Linear Elastic Fracture Mechanics Problems Research Articles

Generalization of Mvcci Approach for Lefm Problems Using Numerical Integration

Modified Virtual Crack Closure Integral (MVCCI) has become a valuable tool for estimation of mixed-mode fracture parameters in Linear Elastic Fracture Mechanics (LEFM) problems. When finite elements with fewer nodes and using polynomial shape functions are employed at the crack tip for fracture analysis, the expressions for MVCCI can easily be derived and many times also be expressed in closed form. For higher order elements they become unwieldy and so a numerical integration scheme is proposed and this can be used with any type of element. The scheme requires a two-step numerical integration: one to evaluate nodal forces exerted by the solid below the crack extension line on the portion above and the second to evaluate Strain Energy Release Rates. This development makes the post-processing for fracture parameters easy and computationally economical for 3-dimensional fracture problems. The technique is used to estimate fracture parameters in certain practical problems using higher order elements and typical numerical results are presented.

Recovery strategies, a posteriori error estimation, and local error indication for second‐order G/XFEM and FEM

AbstractThis article presents a computationally efficient and straightforward to implement a posteriori error estimator for second‐order G/XFEM and FEM approximations. The formulation is based on the recently proposed block‐diagonal Zienkiewicz–Zhu (ZZ‐BD) a posteriori error estimator. The focus is on linear elastic fracture mechanics (LEFM) problems but the proposed error estimator formulation is general and can be adapted to other types of problems such as those involving material interfaces. The proposed ZZ‐BD error estimator is based on a new strategy to recover stress fields from second‐order G/XFEM and FEM approximations. This recovery procedure involves locally weighted projections of raw stresses on an approximation space for discontinuous and singular stress fields. The basis functions for these stress approximations are defined using a low‐order partition of unity together with polynomial, discontinuous, and singular recovery enrichment functions. These singular enrichments are provided by the gradient of G/XFEM enrichments for LEFM problems and are, thus, available in most G/XFEM implementations. They also lead to a more computationally efficient stress recovery procedure than the one in the original ZZ‐BD error estimator. The adopted G/XFEM are optimally convergent with the error in the energy norm of . Numerical experiments show that the error of the recovered stress field converges at the same rate as the discretization error. They also show that the effectivity index of the error estimator is close to the unity and that it provides good local error indicators for mesh adaptivity algorithms. Second‐order FEM for elasticity problems is a special case of the G/XFEM adopted here. This is exploited to define a block‐diagonal ZZ error estimator for the FEM as a special case of the proposed estimator for the G/XFEM. This brings the computational efficiency, simplicity, and accuracy of the ZZ‐BD error estimator, originally developed for the G/XFEM, to second‐order standard FEM.

Phase-field fracture modeling for large structures

The phase-field modeling framework for linear elastic fracture mechanics problems is modified for the purpose of analyzing crack growth in large structures. The phase-field approach to fracture replaces sharp crack surfaces with a diffuse fracture zone that represent the traction-free crack faces. The diffuse fracture zone is characterized by a length scale that appears prominently in the governing partial differential equation that governs the evolution of the phase-field variable. Within numerical calculations the diffuse fracture zone must be resolved with a sufficient number of degrees of freedom in order to obtain accurate solutions. Additionally, all material fracture problems possess a physical process zone length scale that scales with the ratio of the fracture energy to the square of the material strength. For the vast majority of problems governed by linear elastic fracture mechanics, this physical process zone length scale is small, for example it is on the order of microns for aluminum alloys. If the material strength is to be captured, then the most commonly used formulation of the phase-field approach to fracture ties the diffuse phase-field crack length scale to the physical process zone length scale. This presents a challenge for extending such models to large structures that may be on the order of meters in size due to the prohibitive meshing requirements for the phase-field crack length scale. This is especially the case for three-dimensional problems. To address this problem a new form of the phase-field degradation function is introduced that allows for a decoupling of the phase-field length scale from the physical process zone length scale. The behavior and limitations of this new formulation are discussed and illustrated with a series of numerical test cases.

On tracking arbitrary crack path with complex variable meshless methods

This study presents a numerical modelling framework based on complex variable meshless methods, which can accurately and efficiently track arbitrary crack paths in two-dimensional linear elastic solids. The key novelty of this work is that the proposed meshless modelling scheme enables a direct element-free approximation for the solutions of linear elastic fracture mechanics problems. The complex variable moving least-squares approximation with a group of simple complex polynomial basis is applied to implement this meshless model, with which the fracture problems with both stationary or progressive cracks are considered and studied. The effects of choosing different definitions of weighted complex variable error norm and different forms of complex polynomial basis on the computational accuracy of crack tip fields and crack paths prediction are analysed and discussed. Five benchmark numerical examples were studied to demonstrate the superiority of the present complex variable meshless framework over a standard element-free Galerkin method in tracking arbitrary crack paths in two-dimensional elastic solids.

