Planar crack occupying a finite doubly connected region inside an infinite elastic solid

A method is proposed for the approximate solution of the problem of an embedded pressurized planar crack occupying a finite doubly connected region inside an infinite elastic solid. The formulation of the problem produces a system of two integral equations for determining the unknown normal stresses on the plane of the crack outside the crack region, which can be solved using numerical procedures. The proposed method has been applied to obtain the opening mode stress intensity factors K_I, along the boundary lines of an annular crack subjected to a uniform internal pressure.     

Stress intensity factors for an interfacial crack between an orthotropic half-plane bonded to a dissimilar orthotropic layer with a punch

The plane strain problem of determining Stress Intensity Factors (SIF) for a moving interfacial Griffith crack between an elastic orthotropic half-plane and a dissimilar orthotropic layer with a moving punch situated along the boundary of the layer have been considered. The problem is reduced to the solution of three simultaneous singular integral equations with Cauchy-type singularities. Expressions for SIF for the case of a general loading are obtained. Numerical results for some particular cases are also presented graphically.     

Simulation of ellipsoidal particle-reinforced materials with eigenstrain formulation of 3D BIE

Based on the concepts of eigenstrain and equivalent inclusion of Eshelby for inhomogeneity problems, a computational model and its solution procedure are presented using the proposed three-dimensional (3D) eigenstrain formulation of boundary integral equations (BIE) for simulating ellipsoidal particle-reinforced (and/or void-weakened) inhomogeneous materials. In the model, the eigenstrains characterizing deformation behaviors of each particle embedded in the matrix are determined using an iterative scheme with the aid of the corresponding Eshelby tensors, which can be obtained beforehand either analytically or numerically. With the proposed numerical model, the unknowns of the problem appear only on the boundary of the solution domain, since the interface condition between particles/voids and the matrix is satisfied naturally. The solution scale of the inhomogeneity problem can thus be significantly reduced. Using the algorithm, the stress distribution and the overall elastic properties are identified for ellipsoidal particle-reinforced/void-weakened inhomogeneous materials over a representative volume element (RVE). The effects of a variety of factors on the overall properties of the materials as well as the convergence behavior of the algorithm are studied numerically, showing the validity and efficiency of the proposed algorithm.     

Two methods for the numerical solution of Bueckner's singular integral equation for plane elasticity crack problems

The classical collocation and Galerkin methods are used for the numerical solution of singular integral equations of the first kind involving a finite-part integral with a double pole singularity. Such equations appear in plane elasticity crack problems, where they were suggested by Bueckner, and the unknown function in them is proportional to the crack opening displacement function. An application of the proposed methods to the problem of a straight crack under an exponential normal loading distribution is also made and shows the rapid convergence of the obtained numerical results for the stress intensity factors at the crack tips to their theoretical values.     

The dynamic stresses induced by the propagation of skew cracks

Skew crack propagation in an elastic medium due to the application of transient in-plane and anti-plane loads is investigated. For the anti-plane loading the motion of the crack is purely of mode III type, whereas in the case of in-plane loading a mixed mode type of fracture takes place in which both modes I and II occur. The method of solution is numerical and is based on a certain time-dependent transformation which maps the physical plane of the crack into an auxiliary plane in which the crack propagates collinearly with its propagating tip appearing always at the origin of the moving coordinate system. The transformed equations of motion are approximated by an implicit, three-level, finite difference system of equations of second-order accuracy, whose stability analysis is discussed. The reliability of the proposed method of solution is examined in several situations in which analytical results are known, and satisfactory agreement is achieved. Extension to smoothly curving cracks is discussed.     

The laplace transform DBEM for mixed-mode dynamic crack analysis

A boundary element method (BEM) for the two-dimensional analysis of structures with stationary cracks subjected to dynamic loads is presented. The difficulties in modelling the structures with cracks by BEM are solved by using two different equations for coincident points on the crack surfaces. The equations are the displacement and the traction boundary integral equations. This method of analysis requires discretization of the boundary and the crack surfaces only. The time-dependent solutions are obtained by the Laplace transform method, which is used to solve several examples. The influence of the number of boundary elements and the number of Laplace parameters is investigated and a comparison with other reported solutions is shown.     

