Singular Crack Tip Elements Research Articles

A singular crack tip element based on sub-partition and XFEM for modeling crack growth in plates and shells

A singular crack tip element based on sub-partition and XFEM for modeling crack growth in plates and shells

A modified semi-weight function method for stress intensity factor calculation of a vertical crack terminating at the interface

The interfacial crack stress intensity factor (SIF) calculation plays an important role in the safety evaluation and anti-crack design for multilayer structures. The current analytical or numerical methods have difficulty of complicated auxiliary functions or special crack-tip singular elements. In this study, a modified semi-weight function method was proposed based on the submatrix method to calculate the SIFs of the crack vertical to and terminating at the interface of bimaterial (easily appears in compressed air energy storage) and verified by two calculation examples compared with the available literature results. The effects of bimaterial parameters (Ei, υi), relative crack length (a/w2) and outer integration path (r2) on the SIFs were also analyzed. Research results show that the SIFs (K1 and K2) are related to only the ratio of E2/E1 independent of the single E1 or E2, and they are increased with increase of a/w2. The reasonable range of r2 is rs < r2 < a/2 (rs is minimum finite element size). The modified semi-weight function method has advantages of unified auxiliary function for different crack sizes and external loads and simpler calculation process without any crack-tip singular elements over current methods.

Morphological characteristics of crack branching in asphalt mixtures under compression

Crack branching is a multi-crack propagation problem. Singular crack tip elements are frequently encountered, and defining the multi-crack tips’ enrichment functions is difficult. To overcome these drawbacks, this study used an already developed ordinary state-based peridynamics method to simulate crack branching in AC-13, AC-20, and PAC-13 asphalt mixtures. These asphalt mixtures’ digital images were acquired by an X-ray computed tomography method. Next, the Prony series coefficients of the fine asphalt mixtures were obtained by the dynamic modulus test. After that, the fracture energy of the fine asphalt mixtures was measured by the semi-circular bending test. Finally, the measured asphalt mixture properties were input into the ordinary state-based peridynamics method in order to simulate crack branching in the AC-13, AC-20, and PAC-13 asphalt mixtures. The simulation results indicate that the diagonal crack first emanates in asphalt mixtures. Then, the diagonal crack continues to branch into a wing crack, splitting crack, and branched crack. These four types of cracks can generate eight crack branching categories in asphalt mixtures. The crack branching trajectory was quantified by the fractal dimension, whose value ranged from 1.14 to 1.64. The physical significance of the fractal dimension explains the changes of crack branching paths in the asphalt mixtures.

A new crack-tip singular element for cracks in three-dimensional elastic bodies

Symplectic analytical singular element (SASE) is a special crack-tip element in the framework of finite element method (FEM) for modelling cracks. A group of SASEs, which have been developed by the authors, for various crack problems almost in two-dimensional (2D), but not touched yet more complex ones, i.e., in three-dimensional (3D) domain. The underlying reason lies in the fact that analytical symplectic eigen solution for 3D elastic crack problem is not yet available, a 3D SASE thus cannot be constructed through a simple generalization from the 2D SASEs. This study aims to fill out this 3D gap, we thus use a trail displacement field to construct a 3D SASE. The derived strain and stress fields still possess singularity in the vicinity of crack-tip. Finite element formulation is derived based on the minimum total potential energy principle. It is found that some of key features of a 2D SASE still exist in the developed 3D one, e.g., the SIFs can be calculated accurately without any post-processing. Numerical examples are considered to show the accuracy and performance of the proposed element.

Thermal-Mechanical Fracture Analysis Considering Heat Flux Singularity

Abstract Precise modeling of thermoelastic cracks remains challenging due to the fact that both heat flux and stress fields have singularity issue. In the previous studies, the first author proposed different types of symplectic analytical singular element (SASE) for thermal conduction and stress analysis of cracks. It has been demonstrated that these crack-tip elements of which the interior fields are defined by analytical solutions are highly accurate and efficient. However, the thermal mechanical coupling problem of crack cannot be treated with the existing SASEs. The main difficulty is that the analytical solution of the crack problem considering arbitrary temperature distribution is not available. Approximate solution may lead to significant numerical instabilities. Moreover, the construction of a crack-tip singular element for both thermal conduction and stress analysis is complicated and requires more efforts. In this study, the governing symplectic dual equation of thermoelastic crack is restudied. The analytical solution considering arbitrary temperature distribution is obtained in close form which, to the best of the authors' knowledge, has not been found before. Then, the finite element formulation of a new SASE for thermal-mechanical fracture analysis is derived analytically through a variational approach. A two-step analysis procedure is proposed to calculate the mixed mode thermal stress intensity factors (TSIFs)) and the analysis can be done on a fixed finite element mesh. Mesh refinement around the crack tip is unnecessary, and the mixed-mode TSIFs can be solved accurately without any postprocessing.

