Determination of coefficients of crack tip asymptotic fields using the scaled boundary finite element method

Coefficients of the Williams expansion of the linear elastic crack tip asymptotic field can only be evaluated analytically for a few simple cases. Numerical solution is necessary in the general case, and the presence of the singular term presents numerical difficulties. The scaled boundary finite element method is a new semi-analytical approach to computational mechanics developed by Wolf and Song. This paper shows that when the scaling centre is located at the crack tip, the scaled boundary finite element solution converges to the Williams expansion. Consequently the coefficients of the Williams expansion, including the stress intensity factor and the T-stress, can be determined directly without further processing. The technique is applied to several problems for which coefficients of the Williams expansion are available, and close agreement with existing results is obtained with very few degrees of freedom.     

Coefficients of the crack tip asymptotic field for wedge splitting specimens

The stress intensity factor (SIF) and the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are directly computed using a hybrid crack element. Accurate analytical expressions for the first five terms are obtained by fitting the computed data. Numerical results show that the coefficients of terms higher than three are negligibly small, this may explain that the wedge splitting specimen is more stable than other geometries. The first five terms are not sensitive to support conditions. However, for short cracks coefficients of terms, except the SIF, are quite sensitive to the loading arrangement even when the loads are statically equivalent.     

Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field

An improvement to the extended finite element method (XFEM) and generalised finite element method (GFEM) is introduced. It enriches the finite element approximation of the crack tip node as well as its surrounding nodes with not only the first term but also the higher order terms of the linear elastic crack tip asymptotic field using a partition of unity method (PUM). Numerical results show that together with a reduced quadrature rule to the enriched elements, this approach predicts accurate stress intensity factors (SIFs) directly (i.e. without extra post‐processing) after constraining the enriched nodes properly. However, it does not predict accurately the coefficients of the higher order terms. For that a hybrid crack element (HCE) is introduced which is powerful and convenient not only for directly determining the SIF but also the coefficients of higher order terms in the plane linear elastic crack tip asymptotic field. Finally, the general expressions for the coefficients of the second to fifth terms of the linear elastic crack tip asymptotic field of three‐point bend single edge notched beams (TPBs) with span to depth ratios widely used in testing are extended to very deep cracks with the use of the HCE.     

An over‐deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis

An over‐deterministic method has been employed for calculating the stress intensity factors (SIFs) as well as the coefficients of the higher‐order terms in the Williams series expansions in cracked bodies, using the conventional finite element analysis. For a large number of nodes around the crack tip, an over‐determined set of simultaneous linear equations is obtained, and using the fundamental concepts of the least‐squares method, the coefficients of the Williams expansion can be calculated for pure mode I, pure mode II and mixed mode I/II conditions. A convergence study has been conducted to examine the effects of the number of nodes used, the number of terms in Williams expansion and the distance of the selected nodes from the crack tip, on the accuracy of the results. It is shown that the simple method presented in this paper, yields accurate results even for coarse finite element meshes or in the absence of singular elements. The accuracy of SIFs and the coefficients of higher‐order terms are validated by using the available results in the literature.     

Higher order terms of the crack tip asymptotic field for a wedge-splitting specimen

The coefficients of the crack tip asymptotic field of a typical wedge-splitting specimen are computed using a hybrid crack element (HCE), which has the potential to directly calculate not only the stress intensity factor (SIF) but also the coefficients of the higher order terms of the crack tip asymptotic field. The approximate closed-form expression for SIF proposed by Guinea et al. (1996) is calibrated by the results of the HCE. Approximate expressions for the second and third order terms for the wedge-splitting specimen are obtained by fitting the computed data. Numerical results show that the coefficients for terms higher than three are negligibly small, thus the wedge-splitting specimen is more stable than other geometries.     

Higher order terms of the crack tip asymptotic field for a notched three-point bend beam

The coefficients of the first five terms of the crack tip asymptotic field of three-point bend single edge notched beams (TPBs) with span to depth ratios widely used in testing are computed using a hybrid crack element (HCE), which has the potential to directly calculate not only the stress intensity factor (SIF) but also the coefficients of the higher order terms of the crack tip asymptotic field. The general approximate closed-form expression for SIF proposed by Guinea et al. (1998) and the available numerical results for the second T-term are calibrated by the results of the HCE. Approximate analytical expressions for the second, third, fourth and fifth terms for a TPB with a span to depth ratio of 4 and for a single edge notched beam subjected to pure bending are obtained by fitting the computed data. These approximations are then used to predict the general expressions for coefficients of the higher order terms of a TPB with arbitrary span to depth ratio . The accuracy of these general expressions is studied for TPBs with =6, 8 and 12.     

