A non-hypersingular time-domain BIEM for 3-D transient elastodynamic crack analysis

A three-dimensional (3-D) time-domain boundary integral equation method (BIEM) is presented for transient elastodynamic crack analysis. A non-hypersingular traction BIE formulation is used with the crack opening displacements and their derivatives as unknown quantities. A collocation method in conjunction with a time-stepping scheme is developed to solve the non-hypersingular time-domain BIEs. To simplify the analysis and to describe the proper behaviour of the unknown quantities at the crack front, a constant spatial shape function is applied for elements away from the crack front, while a spatial ‘square-root’ crack-tip shape function is adopted for elements near the crack front. A linear temporal shape function is used in the time-stepping scheme. Numerical calculations, have been carried out for penny-shaped and square cracks. Results for the elastodynamic stress intensity factors are presented as functions of the temporal and the spatial variables. For the test examples considered, good agreement between the present results and those of other authors is obtained.     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Dynamic modelling of the flat 2‐D crack by a semi‐analytic BIEM scheme

We present an efficient numerical method for solving indirect boundary integral equations that describe the dynamics of a flat two‐dimensional (2‐D) crack in all modes of fracture. The method is based on a piecewise‐constant interpolation, both in space and time, of the slip‐rate function, by which the original equation is reduced to a discrete convolution, in space and time, of the slip‐rate and a set of analytically obtained coefficients. If the time‐step interval is set sufficiently small with respect to the spatial grid size, the discrete equations decouple and can be solved explicitly. This semi‐analytic scheme can be extended to the calculation of the wave field off the crack plane. A necessary condition for the numerical stability of this scheme is investigated by way of an exhaustive set of trial runs for a kinematic problem. For the case investigated, our scheme is very stable for a fairly wide range of control parameters in modes III and I, whereas, in mode II, it is unstable except for some limited ranges of the parameters. The use of Peirce and Siebrits' ε‐scheme in time collocation is found helpful in stabilizing the numerical calculation. Our scheme also allows for variable time steps. Copyright © 2001 John Wiley & Sons, Ltd.     

Time-domain transient elastodynamic analysis of 3-D solids by BEM

The numerical implementation of the Direct Boundary Element formulation for time-domain transient analysis of three-dimensional solids is presented in a most general and complete manner. The present formulation employs the space and time dependent fundamental solution (Stokes' solution) and Graffi's dynamic reciprocal theorem to derive the boundary integral equations in the time domain. A time-stepping scheme is then used to solve the boundary initial value problem by marching forward in time. Higher order shape functions are used to approximate the field quantities in space as well as in time, and a combination of analytical (time-integration) and numerical (spatial-integration) integration is carried out to form a system of linear equations. At the end of each time step, these equations are solved to obtain the unknown field quantities at that time. Finally, the accuracy and reliability of this algorithm is demonstrated by solving a number of example problems and comparing the results against the available analytical and numerical solution.     

3D crack analysis in functionally graded materials

Elastostatic crack analysis in three-dimensional, continuously non-homogeneous, isotropic and linear elastic functionally graded materials and structures is presented in this paper. A boundary-domain-integral equation formulation is applied for this purpose, which uses the elastostatic fundamental solutions for homogeneous, isotropic and linear elastic materials and involves a domain-integral due to the material’s non-homogeneity. To avoid displacement gradients in the domain-integral, normalized displacements are introduced. The domain-integral is transformed into boundary-integrals over the global boundary of the cracked solids by using the radial integration method. A meshless scheme is developed, which requires only the conventional boundary discretization and additional interior nodes instead of interior cells or meshes. Numerical examples for three-dimensional crack problems in continuously non-homogeneous, isotropic and linear elastic FGMs are presented and discussed, to show the effects of the material gradation on the crack-opening-displacements and the stress intensity factors.     

Time-harmonic crack problems in functionally graded piezoelectric solids via BIEM

In-plane crack analysis of functionally graded piezoelectric solids under time-harmonic loading is performed by using a non-hypersingular traction based boundary integral equation method (BIEM). The material parameters are assumed to vary quadratically with both spatial variables. A frequency dependent fundamental solution, as well as its derivatives and asymptotic expressions, is derived in closed-form by using an appropriate algebraic transformation for the displacement vector and the Radon transform. Numerical results for the stress intensity factors (SIFs) are discussed for different examples. The accuracy of the presented method is checked by comparison with available results from the literature. Investigated are the effects of the inhomogeneity parameters, the frequency of the applied electromechanical load and the geometry of the crack scenario on the K-factors.     

