Extraction Of Stress Intensity Factors Research Articles

Extraction of stress intensity factors of biharmonic equations with free boundary conditions on crack faces (III): SIF of a thin plate subjected to the bending moment

The plate bending problems in the Kirchhoff–Love model can be reduced to biharmonic equations. In the previous papers (Kim et al., 2020; Kim et al., 2022), we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a non-convex (cracked, L-shaped) domain with various boundary conditions such as clamped, simply connected, free, and mixed. In previous papers, we could not show a general SIF formula related to Poisson’s ratio ν contained in the free boundary condition. Moreover, the previous SIF extraction formula obtained for a given Poisson’s ratio requires cumbersome line integrals. In this paper, removing the line integrals in the extraction formulas, we show improved SIF extraction formulas applicable to biharmonic equations with free boundary conditions. Thus, the new, improved SIF extraction formulas for free boundary conditions become more straightforward and can be applied to plate bending problems.

Influence of discontinuities on the fracture behaviour of CNT reinforced composites subjected to thermo-mechanical load using XIGA

This paper is aimed to investigate the fracture behaviour of carbon nanotube (CNT) reinforced composite exposed to the thermo-mechanical environment in the presence of discontinuities using the extended isogeometric analysis (XIGA) method. The study focuses on finding the effects of discontinuities present in a finite plate with a pre-existing crack, on the stress intensity factors (SIFs). The mandatory equivalent mechanical and thermal properties are assessed with the help of various micromechanics models. Two types of CNTs are assumed to be reinforced in the epoxy-matrix: single-walled carbon nanotube (SWCNT) and multi-walled carbon nanotube (MWCNT). The CNT reinforced composite is examined for varying volume percentages of CNTs reinforcement. A comparative study is provided to see the influence of mechanical and coupled thermo-mechanical load on SIFs. Adiabatic crack is taken into account for the computational simulation for thermal loading condition. The interaction integral method is used for the extraction of SIFs. The findings of the test reveal that with the rise in the volume percentage of CNTs, the properties such as fracture energy and fracture toughness also rise. Moreover, the fracture of CNT reinforced composites is delayed with the increased content of CNT. The results establish the profound influence of holes on SIFs than the inclusions.

Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II)

In [13], we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of fΨsk⁎−uΔ2Ψsk⁎ for a cut-off function Ψ and a dual singular function sk⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u, we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ2u=f, we have to use C1-continuous basis functions. For continuous differentiable basis functions, we use either C1-continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities.

The opening mode stress intensity factor extraction method from mixed mode crack using single element strain gage.

The opening mode stress intensity factor extraction method from mixed mode crack using single element strain gage.

A displacement correlation method for stress intensity factor extraction from 3D fractures in anisotropic materials

A numerical method to extract stress intensity factors (SIFs) from 3D fractures in anisotropic materials is presented. The formulation of a displacement correlation method that is valid for linear elastic materials with arbitrary symmetry in their elastic properties is given in detail. An algorithm for the numerical implementation of the method is also given. The methodology can take an approximate displacement field from any numerical method as input to the extraction process. However, to return accurate stress intensity factors an accurate numerical displacement field is required. In this work, this is provided by a Generalized Finite Element Method. Several verification problems show that the anisotropic displacement correlation method is robust and can extract accurate SIFs. The methodology is used in a parametric study of crack–crack interactions in a high-strength aluminum alloy to demonstrate a practical application. More than 200 simulations are run using the Generalized Finite Element Method, which are largely able to be scripted and ran without user intervention after one simulation is set up.

Extraction of Stress Intensity Factors by Using the P-Version Finite Element Method and Contour Integral Method

The stress intensity factors (SIFs) for two-dimensional cracks are extracted using the p-version finite element method (P-FEM) and the contour integral method. Several numerical experiments, e.g., crack initiating from the edge of a circular hole under an unidirectional uniform tension and two equal-length, unequal-length hole–edge cracks, respectively, at a rectangular plate, an inclined centered crack under uniaxial tension at a square plate and a pipeline crack model, are used to demonstrate the accuracy and effectiveness of the approaches. SIFs are presented for the effects of various crack lengths and length–width ratio. Numerical results are analyzed and compared with reference solutions and results obtained by the Voronoi cell finite element method, boundary element method, high-order extended finite element method (high-order XFEM) and commercial finite element software ABAQUS in the available literature. Numerical results are in good agreement with the benchmark problems and show faster convergence rate, higher accuracy and better numerical stability.

