Singular Crack Tip Elements Research Articles

Mixed mode BEM analysis of multiple curvilinear cracks in the general anisotropic solids by the crack tip singular element

We consider multiple curvilinear cracks in the two-dimensional general anisotropic solids and establish a computationally effective technique to determine the stress intensity factors accurately. Each curvilinear crack is represented by a collection of straight crack elements and the crack opening displacement (COD) by the continuous distribution of the dislocation dipoles. The crack element located next to the crack tip is called the crack tip singular element (CTSE), where the known r crack tip opening displacement and the 1/ r stress singularity is mathematically built in the interpolation of the COD using the Chebyshev polynomials. The regular crack elements away from the crack tip use the quadratic polynomial interpolation for the CODs. Simple analytical formulas for the displacement, traction, and the stress intensity factor (SIF) contributions for the CTSE are developed in terms of its own COD coefficients. Since the SIFs are obtained during the main processing no post processing is required; a distinct advantage over the powerful quarter-point element. Numerical results are given for several multiple curvilinear crack problems to demonstrate the accuracy and simplicity of the technique.

Combination of fracture and damage mechanics for numerical failure analysis

A new approach for the analysis of crack propagation in brittle materials is proposed, which is based on a combination of fracture mechanics and continuum damage mechanics within the context of the finite element method. The approach combines the accuracy of singular crack-tip elements from fracture mechanics theories with the flexibility of crack representation by softening zones in damage mechanics formulations. A super element is constructed in which the typical elements are joined together. The crack propagation is decided on either of two fracture criteria; one criterion is based on the energy release rate or the J-integral, the other on the largest principal stress in the crack-tip region. Contrary to many damage mechanics methods, the combined fracture⧹damage approach is not sensitive to variations in the finite element division. Applications to situations of mixed-mode crack propagation in both two- and three-dimensional problems reveal that the calculated crack paths are independent of the element size and the element orientation and are accurate within one element from the theoretical (curvilinear) crack paths.

Analytical formulas for a 2-D crack tip singular boundary element for rectilinear cracks and crack growth analysis

Simple analytical formulas for the displacement, traction, and stress intensity factor for a 2-D crack tip singular boundary element are developed using the continuous distribution of dislocation dipoles and Chebyshev polynomials. The known r crack tip opening displacement and the 1/ r stress singularity at the crack tip is mathematically built into the formulas. In the boundary element implementation the crack tip singular element is placed locally at each crack tip on top of the ordinary non-singular crack elements that cover the entire crack surface. Within the constraint of the rectilinear approximation, any curvilinear cracks can be modeled, including center and edge cracks. Although the quarter-point element is accurate and easy to use, it does not provide the analytical formula for the stress intensity factor which is the key feature of the proposed method. The performance of the crack tip singular element is compared against other crack elements that incorporate the crack tip singularity analytically. A numerical result is given for a crack growth problem where repeated update of the crack shape and calculation of the stress intensity factor are required as the crack grows in a curvilinear path.

Prediction of bimaterial thermal cracking by the boundary element method

A boundary element procedure is developed concerning the prediction of the quasistatic crack growth in uniformly heated bimaterials. This procedure assumes the existence of an initial small crack in one of the two phases, and further cracking progress from this point due to thermal loading. The resulting mixed boundary value problem is solved by applying an incremental boundary-only method in conjunction with the multidomain technique. Fracture characterizing parameters are evaluated utilizing special crack tip singular elements and appropriate formulas. The crack path is predicted using the strain energy release rate criterion, and the mesh is updated at the end of each increment. The presented results are in good agreement with previously reported experimental results and those obtained by the finite element method. Various numerical studies were conducted and interpreted concerning crack-path dependence on individual material property mismatch. © 1998 John Wiley & Sons, Ltd.

