Crack Tip Elements Research Articles

Two-Dimensional Crack Analyses by XFEM Using Crack Tip Elements

The extended finite element method (XFEM) using a crack tip element (TIP element), which is enriched through only the Heaviside function, is applied to crack and its propagation analysis in two-dimensional elastic problems. In the proposed method, two-kind of signed distance functions are utilized in order to express crack geometry implicitly and finite elements, which has interaction with crack, are appropriately partitioned according to the level set values and then integrated numerically for derivation of stiffness matrix. The results by XFEM using TIP elements were compared with those by the conventional XFEM using both the asymptotic bases and the Heaviside function. It was shown that the TIP element provides appropriate stress intensity factors and crack propagation path.

Study of Crack Growth Based on Extended Finite Element Method

A special method based on the extended finite element method is developed for the simulation of dynamic crack growth. It shows great advantages in the simulations of moving crack and mixed mode crack. The extended finite element method for two-dimensional crack is described in this paper. The crack form of the extended finite element in the homogeneous medium is studied in detail, and the internal detail in crack tip element and crack penetration element is analyzed. At last, the displacement mode is generated.

Study of Crack Growth Based on Extended Finite Element Method

A special method based on the extended finite element method is developed for the simulation of dynamic crack growth. It shows great advantages in the simulations of moving crack and mixed mode crack. The extended finite element method for two-dimensional crack is described in this paper. The crack form of the extended finite element in the homogeneous medium is studied in detail, and the internal detail in crack tip element and crack penetration element is analyzed. At last, the displacement mode is generated.

Augmented Cohesive Elements for Efficient Delamination Analyses of Composite Laminates

In this paper, a new type of cohesive element that employs multiple subdomain integration (MSDI) for improved cohesive stress integration accuracy of bonded plate/shell elements has been formulated. Within each subdomain, stress integration is compatible with existing schemes such as Gaussian integration (GI), Newton–Cotes integration, or the mixed Gaussian and subdomain integration (mixed GI&SDI). The numerical accuracy, efficiency, and robustness of this element when employing three integration methods for MSD cohesive stress integration have been evaluated and compared through a benchmark mode-I fracture problem of bonded double-cantilever plates. The MSDI offers at least 50% improvement of numerical accuracy as compared to the best integration method in literature and has the best numerical robustness. This significant improvement pushes the structural mesh size restriction from limiting size of 1/3–1/5 cohesive zone length to 1.5–2 times the cohesive zone length. The formulation is very easy to be implemented into any finite element programs including commercial packages. Furthermore, this formulation enables the use of dual-mesh for delamination analyses of bonded structural shells/plates, which is of practical importance because it greatly reduces the burden of mesh generation for complicated composite structures. It has also been demonstrated that using high-order shell/plate elements can improve the numerical accuracy in general because the nonlinear deformation profile permitted by this type of elements can better describe the nonlinear deformation in the crack-tip element (partially bonded elements).

Application of the Θ-Method to 3D BEM Analysis of Cracks in Piezoelectric Components

A general purpose direct BEM code has been developed for three-dimensional crack problems in piezoelectric structures. Special 3D non-continuous crack tip elements and several techniques for determining crack tip parameters were implemented. To calculate the electromechanical energy release rate for a virtual crack extension, the θ-method is employed, which was originally suggested by BONNET for linear elastic materials. The paper presents the generalization and numerical realization of the θ-method to 3D piezoelectric cracks. The great advantage of the θ-method is the direct computation of energy release rate, whereas the way via K-factors and the IRWIN matrix is more complicated. The efficiency and accuracy of the technique are shown for various example problems by comparing with analytical solutions.

Composite Grid Method on Computation of Stress Intensity Factors of Biomaterial Interface Crack

The stress intensity factors (SIFs) of biomaterial interface crack were analyzed using the Composite Grid Method (CGM). Two different scale grids were adopted: a coarse mesh for the whole area without considering interface crack and a fine mesh for the local singular area considering the impact of interface crack. It solved the global coarse mesh problem and the local fine mesh problem iteratively and obtained the final results. The calculation was carried out by CGM but without the demand of dense mesh near the crack tip and special crack tip elements compared with other methods. The numerical extrapolation method of determining the SIFs to avoid the oscillation singularity of asymptotic fields of elastic interface crack. Reasonable agreement was achieved and efficiency of CGM for such kind of problem was displayed.

Transient dynamic crack analysis in linear magnetoelectroelastic solids by a hypersingular time-domain BEM

Abstract This paper presents a hypersingular time-domain boundary element method (TDBEM) for transient dynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic and linear magnetoelectroelastic solids. Stationary cracks in infinite and finite solids under impact loading are investigated. A combination of the strongly singular displacement boundary integral equations (BIEs) and the hypersingular traction boundary integral equations is used in the present analysis. The Galerkin-method is applied for spatial discretization, while a collocation method is implemented for the temporal discretization. The time-domain fundamental solutions for linear magnetoelectroelastic solids are derived. Both temporal and spatial integrations can be carried out analytically. The line integrals over the unit circle arising in the time-domain fundamental solutions are computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. To ensure a direct and an accurate computation of the dynamic intensity factors (IFs) from the generalized CODs, special crack-tip elements are implemented. Several numerical examples are shown to confirm the accuracy and the efficiency of the present hypersingular time-domain BEM.

