Elastodynamic Crack Problems Research Articles

The continuous-discontinuous cellular automaton method for elastodynamic crack problems

In this paper, the recently proposed continuous-discontinuous cellular automaton method (CDCA) is further developed for elastodynamic problems in 2D elastic solids. A continuity-to-discontinuity enriched function technique is developed for elastodynamics and combined with a cellular automaton formulation; the cellular automaton theory for elastodynamics, in which fast adaptive updating rules for the cell and its neighbors are developed, and the time domain CDCA are proposed. In this method, the cracks are independent of the integral meshes, and no assembled global matrix is needed for the whole process. Time-dependent equations for the cells are discretized by the Newmark method, and rapidly solved by the cellular automaton method. To analyze the fracture behaviors of elastic solids with dynamic loads, mixed-mode dynamic stress intensity factors are obtained by the interaction integral method. Some numerical examples for elastodynamic problems are studied to validate the accuracy and efficiency of the present method, and solutions from analytical method and some other methods are employed to show the high accuracy of the CDCA.

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.

Performance of 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 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.

Analysis of Mode III Elastodynamic Cracked Plane Using the Fractal Two-Level Finite Element Method

In this article, the fractal two-level finite element method, which has mainly been used for static crack problems, is applied to mode III elastodynamic plane crack problems. Using the transformation process in the proposed method, the infinite dimension of the finite element matrices that are assembled for a singular region is made finite in terms of the dynamics stress intensity factors directly and, thus, the computational time and errors can be reduced significantly. The Newmark time integration scheme is then used to obtain the dynamic stress intensity factors for different crack cases. The results from the proposed method are in reasonable agreement with those of classical methods. A parametric study is conducted to investigate the effects of different parameters on the accuracy of the dynamic stress intensity factor.

A fast hierarchical dual boundary element method for three‐dimensional elastodynamic crack problems

AbstractIn this work a fast solver for large‐scale three‐dimensional elastodynamic crack problems is presented, implemented, and tested. The dual boundary element method in the Laplace transform domain is used for the accurate dynamic analysis of cracked bodies. The fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the Laplace parameter. An ACA (adaptive cross approximation) algorithm is used for the population of the low rank blocks and its performance at varying Laplace parameters is investigated. A preconditioned GMRES is used for the solution of the resulting algebraic system of equations. The preconditioners are built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. An original strategy, based on the computation of some local preconditioners only, is presented and tested to further speed up the overall analysis. The reported numerical results demonstrate the effectiveness of the technique for both uncracked and cracked solids and show significant reductions in terms of both memory storage and computational time. Copyright © 2010 John Wiley & Sons, Ltd.

A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.

Time-domain BEM for three-dimensional fracture mechanics

The present paper deals with time-domain analysis of three-dimensional transient dynamic crack problems. The time-domain formulation of the boundary element method for 3-D elastodynamic problems is used. Quarter-point and singular quarter-point elements represent displacements and tractions, respectively, near the crack front. Special attention is paid to integration and algorithms to preserve stability. Cracks in finite and unbounded regions under single and mixed mode dynamic loading conditions are studied. To the authors’ knowledge, no previous BE approach for 3-D elastodynamic crack problems based on the time-domain displacement representation exists.

Elastodynamic Crack Analysis by Boundary Element Method Using Numerical Inversion of Laplace Transform.

In the present study, an alternative strategy for elasto-dynamic crack problems using boundary element method is discussed. The Laplace transformation is applied to time variation in the formulation. Since the numerical solutions are obtained in the Laplace-transformed domain, the numerical-inverse Laplace transformation is adopted to obtain time histories of the solutions. In the present method, the distributions of the displacement and traction in the vicinity of the crack tip are expressed as the superposition of singular terms and regular ones. The singular terms are the analytical solutions including stress intensity factors of Modes I and II, while the regular terms are approximated by the discrete polynomial functions in general BEM formulations. Additionally, the ligament side is discretized by singular-type crack elements in order to increase the accuracy of the traction near the crack tip. It is shown that the dynamic stress intensity factors (DSIFs) of Modes I and II are determined directly without supplementary procedures such as extrapolation or J-integral scheme. The validity of the present method is confirmed through the analysis of mixed mode dynamic problems of crack.