Direct extraction of stress intensity factors for geometrically elaborate cracks using a high-order Numerical Manifold Method

By adopting mathematical and physical covers, the Numerical Manifold Method (NMM) simulates cracks with geometrically elaborate patterns in a unified manner. In this contribution, we propose a high-order NMM together with a direct SIF extraction method for crack modeling. In the proposed framework, SIFs are computed in closed-form upon the solution of the NMM discrete system. This novel formulation avoids cumbersome post-processing procedures and results in naturally decomposed SIFs. In order to meet different accuracy needs, we further generalize the original concept of singular physical covers by defining a parametric enrichment zone, for which high-order asymptotic basis functions derived from Williams’ solution are adopted.Several benchmark examples are presented to validate the excellent performance of the proposed framework. The effect of enrichment zone size as well as enrichment function order on the SIF accuracy is systematically investigated. Furthermore, we discuss the appropriate enrichment scheme as crack tips are in close proximity. The capacity of the proposed framework to address branched and coalescent cracks is demonstrated through the last two examples.

Accuracy and robustness of stress intensity factor extraction methods for the generalized/eXtended Finite Element Method

This paper investigates the accuracy and robustness of three Stress Intensity Factor (SIF) extraction methods for the Generalized/eXtended Finite Element Method (G/XFEM): the Cutoff Function Method (CFM), the Contour Integral Method (CIM), and the Displacement Correlation Method (DCM). Challenges in SIF extraction from G/XFEM solutions using energy-based methods such as the CFM and the Interaction Integral Method are discussed. Numerical studies on problems with stress-free crack surfaces show that the CFM is path independent while the CIM is not. They also show that unless a proper enrichment scheme is adopted, SIFs extracted with the DCM are of low accuracy and sensitive to extraction parameters. Strategies to address both issues with the DCM are presented. This paper demonstrates that the DCM, with proper enrichment of the G/XFEM approximation, has an accuracy and robustness comparable to the CFM at a fraction of the computational cost. It is also shown that the DCM can be applied to problems where SIF extraction using domain integrals is not possible. Several problems aimed at investigating the applicability and accuracy of the various extraction methods are solved.

Stress intensity factor extraction from the enriched partition of unity solution for a cracked linear elastic medium

The stress intensity factor is a useful tool for predicting material failure and describing the stress states of brittle materials. We present a technique to calculate the stress intensity factor for a linear elasticity problem on a cracked domain with an enriched partition of unity method. We use a particular partition of unity function, which is piecewise polynomial and has wide flat-top region. The flat-top area in the partition of unity function helps the displacements and the stress fields in the vicinity of the crack tip to be accurately represented, even with a coarse background mesh. Among other methods for calculating the stress intensity factor, we find that the direct extraction method is the most accurate and efficient one given a relatively coarse background mesh for the enriched partition of unity method.

3-D local mesh refinement XFEM with variable-node hexahedron elements for extraction of stress intensity factors of straight and curved planar cracks

A novel local mesh refinement approach for fracture analysis of three-dimensional (3-D) linear elastic solids is developed, considering both 3-D straight and curved planar cracks. The present local mesh refinement formulation is a combination of the extended finite element method (XFEM), variable-node hexahedron elements, and a posteriori error indicator. Our 3-D formulation using hexahedron elements rigorously embraces a posteriori error estimation scheme, a structural coupling scale-meshes strategy and an enrichment technique. Local mesh refinement is only performed where it is needed, e.g., a vicinity of crack, through an error estimator based on the recovery stress procedure. To treat the mismatching problem induced by different scale-meshes in the domain, a structural coupling scheme employing variable-node transition hexahedron elements based on the generic point interpolation with an arbitrary number of nodes on each of their faces is presented. The 3-D finite element approximations of field variables are enhanced by enrichments so that the mesh is fully independent of the crack geometry. The displacement extrapolation method is taken for the evaluation of linear elastic fracture parameters (e.g., stress intensity factors—SIFs). To show the accuracy and performance of our proposed 3-D formulation, six numerical examples of planar 3-D straight and curved shaped cracks with single and mixed-mode fractures and different configurations are considered and analyzed. The SIFs computed by the developed method are validated with respect to analytical solutions and the ones derived from the conventional XFEM. Associated with an adaptive process, the present 3-D formulation allows the analysts to gain a desirable accuracy with a few trials, which is suited for practices purpose.

