A two-dimensional time-domain boundary element method for dynamic crack problems in anisotropic solids

A time-domain boundary element method (BEM) for transient dynamic crack analysis in two-dimensional, homogeneous, anisotropic and linear elastic solids is presented in this paper. Strongly singular displacement boundary integral equations (DBIEs) are applied on the external boundary of the cracked body while hypersingular traction boundary integral equations (TBIEs) are used on the crack-faces. The present time-domain method uses the quadrature formula of Lubich for approximating the convolution integrals and a collocation method for the spatial discretization of the time-domain boundary integral equations. Strongly singular and hypersingular integrals are dealt with by a regularization technique based on a suitable variable change. Discontinuous quadratic quarter-point elements are implemented at the crack-tips to capture the local square-root-behavior of the crack-opening-displacements properly. Numerical examples for computing the dynamic stress intensity factors are presented and discussed to demonstrate the accuracy and the efficiency of the present method.     

Transient dynamic analysis of interface cracks in layered anisotropic solids under impact loading

Transient elastodynamic crack analysis in two-dimensional (2D), layered, anisotropic and linear elastic solids is presented in this paper. A time-domain boundary element method (BEM) in conjunction with a multi-domain technique is developed for this purpose. Time-domain elastodynamic fundamental solutions for homogenous, anisotropic and linear elastic solids are applied in the present time-domain BEM. The spatial discretization of the boundary integral equations is performed by a Galerkin-method, while a collocation method is adopted for the temporal discretization of the arising convolution integrals. An explicit time-stepping scheme is developed to compute the unknown boundary data and the crack-opening-displacements (CODs). To show the effects of the crack configuration, the material anisotropy, the layer combination and the dynamic loading on the dynamic stress intensity factors and the scattered elastic wave fields, several numerical examples are presented and discussed.     

A time-domain collocation-Galerkin BEM for transient dynamic crack analysis in anisotropic solids

This paper presents a time-domain boundary element method (BEM) for transient elastodynamic crack analysis in homogeneous and linear elastic solids of general anisotropy. A finite crack subjected to a transient loading is investigated. Two-dimensional (2D) generalized plane-strain or plane-stress condition is considered. The initial-boundary value problem is described by a set of hypersingular time-dependent traction boundary integral equations (BIEs), in which the crack-opening displacements (CODs) are unknown quantities. The hypersingular time-domain BIEs are first regularized to weakly singular ones by using spatial Galerkin method, which transfers the derivatives of the fundamental solutions to the unknown CODs and the weight functions. To solve the time-domain BIEs numerically, a time-stepping scheme is developed. The scheme applies the collocation method for temporal discretization of the time-domain BIEs. As spatial shape-functions, two different functions are implemented. For elements away from crack-tips, linear spatial shape-function is used, while for elements near the crack-tips a special ‘crack-tip shape-function’ is applied to describe the local ‘square-root’ behavior of the CODs at the crack-tips properly. Special attention of the analysis is devoted to the numerical computation of the transient elastodynamic stress intensity factors for cracks in general anisotropic and linear elastic solids. Numerical examples are presented to verify the accuracy of the present time-domain BEM.     

Dynamic analysis of interfacial crack problems in anisotropic bi-materials by a time-domain BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study dynamic interfacial crack problems in two-dimensional (2D), piecewise homogeneous, anisotropic and linear elastic bi-materials. The bi-material system is divided into two homogeneous sub-domains along the interface and the traditional displacement boundary integral equations (BIEs) are applied on the boundary of each sub-domain. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. A displacement extrapolation technique is used to determine the complex dynamic stress intensity factors (SIFs). Numerical examples for computing the complex dynamic SIFs are presented and discussed to demonstrate the accuracy and the efficiency of the present time-domain BEM.     

A 2D time-domain collocation-Galerkin BEM for dynamic crack analysis in piezoelectric solids

A time-domain boundary element method (TDBEM) for transient dynamic analysis of two-dimensional (2D), homogeneous, anisotropic and linear piezoelectric cracked solids is presented in this paper. The present analysis uses a combination of the strongly singular displacement boundary integral equations (BIEs) and the hypersingular traction boundary integral equations. The spatial discretization is performed by a Galerkin-method, while a collocation method is implemented for the temporal discretization. Both temporal and spatial integrations are carried out analytically. In this way, only the line integrals over a unit circle arising in the time-domain fundamental solutions are computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is developed to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. Special crack-tip elements are adopted to ensure a direct and an accurate computation of the dynamic field intensity factors (IFs) from the CODs. Several numerical examples involving stationary cracks in both infinite and finite solids under impact loading are presented to show the accuracy and the efficiency of the developed hypersingular time-domain BEM.     

