正交各向异性复合柱圆弧型循环对称界面裂纹

讨论反平面载荷作用下空心复合柱的循环对称圆弧型裂纹问题。复合柱由两极正交各向异性功能梯度弹性层粘接而成。采用分离变量将这个混合边值问题转化为Cauchy核奇异积分方程,并用Lobatto-Chebyshev求积法对积分方程进行数值求解,得到了应力强度因子的数值解。分析了梯度非均匀参数,几何与材料参数变动等对应力强度因子的影响。     

Dynamic singular stresses in glassfiber reinforced plastics with a broken layer at low temperatures

The plane strain dynamic singular stress problem for glassfiber reinforced plastics with a broken layer at low temperatures is considered. With the order of stress singularity around the tip of the crack which is normal to and ends at the interface between orthotropic elastic materials, Laplace and Fourier transforms are used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gauss-Jacobi integration formula. Numerical calculations are carried out, and the dynamic stress intensity factors at different temperatures are shown graphically.     

A general integral equation formulatin for homogeneous orthotropic potential problems

In this paper an integral equation formulation is proposed for the analysis of orthotropic potential problems. The two primary integral equations of the method are derived from the original governing differential equation firstly by rewriting it in a slightly different form and then applying the direct boundary element method formulation. The solution procedure is based on the use of the fundamental solutions for the isotropic potential case and special attention is given to the differentiation of a singular integral which yields an additional term as well as to the evaluation of the resulting Cauchy principal value integral. A simple discretization for the boundary and its interior domain is adopted in order to express the primary integral equations of the method in matrix form. Three examples are presented, the results of which illustrate the satisfactory accuracy of the method. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve such as heat conduction or subsurface flow in anisotropic media and, foremost, to orthotropic and anisotropic elasticity or elastoplasticity.     

正交各向异性厚板振动分析的边界积分方程法

本文采用正交各向异性厚板静力问题的基本解作为边界积分方程的核函数,利用加权残数法建立了正交各向异性厚板振动分析的边界积分方程。文中详细地讨论了边界积分方程的数值处理过程并给出了若干数值算例以论证本文方法的正确性。      

The fundamental solutions of orthotropic shallow shells

Summary A set of partial differential equations governing orthotropic shallow shells with arbitrary quadratic midsurface is reduced into an eight order partial differential equation with one function as unknown. By use of the method of plane wave decomposition this partial differential equation is transformed into an ordinary differential equation. The fundamental solutions for orthotropic shallow shells are presented in the form of a definite integral. The calculation of the fundamental solutions is discussed in detail.     

计入剪切变形的正交各向异性扁壳的基本解

本文采用Ｈｏｒｍａｎｄｅｒ算子法将控制计入剪切变形的正交各向异性扁壳的偏微分方程组转化为含有一个未知函数的高阶偏微分方程，然后，利用平面波分解法将偏微分方程转化为常微分方程来求解。以定积分表示的形式提出了计入剪切变形的正交各向异性扁壳的基本解。     

Stresses in an orthotropic semi-infinite strip due to edge loads

Summary The stresses in an orthotropic elastic semi-infinite strip subject to plane strain are investigated. Symmetrical distributions of surface tractions are prescribed on the sides of the strip, while along the end the boundary conditions are arbitrary. By using an integral transform method the problem is reduced to a singular integral equation. The dependence of the stress singularity and the stress-intensity factors on the orthotropic properties of the strip is investigated. Stress distributions over the strip end are evaluated numerically.With 5 Figures     

Stresses in an orthotropic strip containing A Griffith Crack

The problem of determining the stress and displacement fields in an orthotropic elastic strip containing a Griffith crack situated symmetrically and oriented in a direction normal to the edges of the strip is considered. A general solution in terms of two potential functions is presented. The mixed boundary conditions lead to dual integral equations, which are reduced to Fredholm integral equation of second kind and are solved by the use of Gaussian quadrature formula. Numerical solutions for a fiber-reinforced composite material and some isotropic materials are carried out and the effect of orthotropy on various quantities of physical interest, in fracture mechanics, is discussed.     

Impact response of a cracked orthotropic medium-revisited

Elastodynamics response of an infinite orthotropic medium containing a central crack under impact loading has been investigated. Laplace and Fourier transforms have been employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform domain which has finally been solved by the method of iteration in the low frequency case. Analytic expressions for the stress intensity factors and crack opening displacement are also obtained for low frequency.     

Thermal singular stresses of glassfiber reinforced plastics with surface cracks at cryogenic temperatures

Thermal singular stress problem for glassfiber reinforced plastics with surface cracks at cryogenic temperatures is considered. For the case of the crack which is normal to and ends at the interface between orthotropic elastic materials, the order of stress singularity around the tip of the crack is obtained. Fourier transforms are used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gauss–Jacobi integration formula. Numerical calculations are carried out for the cases of embedded and edge cracks, and the thermal stress intensity factors at different temperatures are shown graphically.     

Dynamic crack propagation analysis of orthotropic media by the extended finite element method

Dynamic crack propagation of composites is investigated in this paper based on the recent advances and development of orthotropic enrichment functions within the framework of partition of unity and the extended finite element method (XFEM). The method allows for analysis of the whole crack propagation pattern on an unaltered finite element mesh, defined independent of the existence of any predefined crack or its propagation path. A relatively simple, though efficient formulation is implemented, which consists of using a dynamic crack initiation toughness, a crack orientation along the maximum circumferential stress, and a simple equation to presume the crack speed. Dynamic stress intensity factors (DSIFs) are evaluated by means of the domain separation integral method. The governing elastodynamics equation is first transformed into a standard weak formulation and is then discretized into an XFEM system of time dependent equations, to be solved by the unconditionally stable Newmark time integration scheme. A number of benchmark and test problems are simulated and the results are compared with available reference results.     