Energy-Based Automatic Mixed-Mode Crack-Propagation Modeling

Based on an energy principle and a virtual crack extension technique, a practical finite-element modeling approach for the automatic mixed-mode two-dimensional (2D) linear elastic fracture mechanics (LEFM crack-propagation analysis has been developed. By decomposing the displacement field obtained from a conventional finite-element analysis into symmetric and antisymmetric displacement fields with respect to the crack, the Mode-I and Mode-II energy release rates can be determined using a generalization of the stiffness derivative method, and the corresponding stress intensity factors can then be calculated. The load at which a crack propagates and the propagation direction can be determined using one of the well-established mixed-mode crack-propagation criteria. Unlike the previously developed approaches to crack propagation, this approach does not require the use of symmetric crack-tip mesh, crack-tip singular elements, which greatly simplifies the subsequent remeshing process to allow for crack propagation. The automatic crack-propagation process is achieved by using a simple and efficient local 2D remeshing algorithm that can be easily programmed. Convergence studies, mesh sensitivity studies, and the practical use of the new approach are presented through various example problems.

Higher-order singular isoparametric elements for crack problems

Standard isoparametric finite elements can be used as singular crack-tip elements in fracture mechanical computations by placing the middle nodes in the neighbourhood of the crack tip. Such elements have already been applied to several plane and three-dimensional problems, so this method can be considered as commonly well accepted. The paper shows that singularity of order r(1 − m)/m is obtained with an nth-order isoparametric element.

Finite element error estimation for crack tip singular elements

A stress-based finite element error estimator is proposed for fracture mechanics analysis involving singular isoparametric crack tip finite elements. In the crack tip region the discretization error is estimated by the L 2 integral norm of the difference between the finite element effective stress function and the known asymptotic effective stress function derived from the Westergaard solution. In the remaining portion of the domain, the L 2 element error norm is based on the difference between the discontinuous finite element effective stress function and the C 0 continuous “smoothed” effective stress function. Numerical convergence studies are presented for the classical edge crack Mode I problem with six different singular and nonsingular element formulations at the crack tip region. The analysis results showed steady monotonic convergence of the L 2 norm error estimator for the critical six-noded triangular and eight-noded quadrilateral singular crack tip elements, whereas poor convergence behaviour is seen in the Zienkiewicz and Zhu stress-based energy norm error estimator applied to the same critical crack tip element. However, monotonic convergence was seen in all global L 2 and energy error norms for all mesh cases.

A direct method for calculating the stress intensity factor in BEM

The proposed algorithm employs singular crack tip elements in which the stress intensity factor appears as a degree of freedom. The additional degrees of freedom are compensated by constraint conditions which originate from imposing continuity across elements and a contour integration formula. The two benchmark problems indicate the proposed algorithm can accurately predict the stress intensity factor and the distribution of the primary and secondary variables in fracture problems.

Modelling of dynamical crack propagation using time-domain boundary integral equations

We study dynamic antiplane cracks in the time domain by the boundary integral equation method (BIEM) based on the integral equation for displacement discontinuity (or crack opening displacement, COD) as a function of stress on the crack. This displacement discontinuity formulation presents the advantage, with respect to methods developed by Das and others in seismology, that it has to be solved only inside the crack. This BIEM is, however, difficult to implement numerically because of the hypersingularity of the kernel of the integral equation. Hence it is rewritten into a weakly singular form using a regularization technique proposed by Bonnet. The first step, following a method due to Sladek and Sladek, consists in converting the hypersingular integral equation for the displacement discontinuity into an integral equation for the displacement discontinuity and its tangential derivatives (dislocation density distribution); the latter involves a Cauchy type singular kernel. The second step is based on the observation that the hypersingularity is related to the static component of the kernel; the static singularity is then isolated and can be expressed in terms of weakly singular integrals using a result due to Bonnet. Although numerical applications discussed in this paper are all for the antiplane problem, the technique can be applied as well to in-plane crack dynamics. The BIEM is implemented numerically using continuous linear space-time base functions to model the COD on the crack. In the present scheme the COD gradient interpolation is discontinuous at the element nodes while the integral equations are collocated at the element midpoints. This leads to an overdetermined discrete problem which is solved by standard least-squares methods. We use the dynamic BIEM to study a set of problems that appear in earthquake source dynamics, including the spontaneous dynamic crack propagation for a very simple rupture criterion. The numerical results compare favorably with the few exact solutions that are available. Then we demonstrate that difficulties experienced with finite difference simulations of spontaneous crack dynamics can be removed with the use of BIEM. The results are improved by the use of singular crack tip elements.

