Investigation Of Singularity Orders And Strength Coefficients For Thermo-elastic V-notches

The V-notched configuration is encountered inevitably in all kinds of engineering structures.The crack,inclusion,and multi-material joint are the special cases of the Vnotch.There is the concentration of the stress and heat flux at the vertex of the V-notch,which is infinite in a mathematical sense and is called as the stress singularity and heat flux singularity.The singular fields of the stress and heat flux around the notch tip might induce crack initiation,which seriously weaken the notched structure.The singularities of the heat flux and stress in the V-notched structures are analyzed in this thesis.The main work and conclusions in this thesis are given as follows,(1)The singularity characteristics of the heat flux and stress for a composite material V-notch are analyzed.By introducing the asymptotic expansion of the physical field around the vertex of a composite material V-notch into elasticity governing equation and heat conduction equation,the singular eigen equations for both plane and antiplane Vnotch are established.The singularity orders of the heat flux and stress,and corresponding eigen angular functions for a composite material V-notch can be obtained after applying the interpolating matrix method to solve the established eigen equations.Both the stress singularity order and heat flux singularity order are obtained synchronously.The eigen angular functions and their first-order derivatives of temperature field and displacement are determined with the same degree of accuracy,which will benefit for calculating the high accuracy of the heat flux and stress.The singularity characteristics of the heat flux and stress for the isotropic material,orthotropic material,and anisotropic material Vnotch are studied respectively.(2)The singularities of the heat flux and stress singularity for a multi-material Vnotch are studied.The interfacial continuity conditions are transformed into a set of differential equations with respect to the singularity orders and eigen angular functions.The singularity orders of the heat flux and stress,and corresponding eigen angular functions for a multi-composite V-notch can be evaluated by solving the eigen equations established in each material domain under the radial boundary conditions and interfacial continuity conditions.The effects of the geometry and material properties on the singularities of heat flux and stress for the multi-composite V-notch,multi-composite junction,and sharp inclusion are investigated respectively.(3)The singularity of the heat flux and stress in a V-notch formed by functionally graded material whose material properties vary with the angle is studied.Based on the asymptotic expansion technique of the physical field near the notch tip,the singularity characteristic equation for functionally graded material V-notch is derived,which is a variable coefficient one.The interpolating matrix method is applied to solve the established characteristic equation.The proposed method not only avoids the inconvenience caused by dividing the functionally graded material into small pieces,but also is suitable for analyzing the singularities of the V-notch whose material properties vary as different functions.It is found that the stress singularity for the V-notch whose elastic modulus,thermal expansivity,and thermal conductivity vary as a reciprocal function is the most serious,while the one whose material parameters vary as a power function is the weakest.(4)The singularity of the heat flux and stress for a propagating V-notch is investigated.Based on the dynamic equilibrium equation of thermoelasticity and the asymptotic expansion of the physical field around the vertex of the V-notch,the thermoelastic characteristic equation of a propagating V-notch is derived.It is found that for a single material V-notch,the stress singularity will be the weakest when the propagating direction is along the bisector of V-notch,whereas it will be the strongest when the propagating direction is perpendicular to the notch bisector.The stress singularity decreases with the increase of Poisson's ratio and density,while it increases with the increase of elastic modulus.For a bi-material V-notch,the first stress singularity decreases with the increase of elasticity modulus ratio,while the second stress singularity increases with it.The strongest stress singularity occurs when the propagating direction is along the harder material region.(5)A new approach based on the finite element method is proposed to determine the singular heat flux and singular stress field in the vicinity of the V-notch tip.A small sector including the singular point is separated from the V-notched structure.The temperature,heat flux,displacement,and the nodal force on the arc boundary,which connects the small sector and the remained structure,are expressed as the linear combinations of the finite series expansion terms,in which the unknown amplitude coefficients are included.The amplitude coefficients in the Williams series asymptotic expansion are calculated by introducing the temperature,heat flux,displacement,and the nodal force on the arc boundary into the finite element system equations built on the remained structure.Finally,the heat flux intensity factors and stress intensity factors are yielded.There are no refined mesh and special element around the notch tip are needed in the proposed method.A good stability and high precision of the stress intensity factor can be yielded even a coarse finite element method grid is discrietized.(6)By combining the characteristic analysis for the V-notch with the conventional finite element method,an over-deterministic method is proposed for calculating the heat flux intensity factors and stress intensity factors.By expressing the displacement values calculated by the finite element method with the singularity eigen analysis parameters,a set of overdetermined equations with respect to the amplitude coefficients are yielded,from which the amplitude coefficients are obtained by the least square method.The numerical results show that the over-deterministic method has a good convergence with respect to the number of nodes.The suggestions are given for the method of selecting nodes.It is found that the displacement field in the vicinity of the V-notch tip can be calculated accurately when only the singular term are considered,whereas the higherorder terms are necessary for the evaluation of the stress with higher precision.