Determination of thermal stress intensity factors for an interface crack in a graded orthotropic coating-substrate structure

The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate is investigated by two different approaches. For the case that most of the material properties in the graded orthotropic coating are assumed to vary as an exponential function, the integral transform and singular integral equation technique is used to obtain some analytical results. In order to analyze the case with more complex material distribution, an interaction integral is presented to evaluate the thermal stress intensity factors of cracked functionally graded materials (FGMs), and then the element-free Galerkin method (EFGM) is developed to obtain the final numerical results. The good agreement is obtained between the numerical results and the analytical ones. In addition, the influence of material gradient parameters and material distribution on the thermal fracture behavior is also presented.     

Orthotropic piezoelectric patches for control of symmetric laminates via an integral equation

Formulation of the problem for the feedback displacement control of a vibrating laminated plate with orthotropic piezoelectric sensors and actuators is given in terms of an integral equation. The objective is to develop a formulation which facilitates the numerical solution to obtain the eigenfrequencies and eigenfunctions of the piezo-controlled plate. The control is carried out via piezoelectric sensors and actuators which are of orthorhombic crystal class mm² with poling in the z direction. The initial formulation of the problem is given in terms of a differential equation which is the conventional formulation most often used in the literature. The conversion to an integral equation formulation is achieved by introducing an explicit Green’s function. Explicit expressions for the kernel of the integral equation are given and the method of solution using the new formulation is outlined. The solution technique involves approximating the integral equation with an infinite system of linear equations and using a finite number of these equations to obtain the numerical results.     

Elasto-plastic Analysis of Two-dimensional Orthotropic Bodies with the Boundary Element Method

The Boundary Element Method (BEM) is introduced to analyze the elasto-plastic problems of 2-D orthotropic bodies. With the help of known boundary integral equations and fundamental solutions, a numerical scheme for elasto-plastic analysis of 2-D orthotropic problems with the BEM is developed. The Hill orthotropic yield criterion is adopted in the plastic analysis. The initial stress method and tangent predictor-radial return algorithm are used to determine the stress state in solving the nonlinear equation with the incremental iteration method. Finally, numerical examples show that the BEM is effective and reliable in analyzing elasto-plastic problems of orthotropic bodies.     

Elastic equilibrium of an interface crack between rigid stamp and orthotropic strip

Rupture of the interface between an absolutely rigid stamp and an orthotropic infinite strip is investigated. A plane elasticity problem for an interface crack formally leads to oscillatory singularities at the crack tip. In order to overcome this nonphysical solution, a model of an interface crack with frictionless contact zones near the crack tips and the corners of the stamp is developed. By using the method of integral Fourier transforms the problem is reduced to a system of three singular integral equations. The system is solved by the method of collocations with the points of collocation chosen at zeros of the Chebyshev polynomials. The stress intensity factors at the crack tips and the stamp corner points are evaluated.     

Incomplete mechanical contact between two elastic half-planes made of orthotropic materials

Summary A planar case in elastic theory is considered for incomplete contact between orthotropic half-planes; complex Kolosov-Muskhelishvili potentials are used. A singular integral equation is derived and solved analytically. The results are analyzed.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 26, No. 3, pp. 65–69, May–June, 1990.     

Stress Intensity Factor of an Edge Crack in Bonded Orthotropic Materials

The article presents the problem of an edge crack under normal point loading terminating perpendicular to the surface of an orthotropic strip of finite thickness which is bonded to another orthotropic half plane. Expressing the displacements and stresses in plane strain condition in terms of harmonic functions, the problem is reduced to a pair of simultaneous integral equations with Cauchy type singularities, which are finally been solved by the Hilbert transform technique. The analytical expression of stress intensity factor (SIF) at the crack tip for large thickness of the strip is calculated, which corresponds to the weight function of a crack under normal loading. The influences of elastic constants of two different orthotropic materials, distinct arbitrary locations of normal point loading on the crack surface and length of the crack on the dynamic SIF are depicted through graphs.     

The contact problem of an orthotropic non-homogeneous elastic half space

The problem of a rigid punch on an elastic half plane with orthotropic and non-homogeneous material is considered. The axes of orthotropy are chosen to coincide with the Cartesian coordinate system in which one axis is parallel to the edge of the half plane and the other is perpendicular to it. Non-homogeneity is introduced in both directions of orthotropy as continuous functions along these directions. Using the Fourier Transform Technique, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically. The formulation of the problem is obtained for a rigid punch with arbitrary shape.     

Application of new fast multipole boundary integral equation method to crack problems in 3D

Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large scale problems. This paper discusses an application of the new version of FMM to three-dimensional boundary integral equation method (BIEM) for crack problems for the Laplace equation. The boundary integral equation is discretised with collocation method. The resulting algebraic equation is solved with generalised minimum residual method (GMRES). The numerical results show that the new version of FMM is more efficient than the original FMM.     

Mode II Macrocrack Initiation in Orthotropic Composite Viscoelastic Plate

Safe loads and initiation time for a straight macrocrack in viscoelastic orthotropic material that is intended to model a fiber composite plate under shear loads is investigated. The composite material is modeled by viscoelastic orthotropic medium. Determination of expression for crack shear displacement as function of time is based on the corresponding elastic solution and the method of operator continued fractions. Initiation time is obtained as a solution of integral equation for the incubation period. Numerical calculations are given for mode II macrocrack initiation.