Determination of characterizing parameters for bimaterial interface cracks using the boundary element method

The complex stress intensity factor, K = K I + iK II , which is used to characterize the near tip fields of a bimaterial interface crack may also be written as K = K 0e iψ , and it has been shown that K 0 is directly related to the strain energy release rate. An efficient means of obtaining this quantity K 0 using the boundary element method (BEM) has been discussed in a recent paper by the present authors. However, some recent studies have suggested that fracture of such interfacial bonds is characterized by critical values of K 0 which may vary with the phase angle ψ. This note describes how ψ may also be determined using the BEM with quarter-point traction singular crack-tip elements. Results of some numerical experiments are presented to demonstrate the accuracy of the approach.

Boundary integral equation fracture mechanics analysis of plane orthotropic bodies

In this paper, a quick and efficient means of determining stress intensity factors, KI and KII, for cracks in generally orthotropic elastic bodies is presented using the numerical boundary integral equation (BIE) method. It is based on the use of quarter-point singular crack-tip elements in the quadratic isoparametric element formulation, similar to those commonly employed in the BIE fracture mechanics studies in isotropic elasticity. Analytical expressions which enable KIand KII to be obtained directly from the BIE computed crack-tip nodal traction, or from the computed nodal displacements, of these elements are derived. Numerical results for a number of test problems are compared with those established in the literature. They are accurate even when only a very modest number of boundary elements are used.

Mixed mode crack propagation in quasi-brittle materials

Crack propagation in rectangular blocks of mortar containing a central notch and subjected to uniaxial compression were studied. Four different notch orientation angles (18°, 36°, 54° and 72°) with respect to the loading direction were used. Holographic Interferometry was used to observe crack initiation and propagation. It was possible to detect crack extensions during the experiments, hence the load vs crack extension curve for each notch orientation angles could be obtained. A separate Holographic Interferometry method was used to measure crack opening and sliding displacements by making four independent observations of holographic fringes from the same holographic plate. Crack initiation theories were employed to study their relative merits for predicting crack initiation angles and loads. A Finite Element Method (FEM) using quarter point singular crack tip element was used to calculate Stress Intensity Factors (SIF) and crack surface displacements for different inclinations and extensions of the propagating cracks. It was found that while crack initiation was predicted well by some of the theories, it was necessary to account for the traction forces on the crack surface before any propagation criterion could be identified. Opening and sliding of the crack faces determined by Holographic Interferometry (HI) and clip gage measurements were used to find the normal and shear traction applied to the propagating crack faces. The SIFs for the traction free cracks were corrected by taking into account normal and shear tractions along the propagating crack. It was concluded that the Maximum Hoop Stress Criterion was reasonable and K I stress intensity factor at the tip of the propagating crack was dominant in the failure mechanism.

Axisymmetric boundary integral equation analysis of interface cracks between dissimilar materials

The numerical boundary integral equation (BIE) method with quadratic quarter-point crack-tip singular elements is used to analyse interface cracks between dissimilar material in axisymmetry. Such crack problems present modelling difficulties using conventional procedures for obtaining the stress intensity factors. This is because of the oscillatorily singular nature of the stresses in the vicinity of the bimaterial interface crack-tip. Analytical expressions for the direct evaluation of the fracture characterising parameters from the BIE numerical results of displacements or tractions are derived. Three different crack problems are investigated, two of which have known solutions in the literature. Excellent agreement between the BIE results and these other established solutions are obtained even with relatively coarse mesh discretisations. The present study illustrates the ease with which the BIE method may be used in the fracture analysis of both straight and curved binaterial interface cracks.