The Analysis of Dynamic Stress Intensity Factor for Semi-Circular Surface Crack Using Time-Domain BEM Formulation

The time-domain BEM was developed to analyze the dynamic stress intensity factor (DSIF) of 3-D elastodynamic crack problems. To simulate the stress singularity along the front of a crack, eight-node isoparametric singular elements were used, and the DSIF for a semi-circular surface crack was firstly calculated based on displacement equation using the time-domain BEM formulation. The new scheme to determine the time step was brought forward. By the dynamic analysis program of time-domain BEM compiled by us, several numerical examples are presented, which demonstrate the unconditional stability and high accuracy of time-domain BEM applied to 3-D elastodynamic crack problems.

Nonsingular traction BIEs for crack problems in elastodynamics

The nonsingular traction BIEs are derived for the Laplace transforms in elastodynamic crack problems. Two different forms of the final nonsingular traction BIEs are received with respect to the leading singularity of the integral kernels involved. In the first one, the traction BIE is derived from the integral representation of stresses which involves hypersingular kernels. In the second way, the partially regularized integral representation of stresses with strongly singular kernels is used as a starting point. The singular integrals are transformed to nonsingular ones by making use of the Stokes theorem.

Contour integrals for mixed-mode crack analysis: effect of nonsingular terms

An integral formulation for computing the nonsingular stresses (NSS) in a cracked body under mixed-mode static and dynamic loads is presented. The reciprocity theorems are applied to find the integral formula. The auxiliary fields are selected to eliminate the singular terms in the asymptotic expansion of the stresses near the crack tip. For elastodynamic crack problems, the integral representation of the NSS is presented in both the time and Laplace transform domain. Required variables along the integration path and region enclosed by the integration contour are obtained from the boundary element analysis. Influence of the NSS on predicting the crack growth direction is investigated for cracks under mixed-mode load conditions.

Exact elastodynamic analysis of some fracture specimen models involving strip geometries

A procedure for analysing a class of transient elastodynamic crack problems is presented. These problems model certain experimental situations which can be used to infer fracture toughness values for materials under stress-wave loadings. In particular, the present analysis provides exact expressions for the elastodynamic stress intensity factor at the tip of a long external crack in a striplike body whose lateral (upper and lower) boundaries are parallel to the crack line. Plane stress/strain conditions are assumed to prevail. In this class of problems, the crack may be situated asymmetrically with respect to the mid-strip line, and various dynamic loadings are considered, including crack face tractions and lateral face displacements. The loadings considered will have an arbitrary time dependence, but will be spatially uniform. The problem analysis is based on integral transforms and an asymptotic usage of the Wiener-Hopf technique. Two useful cases are considered in detail as examples; the symmetrically cracked strip with traction-free lateral boundaries under sudden crack face pressure, and the symmetrically cracked strip with suddenly displaced shear-free lateral boundaries.

Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

Elastodynamic crack problems in an orthotropic medium through complex variable approach

Complex variable technique is employed to solve the elastodynamic problems of (I) two equal collinear Griffith cracks, (II) an infinite row of parallel cracks and (III) a “grid” of cracks of equal arms, propagating with constant speed in a stressed orthotropic medium. Stress and displacement components are represented in terms of holomorphic functions defined in two appropriate complex domains. Stress intensity factors and some asymptotic results are represented and briefly discussed.

