Two-dimensional stress intensity factor computations using the boundary element method

A multidomain boundary element formulation for the analysis of general two-dimensional plane strain/stress crack problems is presented. The numerical results were accurate and efficient. The analyses were performed using traction singular quater-point boundary elements on each side of the crack tip(s) with and without transition elements. Traction singular quarter-point boundary elements contain the correct √r displacement and 1/√r traction variations at the crack tip. Transition elements are appended to the traction singular elements to model the √r displacement variation. The 1/√r traction singularity is not represented with these elements. Current research studies for the crack propagation analysis of quasi-static and fatigue fracture problems are discussed.     

Mathematical simulation of singularities for isoparametric elements of arbitrary orders

For the general quadrilateral isoparametric elements with 4(n?1) nodes (i.e. n nodes along each side), it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by a general but simple rule. This amounts to collapsing the quadrilateral elements into triangular elements around the crack tip and placing the (n?2) mid-side nodes at locations γ²/(n?1)² (γ=1,2,…, n?2) times the length of side emanating from the crack tip. These locations are measured from the crack tip. The known results for n = 3 and n = 4 are thus obtained as special cases.     

On quadratic isoparametric transition elements for a crack normal to a bimaterial interface

The quadratic isoparametric crack‐tip elements proposed by R. E. Abdi and G. Valentin (Computers and Structures 33, 241–248) are reconsidered and a simpler method for calculating the optimal position of the side nodes proposed. Quadratic isoparametric transition elements for an r^λ−1 (0<λ<1) strain singularity are formulated. The effects of these transition elements on the accuracy of the calculated stress intensity factors are shown numerically for a crack normal to and terminating at a bimaterial interface. Finally, layered transition elements are formulated for this case and their effects studied numerically. Copyright © 1999 John Wiley & Sons, Ltd.     

The generation of elements with singularities

A procedure is given for generating two-dimensional conforming singularity elements from standard conforming elements. Three new elements with 0(r-^p) derivative singularities are introduced. The technique is based on the use of elements defined by numerically integrated shape functions. Special quadrature rules are suggested for triangular elements. The degeneration of these elements to crack tip elements and the direct evaluation near the singularity of element quantities, such as the stress intensity factors, is discussed.     

A reappraisal of transition elements in linear elastic fracture mechanics

This article offers a detailed comparison of the transition elements described by P.P. Lynn and A.R. Ingraffea [International Journal for Numerical Methods in Engineering 12,1031–1036] and C. Manu[Engineering Fracture Mechanics 24,509–512]. The source of a numerical phenomenon in using Manu's transitionelement (TE) is explained. The effect of eight-noded TEs with differentquarter-point elements (QPE) on the calculated stress intensity factors (SIFs) isinvestigated. Strain at the crack tip is shown to be singular for any ray emanating from the crack tip within an eight-noded TE, but strain has bothr ^–1/2andr ^–1singularities, withr ^–1/2dominating for large TEs. Semi-transition elements (STEs) are defined and shown to have a marginal effect on the calculated SIFs. Nine-nodedtransition elements are formulated whose strain singularity is shown to be the same as that of eight-noded TEs. Then the effect of eight-noded and nine-noded TEs with collapsed triangular QPEs, and rectangular and nonrectangular quadrilateral eight-noded and nine-noded QPEs, is studied, and nine-noded TEs are shown to behave exactly like eight-noded TEs with rectangular eight-noded and nine-noded QPEs and to behave almost the same with other QPEs. The layered transition elements proposed by V. Murti and S.Valliapan [Engineering Fracture Mechanics 25, 237–258] areformulated correctly. The effect of layered transition elements is shown by two numerical examples.     

Analysis of power type singularities using finite elements

A finite element formulation is described for problems with solution functions known to have local rλ variation (s), 0<λ<1, and thus singular gradients. Special 3-node triangular elements encircle the singularity and focus to share a common node at the singular point. The shape function of each triangle has the appropriate r λ mode and a smooth angular mode expressed in element natural co-ordinates. As with standard elements, the unknowns are the nodal values of the function. Even if the precise angular form of the asymptotic solution is known, the formulation makes no attempt to embed it, but instead piecewise approximates it. This allows assembly of the element coefficient matrix using standard procedures without nodeless variables and bandwidth complications. The conditions of continuity, low order solution capability, and accurate numerical integration of the singularity element are discussed with a view towards establishing the general range of applicability of the formulation. Numerical applications to the elastic fracture mechanics problems of composite bondline cracking and crack branching are discussed.     