Singular integral operators method for anisotropic elastic stress analysis

The singular integral operators method is introduced for the solution of the anisotropic elastic stress analysis, which defines the main feature of the mechanical behaviour of composites. The proposed method depends on the existence and explicit definition of a fundamental solution to the governing partial differential equations. After the determination of the fundamental solution, a real variable boundary integral formula is generated. Two applications are given to the determination of the fracture behaviour of a circular hole subjected to a uniform boundary shear traction in an infinite and orthotropic solid and the fracture behaviour of the same hole in the centre of a rectangular orthotropic plate, subjected to deformation along its sides.     

Fatigue life prediction under cyclic thermal loads using the boundary elements method for two-dimensional problems

This study describes a sub-domain boundary element procedure for predicting the fatigue life of elastic media under cyclic transient thermal loads. The procedure assumes an initial crack in a two-dimensional medium, and evaluates the crack extension along a path defined by the maximum circumferential stress criterion. Partial closure may occur on the growing crack faces due to thermal loading. Appropriate thermal and mechanical boundary conditions are imposed on the numerical model to account for the contact state. The analysis requires solutions to the equations of quasi-static thermo-elasticity, and uses an iterative procedure to detect the contact region. We investigate cases of pure opening or mixed mode fracture, demonstrating the effects of crack closure on the predicted fatigue life. We compare the obtained results with those of other publications.     

The energy generation and transmission in compound elastic cylindrical shells with heavy internal fluid loading—from parametric studies to optimization

The methodology of boundary integral equations is applied for analysis and optimization of the power flows in elastic compound cylindrical shells with heavy internal fluid loading. Two generic model problems are solved and the roles of various physical parameters involved in the problem formulation are assessed. It is shown that the efficiency of an optimization procedure heavily relies on a careful parametric study of wave propagation and correct physical interpretation of its results.     

A matrix formulation to the wave equation with non-local boundary condition

Recently, various sequential numerical schemes have been proposed for the solution of non-classical hyperbolic initial value problems which involve non-local integral terms over the spatial domain. In this paper, we focus on the wave equation with the non-local boundary condition. Two matrix formulation techniques based on the shifted standard and shifted Chebyshev bases are proposed for the numerical solution. Several numerical examples and also some comparisons with another methods will be investigated to confirm the efficiency of this procedure.     

Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves

Solving boundary value problems in plane elastostatics with the aid of integral equations the boundary of the elastic body usually represents the domain of definition of the sought solution as well as of the given boundary values. However, it is also possible to define the solution and the boundary values on two spatially separated curves $\text{S}\text{?}$ and S. In this paper the numerical properties of both methods are investigated and compared and in particular the influence of the distance between $\text{S}\text{?}$ and S is estimated. For these purposes the eigenvalues and eigenfunctions of the operators in the integral equations are determined.     

D/BEM implementation of Robinson's viscoplastic model in creep analysis of metals using a complex variable numerical technique

A new boundary element approach for the solution of time-dependent inelastic problems arising in creeping metallic structures subjected to the combined action of high temperature gradient and quasi-static mechanical loading conditions is investigated. The new approach allows the use of complex variable techniques in the boundary element procedure for the evaluation of stress components as derivatives of the displacement integral equations. This methodology makes faster and more accurate the conventional boundary element method. To validate the efficiency of the proposed method in the implementation of Robinson's viscoplastic model, the results obtained using the present methodology are compared with those obtained by using known analytical and finite element solution for the analysis of a thick-walled internally pressurized cylinder and an experimental cylindrical thrust chamber in plane strain under general thermomechanical loading histories.     

轴向受力梁结构线弹性碰撞问题求解

研究了质点对轴向受力的Euler-Bernoulli梁结构任意位置的横向撞击问题.把撞击体系简化为质量-弹簧体系模型,采用积分变换方法,对撞击体系的控制微分方程、边界条件和连续条件进行Laplace变换,在频域内求得其解析表达式,然后采用Crump逆变换方法进行数值反演,得到时域内的各种动力响应.数值算例给出了撞击力、弯曲应力和剪力随时间变化的曲线,通过与有限元比较验证了本方法的正确性.最后研究了撞击位置、轴向压力、撞击质量、撞击速度和柔度系数等参数对撞击力的影响,得出了一些有用的结论.     