Well-conditioned and optimally convergent second-order Generalized/eXtended FEM formulations for linear elastic fracture mechanics

The Generalized/eXtended Finite Element Method (G/XFEM) has been established as an approach to provide optimally convergent solutions for classes of problems that are challenging for the standard version of the Finite Element Method (FEM). For problems with non-smooth solutions, as those within the Linear Elastic Fracture Mechanics (LEFM) context, the FEM convergence rates are often not optimal and are bounded by the strength of the singularity in the analytical solution. This can be overcome by G/XFEM and many researches have focused on delivering first-order convergent solutions for LEFM problems. The difficulty in obtaining higher-order accurate approximations, however, relies mainly on also controlling the growth rate of the stiffness matrix condition number. In this paper, well-conditioned and optimally convergent second-order G/XFEMs are proposed for LEFM simulations by augmenting second-order Lagrangian FEM approximation spaces. More specifically, two strategies are proposed in order to accurately represent second-order discontinuous functions along a crack. Also, a third strategy that essentially improves the use of linear Heaviside functions in the sense that these enrichments no longer cause linear dependencies among the set of G/XFEM shape functions is proposed. The numerical experiments show the robustness of the formulations presented herein for a set of crack topologies.

Blended isogeometric-finite element analysis for large displacements linear elastic fracture mechanics

Isogeometric analysis plays an important role in solid mechanics, offering the possibility of integrating computer-aided design and structural analysis. However, isogeometric discretizations are not as flexible as standard finite elements ones, especially in the context of crack propagation analysis with continuous and complex remeshing. This paper presents a domain decomposition technique for introducing and propagating cracks in an isogeometrically discretized solid. In this technique, a standard triangular finite element mesh is overlapped to the isogeometric model and the basis functions of both discretizations are modified and blended over a region of the physical domain, leading to a new space of functions. This technique allows the insertion and propagation of discontinuities over an isogeometric discretization without modifying the original mesh. The proposed approach is applied to large and small displacements 2D linear elastic fracture mechanics problems, demonstrating to be a very robust, accurate and versatile tool.

Optimized Mesh-free Analysis for the Singularity Subtraction Technique of Linear Elastic Fracture Mechanics

In linear elastic fracture mechanics, the stress field is singular at the tip of a crack. Since the representation of this singularity in a numerical model raises considerable numerical difficulties, the paper uses a strategy that regularizes the elastic field, subtracting the singularity from the stress field, known as the singularity subtraction technique (SST). In this paper, the SST is implemented in a local mesh-free numerical model, coupled with modern optimization schemes, used for solving two-dimensional problems of the linear elastic fracture mechanics. The mesh-free numerical model (ILMF) considers the approximation of the elastic field with moving least squares (MLS) and implements a reduced numerical integration. Since the ILMF model implements the singularity subtraction technique that performs a regularization of the stress field, the mesh-free analysis does not require a refined discretization to obtain accurate results and therefore, is a very efficient numerical analysis. Mesh-free numerical methods control the accuracy and efficiency of the model through the size of compact supports, the size of integration domains and the distribution of nodes in the body, which are usually heuristically determined through an expensive and time consuming calibration effort. The leading innovation of this paper is the automatic definition of these parameters and the nodal distribution by means of a multi-objective optimization, based on genetic algorithms (GA), with reliable and efficient objective functions. The optimization scheme effectively automates the whole pre-processing phase of a numerical analysis with mesh-free methods. Benchmark problems were analyzed to assess the accuracy and efficiency of the modeling strategy. The results presented in the paper are in perfect agreement with those of reference solutions and therefore, make reliable and robust this mesh-free numerical analysis, coupled with a multi-objective optimization,for linear elastic fracture mechanics problems.

Complex Fourier‐based stress intensity factor analysis of plane elasticity using boundary element theories

AbstractIn this article, the boundary element method is implemented for solving linear elastic fracture mechanics problems, using new complex Fourier shape functions which include constant and linear complex Fourier shape functions. They, unlike the classic Lagrange shape functions, can satisfy the exponential and trigonometric function fields in addition to the polynomial ones. Six numerical examples are provided to investigate the new shape functions and also, the stress intensity factors (SIF) are calculated for each so that the accuracy of the present study is illustrated. The results are compared with those obtained by classic Lagrange ones and the existing methods.