Linear elastic fracture simulation directly from CAD: 2D NURBS-based implementation and role of tip enrichment

We propose a method for simulating linear elastic crack growth through an isogeometric boundary element method directly from a CAD model and without any mesh generation. To capture the stress singularity around the crack tip, two methods are compared: (1) a graded knot insertion near crack tip; (2) partition of unity enrichment. A well-established CAD algorithm is adopted to generate smooth crack surfaces as the crack grows. The M integral and $$J_k$$ integral methods are used for the extraction of stress intensity factors (SIFs). The obtained SIFs and crack paths are compared with other numerical methods.

Transient analysis of stationary interface cracks in orthotropic bi-materials using oscillatory crack tip enrichments

The problem of an interface crack between two orthotropic layers under dynamic loading is analyzed. Special crack tip enrichment functions are incorporated into the standard finite element shape functions to exactly reproduce oscillatory stress and displacement fields near the tip of the interface crack. Moreover, kinematics of displacement and its gradients across the crack face and material interface are also modeled by partition of unity enrichments. Special attention is given to extraction of stress intensity factors by utilizing a proper form of the interaction integral for orthotropic bi-materials. Advantages of this method of extracting stress intensity factors over the conventional displacement extrapolation technique are discussed. Several bi-material configurations with both vanishing and non-vanishing oscillatory indexes are solved using the interaction integral and the results are compared with the available data in the literature. Effects of employing oscillatory crack tip enrichments and validation of the path-independent J-integral are also discussed.

Extraction of stress intensity factors from Irwin's integral using high‐order XFEM on triangular meshes

SummaryThis paper is dedicated to Professor Ted Belytscho's 70th birthday. His seminal contributions to Fracture Mechanics and the extended finite element method have significantly inspired us in this work. The current paper extends our recent work on the extraction of stress intensity factors (SIFs) from Irwin's integral, using a high‐order extended FEM (XFEM) formulation. By matching leading r terms (r being the distance from the crack tip to any other material point) in XFEM with analytical expansion of Irwin's Integral, a closed‐form solution for SIFs is obtained. Hence, SIFs can directly be obtained upon solution of the XFEM discrete system, which were shown to be accurate on structured rectangular meshes. However, extension to unstructured quadrilateral elements may not be trivial. To this end, we extend the formulation to triangular elements, where significant simplification of the closed‐form expression is obtained. Moreover, the formulation is directly applicable to unstructured meshes without extra work. Hence, this paper is considered a significant step toward automating and extending this method to more general applications. Numerical results show that accurate and consistent SIFs can be obtained on some pure mode and inclined crack benchmark problems, on structured and unstructured meshes. Examples of a crack approaching a hole and two cracks approaching each other are also investigated. The latter illustrates the advantage of this method over a J‐integral class of methods, as SIFs can still be calculated when cracks are in proximity and no remeshing is required. Hence, potentially, this method can address crack coalescence, branching and proximity to boundaries more rigorously. Copyright © 2014 John Wiley & Sons, Ltd.

Extraction of stress intensity factors for 3D small fatigue cracks using digital volume correlation and X-ray tomography

This paper describes a methodology used to compute stress intensity factor values along the curved front of a fatigue crack inside a nodular cast iron. An artificial defect is introduced at the surface of a small sample. The initiation and growth of a fatigue crack from this defect during constant amplitude cycling is monitored in situ by laboratory X-ray tomography. The method for processing the 3D images in order to compute SIF values is described in detail. The results obtained show variations of the stress intensity factor values along the crack front.

Modified crack closure integral technique for extraction of SIFs in meshfree methods

A new approach for extracting stress intensity factors (SIFs) by the element-free Galerkin (EFG) class of methods through a modified crack closure integral (MCCI) scheme is proposed. Its primary feature is that it allows accurate calculation of mode I and mode II SIFs with a relatively simple and straightforward analysis even when a coarser nodal density is employed. The details of the adoption of the MCCI technique in the EFG method are described. Its performance is demonstrated through a number of case studies including mixed-mode and thermal problems in linear elastic fracture mechanics (LEFM). The results are compared with published theoretical solutions and those based on the displacement method, stress method, crack closure integral in conjunction with local smoothing (CCI–LS) technique, as well as the M-integral method. Its advantages are discussed.

