Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity
Introduction
The elastic displacement, strain and stress fields in the vicinity of the tip of a plane crack with traction-free faces subjected to arbitrary loading have been obtained by the eigenfunction expansion method first introduced by Williams [1] and further extended by Sih and Liebowitz [2] using the complex function approach of Muskhelishvili [3]. The first (i.e. the singular) term of the crack tip asymptotic field is controlled by the stress intensity factor (SIF) which has been used for years as the single controlling parameter for the initiation and propagation of a crack in brittle materials. However, recent studies show higher order terms of the asymptotic field are of great relevance to predicting the constraint of elastic–plastic crack tip fields [4], [5], [6], [7], [8] and to interpreting the size effect of quasi-brittle materials [9], [10].
The use of SIF in examining the crack stability requires an accurate evaluation of the SIF for the structural geometry, loading and boundary conditions in question. Since the analytical solutions only exist for a few simple cases, numerical techniques play an important role in complex practical situations. As a major tool in the general field of stress analysis, the finite element (FE) method would at first sight appear to be an ideal method. Unfortunately, the FE solution converges very slowly if conventional elements that do not include stress singularities properly are used. Tong and Pian [11] have shown that, in general, the convergence rate for the FE method is dominated by the nature of the solution near the point of singularity. The regular high accuracy element using high order polynomials as interpolation functions cannot improve the rate of convergence. They also showed that the error from the elements immediately adjacent to the point of singularity is of the same order as that of the remainder of the elements. Therefore, the use of finer elements cannot improve the situation either. In order to improve the convergence rate of the FE solution, various attempts have been made to include the required crack tip singularity in the element formulation, as has been extensively reviewed by Gallagher [12], Fawkes et al. [13], Owen and Fawkes [14], and Liebowitz and coworkers [15], [16]. Recently, singular p-version FEs [17], enriched element-free Galerkin methods and extended FE methods that include the crack tip singular solution directly [18] and by the partition of unity method [19], respectively, have also been employed to determine the SIF. These methods match the exact crack tip field to the solution of the surrounding FEs.
A direct matching of the displacement or the stress field does not work because the crack tip field cannot be regarded solely as a displacement or a stress constraint. It is still not satisfactory to match an exact solution if area integrations within the region cannot be avoided. From this point of view, among the various elements available for plane crack problems, the optimum one seems to be the hybrid crack element (HCE) introduced by Tong et al. [20]. In its formulation, truncated asymptotic crack tip displacement and stress expansions are used within the HCE enclosing the crack tip, while interelement boundary displacements are made compatible with the surrounding regular elements. The HCE represents a crack by only one super-element which is connected compatibly with the surrounding elements. The SIF is obtained directly without the use of the J-integral, or other energy-related variables. Moreover, only numerical integrations along the boundary far from the singular crack tip are necessary, so that standard quadrature rules can be directly adopted. Other crack elements use the crack tip as one node, so that more complicated quadrature methods are necessary [14], [21]. Lin et al. [22] show that the accuracy of the SIF computed with HCE is insensitive to the size of the super-element. The above advantages of the HCE have been appreciated by many researchers. It has been extended to anisotropic materials by Tong [23] and Lin and Tong [24], to unidirectional composites by Khalil et al. [25], to interface cracks in isotropic and anisotropic bimaterials by Lin and Mar [26], and Lee and Gao [27], respectively, while Piltner [28], Ghosh and Mukhopadhyay [29] and Zhang and Katsube [30] have extended it to heterogeneous materials with voids and/or inclusions. However, doubts have been cast on the accuracy of the results by the investigations of Owen and Fawkes [14]. Also, Lee and Gao [27] used element stress parameters whose number violated the theory of hybrid element. Likewise, the published literature is full of ambiguities in the derivation of the simplified variational principle for formulating the HCE because it is based on the energy functional of the system, and/or does not consider properly the boundary conditions of the element. Nevertheless, as the advantages offered by this element are considerable, it will be extensively evaluated and improved in the present paper.
