Study Of Planar V-notch SIFs By The Finite Element Method With And Engineering Applications

Civil engineering structures or components are usually accompanied by various defects in application,such as cracks,notches,etc.Under the external load,stress concentration can occur at the tip,leading to fracture of the whole structure,which greatly endangers people’s life and property.Compared with cracks,the stress field and displacement field at the tip of V-notches are more complex,resulting in a cumbersome analysis and low solution accuracy,making it difficult to accurately assess the safety and stability of V-notches in practical engineering.Therefore,it is of great significance to carry out a highly accurate and efficient numerical algorithm research on the fracture behavior of V-notches under external load.Combined with the partition Müller acceleration method to solve the V-notch feature equation,the corresponding feature value array is obtained and the selection rule of the feature value array is established.Based on generalized parameter finite element method,Williams element（W element）for the analysis of Stress Intensity Factors（SIFs）is derived and established,which can obtain SIFs with high accuracy by solving the integral control equation and substituting the corresponding eigenvalues and generalized parameters into the equation.This paper utilizes the advantage of the W element and the following research on the fracture of planar elastomers with V-notches:（1）The Williams series stress functions and characteristic equations are derived for the planar V-notch,and the corresponding eigenvalue array was obtained by solving the V-notch characteristic equation using the partition Müller acceleration method.Then,the effects of repeated eigenvalues on higher order displacement fields and eigenvalue specific solutions on rigid body displacements are investigated,and rules for the selection of V-notch eigenvalue arrays are established:For conjugate complex eigenvalues only the real part of one is retained;for real eigenvalues,except for the smallest eigenvalue,all real eigenvalues are taken as the smaller value in the same solution interval,and the special solutionsλ₀^I=λ₀^II=0 and ₂λ^II=1 need to be added.Based on the above,the Williams series displacement field for a planar V-notch is derived,and the correctness of the displacement field derivation.（2）Based on the Williams series of the V-notch tip displacement field and the finite element method with generalized degrees of freedom,theory of W element for the analysis of SIFs is established.Combined with the comparative analysis of arithmetic cases,it is proved that the theory of this paper is applicable to the solution of SIFs for in-plane cracks and V-notch tips,and is insensitive to load type,notch depth,notch opening angle and relative eccentricity position,and has high precision and universality.On this basis,the proposed values of the three important parameters of the W element and the size of the singular region are investigated,and it is shown that W element is insensitive to the relative size of singular zone and mesh encryption and when the three key parameters of W element are taken as m=10,α=0.9 and n=300,the calculation results are of convergence and high accuracy.Based on this,the empirical formula for calculating mode I nondimensional SIFs from V-notch tips subject to tension is given.（3）ANSYS finite element software cannot directly calculate V-notch tip SIFs,and when the W element calculates the V-notch tip SIFs,the pre-processing process is troublesome.In response to this problem,the co-simulation analysis of W element and ANSYS for planar V-notch fracture problems is established,in which the pre-processing process is optimized and the computational efficiency of the W element is improved.Based on the co-simulation analysis,the V-notch at the weld toe of three different shapes of weld structures in the T-weld commonly found in practical engineering is studied.The practicality of the W element in engineering applications is verified and the corresponding engineering recommendations are given:When the weld is subjected to external loads,the concave weld can effectively improve the stress concentration at the weld toe.