Smoothed floating node method for modelling 2D arbitrary crack propagation problems

In this work, Floating Node Method (FNM), first developed for fracture modelling of laminate composites, is coupled with cell-wise strain Smoothed Finite Element Method (SFEM) for modelling 2D linear elastic fracture mechanics problems. The proposed method is termed as Smoothed Floating Node Method (SFNM). In this framework, FNM is used to represent the kinematics of crack and the crack front inside the domain without the requirement of remeshing and discontinuous enrichment functions during crack growth. For smoothing, a constant smoothing function is considered over the smoothing domains through which classical domain integration changes to line integration along each boundary of the smoothing cell, hence derivative of shape functions are not required in the computation of the field gradients. The values of stress intensity factor are obtained from the SFNM solution using domain based interaction integral approach. Few standard fracture mechanics problems are considered to check the accuracy and effectiveness of the proposed method. The predictions obtained with the proposed framework improves the convergence and accuracy of the results in terms of the stress intensity factors and energy norms.

A generalized 2D Bézier-based solution for stress analysis of notched epoxy resin plates reinforced with graphene nanoplatelets

In this study, a robust Bezier-based multi-step method is extended to accurately solve the governing fourth-order complex partial differential equation (PDE) in linear elastic fracture mechanics (LEFM) problems. The Bezier technique was first introduced by the authors to solve initial value problems in one dimension. Now, the method is further extended to simultaneously solve Boundary Value Problems (BVPs) in orthogonal directions. To examine the accuracy and performance of the present method, the first-mode normalized stress intensity factor (SIF) of a 2D epoxy resin plate having an initial edge crack and reinforced with randomly oriented graphene nanoplatelets (GnP) is determined and compared with the associated exact analytical solution using the Bayesian statistical analysis. Besides, the impact of GnP aspect ratio on the normalized crack opening displacement (COD) of the reinforced matrix is elaborated for the first time in the literature. Results of the present study suggest that GnPs with maximum aspect ratio are most effective to enhance elastic properties of the plate and potentially limit the edge crack propagation. Specifically, inclusion of 0.5 and 1.0% of needle-shaped GnPs in a notched epoxy resin plate decrease the maximum normalized COD by 33 and 50%, respectively, while for square-shaped GnPs, these reductions are limited to 20 and 37%, respectively.

A new insight into analysis of linear elastic fracture mechanics with spherical Hankel boundary elements

In this research, the solution of linear elastic fracture mechanics problems when using spherical Hankel shape functions in the boundary element method is investigated. These shape functions benefit from the advantages of Bessel function fields of the first and second kind in addition to the polynomial ones, which makes them a robust and efficient tool for interpolation. Among the most important features of Hankel shape functions, the infinite piecewise continuity and the utilization of complex space with no influence on the imaginary part of the partition of unity property can be pointed out. The calculation of the stress intensity factors (SIFs) is carried out in five numerical examples so that the accuracy of the present study can be illustrated. Comparisons are made between the results of using spherical Hankel shape functions, the experimental ones, and those obtained by classical BEM with Lagrange shape functions. According to the results of these comparisons, the proposed spherical Hankel shape functions present more accurate and reliable results in comparison with the classic Lagrange ones.

A simple, first-order, well-conditioned, and optimally convergent Generalized/eXtended FEM for two- and three-dimensional linear elastic fracture mechanics

This paper proposes a first-order Generalized/eXtended Finite Element Method (G/XFEM) for 2-D and 3-D linear elastic fracture mechanics problems. The conditioning of the method is of the same order as the standard FEM and it is robust with respect to the position of the mesh relative to 2-D or 3-D fractures. The method achieves an optimal rate of convergence in energy norm even when the solution of the problem in the neighborhood of the fracture front is not contained in the spaces spanned by the adopted singular enrichments. Control of conditioning associated with Heaviside enrichment functions is achieved by shifting them by their nodal values, combined with a diagonal pre-conditioner. In the case of singular enrichment functions, a simple extension of the enrichment shifting concept, denoted as discontinuous-shifting, is adopted. These enrichment modifications are space-preserving—the solution space spanned by the modified enrichments is the same as the space spanned by the unmodified ones. This is often not the case for conditioning control strategies proposed in the literature. The computational cost of these enrichment modifications is negligible and their implementation in existing G/XFEM software is straightforward. The optimal convergence, well-conditioning, and robustness of the method are numerically illustrated with the aid of representative 2-D and 3-D problems, including the case of a non-planar fracture with a curved fracture front.