Numerical evaluation of stress intensity factor and J integral in three-point bend specimen

The present paper attempts to evaluate the fracture mechanics parameters, the stress intensity factor (K) and Rice's energy integral (J) in plane strain conditions for three-point bend specimens. Both the parameters have been evaluated by the FEM using higher order isoparametric elements (i.e. quadratic elements). The crack tip elastic singularity (1/√r) has been taken into account by the use of the special crack tip elements of degenerate triangular element type as well as the fine eight-noded isoparametric plane elements. The stress distribution has been compared with the Westergaard solution in the vicinity of the crack. The K and J values have also been-compared with the theoretical results.     

An assessment of crack tip singularity models for use with isoparametric elements

The use of finite element methods to analyse fracture problems is complicated by the stress field singularity which exists at the crack tip. The two most successful methods of approach would appear to be the so-called energy technique and the singularity function formulation. The necessity for extremely fine meshes in the crack tip region can be overcome by the use of special elements which incorporate the required stress singularity in their formulation. The aim of this paper is to develop various promising singularity function elements and assess their performance in the solution of standard test problems. These elements are based on the eight node parabolic isoparametric element; this being the most popular element in general use. Such crack tip elements may be readily incorporated into a mesh of standard isoparametric elements permitting numerical fracture studies to be undertaken without extensive mesh regeneration or refinement. In particular elements based on the use of distorted shape functions, standard shape functions, analytic solutions, a superposition process and a hybrid technique are considered. Test problems of both single and combined mode fracture are employed in the assessment of each model.It is also demonstrated that the hybrid element is a special case of the boundary integral method, and suggestions are made for possible future development.     

Singular boundary elements for the analysis of cracks in plane strain

Straight and curved cracks are modelled by direct formulation boundary elements, of geometry defined by Hermitian cubic shape functions. Displacement and traction are interpolated by the Hermitian functions, supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance r from the crack tip, and subdominant modes in which it is proportional to r^1·5. The singular functions extend over many boundary elements on each crack face. A nodal collocation scheme is used, in which additional boundary integral equations are obtained by differentiation of the equation obtained from Betti's theorem. The hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack. Examples are shown of the analysis of buried and edge cracks, to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements.     

A multilayer hybrid-stress finite element analysis of a through-thickness edge crack in A ± 45° laminate

A solution is given for the three-dimensional stress field near a through-thickness edge crack in a thin ± 45° laminate having elastic ply moduli typical of graphite/epoxy. The stress distribution was obtained by a three-dimensional multilayer finite element analysis based on the hybrid stress model, formulated through the minimum complementary energy principle. The results indicate that the in-plane stresses of each individual ply follow the classical $\text{1}\text{√r}$ stress singularity, but that the shape of isostress contours in the crack tip region is strongly distorted from predictions based on two-dimensional anisotropic fracture mechanics theory. The interlaminar shear stresses increase rapidly as the crack tip is approached, but are restricted to a local region around the crack tip and flanks. The interlaminar normal stress is assumed to be negligible in the formulation of the analysis.     

On the computation of two-dimensional stress intensity factors using the boundary element method

General two-dimensional linear elastic fracture problems are investigated using the boundary element method. The √r displacement and 1/√r traction behaviour near a crack tip are incorporated in special crack elements. Stress intensity factors of both modes I and II are obtained directly from crack-tip nodal values for a variety of crack problems, including straight and curved cracks in finite and infinite bodies. A multidomain approach is adopted to treat cracks in an infinite body. The body is subdivided into two regions: an infinite part with a finite hole and a finite inclusion. Numerical results, compared with exact solution whenever possible, are accurate even with a coarse discretization.     

Analysis of three-dimensional interface cracks using enriched finite elements

Many important interface crack problems are inherently three-dimensional in nature, e.g., debonding of laminated structures at corners and holes. In an effort to accurately analyze three-dimensional interface fracture problems, an efficient computational technique was developed that utilizes enriched crack tip elements containing the correct interface crack tip asymptotic behavior. In the enriched element formulation, the stress intensity factors K _I, K _II, and K _III are treated as additional degrees of freedom and are obtained directly during the finite element solution phase. In this study, the results that should be of greatest interest are obtained for semi-circular surface and quarter-circular corner cracks. Solutions are generated for uniform remote tension and uniform thermal loading, over a wide range of bimaterial combinations. Of particular interest are the free surface effects, and the influence of Dundurs’ material parameters on the strain energy release rate magnitudes and corresponding phase angles.     

Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

On the use of isoparametric finite elements in linear fracture mechanics

Quadratic isoparametric elements which embody the inverse square root singularity are used in the calculation of stress intensity factors of elastic fracture mechanics. Examples of the plane eight noded isoparametric element show that it has the same singularity as other special crack tip elements, and still includes the constant strain and rigid body motion modes. Application to three-dimensional analysis is also explored. Stress intensity factors are calculated for mechanical and thermal loads for a number of plane strain and three-dimensional problems.     

On J-integral and potential energy variation

The divergence theorem has been used in a region containing the crack tip to derive the J-integral from the potential energy variation in most fracture mechanics books. Such a derivation is flawed because of the crack tip stress singularity. The present study describes a rigorous and straightforward derivation of the J-integral from the potential energy variation with crack extension by carefully addressing the effect of the crack tip singularity.     

Transition elements matching the near crack-tip strain field

New transition elements used with the collapsed triangular singular elements are constructed by using the improved isoparametric transformations. Without 1/r singular terms, the new transition elements' strain fields contain only % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+% cadaGcaaqcaawaaiaadkhaaKqaGfqaaaaa!3B47!\[1/\sqrt r \] singularities and match the Williams solution quite well near the crack tip. It is convenient for the new transition elements to be constructed and introduced in the general purpose finite element programs by adding some modifications. Numerical results show that the transition elements possess good properties and are worth being applied to linear fracture computations.     

The mechanics of self-similar and self-affine fractal cracks

In this paper we study the mechanical attributes of the fractal nature of fracture surfaces. The structure of stress and strain singularity at the tip of a fractal crack, which can be self-similar or self-affine, is studied. The three classical modes of fracture and the fourth mode of fracture are discussed for fractal cracks in two-dimensional and three- dimensional solid bodies. It is discovered that there are six modes of fracture in fractal fracture mechanics. The J-integral is shown to be path-dependent. It is explained that the proposed modified J-integrals in the literature that are argued to be path-independent are only locally path-independent and have no physical meaning. It is conjectured that a fractal J-integral should be the rate of potential energy release per unit of a fractal measure of crack growth. The powers of stress and strain singularities at the tip of a fractal crack in a strain-hardening material are calculated. It is shown that stresses and strains have weaker singularities at the tip of a fractal crack than they do at the tip of a smooth crack.     

Variable power singular interface elements for a crack normal to the bimaterial interface

Zero thickness crack tip interface elements for a crack normal to the interface between two materials are presented. The elements are shown to have the desired r^λ−1 (0 < λ < 1) singularity in the stress field at the crack tip and are compatible with other singular elements. The stiffness matrices of the quadratic and cubic interface element are derived. Numerical examples are given to demonstrate the applicability of the proposed interface elements for a crack perpendicular to the bimaterial interface.     

A FRACTURE CRITERION FOR CRACKS UNDER MIXED-MODE LOADING

Abstract A fracture criterion is proposed, based on maximum energy release rates at the tips of short kinks when the main cracks are subjected to mixed mode loading. The criterion differs from existing energy based criteria in that the fracture toughness, g_c, is not independent of the stress mode prevailing in the region of the tip of the kink but is a function of the ratio of the mode II to mode I stress intensity factors at the tip of the kink, i.e., g_c is determined directionally by an elliptical region with major and minor axes equal to the fracture resistances of the material, K_Ir and K_IIr, for pure mode I and pure mode II, respectively. Points inside the elliptical region are considered safe. When K_IIr is equal to K_Ir the ellipse degenerates into a circle and the fracture criterion reverts to the existing familiar maximum energy release rate criterion based on a single value of the fracture toughness, irrespective of the active mode prevailing in the region at the tip of the kink. In this case, under pure shear (mode II) applied load, K_II, the angle of inclination of the fracture crack extension to the main crack, α, is in the region of ?76°, in general agreement with previous well established results. However, when the ratio r (=K_IIrK_Ir) is less than r′ (=0.82, approximately) a different pattern emerges and, in particular, under pure mode II load, the crack advance is co-planar with the main crack, i.e., in mode II. A lower transition value r″ (=0.582, approximately) was also detected under pure mode I applied load. Thus for values of r≥r″, the crack extension is in pure mode I and is co-planar with the main crack but when r < r″, the crack branches out at an angle (which can be positive or negative) in mixed modes I/II crack extension. Some implications of these results are discussed.