基于光滑-扩展有限元法的裂纹扩展研究

Research on Crack Propagation Based on Smooth-extended Finite Element Method

为了研究裂纹扩展问题,将光滑有限元法(S-FEM)与扩展有限元法(X-FEM)相结合,形成光滑-扩展有限元法(S-XFEM)算法及程序。首先基于光滑-扩展有限元法理论,得到裂纹应力强度因子计算方法,编写适用于一般有限元法的通用程序,对含有中心裂纹的无限大平板受单向拉伸的模型进行分析。通过改变网格尺寸和裂纹角度分析对裂纹应力强度因子的影响,并将该算法所得的Ⅰ型、Ⅱ型以及Ⅰ型、Ⅱ型复合裂纹的应力强度因子数值与理论数值进行对比。随着网格尺寸的不断减小,利用S-XFEM所得的I型、Ⅱ型的应力强度因子值不断逼近于精确解;随着裂纹度数的增加,Ⅰ型的应力强度因子值不断增大,Ⅱ型裂纹的应力强度因子值随角度先增大后减小,并且角度越大,裂纹越接近于Ⅰ型裂纹。可见,利用光滑-扩展有限元法计算所得的应力强度因子具有较高的精度和良好的适应性。其次,在此基础上,引入裂纹扩展判据,利用MATLAB编程对含侧边裂纹板和含孔洞梁四点弯曲实验模型进行裂纹扩展路径研究。结果表明:通过模拟算例所得的裂纹扩展路径结果与已有文献所得的裂纹扩展路径一致。采用光滑-扩展有限元方法进行裂纹扩展研究,相较于利用其他裂纹扩展研究的方法,不需要对裂纹附近区域的网格进行重划分或加密,使得分析过程更加简单,计算效率明显提高。

In order to study the crack propagation problem,the smooth finite element method(S-FEM)was combined with the extended finite element method(X-FEM)to form the smooth-extended finite element method(S-XFEM)algorithm and program.Firstly,based on the theory of S-XFEM,the calculation method of crack stress intensity factor was formed,and the general program suitable for general finite element method was written to analyze the unidirectional tensile model of an infinite plate with a central crack.By changing the mesh size and crack angle to analyze the influence of crack stress intensity factor,the stress intensity factor values of type I,type II and type I,II composite cracks obtained by the algorithm were compared with the theoretical values.With the decrease of mesh size,the stress intensity factor values of type I and type II obtained by S-XFEM approached to the exact solution continuously.With the increase of crack degree,the stress intensity factor value of type I increased continuously,and the stress intensity factor of type II increased first and then decreased with the angle.The larger the angle was,the closer the crack was to the type I crack.Therefore,the stress intensity factor calculated by the S-XFEM had high accuracy and good adaptability.Secondly,on this basis,the crack propagation criterion was introduced,and the MATLAB programming was used to study the crack propagation path of four-point bending experimental model of the plate with side cracks and the beam with holes.The results showed that the crack propagation path obtained by this example was consistent with that obtained by references.Finally,in this paper,S-XFEM was used to study the crack propagation,compared to using other methods for crack propagation research,it was not necessary to re-divide or encrypt the grid in the area near the crack,which would make the analysis process more simple,improve the computational efficiency.