${\dot \varepsilon_{_0}}$/s−1 17 600 122 1.506 0.177 0.12 0.016 1.0 1.0 Constitutive model parameters cp/(J·kg−1·K−1) Tm/K Tr/K D1 D2 D3 D4 D5 134 1 723 300 2.0 0 0 0 0 Equation of state parameters Cs/(m·s−1) S1 S2 S3 γ0 A0 4 029 1.23 0 0 1.54 0.4
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Table 6. Values of various parameters for RHA[35] (JC model)
Constitutive model parameters ρc/(kg·m−3) G/GPa A1/GPa B1/GPa N1 C1 M1 ${\dot \varepsilon _{_{0}}}$/s−1 7 800 77 0.792 0.51 0.26 0.014 1.03 1.0 Constitutive model parameters cp/(J·kg−1·K−1) Tm/K Tr/K D1 D2 D3 D4 D5 477 1 793 294 0.05 3.44 −2.12 0.002 0.61 Equation of state parameters Cs/(m·s−1) S1 S2 S3 γ0 A0 4 569 1.49 0 0 2.17 0.460
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