Coefficients of the crack tip asymptotic field for a standard compact tension specimen

The coefficients of the crack tip asymptotic field of a standard compact tension (CT) specimen are computed using a hybrid crack element (HCE). It allows the direct calculation (without post-processing) of not only the stress intensity factor (SIF) but also the coefficients of higher order terms of the crack tip asymptotic field. Approximate closed-form expressions for the first five terms for the CT specimen that are accurate for shallow to very deep cracks are obtained by fitting the computed data. The SIF formula proposed by Brown and Srawley (1966) is shown to be accurate when the crack length to depth ratio () ranges from 0.35 to 0.75. The formula proposed by Newman (1974) and Srawley (1976) is accurate for 0.15. However, the accuracy of available formulas for the second T-term in the literature is quite disappointing. Numerical results also show that, unlike the notched three-point bend beam and the wedge splitting specimen, the second T-term of the CT specimen is always positive.     

Oscillatory crack tip triangular elements for finite element analysis of interface cracks

In this paper a special crack tip element has been developed in which displacements and stresses have the same behaviour as those of bi‐material interface cracks with open tips. The element degenerates into a traditional triangular quarter point element in cases of homogeneous cracks. An isoparametric co‐ordinate system (ρ, t) is defined in this study, and numerical techniques using these co‐ordinates to evaluate Jacobian matrices, shape function derivatives, and element stiffness matrices are developed. Also, equations calculating the complex stress intensity factor using displacements are obtained in this study. Numerical results are in good agreement with known analytical solutions in two examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Direct determination of asymptotic structural postbuckling behaviour by the finite element method

Application of the finite element method to Koiter's asymptotic postbuckling theory often leads to numerical problems. Generally it is believed that these problems are due to locking of non-linear terms of different orders. A general method is given here that explains the reason for the numerical problems and eliminates these problems. The reason for the numerical problems is that the postbuckling stresses are inaccurately determined. By including a local stress contribution, the postbuckling stresses are calculated correctly. The present method gives smooth postbuckling stresses and shows a quick convergence of the postbuckling coefficients. © 1998 John Wiley & Sons, Ltd.     

Continuum shape sensitivity analysis of mixed-mode fracture using fractal finite element method

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, K_I and K_II, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or K_I and K_II. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.     

Numerical simulation of crack problems using triangular finite elements with embedded interfaces

The aim of this paper is to propose numerical aspects for the modeling of discrete cracks in quasi-brittle materials using triangular finite elements with an embedded interface based on the formulation in [Computational Mechanics 27 (2001) 463]. The kinematics of the discontinuous displacement field and the variational formulation applied to a body with an internal discontinuity is given. The discontinuity is modeled by additional global degrees of freedom and the continuity of the displacement jumps across the element boundaries is enforced. To show the performance of the model, a single element test and two examples for mode-I dominated fracture, namely a tension test and a three-point bending beam, are discussed.     

Numerical simulation of stress-strain state near crack tip in a compact tensile specimen

A computational scheme has been developed and a numerical simulation of the stress-strain state near the crack tip is performed at different levels of the stress intensity factor using a compact tensile specimen as an example. The authors analyze the influence of the finite element size near the crack tip and compare the results obtained in different codes (software packages) for different crack geometries. __________ Translated from Problemy Prochnosti, No. 1, pp. 134–140, January–February, 2009.     

Goal‐oriented adaptive refinement for phase field modeling with finite elements

Phase field modeling is very often performed with the finite‐difference method for equally spaced grids. Typically its solutions are highly non‐homogenous; and, therefore, non‐equally spaced grids with dense meshes at interfaces between different phases and coarse meshes in homogenous regions would be more advantageous with respect to both, efficiency and reliability of the numerical solutions. To this end, in the present work, an adaptive strategy with finite elements for phase field modeling is adopted, where the time step and the grid size are selected on the basis of goal‐oriented error estimation. In order to account for nonlinearity of the variational equations, we introduce a secant form for the dual problem, which for practical purposes is approximated by a tangent form. In a numerical example, we investigate transformation and retransformation for a two‐phase system in a square region subjected to thermal loading. Copyright © 2013 John Wiley & Sons, Ltd.     

