Analysis Of Stress Singularity Of Elastic And Plastic V-notches And Extended Boundary Element Analysis Of Interfacial Strength Of V-notched Structures

Based on the review of the analytic methods for the singularity orders of V-notches and interfacial strength of V-notched structures, the singularity analysis for the elastic and plastic V-notches by the interpolating matrix method and the mechanical field analysis at the V-notch tip of V-notched/cracked structures by the boundary element method are proposed in this thesis. A new approach named the extended boundary element method (XBEM) is established in order to effectively solve elastic or plastic singular stress fields in V-notch/crack tip region. The corresponding calculation program for the XBEM is developed. The main work and contribution of this thesis are given as follows:1A new way of analyzing the heat flux singularity of the V-notched structures is proposed by the use of the interpolating matrix method. Based on the asymptotic expression of the temperature field near the notch tip, a characteristic analysis method for calculating the singularity orders of heat flux density at the V-notch/crack tip is proposed. After introducing the expression of temperature field into the differential equation of the steady-state heat conduction problem, the governing equations are transformed into a set of non-linear characteristic ordinary differential equations with the singularity orders. By adopting the variable substitution technique, the established non-linear characteristic equations are transformed into a set of linear ones. Then the interpolating matrix method is adopted to solve the established equations so that the leading heat flux singularity orders and the associated angular functions are obtained.2The interpolating matrix method is proposed to analyze the stress singularity at the tip of composite V-notch. Based on the asymptotic extension of the displacement field at the composite notch tip, the governing equations for the notch subjected to the anti-plane loading are transformed into the characteristic differential equations with the notch singularity orders. A transformation is applied to converting these equations into a set of linear characteristic ordinary different equations. Then the interpolating matrix method is used to solve the established equations for obtaining the notch singularity orders. The single material notch, bi-material notch and the notch terminated at the bi-material interface are successively studied by the present method. The examples indicate that the present method can provide all the leading stress singularity orders synchronously. Although the eigenpairs corresponding to the non-singular orders in the asymptotic extension do not yield the singular stress components for the notched structures, they are the indispensable extension terms for evaluating the complete stress field in the notch tip region.3The difficulty of the analysis of stress singularity at three-dimensional (3-D) V-notch notch tip is tackled by the interpolating matrix method. After the expression of displacement asymptotic expansion in the notched root zone of3-D structures with column notch is introduced into the linear elasticity governing equation, the eigenvalue problem of ordinary differential equations with the stress singularity orders for3-D V-notch root zone are established by a series of deduction. Then by applying the interpolating matrix method to solve the established equations, all the leading stress singularity orders and the associated displacement/stress angular eigenfunctions can be achieved simultaneously. The numerical results show that some of the singularity orders at the3-D notch problem are converging to the theoretic solutions of the plane strain V-notch problem. However, the number of the singularity orders for3-D V-notch is more than one of2-D plane strain V-notch. If the plane strain theory is used to predict the stress singularity orders of3-D V-notch, some important terms in the asymptotic expansion will be lost. One of the important advantages of the present method is that the computed results of the angular eigenfunctions and their derivative functions corresponding to each asymptotic expansion term have the same order of accuracy. Another advantage is that all the useful eigenpairs in the asymptotic expansion can be yielded at the same time. Moreover the interpolating matrix method takes a small amount of computation to solve the eigenvalue problems and is easily used. These advantages are very beneficial to solve the stress field and temperature gradient in V-notches and cracks tip region subsequently.4A new numerical method named the extended boundary element method (XBEM) is established in the present thesis, which is used to analyze the displacement and stress field in the linear elastic plane notched/cracked structures and simulate the crack propagation beginning from the notch/crack tip. First of all, the displacement and stress fields in the characteristic radius region from a notch tip are expressed by the Williams asymptotic expansions. After the series expansion is substituted into the governing equation in elasticity, the stress singularity order and the associated angular functions can be obtained by solving the characteristic differential equations. Because there is no stress singularity in the remained region after the V-notch tip region is dug out, the conventional boundary element method (CBEM) can be used to analyze it. Hence all the leading unknown coefficients in the Williams series expansion and the complete stress field for the notched structures can be calculated by applying CBEM and combining with the results of the former eigenanalysis for the notch tip region. Here this is called eXtended Boundary Element Method (XBEM). The singular stress terms together with the non-singular stress terms in the tip region can be conveniently obtained. Then the effect of the non-singular stress on the fracture toughness and the critical loading for the centrally sharp V-notched specimen problem are discussed in detail. The numerical results show that the predicted critical loading and fracture toughness of V-notched structures when the non-singular stress is taken into consideration are more accurate than the predicted ones only considering the singular stress by comparing with the experimental results. Based on the consideration of the contribution of non-singular stress term and the maximum circumferential stress criterion of brittle fracture, the crack initiation extended direction from V-shaped notch/crack tip in a semicircular bending specimen can be determined by the XBEM. The strategy for XBEM tracking the crack propagation process is given. The numerical examples show that the XBEM is correct and effective for simulating the propagation process on plane crack.5A new approach of analyzing the stress singularity of the plane V-notches and cracks in power law hardening materials is proposed. Firstly, the asymptotic displacement field in terms of radial coordinates in the notch tip region is adopted. As soon as the notch tip region appears the plastic deformation, the Von Mises yield criterion and total strain plastic theory are adopted. By introducing the displacement expressions into the governing differential equations of the plastic theory, a set of the eigenvalue problem of nonlinear ordinary differential equations with the stress singularity orders and the associated eigenfunctions are proposed after a series of derivation. Then the interpolating matrix method is used to solve the eigenvalue problem by an iteration process. Several leading plastic stress singularity orders of plane V-notches and cracks can be obtained at a time. Simultaneously, the associated displacement and stress eigenvectors in the notch tip region have been determined as the same degree of accuracy with the corresponding singularity order. There are three to five significant figures in the computed values of the first three stress singularity orders obtained by the interpolating matrix method. Few of the published literatures gave out the effective and creditable second stress singularity order for the plane plastic V-notch.6The extended boundary element method (XBEM) is proposed to determinate the singular stress field of the V-notched structures with local plastic deformation near notch tip region. Firstly, the elastic-plastic V-notched structure is divided into two parts, a small region around the V-notch tip and the remaining structure without the small region. Consider that the small region around the V-notch tip yields the plastic deformation due to the stress concentration. Secondly, based on the computed results of the multiple plastic stress singularity orders and the associated eigenfunctions in the singularity analysis for the small region, the displacement and stress components in the small region are expressed as the linear combinations of the finite terms of the series expansion with the leading singularity orders. The remaining structure is considered as linear elastic region, where the discrete boundary integral equations (BIE) can be established along the boundary of the remaining structures, included the connection border with the small region. Then the results of the eigen analysis for the stress singularity in the notch tip region are combined together with the discrete BIE, which is called XBEM. The XBEM can determine the elastic-plastic displacement and stress fields of the V-notched structures and the multiple plastic stress intensity factors at the notch tip.The stress solutions obtained by the XBEM is in agreement with analytical characteristics of local plastic singular stress field at the notch tip. Hence the XBEM provides an effective new way to investigate the fatigue fracture and crack propagation process of elastic-plastic V-notched and cracked structures.