Transient response of a semi-permeable crack between dissimilar anisotropic piezoelectric layers by time-domain BEM

This paper studied the transient response of a semi-permeable crack between two dissimilar anisotropic piezoelectric layers by a time-domain BEM with a sub-domain technique and an iterative process for the non-linear crack-face boundary conditions. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. A universal matrix-form displacement extrapolation formula and its explicit formula are used to determine the dynamic intensity factors. Several examples are presented and discussed to show the effects of the electrical crack-face boundary condition on dynamic intensity factors.     

Transient elastodynamic analysis of nonhomogeneous anisotropic plane bodies

A boundary-only BEM procedure is employed to solve the transient dynamic analysis of nonhomogeneous anisotropic plane elastic bodies. The response of such bodies is governed by two coupled linear, second-order hyperbolic PDEs with spatially dependent coefficients. The lack of a reliable 2D time-domain elastodynamic fundamental solution is overcome using the principle of the Analog Equation, a method by which the equations of motion of the problem are substituted by two coupled quasi-static Poisson-type equations having as nonhomogeneous terms the components of a fictitious time-dependent load distribution in the specified domain. The standard BEM is employed for the solution of the substitute equations. To avoid the appearance of the domain integral in the integral representation of the solution, the fictitious load distribution is approximated by multiquadrics with unknown time-dependent expansion coefficients, which are calculated at discrete timepoints by collocating the equations of motion at a predefined set of domain interpolation nodes. The obtained numerical results by the proposed method demonstrate its stability and accuracy over other numerical methods.     

Transient response of an insulating crack near to the interface between two piezoelectric half-planes under electromechanical impacts by BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study transient response of an insulating crack near to the interface between two anisotropic piezoelectric half-planes under electromechanical impacts. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the crack tip. A displacement extrapolation technique is used to determine the dynamic stress intensity factors (DSIFs) and the dynamic electrical displacement intensity factor. Numerical examples are presented to show the effects of load combination, geometric configuration and material combination on dynamic intensity factors and dynamic energy release rate.     

A comparative study of three BEM for transient dynamic crack analysis of 2-D anisotropic solids

Three different boundary element methods (BEM) for transient dynamic crack analysis in two-dimensional (2-D), homogeneous, anisotropic and linear elastic solids are presented. Hypersingular traction boundary integral equations (BIEs) in frequency- domain, Laplace-domain and time-domain with the corresponding elastodynamic fundamental solutions are applied for this purpose. In the frequency-domain and the Laplace-domain BEM, numerical solutions are first obtained in the transformed domain for discrete frequency or Laplace-transform parameters. Time-dependent results are subsequently obtained by means of the inverse Fourier-transform and the inverse Laplace-transform algorithm of Stehfest. In the time-domain BEM, the quadrature formula of Lubich is adopted to approximate the arising convolution integrals in the time-domain BIEs. Hypersingular integrals involved in the traction BIEs are computed through a regularization process that converts the hypersingular integrals to regular integrals, which can be computed numerically, and singular integrals which can be integrated analytically. Numerical results for the dynamic stress intensity factors are presented and discussed for a finite crack in an infinite domain subjected to an impact crack-face loading.     

Time-domain BEM for transient interfacial crack problems in anisotropic piezoelectric bi-materials

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study transient response of interfacial cracks in piecewise homogeneous, anisotropic and linear piezoelectric bi-materials under electrical and mechanical impacts. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. To determine the real or complex dynamic stress intensity factors and the dynamic electrical displacement intensity factor of the interfacial cracks, an explicit extrapolating formula in a typical state of the crack plane perpendicular to the poling direction is presented in this paper. Numerical examples are presented; and the effects of the load combination and material combination on dynamic intensity factors and dynamic energy release rate are discussed.     

A hypersingular time‐domain BEM for 2D dynamic crack analysis in anisotropic solids

A hypersingular time‐domain boundary element method (BEM) for transient elastodynamic crack analysis in two‐dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack‐faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time‐stepping scheme is obtained to compute the unknown boundary data including the crack‐opening‐displacements (CODs). Special crack‐tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time‐domain BEM. Copyright © 2008 John Wiley & Sons, Ltd.     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.     

Time-domain BEM for 3-D transient elastodynamic problems with interacting rigid movable disc-shaped inclusions

Formulation of time-domain boundary element method for elastodynamic analysis of interaction between rigid massive disc-shaped inclusions subjected to impinging elastic waves is presented. Boundary integral equations (BIEs) with time-retarded kernels are obtained by using the integral representations of displacements in a matrix in terms of interfacial stress jumps across the inhomogeneities and satisfaction of linearity conditions at the inclusion domains. The equations of motion for each inclusion complete the problem formulation. The time-stepping/collocation scheme is implemented for the discretization of the BIEs by taking into account the traveling nature of the generated wave field and local structure of the solution at the inclusion edges. Numerical results concern normal incidence of longitudinal wave onto two coplanar circular inclusions. The inertial effects are revealed by the time dependencies of inclusions’ kinematic parameters and dynamic stress intensity factors in the inclusion vicinities for different mass ratios and distances between the interacting obstacles.     