Stress and velocity in 2D transient elastodynamic analysis by the boundary element method

This paper is mainly concerned with the development of integral equations to compute stress and velocity components in transient elastodynamic analysis by the boundary element method. All expressions required are presented explicitly. The boundary is discretized by linear isoparametric elements whereas linear and constant time interpolation are assumed, respectively, for the displacement and traction components. Time integration is carried out analytically and the resulting expressions are presented. An assessment of the accuracy of the results provided by the present formulation can be seen at the end of the article, where two examples are presented.     

Axisymmetric elastodynamic response of a flat annular crack to normal impact waves

The axisymmetric response of a flat annular crack in an infinite medium subjected to normal impact load is investigated in this study. A step stress is applied to the crack surface. The singular solution is equivalent to solutions of the problem of diffraction of normally incident tension wave by a flat annular crack, and the problem of the sudden appearance of a flat annular crack in a uniform tensile stress field. Laplace and Hankel transforms are used to reduce the problem to the solution of a set of triple integral equations in the Laplace transform domain. These equations are solved by using a integral transform technique and the result is expressed in terms of a singular integral equation of the first kind with the kernel which is improved by means of a contour integration on the Riemann surface. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results of the dynamic stress intensity factor are obtained to show the influence of inertia, the ratio of the inner radius to the outer one and Poisson's ratio on the load transmission to the crack tip.     

A general 3D BEM/FEM coupling applied to elastodynamic continua/frame structures interaction analysis

This paper presents a procedure for coupling general finite element models with three‐dimensional bodies modelled by the Boundary Element Method (BEM). Shells, plates and frames are modelled by the Finite Element Method (FEM) and coupled to the BEM domain directly or by means of rigid blocks. The coupling is used for the analysis of buildings connected to half‐space by means of rigid footings, piles or plates in bending and other problems where combinations of different types of sub‐domains are required, composite domains for instance. Several numerical examples are analysed to demonstrate the robustness and accuracy of the proposed scheme. Copyright © 1999 John Wiley & Sons, Ltd.     

An elastodynamic fracture analysis in a centre-cracked plate

A two-dimensional linear elastodynamic analysis of crack initiation and fast crack propagation in a centre-cracked plate, subjected to constant tension is presented. The analysis is performed using the previously developed SMF2D code in its generation mode. The experimentally measured crack tip motion, as well as the specimen's geometry and its material characteristics serve as input to the simulation. The dynamic stress intensity factor, the dynamic energy release rate, and the various energy distributions are subsequently evaluated. Special attention is given to the influence of the energy supplied to the body during the fracture process due to the work done by the external tractions.     

A regularized boundary integral equation method for elastodynamic crack problems

This paper presents a double layer potential approach of elastodynamic BIE crack analysis. Our method regularizes the conventional strongly singular expressions for the traction of double layer potential into forms including integrable kernels and 0th, 1st and 2nd order derivatives of the double layer density. The manipulation is systematized by the use of the stress function representation of the differentiated double layer kernel functions. This regularization, together with the use of B-spline functions, is shown to provide accurate numerical methods of crack analysis in 3D time harmonic elastodynamics.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Orthogonal meshless finite volume method applied to elastodynamic crack problems

An orthogonal meshless finite volume method has been presented to solve some elastodynamic crack problems. An orthogonal weighted basis function is used to construct shape function so there is no problem of singularity in this new form. In this work, for three-dimensional dynamic fracture problems, a new displacement function is used at the tip of the crack to give a new OMFVM. When the new OMFVM is used, the singularity of the stresses at the tip of the crack can be shown to be better than that in the primal OMFVM. High computational efficiency and precision are other benefits of the method. Solving some sample crack problems of thin-walled structures show a good performance of this method.     