Fracture mechanics based predictions of initiation and growth of multi-level delaminations in a composite specimen

Design and manufacture of composite test specimens with two embedded delaminations subjected to transverse tension is described in this paper. The behaviour of the specimens has been investigated through experimental testing and finite element analyses. Fracture mechanics approaches were applied to predict the initiation and growth of delaminations and the degradation of the load carrying capacity of the specimens. Models were generated in the FE software programs MSC.Nastran and Abaqus. The MSC.Nastran model was used to calculate strain energy release rates employing a crack tip element methodology. The Abaqus model was evaluated using the virtual crack closure technique. The accuracy of numerical solutions was established by qualitative and quantitative comparisons with the experimental measurements, the results obtained from the ultrasonic non-destructive inspections and microscopic examination. It was shown that fracture mechanics can be used to accurately predict the failure initiation locations and failure loads of composite specimens containing multi-level delaminations.

A novel variable power singular element in G space with strain smoothing for bi-material fracture analyses

This paper aims to formulate a triangular five-node (T5) singular crack-tip element in G space with strain smoothing to simulate an r λ−1 (0< λ<1) stress singularity for bi-material fracture analyses. In the present formulation, a direct point interpolation with a proper fractional order of extra basis functions is specially employed to construct variable power type singular shape functions that are in a G 1 space. Within strain smoothing, the singular terms of functions as well as mapping procedures are no longer necessary to compute the stiffness matrix. Furthermore, thanks to the point interpolation, the proposed singular element eliminates the need to shift the position of the side nodes adjacent to the crack-tip, and is thus quite straightforward and easily implemented in existing codes. The effectiveness of the present singular element is demonstrated via numerical examples of a wide range of material combinations and boundary conditions.

EFG Virtual Crack Closure Technique for the Determination of Stress Intensity Factor

The aim of this paper is to introduce a virtual crack closure technique based on EFG method for thread-shape crack. The cracked component is discretized and the displacement field is determined using a coupled FE/EFG method, by which EFG nodes are arranged in the vicinity of crack tip and FE elements in the remain part in order to improve computational efficiency. Two typical parameters, nodal force and crack opening displacement attached to crack tip are calculated by means of setting up an auxiliary FE zone around crack tip. Strain energy release rate (SERR), further stress intensity factor (SIF) are determined by the two parameters. The method to calculate SIF is named as virtual crack closure technique based on EFG method. It is showed by several numerical examples that using the method presented in this paper, SIF on the crack tip can be obtained accurately.

Effects of residual thermal stresses on the debond characterization of sandwich beams

The crack tip element method is applied to the formulation of energy release rates associated with face sheet debonding of sandwich beams with residual thermal stresses using the bi-layer Timoshenko beam model. Special attention is paid to the energy release rates of DCB and MMB tests for debond characterization. Mode-I and mode-II energy release rates are formulated including residual thermal stresses. The derived analytical results are verified by comparison with finite element analysis. The effect of residual thermal stresses on the evaluated apparent fracture toughness, resulting from the neglect of residual thermal stresses in the standard evaluation methods, is discussed.

Simulation of three-dimensional fatigue crack propagation using enriched finite elements

A three-dimensional methodology for simulation of fatigue crack propagation is presented. The method is leveraged by the use of enriched crack tip elements to compute the mixed-mode stress intensity factors. The crack growth model used and the crack propagation life calculation are also described. As examples, fatigue crack propagation of a mode-I surface crack and crack advancements of mixed-mode surface cracks are simulated. The predicted results showed excellent agreement with experimental data from the literature. Thus, it is concluded that the crack propagation method developed allows efficient and accurate simulation of three-dimensional fatigue crack propagation problems.

Fracture Mechanics Analysis of Interfacial Crack in Anisotropic Bi-Materials

This paper presents a single-domain boundary element method (BEM) for linear elastic fracture mechanics analysis in the two-dimensional anisotropic bi-materials. In this formulation, the displacement integral equation is applied on the outer boundary only, and the traction integral equation is applied on one side of the crack surface only. A special interfacial crack-tip element was introduced to capture exactly the oscillatory behavior. The computer program with the FORTRAN code has been developed to effectively calculate the stress intensity factors (SIFs) of an interfacial crack within anisotropic bi-materials. This BEM program has been verified having a good accuracy with the previous researches. Furthermore, by analyzing the different anisotropic degree of interfacial crack in an infinite domain, we found that the stress intensity factors of interfacial crack tips had apparent influence by the geometry forms of cracks and media with different anisotropic degrees.