Extraction of Stress Intensity Factors for the Simulation of 3-D Crack Growth with the Generalized Finite Element Method

Two methods for the extraction of Stress Intensity Factors (SIFs) from three-dimensional (3-D)problems are presented: the Contour Integral Method and the Cutoff Function Method. The formula-tions are tailored for the Generalized Finite ElementMethod andmixed-mode 3-D propagating cracks.The case of crack faces loaded by prescribed tractions is also considered. Another contribution of thispaper is a procedure to control the noise of extracted SIFs based on theMoving Least SquaresMethod.The proposed approach provides a continuous and smooth approximation of 3-D SIF functions foreach fracture mode. Numerical experiments demonstrating the accuracy and robustness of the pro-posed methodology are presented. They include 3-D mixed-mode fatigue crack growth simulationsand the case of a pressurized crack.

Convergence of domain integrals for stress intensity factor extraction in 2‐D curved cracks problems with the extended finite element method

SUMMARYThe aim of this study is the analysis of the convergence rates achieved with domain energy integrals for the computation of the stress intensity factors (SIF) when solving 2‐D curved crack problems with the extended FEM (XFEM). Domain integrals, specially the J‐integral and the interaction integral, are widely used for SIF extraction and provide high accurate estimations with FEMs. The crack description in XFEM is usually realized using level sets. This allows to define a local basis associated with the crack geometry. In this work, the effect of the level set local basis definition on the domain integral has been studied. The usual definition of the interaction integral involves hypotheses that are not fulfilled in generic curved crack problems, and we introduce some modifications to improve the behavior in curved crack analyses. Despite the good accuracy of domain integrals, convergence rates are not always optimal, and convergence to the exact solution cannot be assured for curved cracks. The lack of convergence is associated with the effect of the curvature on the definition of the auxiliary extraction fields. With our modified integral proposal, the optimal convergence rate is achieved by controlling the q‐function and the size of the extraction domain. Copyright © 2013 John Wiley & Sons, Ltd.

A hybrid element approach to three dimensional problems of cracked anisotropic multi-material

This work deals with the extraction of stress intensity factors in the three dimensional problems. In order to have a good estimation of them, a special class of mixed finite element, named “Metis” element (mongrel element), coupled to a code evaluating overstress, is used. After a brief explanation of the asymptotic model leading to stress/displacement in the vicinity of a delamination crack or a singular line at the junction between different materials, a simple and accurate numerical method is proposed. Next, the singular part is included in Metis element to estimate simultaneously the structural variables and the stress intensity factors. Finally, a numerical example on a bimaterial is illustrated and compared with a case found in the literature.

Stress intensity factors calculation in anti-plane fracture problem by orthogonal integral extraction method based on FEMOL

For an anti-plane problem, the differential operator is self-adjoint and the corresponding eigenfunctions belong to the Hilbert space. The orthogonal property between eigenfunctions (or between the derivatives of eigenfunctions) of anti-plane problem is exploited. We developed for the first time two sets of radius-independent orthogonal integrals for extraction of stress intensity factors (SIFs), so any order SIF can be extracted based on a certain known solution of displacement (an analytic result or a numerical result). Many numerical examples based on the finite element method of lines (FEMOL) show that the present method is very powerful and efficient.

Extraction of stress intensity factors from generalized finite element solutions

The performance of several superconvergent techniques to extract stress intensity factors (SIFs) from numerical solutions computed with the generalized finite element method is investigated. The contour integral, the cutoff function and the J-integral methods are considered. An implementation of the extraction techniques based on a sequence of mappings that are independent of the underlying solution method or discretization is proposed. It is shown that this approach is suitable for virtually any mesh-free or mesh-based solution method. Several numerical examples demonstrating the convergence of the computed SIF and the flexibility of the proposed implementation are presented. The path independence of the extraction methods is also investigated. Numerical experiments demonstrate that the contour integral and the cutoff function methods are more robust than the J–integral method with the CFM being the most accurate.