On the numerical integration in generalized/extended finite element method analysis for crack propagation problems

PurposeThe purpose of this paper is to evaluate some numerical integration strategies used in generalized (G)/extended finite element method (XFEM) to solve linear elastic fracture mechanics problems. A range of parameters are here analyzed, evidencing how the numerical integration error and the computational efficiency are improved when particularities from these examples are properly considered.Design/methodology/approachNumerical integration strategies were implemented in an existing computational environment that provides a finite element method and G/XFEM tools. The main parameters of the analysis are considered and the performance using such strategies is compared with standard integration results.FindingsKnown numerical integration strategies suitable for fracture mechanics analysis are studied and implemented. Results from different crack configurations are presented and discussed, highlighting the necessity of alternative integration techniques for problems with singularities and/or discontinuities.Originality/valueThis study presents a variety of fracture mechanics examples solved by G/XFEM in which the use of standard numerical integration with Gauss quadratures results in loss of precision. It is discussed the behaviour of subdivision of elements and mapping of integration points strategies for a range of meshes and cracks geometries, also featuring distorted elements and how they affect strain energy and stress intensity factors evaluation for both strategies.

2-D Crack propagation analysis using stable generalized finite element method with global-local enrichments

In this paper, the technique of the Stable Generalized Finite Element Method (SGFEM) is applied to the numerically constructed functions of the Generalized Finite Element Method with Global-Local Enrichments (GFEMgl). The application of the resulting approach, named SGFEMgl, is expanded here to 2-D quasi-static crack propagation problems. Crack growth is performed by a two-scale strategy, using local problems generated at each propagation step – whose solutions enrich a single global problem defined on a coarse mesh. Stress Intensity Factors (SIFs) computed along crack growth, strain energy measures, performance in blending elements and the condition number are used to study the accuracy and conditioning of SGFEMgl. The method is compared with the standard GFEMgl. Numerical experiments demonstrate remarkable accuracy of SGFEMgl in linear elastic fracture mechanics problems, considering crack opening modes I and II. Convergence rates analyses also show the superiority of the method, especially with the use of geometrical enrichments.

A comparison between some fracture modelling approaches in 2D LEFM using finite elements

The finite element method has been widely used to solve different problems in the field of fracture mechanics. In the last two decades, new methods have been developed to improve the accuracy of the solution in 2D linear elastic fracture mechanics problems, such as the extended finite element method (XFEM) or the phantom node method (PNM). The goal of this work is to quantify the differences between some numerical approaches: standard finite element method (FEM), mechanical property degradation, interelemental crack method with multi-point constraints, XFEM and PNM. We explain the different techniques analysed together with their advantages and disadvantages. We compare these numerical techniques to model fracture using problems of reference with known solutions, evaluating their behaviour in terms of convergence with respect to the element size and accuracy of the stress intensity factor (SIF), stresses ahead the crack tip and crack propagation prediction. Some of the new techniques have shown a better accuracy in SIF calculation or stress fields ahead the crack tip and other lead to high errors in local results estimations. However, all methods reviewed here can predict crack propagation for the problems of reference of this work, showing good accuracy in crack orientation prediction.

Numerical determination of stress intensity factors: J-integral and modified virtual crack closure technique

Together with the development of numerical tools for stress-strain analysis, approaches to deal with fracture mechanics problems have been the object of continuous improvement. For linear-elastic fracture mechanics problems, the determination of Stress Intensity Factors (SIF) is available as a post-processing possibility in most commercial finite element software packages such as Ansys and Abaqus, allowing the users to perform straightforward assessments of cracked mechanical parts or structures. However, available techniques for SIF determination in post-processing of these commercial solutions are almost limited to the so-called displacement extrapolation and J-integral techniques. In the present work, the implementation of the J-integral technique is revised considering a new finite element software, labeled as Omicron. Furthermore, a more recent technique, the modified Virtual Crack Closure Technique (mVCCT) is also implemented and evaluated. For results assessment, finite element models were built in Omicron and SIFs were determined considering the J-integral and mVCCT approaches. From this study, it is concluded that the mVCCT implementation is simpler than the J-integral and it allows to determine accurately SIFs. Nevertheless, this technique is not straightly available in the current versions of the commercial packages for finite element modeling, which require separate post-processing for SIF determination.