Determining stress intensity factors from smoothing finite-element representation of photomechanical data

The ability of the smoothing finite-element method to determine stress intensity factors from very little photomechanical data is demonstrated. Measured photomechanical data from interior regions behind the crack-tip are combined with special smooth crack-tip singular elements to extract the stress intensity factors. The method reduces the experimental scatter and is unaffected by rigid-body motion. The accuracy and practicality of the method are illustrated by determining the stress-intensity factor of a compact-tension specimen using only two elements and 16 moiré-measured displacement values.

Three-dimensional stress intensity factor analysis of a surface crack in a high-speed bearing

The boundary element method is applied to calculate the stress intensity factors of a surface crack in the rotating inner raceway of a high-speed roller bearing. The three-dimensional model consists of an axially stressed surface cracked plate subjected to a moving Hertzian contact loading. A multi-domain formulation and singular crack-tip elements were employed to calculate the stress intensity factors accurately and efficiently for a wide range of configuration parameters. The results can provide the basis for crack growth calculations and fatigue life predictions of high performance rolling element bearings that are used in aircraft engines.

Use of analytically decoupled near-tip displacement solutions for calculating the decoupled weight functions of mixed fracture modes

A new approach for the efficient finite element evaluation of decoupled weight functions is presented in this paper for mixed-mode 2-D cracks. This new finite element approach uses the analytically decoupled near-tip displacement solutions (ADNDS), which are incorporated into a single colinear virtual crack extension (VCE) technique with singular crack-tip elements in the immediate crack-tip vicinity. The decoupled near-tip displacement fields are achieved by applying the analytical near-tip displacement solutions for an infinite medium to the singular elements with either K I = 1.0 and K II = 0 or K I = 0 and K II = 1.0 for the efficient finite element evaluation of Mode I and Mode II decoupled weight functions ( h I(II ) respectively. The validity of using the ADNDS approach for efficient calculation of the decoupled weight functions at all locations within a flawed Isotropic structure of interest is assessed by direct comparison with the earlier weight function results. These prior results on decoupled weight functions of a mixed-mode isotropic crack are obtained with the symmetric mesh approach in the crack-tip vicinity for the same crack geometry under identical constraint conditions. A remarkable agreement in the computed weight function vectors has been found for the ASTM three-point bending specimens of different crack lengths with two different approaches (symmetric mesh approach versus ADNDS approach). However, elimination of the symmetric mesh requirement in the crack-tip vicinity of the new analytical displacement approach can facilitate further extension of the explicit weight function evaluation to the anisotropic crack.

Using generalized variational principles to resolve the St. Venant's torsional bar with a crack

According to generalized variational principles suitable for linear elastic incompatible displacement elements given by Professor Chien Weizang, using crack tip singular element and isoparametric surrounding element given by the author of this paper, we will study the St. Venant's torsional bar with a radial vertical crack and compare the present computed results with the results of reference [2]. The present computed results show that, using the method provided in this paper, satisfactory convergent solution can be obtained under lower degree of freedom.

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.

Modelling the HRR-singularity with akin and conventional singular elements

A single edge notched plate of elastic-plastic material has been analysed using different finite element meshes and different singular crack tip element. Especially the Akin element was of interest since it gives a possibility to model the HRR-singularity.

Crack growth initiation in elastic-plastic materials

The problem of initiation of crack growth in an elastic-plastic material under idealized conditions of plane stress or plane strain was studied. The stress analysis of the cracked plate was performed by using an elastic-plastic finite element computer program based on the J2 flow theory of plasticity and a singular crack tip element. A method for determining the critical quantities at the moment of crack growth was presented. The strain energy density failure criterion was used according to which the crack starts to grow when material elements ahead of the crack absorb a critical amount of stored strain energy density which remains after subtracting the amount of irreversible energy dissipated by permanent deformation. The critical stress and the plastic zone at crack initiation were determined. No restrictions regarding the amount of plastic deformation at the crack tip and the form of the stress-strain diagram of the material in tension were imposed in the analysis. Results were given for a center cracked plate specimen for various crack lengths and ultimate stresses of the material in tension. The dependence of the critical stress on these factors and on the idealized conditions of plane stress or plane strain was established.