The Wiener-Hopf technique in elastodynamic crack problems with characteristic lengths in loading

This paper describes a solution procedure for analyzing the elastodynamic fields of a semi-infinite crack in an otherwise unbounded solid. The deformations are caused by a pair of concentrated normal impact forces which are symmetric with respect to the crack plane and attain certain fixed characteristic lengths. The crucial steps in the analysis are based on integral transforms and the direct application of the Wiener-Hopf technique as if there were no fixed characteristic length. Exact expressions, which are the sum of a number of finite range integrations, are obtained for the resulting mode I transient stress intensity factors as functions of time. For comparison to previous works we also consider the particular situation when the pair of concentrated forces located a distance away from the crack tip are acting on the crack faces. The formulation departs significantly from the superposition argument used in previous works. The present formulation allows the solution to be presented in a more straightforward manner, and has potential with regard to other elastodynamic crack problems. The numerical results are presented for various characteristic lengths. It is shown that the behaviors of stress intensity factors, for the cases where the concentrated forces are not acting on the crack faces, experience finite discontinuities when direct longitudinal and transverse waves emitted from the loading position arrive at the crack tip. These are quite different from those where the concentrated forces are acting on the crack faces.

Evaluation of a finite element based crack-closure method for calculating static and dynamic strain energy release rates

A finite element based crack-closure integral for determining strain energy release rates for two-dimensional elastostatic and elastodynamic crack problems in finite bodies is evaluated. Both the J- integral and crack-closure integral are used to obtain strain energy release rates from the finite solution. Comparison with existing solutions shows that both methods yield very good results. However, the crack-closure integral can directly yield strain energy release rates and stress intensity factors for separate fracture modes in mixed-mode problems, and is very convenient to use in propagation crack problems.

On the application of the optical method of caustics to the investigation of transient elastodynamic crack problems: Limitations of the classical interpretation

Several possible sources of inaccuracy that occur in the classical interpretation of caustics patterns generated during transient crack growth in elastic materials are examined using a ‘Bifocal Caustics’ set-up and a new full field optical technique called ‘Coherent Gradient Sensing’. During unsteady dynamic crack growth, strict K d I-dominance is generally absent, especially at times close to crack initiation and arrest, even in regions outside the crack-tip 3-D zone where plane stress conditions persist. In such cases a truly transient higher order expansion is found to be essential for correctly describing stress fields outside the 3-D zone.

On the path independent [formula omitted] integral in nonlinear and dynamic fracture mechanics

Recently the authors have derived new types of path independent integrals in which the theoretical limitations of the so-called J integral are overcome. For elastodynamic crack problems, a path independent integral J' (dynamic J integral) which has the physical meaning of energy release rate was derived. Later the J' integral was extended to a more general form of the path independent integral T∗ (generalized J integral) which is valid for any constitutive relation under quasi-static as well as dynamic conditions. This paper presents recent developments in a series of studies on the T∗ integral in nonlinear and dynamic fracture mechanics. The first study concerns the behavior of the T∗ integral in an elastoplastic dynamic fracture initiation test specimen. The T∗ integral exhibits excellent path independency despite the impact loadings and large plastic deformations. The second study reveals the invariance of the T∗ integral with respect to the shapes and sizes of the fracture process zone. The last study shows the methodology of a direct measurement of the T∗ integral using the method of caustics (shadow pattern method). These results serve to illustrate the validity of the T∗ integral as a unified crack-tip parameter for various types of fracture mechanics problems.

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.

Computation of dynamic stress intensity factors by the time domain boundary integral equation method-I. Analysis

Application of the direct time domain boundary integral equation method (BIEM) to the solution of a number of elastodynamic crack problems is presented. In this part I, we describe the analysis which includes the basic governing differential equations, their transformation to boundary integral equations and the numerical solution procedure of the discretized form of the BIE. In addition to the usual constant interpolation in space and time of tractions and displacements on the boundary, a new boundary element which incorporates quadratic variation in space and linear variation in time (of the traction and displacements) is developed. These two boundary elements are implemented in a computer code and are used in the examples described in part II. A consistent method of estimation of the dynamic stress intensity factors in conjunction with the two types of the elements mentioned is described. No special singular crack tip elements are used other than the quadratic isoparametric quarter-point boundary elements for modeling the crack tip singularities. Subsequent results show that the numerical solution is accurate and has good convergence properties with regard to mesh refinement and time step size.