Study On Stress Singularity Of V-notch In Orthotropic Composites By Differential Quadrature Method

In the past,scholars have made more and more in-depth research on ordinary materials,and the comprehensive performance of composites has been further improved.In real life,from precision micro electronic devices to aerospace equipment,there are a large number of components with V-notch due to the need of function.According to the analysis of linear elasticity theory,the stress near the tip of V-notch will increase sharply,that is,stress singularity will occur.Therefore,solving the problem of stress singularity of V-notch and ensuring that the structure can continue to serve has important engineering application value.Differential quadrature mehtod（DQM）is a basic theory proposed by Bellman in 1971.This method has the advantages of simple formula,convenient programming,less calculation and high precision.Due to the development of fast computer and simulation technology,it has attracted the attention of scholars and is widely used in the anal ysis and research of dynamic characteristics of engineering structures.In this paper,DQM is applied from the field of structural dynamics to the field of engineering fracture mechanics for the first time in China to analyze the stress singularity of V-notch in isotropic and orthotropic materials.The main work of this paper is as follows:（1）The differential quadrature method is proposed to analyze the stress singularity of antiplane V-notch in isotropic single-phase materials and two-phase materials.Based on the linear elastic theory and the series asymptotic expansion of Williams displacement equation,the typical terms are substituted into the stress-strain relationship and the stress balance equation of anti plane problem,and the calculation of the stress singularity index of anti plane V-notch is transformed into the eigenvalue problem of ordinary differential equations under specific boundary conditions.The numerical results show that the stress singularity index at the end of antiplane V-notch of isotropic single-phase and two-phase materials obtained by DQM is completely consistent with the exact solution and the calculation results in the existing literature,which proves the accuracy and feasibility of differential quadrature method in calculating the stress singularity index at the end of antiplane V-notch.In addition to DQM,the first-order stress singularity index at the tip of anti plane notch is obtainedλ₁ In addition to,the first several high-order nonsingular indices can be obtained at the same timeλ₂ andλ₃。For the complete stress field of bimaterial anti plane V-notch,althoughλ₂ andλ₃ These nonsingular parameters are indispensable physical quantities to describe the complete stress field at the notch tip.（2）The differential quadrature method is proposed to analyze the stress singularity of plane V-notch in isotropic single-phase materials and two-phase materials.Based on the series asymptotic expansion of Williams radial r of the displacement field at the notch root,it is substituted into the constitutive equation of elasticity and the equilibrium equation of plane problem.The linear elastic theoretical differential equation of plane joint is transformed into a kind of eigenvalue problem of ordinary differential equations.According to the calculation formula of differential quadrature method theory,a new solver DQMEI is compiled in FORTRAN to solve the eigenvalue problem of general ordinary differential equations.Finally,the stress singularity index of plane V-shaped notch is analyzed by DQMEI.At the same time,the displacement and stress characteristic functions corresponding to the micro sector area at the root of plane V-shaped notch are also obtained.These characteristic functions are indispensable physical quantities in analyzing the complete stress field and stress intensity factor of plane V-shaped notch.（3）The differential quadrature method is proposed to analyze the stress singularity of antiplane V-notch in orthotropic single-phase and two-phase materials.In the polar coordinate system,according to the basic theories of elasticity and fracture mechanics,the calculation of the stress singularity index at the end of the antiplane V-notch of single-phase composites and two-phase composites is transformed into the eigenvalue problem of the system of variable coefficient differential equations about the stress singularity index and its corresponding displacement characteristic function.According to the basic theory of differential quadrature method,the nonlinear The eigenvalue problem of variable coefficient differential equations is transformed into the eigenvalue problem of standard generalized algebraic equations.Finally,the stress singularity index and displacement eigenfunction at the end of antiplane V-notch in orthotropic materials are calculated at one time by QR method.The calculation results show that the opening angle of V-shaped notchα、the shear modulus ratio G₁₃/G₂₃ and boundary conditions have obvious effects on the stress singularity of anti plane V-notch.（4）The differential quadrature method is proposed to analyze the stress singularity of two-dimensional and three-dimensional V-notch in orthotropic materials.Based on the series asymptotic expansion of the displacement field in the micro sector at the root of the notch,it is substituted into the two-dimensional and three-dimensional stress balance equations of elasticity,so that the calculation of the stress singularity index in the two-dimensional and three-dimensional V-shaped notch root of orthotropic materials is transformed into a set of eigenvalue problems of variable coefficient and nonlinear ordinary differential characteristic equations.Based on the basic theory of differential quadrature method,The calculation of eigenvalues of differential equations is transformed into the eigenvalue solution of standard generalized algebraic equations.This method avoids the difficulty of solving a set of transcendental equations by iterative method,and all the singular eigenvalues of plane stress and anti plane and their characteristic angle functions can be solved at the same time.For the three-dimensional V-notch of orthotropic materials,an effective discrimination method is given to determine whether the stress singularity index calculated by the differential quadrature method is the plane stress singularity index or the anti plane singularity index.