Fundamental-solution-based hybrid finite element with singularity control for two-dimensional mixed-mode crack problems

In this work, a fundamental-solution-based hybrid finite element method is presented for modeling mixed-mode cracks in two-dimensional (2D) isotropic elastic media. In the method, a double-variable hybrid functional for each element is formulated to derive element stiffness equation in terms of nodal displacements, which includes line integrals along the element boundary only. The element interior displacement field is approximated using a linear combination of displacement fundamental solutions at different source locations, while the independent element frame displacement field is approximated by one-dimensional shape function interpolation. To correctly model the behavior of crack-tip displacement, the discontinuous quarter-point crack-tip singular hybrid element formulation is developed, so that crack tip stress intensity factors can be easily evaluated by the near-tip displacement method. Three numerical examples of internal cracks in 2D elastic domains are presented to show the efficiency of the proposed method.

Static fracture and modal analysis simulation of a gas turbine compressor blade and bladed disk system

This paper presents a methodology for conducting a 3-D static fracture analysis with applications to a gas turbine compressor blade. An open crack model is considered in the study and crack-tip driving parameters are estimated by using 3-D singular crack-tip elements in ANSYS $$\circledR $$ . The static fracture analysis is verified with a special purpose fracture code (FRANC3D). Once the crack front is perfectly defined and validated, a free vibration study is conducted by analyzing the natural frequencies and modeshapes for both a single blade and bladed disk system. Taking advantage of high performance computing resources, a high fidelity finite element model is considered in the parametric investigation. In the fracture simulation, the influence of the size of a single edged crack as well as the rotational velocity on fracture parameters (stress intensity factors and J-Integral) are evaluated. Results demonstrate that for the applied loading condition, a mixed mode crack propagation is expected. In the modal analysis study, increasing the depth of the crack leads to a decrease in the natural frequencies of both the single blade and bladed disk system, while increasing the rotational velocity increases the natural frequencies. The presence of a crack also leads to mode localization for all mode families, a phenomenon that cannot be captured by a single blade analysis.

Finite Element Modeling of Fatigue Crack Propagation Using a Self Adaptive Mesh Strategy

A new finite element model has been developed to predict fatigue crack growth in arbitrary 2D geometries under constant amplitude loading. The purpose of this model 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 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 correlation 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 has to 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. Verification of the predicted fatigue life is validated with relevant experimental data and numerical results obtained by other researchers. The comparisons show that this model is capable of demonstrating the fatigue life prediction results as well as the fatigue crack path satisfactorily. Copyright © 2008 Praise Worthy Prize S.r.l. - All rights reserved.

S-FEM FOR FRACTURE PROBLEMS, THEORY, FORMULATION AND APPLICATION

This paper briefs some of recent works on applications of the smoothed finite element method (SFEM) to fracture problems. In the SFEM formulation for simulating the singular filed near the crack-tip, it is known that one can directly enrich the filed by adding in proper terms needed to simulate the singularity, which can be done in a number of ways (see the SFEM book). In this work a generalized technique called the "enriched linear PIM" for constructing shape functions is used to formulate a special "five-node (T5) singular crack-tip element" that can produce a proper order of stress singularity near the crack-tip. One layer of T5 crack-tip singular elements are used in a basic mesh of linear elements leading to a very simple model that can be crated automatically for complicated geometry. Because the SFEM uses weakened weak formulations, such a simple model does not require transition elements and the compatibility is ensured. In addition, the singular terms of functions as well as mapping procedures are no longer necessary to compute the stiffness matrix. Thus, the singular SFEM method is straightforward and can be easily implemented in the existing codes. The stress intensity factors of mix-modes can also be easily evaluated by an appropriate treatment during the domain form of the interaction integral, due to the use of simple triangular mesh. The effectiveness of the present singular T5 element is demonstrated via benchmark examples.

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.

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 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.

A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks

The recently developed edge-based smoothed finite element method (ES-FEM) is extended to the mix-mode interface cracks between two dissimilar isotropic materials. The present ES-FEM method uses triangular elements that can be generated automatically for problems even with complicated geometry, and strains are smoothed over the smoothing domains associated with the edges of elements. Considering the stress singularity in the vicinity of the bimaterial interface crack tip is of the inverse square root type together with oscillatory nature, a five-node singular crack tip element is devised within the framework of ES-FEM to construct singular shape functions. Such a singular element can be easily implemented since the derivatives of the singular shape term \({(1/\sqrt r)}\) are not needed. The mix-mode stress intensity factors can also be easily evaluated by an appropriate treatment during the domain form of the interaction integral. The effectiveness of the present singular ES-FEM is demonstrated via benchmark examples for a wide range of material combinations and boundary conditions.

