A Phase-Field Model for Interfacial Fracture in 2D Quasicrystal Bimaterials
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摘要: 针对二维十次准晶双材料的界面断裂问题，建立了用于预测其裂纹扩展路径的相场分析模型. 首先，引入界面相场将离散界面转化为连续分布界面，并获得了界面相场问题的控制方程和边界条件. 利用有限元方法对控制方程进行离散，并求解获得连续分布的界面相场，从而实现了对界面材料参数的弥散处理，消除了材料参数在界面处的奇异性. 其次，基于Francfort-Marigo变分原理建立了二维准晶双材料的控制方程，并采用交错求解方案求解其相场分布. 在数值算例中，通过与现有文献进行对比，证明了该方法的正确性，并研究了相位子场对裂纹扩展路径的影响，以及多裂纹情况的演化规律.Abstract: A phase-field model for the interfacial fracture of 2D decagonal quasicrystal (QC) bimaterials was proposed to predict the crack propagation path. Firstly, the discrete interface was transformed into a smeared interface through introduction of an interface phased field, and therefore the interface phase field governing equations and corresponding boundary conditions were obtained. The continuous distribution of the interface phased field was obtained with the finite element method, and in turn the singularity of material properties at the interface was eliminated. Subsequently, the governing equations for 2D QC bimaterials were obtained based on the Francfort-Marigo variational principle, and solved with the staggered solution scheme. In numerical examples, the present results were compared with existing references and excellent agreements were observed. In addition, the effects of the phason field on the crack propagation path were investigated, with the evolution of multiple cracks explored.
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图 6 含单边裂纹的正方形准晶的几何尺寸和边界条件(单位: mm)
Figure 6. Geometry and boundary conditions of the edge-cracked square quasicrystal (unit: mm)
图 8 三点弯曲问题的几何模型及边界条件(单位: mm)
Figure 8. Geometry and boundary conditions for the 3-point bending test (unit: mm)
图 9 反力-位移曲线和裂纹演化过程
Figure 9. The reaction force-displacement curve and the crack evolution
图 10 含单边裂纹的准晶双材料的几何模型及边界条件(单位: mm)
Figure 10. Geometry and boundary conditions of the edge-cracked quasicrystal bimaterials (unit: mm)
图 11 θ=30°时的反力-位移曲线和裂纹扩展路径
Figure 11. Reaction force-displacement curves and crack patterns for θ=30°
图 12 θ=45°时的反力-位移曲线和裂纹扩展路径
Figure 12. Reaction force-displacement curves and crack patterns for θ=45°
图 13 θ=60°时的反力-位移曲线和裂纹扩展路径
Figure 13. Reaction force-displacement curves and crack patterns for θ=60°
图 14 不同Gi/Gc情况下的反力-位移曲线
Figure 14. Reaction force-displacement curves obtained from different Gi/Gc ratios
图 16 含双边裂纹的准晶双材料的几何模型及边界条件(单位: mm)
Figure 16. Geometry and boundary conditions of the double-notches quasicrystal bimaterials (unit: mm)
图 17 含双边裂纹的准晶双材料受拉时的反力-位移曲线
Figure 17. The reaction force-displacement curve of tension test of the double-notches quasicrystal biomaterials
parameter QC-1 QC-2 phonon elastic modulus C11/GPa 234.3 200 phonon elastic modulus C12/GPa 57.34 100 phonon elastic modulus C66/GPa 88.45 50 phason elastic modulus K1/GPa 122 50 phason elastic modulus K2/GPa 24 20 phonon-phason coupling coefficient R1/GPa -1.1 10 phonon-phason coupling coefficient R2/GPa 0.1 10 critical energy release rate Gc/(kN/mm) 5.56×10-7 1.99×10-6 
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