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

We introduce a novel enriched Boundary Element Method (BEM) and Dual Boundary Element Method (DBEM) approach for accurate evaluation of Stress Intensity Factors (SIFs) in crack problems. The formulation makes use of the Partition of Unity Method (PUM) such that functions obtained from a priori knowledge of the solution space can be incorporated in the element formulation. An enrichment strategy is described, in which boundary integral equations formed at additional collocation points are used to provide auxiliary equations in order to accommodate the extra introduced unknowns. In addition, an efficient numerical quadrature method is outlined for the evaluation of strongly singular and hypersingular enriched boundary integrals. Finally, results are shown for mixed mode crack problems; these illustrate that the introduction of PUM enrichment provides for an improvement in accuracy of approximately one order of magnitude in comparison to the conventional unenriched DBEM.     

A doubly iterative BEM for solving crack closing in an elastic body

An algorithm for the two-dimensional elastic bodies with closed crack and defects loaded by a moving contact elastic body is proposed using a sub-regional boundary element method. Since the extent and status of the contact surface of two elastic bodies and the crack within the body are all not known in advance, a doubly iterative contact algorithm has been presented. The BEM program for solving the closed crack cases has been developed, some numerical examples have been calculated, and the results of the center crack cases are in good agreement with the analytical solution in classical fracture mechanics. In the condition of friction and non-friction, some coupling computational results of the SIF for the closed crack cases, which have different angles and loaded by a moving contact elastic body, have also been obtained by numerical computation. The cracked plate subjected to the moving contact elastic body has been analyzed under the conditions of drilling hole near the crack.     

Fracture analysis of plane piezoelectric/piezomagnetic multiphase composites under transient loading

The transient response of cracked composite materials made of piezoelectric and piezomagnetic phases, when subjected to in-plane magneto-electro-mechanical dynamic loads, is addressed in this paper by means of a mixed boundary element method (BEM) approach. Both the displacement and traction boundary integral equations (BIEs) are used to develop a single-domain formulation. The convolution integrals arising in the time-domain BEM are numerically computed by Lubich’s quadrature, which determines the integration weights from the Laplace transformed fundamental solution and a linear multistep method. The required Laplace-domain fundamental solution is derived by means of the Radon transform in the form of line integrals over a unit circumference. The singular and hypersingular BIEs are numerically evaluated in a precise and efficient manner by a regularization procedure based on a simple change of variable, as previously proposed by the authors for statics. Discontinuous quarter-point elements are used to properly capture the behavior of the extended crack opening displacements (ECOD) around the crack-tip and directly evaluate the field intensity factors (stress, electric displacement and magnetic induction intensity factors) from the computed nodal data. Numerical results are obtained to validate the formulation and illustrate its capabilities. The effect of the combined application of electric, magnetic and mechanical loads on the dynamic field intensity factors is analyzed in detail for several crack configurations under impact loading.     

Discretization error due to the identity operator in surface integral equations

We consider the accuracy of surface integral equations for the solution of scattering and radiation problems in electromagnetics. In numerical solutions, second-kind integral equations involving well-tested identity operators are preferable for efficiency, because they produce diagonally-dominant matrix equations that can be solved easily with iterative methods. However, the existence of the well-tested identity operators leads to inaccurate results, especially when the equations are discretized with low-order basis functions, such as the Rao-Wilton-Glisson functions. By performing a computational experiment based on the nonradiating property of the tangential incident fields on arbitrary surfaces, we show that the discretization error of the identity operator is a major error source that contaminates the accuracy of the second-kind integral equations significantly.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

A boundary integral method applied to plates of arbitrary plan form

An integral equation method for the solution of thin elastic plates of arbitrary plan form has been presented. The method involves embedding the real plate in a fictitious plate for which the Green's function is known. An unknown load vector is then introduced on the boundary of the real plate (line load and line normal moment). The deflection field due to both known transverse and unknown boundary loads can then be found everywhere by superposition. Satisfaction of the boundary conditions on the real plate results in a vector integral equation in the unknown boundary vector.In concept, any consistent set of boundary conditions will yield a solution. Practically, boundary conditions requiring higher derivatives of the deflection are both very cumbersome and yield singularities in the integral equations which cause numerical difficulties. For these reasons only clamped boundary conditions are treated numerically in the present paper.For interior bending moments and deflections (greater than distances of the order of one boundary subdivision from the boundary) the method is both highly accurate and inexpensive. Errors right on the boundary, e.g. the clamping moment in the clamped boundary condition case, can be appreciable, however. While this can be improved by a more sophisticated treatment of the unknown boundary vector in the numerical solution (increased expense) it is shown in the paper that a simple boundary extrapolation procedure gives excellent accuracy there.