Modeling of crack propagation via an automatic adaptive mesh refinement based on modified superconvergent patch recovery technique

In this paper, an automated adaptive remeshing procedure is presented for simulation of arbitrary shape crack growth in a 2D finite element mesh. The Zienkiewicz–Zhu error estimator is employed in conjunction with a modified SPR technique based on the recovery of gradients using analytical crack-tip fields in order to obtain more accurate estimation of errors. The optimization of crack-tip singular finite element size is achieved through the adaptive mesh strategy. Finally, several numerical examples are illustrated to demonstrate the effectiveness, robustness and accuracy of computational algorithm in calculation of fracture parameters and prediction of crack path pattern.

A finite element modelling for directly determining intensity factors of piezoelectric materials with cracks

Recently, authors(Cao et al., Acta Aeronautica et Astronautica Sinica 25(5): 470–472, 2004) extended the singular crack element originally introduced by Wang et al. (Eng Fract Mech 37(6):1195–1201, 1990) for evaluating the stress intensity factors (SIFs). Extensive studies have proved the versatility and accuracy of the element. This study is to show the versatility of the element for piezoelectric materials. In this paper, electric potential and displacement fields near a crack tip of piezoelectric materials are first used to construct a finite element version for directly determining intensity factors of piezoelectric materials with cracks. A singular finite element is constituted and a new method to calculate intensity factors of piezoelectric materials with cracks is obtained without any post-processing procedures. Detailed derivations are given and the results obtained with present method are good agreement with those of theoretical results, the FEM data by ANSYS and singular electromechanical crack tip elements. The results to the different selections of the structural dimensions are carried out. Numerical examples demonstrate the accuracy and validity of the novel element of present method.

Finite Element Simulation of Fatigue Life Estimation and Crack Path Prediction of Two-dimensional Structures Components

The main objective of this work is to analyse fatigue crack propagation of two-dimensional structures under constant amplitude loading using the developed adaptive mesh finite element. The finite element mesh is generated using the advancing front method. The adaptive remeshing process is carried out based on the posteriori stress error norm scheme. The stress intensity factors are estimated by employing the displacement extrapolation technique facilitated by construction of singular crack tip elements. The crack propagation is modelled by the splitting node approach and the trajectory follows the successive linear extensions of each crack increment. The propagation process is driven by linear elastic fracture mechanics approach. In the analysis, the resulted stress intensity factor range at each of crack increments is recorded. Fatigue life cycle can be predicted immediately after completion of the recording by integrating the generalized Paris’ equation. The procedure is applied to the fatigue analysis ...

Finite Element Analyses of Cracks in Piezoelectric Structures

A review is given about FEM-techniques to compute the coupled electromechanical boundary value problem of cracks in piezoelectric structures under static and dynamic loads. To calculate the relevant fracture parameters very precisely and efficient, the following numerical techniques are presented: i) Special singular crack tip elements, ii) Modified crack closure integral, iii) Computation of electromechanical J-integral and iv) Usage of interaction integrals. Special emphasis is devoted to different electric crack face boundary conditions. The accuracy, efficiency and applicability of these techniques are examined by various example problems and discussed with respect to their advantages and drawbacks for practical applications.

Finite element analyses of cracks in piezoelectric structures: a survey

Piezoelectric materials have widespread applications in modern technical areas such as mechatron- ics, smart structures or microsystem technology, where they serve as sensors or actuators. For the assessment of strength and reliability of piezoelectric structures under combined electrical and mechanical loading, the existence of cracklike defects plays an important role. Meanwhile, piezoelectric fracture mechanics has been established quite well, but its application to realistic crack configurations and loading situations in piezoelec- tric structures requires the use of numerical techniques as finite element methods (FEM) or boundary element methods (BEM). The aim of this paper is to review the state of the art of FEM to compute the coupled electro- mechanical boundary value problem of cracks in 2D and 3D piezoelectric structures under static and dynamic loading. In order to calculate the relevant fracture parameters very precisely and efficiently, the numerical treatment must account for the singularity of the mechanical and electrical fields at crack tips. The follow- ing specialized techniques are presented in detail: (1) special singular crack tip elements, (2) determination of intensity factors KI-KIV from near tip fields, (3) modified crack closure integral, (4) computation of the electromechanical J -integral, and (5) exploitation of interaction integrals. Special emphasis is devoted to a realistic modeling of the dielectric medium inside the crack, leading to specific electric crack face boundary conditions. The accuracy, efficiency, and applicability of these techniques are examined by various example problems and discussed with respect to their advantages and drawbacks for practical applications.

New boundary element method for torsion problems of cylinder with curvilinear cracks

The Saint-Venant torsion problems of a cylinder with curvilinear cracks were considered and reduced to solving the boundary integral equations only on cracks. Using the interpolation models for both singular crack tip elements and other crack linear elements, the boundary element formulas of the torsion rigidity and stress intensity factors were given. Some typical torsion problems of a cylinder involving a straight, kinked or curvilinear crack were calculated. The obtained results for the case of straight crack agree well with those given by using the Gauss-Chebyshev integration formulas, which demonstrates the validity and applicability of the present boundary element method.