Research Of Curved Crack Propagation In FGM Structure Based On The Finite Element Method With Generalized Degrees Of Freedom

When the direction of the complex load changes,the crack propagation angle changes,which makes the crack in the actual engineering usually behave as a curved crack.Due to the non-uniformity of the properties of Functionally Gradient Material?abbreviated as FGM?structures with cracks,the stress intensity factors?abbreviated as SIFs?of each tip of cracks are relatively difficult to be solved efficiently,precisely and simultaneously under complex loads.Therefore the results of researching on the crack propagation of FGM structure on the basis of SIFs becomes ever more comple.The research has been carried out focusing on the solution of the crack tip SIFs of FGM structure and the crack propagation problem based on the SIFs:?1?Because the boundary conditions of the Williams series displacement field can not be directly used for the curved crack,the curved crack in the singular region of the crack tip is equivalent to the broken line crack.Based on the finite element method with generalized degrees of freedom,the Williams element with generalized degrees of freedom?abbreviated as W element?calculation format for SIFs analysis of the crack tip of homogeneous material structure under complex load is derived in detail.The spatial arbitrary load can be decomposed into the coupling action of in-plane and out-of-plane loads,which makes the control equation increase with the increase of node degrees of freedom,resulting in the exponential growth of the required computer memory with the number of nodes of the element,resulting in the phenomenon of slow calculation or even crash.The control equation can be effectively reduced by compressing the total stiffness and combining the improved LU decomposition method.The calculation efficiency is improved and all SIFs values of crack tip are obtained at the same time.The displacement field of W element contains parameters directly related to SIFs,which can avoid the defect that the traditional finite element method needs tedious post-processing and human error.It lays a theoretical foundation for the study of fracture parameters of FGM structure with cracks?2?Considering the unidirectional gradient change of the elastic modulus of the structure,the elastic modulus E / shear modulus G with the coordinate change is taken into the elastic matrix by using the piecewise exponential method.Then based on the finite element method with generalized degrees of freedom,the heterogeneous W element with generalized degrees of freedom?abbreviated as heterogeneous W element?for SIFs analysis of the crack tip of FGM structure with curved crack under complex load is established.Because of the nonuniformity of the material properties,the stiffness matrix of all the elements in the singular region and the conventional region around the crack tip is a function of the material parameters.The generalized parameter single stiffness transformation needs to be carried out one by one.The matrix explosion problem also exists in the total stiffness integration.The improved LU decomposition method greatly improves the calculation efficiency and can directly determine the SIFs of the crack tip of FGM structure.The analysis shows that the solution of SIFs of FGM structure with curved crack based on heterogeneous W element and improved LU decomposition method is highly accurate and efficient,which provides a new idea for accurately simulating the crack propagation of FGM structure.?3?The relationship between the crack growth angle ?₀ and generalized parameters is established,and the improved maximum circumferential stress criterion is improved.After plugging the Williams series displacement field into maximum hoop stress criterion,the relationship between the crack growth angle ?₀ or the criterion of crack propagation and the generalized parameters is derived.The value of the generalized parameters,which is solved by using the heterogeneous W element established in this paper,is introduced to the expression between the crack growth angle ?₀ and the generalized parameters.If the crack propagates,the crack initiation load is predicted,and the crack is assumed to propagate in a straight line with equal steps.This method can avoid the error accumulation caused by the common finite element method solving SIFs at the crack tip many times,and simulate the crack propagation path more truly.The results show that the crack propagation direction and the crack initiation load can be changed by changing the distribution or the gradient of elastic modulus,or the type of material.A new method is provided for the study of crack propagation in FGM structure with crack.