The importance of the higher order terms calls for highly accurate numerical methods for determining not only the SIF but also the coefficients of the higher order terms. Various methods have been applied in the past to compute the second, the so-called T-term for standard test specimens. Larsson and Carlsson [4] determined the T-term with the aid of two elastic FE solutions. The first was obtained from a boundary layer formulation; the second was obtained for the actual specimen geometry and loading configuration with the load level being taken to be the one which would yield the same value of KI at the crack tip as the imposed KI in the boundary layer problem. Leevers and Radon [31] used a variational method to incorporate directly the eigenfunctions of the Williams expansion. Kfouri [32] employed a FE elastic solution for a specimen under given loading conditions, and an auxiliary elastic solution which was constructed by superposing the point force solution to the FE result. This point force acts at the tip of a semi-infinite crack in the direction of the crack plane. The elastic T-term was then computed from a formula which relates T to the values of the J-integral evaluated from the FE solution and the auxiliary solution, respectively. Sham [33] developed a second order weight function based on a work-conjugate integral and evaluated it using the FE method. Recently, Fett [34], [35], [36] used the boundary collocation method (BCM) to calculate the second T-term. The methods introduced by Larsson and Carlsson [4], Kfouri [32] or Sham [33] are not straightforward to be extended to the evaluation of the terms of order higher than two. As Leevers and Radon [31] and Fett [34], [35], [36] used the truncated Williams expansion to model the whole specimen, a large number of terms is needed for obtaining satisfactory accuracy, which generally results in ill-conditioned algebraic equations, requiring special techniques for their solution. Fett [36] calculated the higher order terms of some typical specimens, however, it is not known whether the method introduced by Leevers and Radon [31] gives stable higher order terms. Since the BCM is suitable only for relatively simple geometry and/or loading conditions, it is valuable to develop or extend some universal FEs to predict stable higher order terms that are suitable for simple as well as complicated geometries and/or loading conditions.
In this study, the HCE originally introduced by Tong et al. [20] will be improved to have p-adaptivity and will be applied to calculate not only the SIF but also the coefficients relevant to higher order terms of the crack tip asymptotic field. In order to clear the many ambiguities in published literature, the simplified variational principle for formulating the HCE will be derived from modified principles with relaxed continuity requirements [37]. An n-node polygonal element with p-adaptivity will be formulated for determining the SIF, the T-term, as well as the coefficients of the terms higher than order two. The parameter matching condition will be discussed according to the theory of hybrid element [38], [39]. Several confirmative numerical tests on the parameter matching condition, the Gauss integration order, convergence, size sensitivity, and the application of the HCE to short cracks, will be presented and discussed. The known results of the SIF, the second T-term and terms higher than order two will be used to check the accuracy of the element.
Section snippets
Variational fundamentals
The discussion to follow is restricted to two-dimensional linear elasticity. The simplified variational principle for formulating the HCE can be derived either from the Hellinger–Reissner principle [14], [20], [24] or from the modified complementary principle [39]. We will show that it can also be derived from the modified potential principle. Our discussion will start from modified functionals of an individual element with relaxed continuity requirements. It will clear ambiguities in the
Element formulation
The functional (10) can be written in matrix form asIn the formulation of elements using the simplified functional (10) or (15), the assumed element displacement and stress fields, ui and σij, should meet the equations of equilibrium (1) and the stress–displacement relations (2). Thus the internal displacements and stresses may be written in a matrix form asMaking use of , , we have the boundary tractionsin whichThe interelement
Hybrid crack element with p-adaptivity
Note that in the formulation of the HCE a compatible displacement field is assumed only along the element boundaries instead of the entire element, thus an n-node polygonal element with p-adaptivity can be easily formulated.
The available crack tip asymptotic field is used directly in assuming the element displacement and boundary tractions (16) and (18). If a crack with traction-free faces lies on the negative x-axis, and the polar coordinates centred at the crack tip are designated r and ϑ (ϑ
Numerical tests
Numerical tests on the application of the HCE for mode I cracks are presented and discussed in this section. As only terms relevant to an1 in , , , , are retained, these coefficients will henceforth be simply designated an for brevity. The number of element stress parameters, nβ, is obviously equal to the number of terms retained in the crack tip asymptotic field.
A single edge cracked panel (SEC) of width b and height 2h containing a crack of length c in the middle of one side, subjected to a
Conclusions
This study has shown that the HCE introduced by Tong, Pian and Lasry can be improved to be an n-node polygonal element with p-adaptivity and made into a powerful and convenient element not only for determining the widely used SIF in plane crack problems but also the coefficients of higher order terms in the crack tip asymptotic field. The accuracy of the element is not diminished even when very short cracks are considered. Numerical tests show high convergence rate of the computed higher order
Acknowledgements
Financial support from the EPSRC under grant number GR/M 78106 is gratefully acknowledged. We would also like to thank the reviewers for bringing the references [7], [8] and [35], [36] to our attention. We are particularly grateful to the anonymous reviewer for providing the BCM results that we have cited in the text.
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