A frequency-domain BEM for 3D non-synchronous crack interaction analysis in elastic solids

A frequency-domain boundary element method (BEM) is presented for non-synchronous crack interaction analysis in three-dimensional (3D), infinite, isotropic and linear elastic solids with multiple coplanar cracks. The cracks are subjected to non-synchronous time-harmonic crack-surface loading with contrast frequencies. Hypersingular frequency-domain traction boundary integral equations (BIEs) are applied to solve the boundary value problem. A collocation method is adopted for solving the BIEs numerically. The local square-root behavior of the crack-opening-displacements at the crack-front is taken into account in the present method. For two coplanar penny-shaped cracks of equal radius subjected to non-synchronous time-harmonic crack-surface loading, numerical results for the dynamic stress intensity factors are presented and discussed.     

地面谐运动激励下任意外形水坝稳态响应的双边界元解

本文基于有限水深带形域势流问题的基本解和二维线弹性力学问题的Kelvin解,建立了坝库系统在谐激励下稳态响应的双边界积分方程.推导过程中,利用了Nardini和Brebbia方法将分布惯性力项的体积分化为相应的边界积分.然后通过边界元离散技术,针对两个不同型式的坝体计算了作用在界面上的水动压力分布,其中一个算例的结果和已有的有限元解作了比较.     

The analog equation method for large deflection analysis of membranes. A boundary-only solution

 In this paper the analog equation method (AEM) is applied to nonlinear analysis of elastic membranes with arbitrary shape. In this case the transverse deflections influence the inplane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear. The present formulation, being in terms of the three displacements components, permits the application of geometrical inplane boundary conditions. The membrane is prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of the membranes. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 21 November 2000     

求解水下非圆弹性环声散射问题的一种半解析方法

基于传递矩阵法、齐次扩容精细积分法和复数矢径虚拟边界谱方法 ,提出了一种求解水下非圆弹性环声散射问题的半解析方法。该方法具有以下几个优点 :(1)采用复数矢径虚拟边界谱方法 ,不仅能保证在全波数域内Helmholtz外问题解的唯一性 ,而且由于虚拟源强密度函数采用 Fourier级数展开 ,克服了用单元离散解法不能用于较高频率范围的缺点 ;(2 )采用齐次扩容精细积分法求解非圆弹性环的状态微分方程 ,其计算结果具有很高的精度 ;(3)耦合方程不需要交错迭代求解 ,提高了计算效率。文中给出了两个典型非圆弹性环在平面声波激励下的声散射算例 ,计算结果表明本文方法是一种求解二维非圆弹性环声散射问题非常有效的半解析法。     

Time discontinuous linear traction approximation in time-domain BEM scalar wave propagation analysis

In the present paper the traditional BEM formulation for time-domain scalar wave propagation analysis is extended to a new class of problems. A procedure to consider linear time interpolation for boundary tractions is worked out. Time discontinuities are included by adding to the standard BEM equation the integral equation for velocities. Numerical examples are presented in order to assess the accuracy of the proposed formulation. © 1998 John Wiley & Sons, Ltd.     

Transient dynamic elastoplastic analysis by the time-domain BEM formulation

This paper presents a time-domain BEM formulation applied to the solution of transient dynamic elastoplastic problems. The initial stress approach is adopted to solve the elastoplastic problem. Linear time variation is assumed for the displacements and initial stress components whereas traction components are assumed to have a constant time variation. Boundary discretization employs linear elements and the part of the domain where plastic deformation is expected to occur is discretized by employing linear triangular cells. Time integrals are computed analytically, boundary integrals are computed numerically and the domain integrals are computed by following a semi-analytical procedure. A numerical example is presented and the results are compared with another BEM formulation.     

Einführung in moderne Galerkin-Randeiementmethoden mit einer Anwendung aus dem Maschinenbau

The Galerkin-type boundary element method (BEM) is an discretization procedure for integral equations, represents itself however compared with classical integral equation methods as an universal tool for the solution of practical engineering problems and can be coupled very easily with finite element substructures. The BEM, whose main advantage lies in the fact that only a surface mesh must be generated, is superior to FEM in special applications, i.e. in elastostatics (notch problems) and fracture mechanics. In this paper the individual steps to solving an elliptical boundary value problem of 3-D linear elasticity theory by way of an equivalent system of boundary integral equations will be explained. For the mathematical investigation of elliptical differential equations and integral equations, the theory of Sobolev spaces has proved to be especially suitable. Basic terms to Sobolev spaces will be introduced so that the reader does not have to refer to textbooks for new terms. The transformation of elliptical boundary value problems to systems of singular and hypersingular integral equations will be explained with help of a Calderón projector, which is defined by using fundamental solutions. The discretization of the obtained integral equations with the Galerkin-type BEM will be presented. Finally the approximation of non-linear problems by using the Galerkin-type BEM will be shown. A numerical test for a strength problem will be discussed shortly.