The Sub-domain Extended Boundary Element Method Of Analyzing Elastic Plastic Stress Fields And The Crack Propagations For V-notched Structures

According to the theory of linear elasticity and elastic-plasticity,the numerical methods are difficult to calculate the complete stress fields and the crack propagation of the V-notched/cracked structure.In this thesis,the sub-domian extended boundary element method(XBEM)is proposed to determine the complete stress fields of the V-notched/cracked structure.The main idea of XBEM is to divide the V-notched/cracked structure into a small sector around the V-notch/crack tip and the outer sector.The displacement and stress fields in the small sector are expressed as the asymptotic series expansions with respect to the radial coordinate from the tip,The outer sector is anlayzed by the conventional boundary element method.Consequently,the complete displacement and streess fields of the V-notched/cracked structure are solved by combining the boundary element analysis and the asymptotic series expansions of the stress field.The corresponding calculation program of the sub-domain XBEM is developed.And the numerical examples are given to demonstrate the effectiveness and accuracy of the sub-domain XBEM.The main work and contribution of the present thesis are given as follows:1.Based on the linear elastic theory,the basic theory and governing equations of XBEM for analyzing the complete displacement and stress fields of plane V-notched/cracked structure are given.Then the complete displacement and stress fields of plane V-notched/cracked structures under combined loads calculated by XBEM are obtained.The accuracy of XBEM and the effective region of the stress asymptotic series expansion in the tip region are discussed with different truncated number and different radius of the small sector around the tip.For the purpose,three benchmark examples are presented to show that the effective region of the stress asymptotic series expansion.And the accuracies of the XBEM solutions increase as the truncated number increases.The solutions of the XBEM by taking the term number N=8 are accurate enough to act as standard solution of the full stress fields of the notch/crack plates.2.The sub-domain XBEM is proposed to analyze the displacement and stress fields for V-notched/cracked structures in a bonded bi-material.According to the different Young's modulus ratios in the two material regions,the reasonable terms in the displacement asymptotic series expansions in the two small sectors around the tip are chosen to obtain the corresponding stress and displacement egienfunctions in the two tip regions with different materials.On the other hand,the conventional boundary element method is used to analyze the outer regions removed the two small sectors.Consequently,the whole displacement and stress fields of the V-notched/cracked structures are obtained by combining the boundary element analysis in outer regions and asymptotic series expansions of the tip regions in the bi-material.The computed results show the accuracy and effectiveness of the XBEM for determining the stress fields of the V-notched/cracked structures in a bonded bi-material with the various Young's modulus ratios.The strategies of choosing the stress and displacement egienfunctions proposed in the present thesis can provide references for other methods analyzing the bi-material V-notched/cracked structures.3.The sub-domain XBEM is developed to predict the crack propagation paths of the multi-crack structures.The analysis procedure is as follows.Firstly,the whole displacement and stress fields of the multi-crack structure are calculated by sub-domain XBEM.Secondly,based on the maximum circumferential stress criterion,the crack initiation angles are obtained by considering the contributions of non-singular stress terms in the asymptotic series expansions.Thirdly,the cracks propagate forward along the crack initiation angles to form a new multi-crack structure.In this process,an adaptive mesh technique is established to discretize boundary and the sub-domain XBEM is used to analyze the newly formed multi-crack structure repeatedly.Finally,the crack propagation paths of the multi-crack structure are obtained.4.The sub-domain XBEM is developed to determine the complete elastic-plastic stress fields of the V-notched/cracked structures and the plastic zones near the notch/crack tips.Firstly,the V-notched/cracked structure is divided into two parts,which consists of a small region around the notch/crack tip and the outer region without the tip region.The small region is considered to be a plastic zone,while the outer region is an elastic zone.Based on the results of the eigen-analysis for the singular plastic stress functions in the tip region,the sub-domain XBEM is used to evaluate the whole displacement and stress fields of the notched/cracked structures.Thirdly,by comparing the von-Mises stresses on the boundary of the predicting plastic zone with the yield stress of the material,the assumed plastic zone around the notch tip is scaled along the radial direction from the tip to form a new V-notched/cracked structure.In the process of modifying the tip plastic zone,the sub-domian XBEM is used to analyze the newly formed V-notched/cracked structure again and again.Finally,the real plastic zone of the V-notched/cracked structure is obtained by the iteration analysis.Here,the present sub-domain XBEM should give the accurately plastic zone shapes near the V-notched/cracked tips.Meanwhile the whole elastic-plastic displacement and stress fields of V-notched/cracked structures are determined in the sub-domain XBEM analyses.Such as one of three benchmark examples,the plastic zone of mode I crack calculated by sub-domain XBEM is like a typical apple shape and especially there is"Carpopodium" plastic zone along the wedge.The computed results from the present sub-domain XBEM demorstrate that the plastic zone shapes near the tip obtained by the published literatures for the V-notched/cracked structures are not accurate.Of course,the plastic zones in the tip regions predicted by linear elastic thorey are not correct.5.Three dimensional(3-D)sub-domian XBEM is first proposed to evaluate the complete displacement and stress fields of 3-D V-notched/cracked structures based on the linear elastic theory.The analysis idea is the same as above.The 3-D V-notched/cracked structure is divided into two parts,which are a small sectoral column around the notch/crack tip and the outer region without the tip sectorial column.Then,the eigen-anaylsis results of the asymptotic series expansions of the singular stress fields in the notch tip region are assembled with the boundary integral equation of the outer region without stress singularity.Hence,the complete displacement and stress fields of both the tip region and outer region are accurately obtained.Consequentely,the fracture parameters and the propagation path of 3-D cracked structure can be determined by sub-domain XBEM.The sub-domian XBEM proposed in the present thesis accurately evaluate the complete displacement and stress fields of the 2-D and 3-D V-notched/cracked structures and the plastic stress fields of the plane V-notched/cracked structures,which provide a new tool for analyzing the whole stress fields and failure propagation paths of the general cracked structures.