Quasi-static analysis of mixed-mode crack propagation using the meshless local Petrov–Galerkin method

The efficiency of classical truly meshless local Petrov–Galerkin method with linear test function approximation in the crack growth problems of complex configurations is investigated in this article. Several unique and multiple (two) cracks are examined to show the accuracy of the proposed meshfree method by comparing the crack path with boundary element, finite element, element-free Galerkin and experimental results. Crack propagation under both mechanical and thermal loads and also mode I and mixed-mode conditions are evaluated. Effect of the functionality of material properties on crack paths and sensitivity of crack growth to boundary conditions are investigated. Classical maximum circumferential stress criterion is employed into the formulation of the MLPG method to predict crack growth direction. Despite the common numerical studies, a simple and straightforward automatic increment size determination method is introduced and implemented in the MLPG method. Besides, regular domain point distribution in the vicinity of the crack tip without the increased density of points is used. Various test problems show the computational accuracy and applicability of this method to linear elastic fracture mechanics problems.

A polygonal XFEM with new numerical integration for linear elastic fracture mechanics

The extended finite element method (XFEM) has become a powerful and effective technique for modeling fracture problems without remeshing. In this paper, we introduce a novel and effective computational approach that is based on polygonal XFEM (named as PolyXFEM) for the analysis of two-dimensional (2D) linear elastic fracture mechanics problems. The PolyXFEM is equipped with a new numerical integration technique that uses the concept of Cartesian transformation method (CTM) over polygonal domains. The underlying idea of the CTM is to transform a domain integral into a boundary integral and a one-dimensional integral. This is computationally more efficient compared to the two-level mapping integration commonly used on polygons. Furthermore, the PolyXFEM is effective in modeling crack problems due to the local enrichment based on the partition of unity method. Our numerical results show that improvements in accuracy are reached thanks to the higher-order of the polygonal shape functions of the PolyXFEM. In addition, efficiency in computational time is achieved because of the usage of the CTM integration scheme, which appears to be quite suitable for polygonal domains. To demonstrate the accuracy and performance of the developed PolyXFEM approach, several numerical examples for 2D linear elastic fracture problems are considered, in which static stress intensity factors and crack propagation are investigated and compared with reference solutions. The convergence and mesh independence of the PolyXFEM for crack analysis is also analyzed.

Interface solutions of partial differential equations with point singularity

Abstract Partial differential equations (PDEs) such as those that describe linear elastic fracture mechanics admit analytic solutions with low regularity at crack tips. Existing numerical methods for such PDEs suffer from the constraint of this low regularity and fail to deliver an optimal high-order rate of convergence. We approach this problem by (i) choosing an artificial interface to enclose the center of low regularity and (ii) representing the solution in the interior of artificial interface as unknown linear combination of known low-regularity modes of the solution. This results in an interface formulation of the original PDE, and the linear combination is represented in the interface conditions. By enforcing the smooth component of the numerical solution in the interior domain to be approximately zero, a least squares problem is obtained for the unknown coefficients. The solution of this least squares problem will provide the approximate interface conditions for the numerical solution of the PDE in the exterior domain. The potential of our interface formulation is favorably demonstrated by numerical experiments on one- and two-dimensional Poisson equations with low-regularity solutions. High-order numerical solutions of unknown coefficients and PDEs are obtained. This proves the potential of the proposed interface formulation as the theoretical basis for solving linear elastic fracture mechanics problems. We address the relations between our interface formulation and domain decomposition methods, as well as a regularization strategy for the Poisson–Boltzmann equation with singular charge density.

Computational fracture mechanics: An overview from early efforts to recent achievements

AbstractAn overview of the state of the art of computational fracture mechanics is presented, starting from early efforts and going all the way up to recent achievements. Some specific aspects of linear elastic and elastic plastic fracture mechanics problems have been discussed, including the fact that even for a static loading, numerical simulation is not a simple task because of complex geometries, material nonlinearity, and heterogeneity, and especially if crack growth is considered. Therefore, micromechanical modelling of elastic‐plastic crack growth is presented as new and promising approach to overcome some of the shortages of traditional approach. Besides static loading, couple of other important practical problems are tackled, like fatigue crack growth, with remaining life in the focus of investigation, using the empirical laws for crack growth rates. Numerical simulation of fatigue crack growth is inherently complex both because of complex material damage processes and lack of sound theoretical basis to define them. Therefore, combination of theoretical, experimental, and numerical approach is presented here to get reliable and efficient estimation of life under fatigue loading.