Precise 3D crack growth simulations

The continuous growth of 3D cracks under cyclic loading conditions is considered within a discrete simulation procedure. It is performed within the framework of linear elastic fracture mechanics. An incremental procedure is applied to consider the non-linear behavior of crack growth within the simulation. In each increment the direction and magnitude of the crack propagation for each point along the crack front are needed to define the new crack front. Within the present context the crack deflection results from the maximum tangential stress criterion and the crack extension is obtained by the evaluation of a crack propagation rate. To simulate the crack propagation as exactly as possible the evolution of the stress field between two consecutive crack fronts is taken into account. The analysis of the changing stress field is utilized for optimization of the predicted crack fronts. The whole procedure is realized in terms of a predictor–corrector scheme. Numerical examples are presented to demonstrate the benefits of this concept.     

A variational boundary integral equation method for an elastodynamic antiplane crack

This paper investigates the transient wave scattering by a crack by means of the Boundary Integral Equation Method (BIEM). The author has developed a new formulation to solve the BIE for the Crack Opening Displacement (COD). The resolution is done directly in the time domain. The solution is represented by means of a retarded double layer potential, and the resulting BIE, with the COD as unknown, has a hypersingular kernel. The corresponding difficulty is overcome by using a variational method. We present the application of this method to an antiplane crack, describe the approximate problem and finally give some numerical results.     

Strain energy density fracture criterion in elastodynamic mixed mode crack propagation

Sih's fracture criterion based on strain energy density, S, for mixed mode crack extension under static loading is extended to dynamic mixed mode, K_I and K_II, crack propagation. Influence of the second order term, σ_ox, which represents the non-singular constant stress acting parallel to the direction of crack propagation, on the S distribution surrounding the crack tip, is demonstrated. Numerical studies show that positive σ_ox enhances the fracture angle and negative σ_oxreduces the fracture angle irrespective of the sign of K_II/K_I, when S is measured at a critical distance r_c from the crack tip. This fracture criterion is verified by the crack curving results of dynamic photoelastic fracture specimens. Omission of σ_ox term leads to predicted fracture angles which are at variance with experimental data.     

Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids

An advanced computational method for transient heat conduction analysis in 3-D axisymmetric continuously nonhomogeneous functionally graded materials (FGM) is proposed. The analysed domain is covered by small circular subdomains. On each subdomain local boundary integral equations for the transient heat conduction problem are derived in the Laplace transform domain. The meshless approximation based on the moving least-squares method is employed for the numerical implementation. The Stehfest algorithm is applied for the numerical Laplace inversion to obtain the temporal variation. Numerical results are presented for finite full and hollow cylinders with an exponential variation of material parameters with spatial coordinates. The authors acknowledge the support by the Slovak Science and Technology Assistance Agency registered under number APVT-51-003702, and the Project for Bilateral Cooperation in Science and Technology supported jointly by the International Bureau of the German BMBF and the Ministry of Education of Slovak Republic under the project number SVK 01/020.     

Surrogate modeling of 3D crack growth

This paper presents the development of a surrogate modeling technique for efficient non-planar fatigue crack growth analysis in mechanical components under multi-axial loading. Non-planar crack fronts are freely deformable space curves and require a high-dimensional representation. The large number of Cartesian co-ordinate variables involved in crack front representation makes it prohibitively expensive to train surrogate models for crack growth. Therefore, in our previous work, the crack shape was approximated using a planar parametrized representation. However, the parametrized representation limits the choice of crack shapes that can be considered. This paper presents the development of a non-parametric crack shape representation that allows for construction of a surrogate model for non-planar crack growth with complex crack shapes. The surrogate model is trained using a few runs of high-fidelity 3D simulations and predicts the evolution of a non-planar crack front under a given multi-axial, variable amplitude load history. We first parametrize the crack fronts as 3D spline curves with a fixed number of nodes. Instead of modeling the crack growth in this high dimensional data space, we project the data to a lower dimensional space using Principal Component Analysis (PCA) and then model the crack growth in this lower dimensional space. Finally, the predicted crack fronts are recovered using PCA back to the original data space. The proposed crack representation, growth modeling and recovery are illustrated using training points gathered from high-fidelity 3-D finite element simulations of non-planar crack growth in a cylindrical component similar to a rotorcraft mast, and the ability of the surrogate model to accurately predict the evolution of the crack growth over entire load histories is demonstrated.     

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.