Study Of Extended Boundary Element Method For Analysis Of Notch Strength Of Three-Phase Materials

In the current economic level and rapid development of technology today,the appearance of notches in the engineering structure is unavoidable,resulting in stress concentration at the tip of the notch,the material is prone to fracture damage at the tip of the notch,which will have an impact on the performance of the material itself and will further produce structural instability.Therefore,measures are needed to prevent such fractures from occurring.In the paper,the emphasis was placed on the characterization of stress singularity in three-phase material notches and the solution of completely singular stress fields at the end of the notches,and this work is of great importance for the safety assessment of engineering structures.In this paper,we establish the extended boundary element method(XBEM)to study the V-notch structure of three-phase materials and divide the V-notch structure into the stress singularity region with a tiny sector at the end and the residual structure without stress singularity.First,the stress singularity characteristic analysis of the notched end is performed by the interpolation matrix method,then,the remaining structure without stress singularity is analyzed by applying the conventional boundary element method,and finally,the stress intensity factor and the complete singular stress field of the V-notch notched structure of the three-phase material can be obtained by organically coupling the characteristic analysis with the conventional boundary element method analysis.The main research of this paper is as follows:(1)Firstly,the stress singularity characteristics of the isotropic threephase material with a planar V-notch are analyzed.The displacement field in the V-notch tip region of the three-phase material is expressed by the asymptotic expansion of the radial coordinate r.The asymptotic expression of the displacement field is then transformed into the solution of the eigenvalues of the nonlinear system of ordinary differential equations.Finally,the eigenvalue solution of the nonlinear system of ordinary differential equations is converted into the eigenvalue solution of the standard generalized algebraic system of equations by the interpolation matrix method(IMM),so that several stress characteristic indices before the planar V-notch of the isotropic three-phase material,as well as the corresponding displacement and stress characteristic angle functions,can be calculated.(2)Based on the results of the analysis of the stress singularity characteristics of the planar V-notch of the three-phase material,the peripheral structure of the notch is then analyzed using boundary elements.The basic theory and controlling equations of the extended boundary element method to analyze the complete displacement and stress field of the V-shaped notch structure of three-phase material is given,and then the problem of the full-region stress field of the V-shaped notch of planar single-phase material and two-phase material,especially the stress field near the tip region,is analyzed by establishing the extended boundary element method.The stress intensity factors of Vshaped notches in single-phase and dual-phase materials were calculated,the effect of the grade truncation term on the stress intensity factor was analyzed,and the variation law of the stress intensity factor at the end of the notched with the depth of the notch and the notch tension angle was studied.And the complete singular stress fields at the cut ends of singlephase and two-phase materials are given by XBEM.(3)Based on XBEM analysis of planar V-shaped notches in singlephase and two-phase materials,XBEM analysis was then established to analyze the stress intensity factor at the tip of V-shaped notches in tilted three-phase materials,double-sided cracked three-phase materials,and the stress field in the tip region.The effects of notch depth,notch tension angle,elastic modulus ratio,and tilt angle on the stress intensity factor and complete stress field of the three-phase material structure are analyzed for tilted three-phase material and two-sided cracked threephase material,and the effective range of stress asymptotic field solutions in the tip region is explored.And give the tilted three-phase material,double-sided cracked three-phase material V-shaped cutting-edge circular stress field.The extended boundary element method proposed in this paper analyzes the stress singularities and stress fields of V-shaped notched structures of planar isotropic three-phase materials,especially solving the difficult problem of analyzing the complete stress field in the tip region,thus establishing a new path for the damage extension analysis of multicracked structures.