Delamination of layered structures on elastic foundation

This study presents an interface fracture mechanics analysis of delamination of a layered beam resting on a Winkler elastic foundation subject to general mechanical loads. A crack tip element on elastic foundation model is established first, through which, two concentrated forces existing at the crack tip are determined in closed-form. Then total energy release rate of the crack can be expressed in term of these two forces. By using available numerical results in the literature, the phase angle of the total energy release rate is also obtained. To verify the validity and accuracy of the solutions, debonding of a bonded overlay from the base structure resting on a Winkler elastic foundation is analyzed using the present solution. Comparisons with the baseline results obtained by finite element analysis suggest that the present analytical solution provides an excellent estimation of the total energy release rate and its phase angle for interface cracks in layered structure on elastic foundation. This study provides an approximated analysis of the debonding of a thin overlay debonding from the concrete pavement, where the effect of the base structure is simplified by a Winkler elastic foundation. This solution can also be used to analyze other similar delamination problems, such as local delamination in laminated composites, and face sheet delimitation in sandwich beams.

Determination of Second-Order term Coefficients for the Inclined Crack in Orthotropic Plate using Singular Finite Elements

The finite element method with quarter-point crack-tip elements is used and a simple formula for obtaining the coefficients of the second-order terms in the series expansion for near crack tip stresses in orthotropic materials under biaxial loading is presented. This formula is obtained by comparing the variation of the displacements along the crack tip element with the elastic field solution for the crack tip. Numerical examples are given for the validity of the present formulation. The results obtained are compared with the theoretical ones and a good agreement between the two solutions is obtained.

A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media

Abstract The recently developed edge-based smoothed finite element method (ES-FEM) is extended to fracture problems in anisotropic media using a specially designed five-node singular crack-tip (T5) element. In the formulation of singular ES-FEM, only the assumed displacement values (not the derivatives) on the boundaries of the smoothing domains are needed. Thus, a layer of T5 crack-tip element is devised to construct “singular” shape functions via a simple point interpolation with a fractional order basis, without mapping procedure. The effectiveness of the present singular ES-FEM is demonstrated by intensive examples for a wide range of degrees of anisotropy.

A partly and fully cracked triangular XFEM element for modeling cohesive fracture

This paper discusses the build-up of a partly cracked cohesive crack tip element. The crack tip element is based on the principles of the eXtended Finite Element Method (XFEM) and is of Linear Strain Triangle (LST) type. The composition of the enrichment has been in focus to achieve as complete a description as possible on both sides of the crack. The stress accuracy within the crack tip element has been improved to a level, so that the crack tip stresses can be evaluated locally without introducing nonlocal averaging in the surroundings of the crack tip. The partly cracked element can also be applied as a fully cracked element with a few restrictions in the displacement field. The performance of the developed element has been tested in three examples. One example is an infinite sheet with an initial flaw in pure tension, where a semianalytical solution exists. The two other examples are the two benchmark tests: Three-point Bending Test (TPBT) and the Four-point Shear Beam (FPSB). The partly cracked element performs well, and captures the overall structural response even for coarse meshes in all the three examples. Copyright © 2010 John Wiley & Sons, Ltd.

Smoothed nodal forces for improved dynamic crack propagation modeling in XFEM

AbstractImprovements in numerical aspects of dynamic crack propagation procedures by the extended finite element method are described and studied. Using only the discontinuous enrichment function in XFEM gives a binary description of the crack tip element: it is either cut or not. We describe a correction force to modify the forces to smoothly release the tip element while the crack tip travels through the element. This avoids creating spurious stress waves and improves the accuracy of the stress intensity factors during propagation by decreasing the oscillations. Copyright © 2010 John Wiley &amp; Sons, Ltd.

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

AbstractIt is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Finite element procedures for the numerical simulation of fatigue crack propagation under mixed mode loading

This paper addresses the numerical simulation of fatigue crack growth in arbitrary 2D geometries under constant amplitude loading by the using a new finite element software. The purpose of this software is on the determination of 2D crack paths and surfaces as well as on the evaluation of components Lifetimes as a part of the damage tolerant assessment. Throughout the simulation of fatigue crack propagation an automatic adaptive mesh is carried out in the vicinity of the crack front nodes and in the elements which represent the higher stresses distribution. The fatigue crack direction and the corresponding stress-intensity factors are estimated at each small crack increment by employing the displacement extrapolation technique under facilitation of singular crack tip elements. The propagation is modeled by successive linear extensions, which are determined by the stress intensity factors under linear elastic fracture mechanics (LEFM) assumption. The stress intensity factors range history must be recorded along the small crack increments. Upon completion of the stress intensity factors range history recording, fatigue crack propagation life of the examined specimen is predicted. A consistent transfer algorithm and a crack relaxation method are proposed and implemented for this purpose. Verification of the predicted fatigue life is validated with relevant experimental data and numerical results obtained by other researchers. The comparisons show that the program is capable of demonstrating the fatigue life prediction results as well as the fatigue crack path satisfactorily.