Coupling of the meshfree and finite element methods for determination of the crack tip fields

This paper develops a new concurrent simulation technique to couple the meshfree method with the finite element method (FEM) for the analysis of crack tip fields. In the sub-domain around a crack tip, we applied a weak-form based meshfree method using the moving least squares approximation augmented with the enriched basis functions, but in the other sub-domains far away from the crack tip, we employed the finite element method. The transition from the meshfree to the finite element (FE) domains was realized by a transition (or bridge region) that can be discretized by transition particles, which are independent of both the meshfree nodes and the FE nodes. A Lagrange multiplier method was used to ensure the compatibility of displacements and their gradients in the transition region. Numerical examples showed that the present method is very accurate and stable, and has a promising potential for the analyses of more complicated cracking problems, as this numerical technique can take advantages of both the meshfree method and FEM but at the same time can overcome their shortcomings.     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

Applicability of the crack tip element analysis for damage prediction of composite T-joints

The expanding application of polymeric composite materials in the Aerospace industry has led to the extension of its application to other industries such as the marine industry. A typical joint between the hull and bulkhead used in a monocoque structure is known as a T-joint. It consists of composite overlaminates over a shaped fillet to allow the transmission of direct and membrane shear stresses. The CTE (crack tip element) method offers the capability to provide accurate results with minimum computational resources. It is also an excellent damage prediction tool for composite laminates where oscillatory singularity exists at the crack tip. This paper describes the application of the CTE method for damage prediction of the T-joint. Issues involved in the current modeling approach and recommended solutions are discussed.     

An overview of a hybrid crack element and determination of its complete displacement field

The hybrid crack element (HCE) is one of the most accurate and convenient finite elements for the direct calculation of the stress intensity factor (SIF) and coefficients of the higher order terms of the Williams expansion. It is formulated from a simplified variational functional using truncated asymptotic crack tip displacement and stress expansions and interelement boundary displacements compatible with the surrounding regular elements. However, the exclusion of the rigid body modes in the truncated asymptotic displacements creates jumps between these displacements and element compatible boundary displacements. In this study, an overview of the HCE is given. Furthermore, the rigid body modes excluded in its formulation are recovered by minimizing the jumps via a least squares method. Limitations of the boundary collocation method (BCM) widely used for predicting these terms, as well as the complete displacements are also investigated.     

Modeling of interfacial debonding crack in particle reinforced composites using Voronoi cell finite element method

In this paper, Voronoi cell finite element method (VCFEM), introduced by Ghosh and coworkers (1993), is applied to describe the matrix-inclusion interfacial debonding for particulate reinforced composites. In proposed VCFEM, the damage initiation is simulated by partly debonding of the interface under the assumption of the critical normal stress law, and gradual matrix-inclusion separations are simulated with an interface remeshing method that a critical interfacial node at the crack tip is replaced by a node pairs along the debonded matrix-inclusion interface and a more pair of nodes are needed to be added on the crack interface near the crack tip in order to better facilitate the free-traction boundary condition and the jumps of solution. The comparison of the results of proposed VCFEM and commercial finite element packages MARC and ABAQUS. Examples have been given for a single inclusion of gradually interfacial debonding and for a complex structure with 20 inclusions to describe the interfacial damage under plane stress conditions. Good agreements are obtained between the VCFEM and the general finite element method. It appears that this method is a more efficient way to deal with the interfacial damage of composite materials. The financial support by the Special Funds for the National Major Fundamental Research Projects G19990650 and the National Natural Science Foundation of China No. 59871022 are gratefully acknowledged.     

Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model

A hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations including nonlinear Poisson-Boltzmann equation and diffusion-reaction equation. The nonlinear force term in the Poisson-type equation is frozen by introducing the imaginary terms at each Picard iteration step, and then the induced Poisson problem is solved by the present hybrid finite element model involving element boundary integrals only, coupling with the particular solution method with radial basis function interpolation. The numerical accuracy of the present method is investigated by numerical experiments for problems with complex geometry and various nonlinear force functions.     

Elimination of spurious static modes in meshes of hybrid compatible triangular and tetrahedral finite elements

In the assumed displacement, or primal, hybrid finite element method, the requirements of continuity of displacements across the sides are regarded as constraints, imposed using Lagrange multipliers. In this paper, such a formulation for linear elasticity, in which the polynomial approximation functions are not associated with nodes, is presented. Elements with any number of sides may be easily used to create meshes with irregular vertices, when performing a non‐uniform h‐refinement. Meshes of non‐uniform degree may be easily created, when performing an hp‐refinement. The occurrence of spurious static modes in meshes of triangular elements, when compatibility is strongly enforced, is discussed. An algorithm for the automatic selection, based on the topology of a mesh of triangular elements, of the sides in which to decrease the degree of the approximation functions, in order to eliminate all these spurious modes and preserve compatibility, is presented. A similar discussion is presented for the occurrence of spurious static modes in meshes of tetrahedral elements. An algorithm, based on heuristic criteria, that succeeded in eliminating these spurious modes and preserving compatibility in all the meshes of tetrahedral elements of uniform degree that were tested, is also presented. Copyright © 2001 John Wiley & Sons, Ltd.