Thermal Shock Crack Propagation of Alumina Simulated With the Phase-Field Method Under Temperature-Dependent Damage Criteria
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摘要:
氧化铝陶瓷材料的力学性能受温度影响显著，因此使用相场法模拟热冲击裂纹的扩展时有必要考虑损伤判据的温度相关性。在现有热力学相场模型的基础上通过引入温度相关性损伤判据，修正了相场模型的控制方程。利用该模型对氧化铝陶瓷热冲击实验进行有限元模拟，并将模拟结果与氧化铝热冲击实验结果和不考虑温度相关性损伤判据的有限元模拟结果进行对比。结果表明，通过引入温度相关性损伤判据，可实现对热冲击裂纹的萌生和扩展过程更合理的模拟。
Abstract:The mechanical properties of alumina ceramic materials are significantly affected by temperature, so it is necessary to consider the temperature dependence of the damage criteria during the simulation of thermal shock crack propagation with the phase-field method. Based on the existing thermodynamic phase-field model, the governing equations for the phase-field model were modified through introduction of the temperature-dependent damage criterion. Then the revised phase-field model was used to simulate the thermal shock experiment of alumina ceramics, and the simulation results were compared with the experimental results and the finite element simulation results without temperature-dependent damage criteria. The results show that, introduction of the temperature-dependent damage criterion helps more reasonably simulate the initiation and propagation process of thermal shock crack.
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Key words:
- alumina ceramic /
- thermal shock /
- phase-field method /
- finite element simulation
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图 1 x=0处尖锐和弥散裂纹拓扑：(a)尖锐裂纹；(b) 弥散裂纹，用长度尺度l表示裂纹相场
$ \bar{d} $ Figure 1. Topologies of sharp and diffuse cracks at x=0: (a) the sharp crack; (b) phase field
$ \bar{d} $ of the diffuse crack denoted by length scale l
图 4 裂纹相场和温度场随时间的演化
Figure 4. Evolution of the crack phase field and the temperature field with time
图 5 水淬实验结果与有无考虑温度相关性损伤判据模拟结果的对比：(a) Chu等[13]的模拟结果；(b)实验结果[6]；(c) 本文模拟结果
Figure 5. Comparison between water quenching experimental results and simulation results with or without temperature-dependent damage criteria: (a) simulation results of Chu et al [13]; (b) the experimental results [6]; (c) simulation results of this paper
图 6 圆盘热冲击问题的几何模型和热边界条件（红色区域表示外加热流）
Figure 6. The geometric model and thermal boundary conditions for thermal shock problems of disks (the red area represents applied heat flow)
图 7 圆盘辐射加热实验结果和模拟结果
Figure 7. Experimental results of radiation heating and simulation results of the phase field method
图 8 圆盘辐射加热实验结果和模拟结果： (a)实验结果；(b) 考虑温度相关性临界能量释放率Gc(T)的模拟结果；(c) 不考虑温度相关性临界能量释放率Gc的模拟结果
Figure 8. Experimental and simulation results of disk radiation heating: (a) the experimental results; (b) the simulation results with temperature-dependent critical energy release rate Gc(T); (c) the simulation results without temperature-dependent critical energy release rate Gc
图 9 不同热流密度下的裂纹和裂纹尖端温度随时间的演化：(a)
$ \gamma =7.5\times {10}^{5}\;\mathrm{k}\mathrm{W}/{\mathrm{m}}^{3} $ ；(b)$\gamma =1\times {10}^{6}\;\mathrm{k}\mathrm{W}/{\mathrm{m}}^{3}$ Figure 9. Evolution of crack and crack tip temperatures with time under different heat fluxes: (a)
$ \gamma =7.5\times {10}^{5}\;\mathrm{k}\mathrm{W}/{\mathrm{m}}^{3} $ ; (b)$\gamma =1\times {10}^{6}\;\mathrm{k}\mathrm{W}/{\mathrm{m}}^{3}$ material parameter value or expression elastic modulus $ E/{\text{GPa}} $ $340 - 2.54T{\exp({ - {T_{\rm{m} } } }/T}) + 1.9( {T - 0.363{T_{\rm{m} } } + | {T - 0.363{T_{\rm{m} } } } |} ){\exp ({ { { - {T_{\rm{m} } } } / T} } )}$ density $ \rho /( {{{\text{g}} / {{\text{c}}{{\text{m}}^{\text{3}}}}}} ) $ 6.119 Poisson’s ratio $ \upsilon $ 0.22 melting point $ {T_{\rm{m}}}/{\text{K}} $ 2327.15 thermal conductivity $ k/( {{{\text{W}} / {( {{\text{m}} \cdot {\text{K}}} )}}} ) $ $ 210.75\ln ( T ) - 746.28 $ heat capacity $ C_p/( {{{\text{J}} / {( {{\text{kg}} \cdot {\text{K}}} )}}} ) $ $60.225 - 0.011\;28T + 1.244\;56 \times {10^{ - 6} }\times{T^2}$ thermal expansion $ \alpha /( {{{\text{1}} / {\text{K}}}} ) $ $( {6.52 + 6.811\;4 \times { {10}^{ - 4} }\times T} ){\text{ } } \times {10^{ - 6} }$ 
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material parameter value or expression elastic modulus $ E{\text{/GPa}} $ $380 - 2.54T{\exp ({ { { - {T_{\rm{m} } } } / T} } }) + 1.9( {T - 0.363{T_{\rm{m} } } + | {T - 0.363{T_{\rm{m} } } } |} ){\exp( { { { - {T_{\rm{m} } } } / T} } )}$ density $ \rho /( {{{\text{g}} / {{\text{c}}{{\text{m}}^{\text{3}}}}}} ) $ 3.90 Poisson’s ratio $ \upsilon $ 0.25 melting point $ {T_{\rm{m}}}/{\text{K}} $ 2327.15 thermal conductivity $k/( { { {\text{W} } / ({ {\text{m} } \cdot {\text{K} } } }} ) )$ $31.06 - 0.113\;8 T + 2.95 \times {10^{ - 4} } \times {T^2} - 4.43 \times {10^{ - 7} } \times {T^3}$ heat capacity $ C_p/( {{{\text{J}} / {( {{\text{kg}} \cdot {\text{K}}} )}}} ) $ $60.225 - 0.011\;28T + 1.244\;56 \times {10^{ - 6} }\times {T^2}$ thermal expansion $ \alpha /( {{1 / {\text{K}}}} ) $ $60.225 - 0.011\;28T + 1.244\;56 \times {10^{ - 6} }\times {T^2}$
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