Irregular Williams Element With Generalized Degrees Of Freedom For Crack Propagation

The finite element method is one of the commonly used numerical method for structural fracture analysis.With the development of computer technology,the use of finite element software to solve the crack tip stress intensity factor has become increasingly mature.However,there is still dependence on the discrete technology of the grid in the solution process.Williams element with generalized degrees of freedom(abbreviated W element)can directly solve the crack tip stress intensity factors with less discrete elements,and it is insensitive to the size of the singular region.But the existing research is limited to the square singular region regular W element,requires the crack tip locates in center of the square,and the singular region is discretized into eight congruent sector elements.Based on that,this paper proposes an analytical model in which the cracks are relatively independent of the grid,takes the element where the crack tip is located as the singular region,the W element is embedded in the singular region.The arbitrariness of the position of the crack tip makes the W element along the circumferential direction of the crack tip irregular,and then studied the sensitivity of the irregular W element to the shape and size of the singular region and the effect of the element number of circumferential discrezation.According to the coordinates of the crack tip and the coordinates of the four corner points of the singular region,the irregular W element can be automatically formed.The stress intensity factor of the straight crack propagation in a fixed step is further analyzed,which provides a new idea for crack propagation.The main research contents and conclusions are as follows:Firstly,the Williams stress and displacement fields in polar and cartesian coordinates are derived,the W element is summerrized as a finite element special embedded element.It is proved that the W element does not depend on the shape of the singular region,and the calculation format of the stress intensity factor for the W element is improved,then a solving program is written.The example verifies that:the W element solution agrees well with the singular element solution and XFEM solution,and it has a high accuracy.Proves that the W element method has simpler theory derivation,more convenient grid dispersion and higher precision.Secondly,with the W element solving program and the grid discrete technology of the ABAQUS,takes the plane crack problem as basic model,search on the regular W element with crack tips in the center of the rhombus,rectangular,circular,and elliptical singular region.Analyze the influence factors such as size,a,m,n,N of corresponding singular region.Suggest that m=9,n=300,?=0.9,N=8,and S>1/16a for the circular singular region,OjB:OjA>0.4 for the other singular shape.The example verifies that the precision of the W element is independent of the shape and the size of the singular region.Thirdly,establishes an analytical model of irregular W element which the element number of circumferential discrezation is 6 and 5,the accuracy sensitivity of it is studied.The range of the value of the crack tip relative to the geometrical center of the singular region is given:5%is the error control line,suggest that while N=6,|?|?0.72,|?|?0.48;while N=5,??[-0.56,0.4),??[-0.24,0.4].The high precision of irregular W element is ensured and the correctness of the reference range is verified.Finally,set up two crack propagation model of propagation within element and between element,based on the irregular W element with the element number of circumferential discrezation is 6 and 5.The quasi-static extension of planar cracks was simulated by examples,the feasibility and high precision of the W element in the crack propagation application are verified.According to the crack tip position reference range,the calculation error can be well eliminated.In addition,the discrete grid of irregular W element with crack propagation is proposed.The crack propagation mode in the element can maintain a high solution efficiency and adapt to the crack propagation trend;the mode of crack propagation between elemen can maintain a high solution accuracy and is suitable for studies with high precision requirements.