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This paper presents the application of a two-scale generalized finite element method (GFEM) which allows for static fracture analyses as well as fatigue crack propagation simulations involving the interaction of multiple crack surfaces on fixed, coarse finite element (FE) meshes. The approach is based on the use of numerically-generated enrichment functions computed on-the-fly through the use of locally-defined boundary value problems (BVPs) in the regions of existing mechanically-short cracks. The two-scale GFEM approach is verified against analytical reference solutions as well as alternative numerical approaches for crack interaction problems, including the coalescence of multiple crack surfaces. The numerical examples demonstrate the ability of the proposed approach to deliver accurate results even in scenarios involving multiple, interacting discontinuities contained within a single computational element. The proposed approach is also applied to a crack shielding/crack arrest problem involving two propagating crack surfaces in a representative panel model similar in complexity to that which may be of interest to the aerospace community.

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... In unnotched fatigue specimens, the main difference between the behavior of early damaging short cracks and long crack behavior is that short cracks exhibit teamwork behavior, while a long crack acts individually. [31][32][33][34][35] The surface yield stress, detected by X-ray diffraction, is significantly less than the bulk yield stress. 36 Further, the surface yield stress is sensitive to the surface roughness, and thus, the fatigue strength is influenced. ...

... As a result, each crack behaves uniquely, given rise to experimentally observed scatter. [39][40][41][42] Via replica and acoustic microscopy observations, eg, previous studies, [31][32][33]35,38,42,43 the interaction and evolution of short fatigue cracks and their coalescence were clarified. However, the number of static images, typically = 7 to 15 for each specimen, is too small to describe all the kinematic events possibly taken place over the specimen surface throughout its life. ...

... Such observations are common in relevant published experimental works. 34,35,60,61 The fatigue limit of a material is defined as the maximum stress range below which there is no fracture after the application of an infinite number of loading cycles. According to this definition, Δσ F = 2σ F = 332 MPa for the present material. ...

A fracture mechanics numerical model is developed to simulate the collective behavior of growing short fatigue cracks originating from the surface of unnotched round specimens made of a two‐phase alloy. The specimen surface roughness is considered resembling microcracks of different sizes and locations along the minimum specimen circumference. Material grains of different phases, sizes, and strengths are randomly distributed over that circumference. Variations in mechanical and microstructural features of grains are randomly distributed. Possible activities of surface cracks are predicted against loading cycles till either fracture occurs or all existing cracks become nonpropagating. The material's S‐N curve and fatigue limit can, thus, be assessed. Published experimental data on ferritic‐pearlitic steel specimens in push‐pull constant amplitude loading (CAL) were utilized. Different specimens were randomly configured and virtually tested. Comparison of experimental results and corresponding predictions validates the model, which, further, recognizes the effect of surface roughness, specimen size, and mean stress on lives.

... In unnotched fatigue specimens, the main difference between the behavior of early damaging short cracks and long crack behavior is that short cracks exhibit teamwork behavior, while a long crack acts individually. [31][32][33][34][35] The surface yield stress, detected by X-ray diffraction, is significantly less than the bulk yield stress. 36 Further, the surface yield stress is sensitive to the surface roughness, and thus, the fatigue strength is influenced. ...

... As a result, each crack behaves uniquely, given rise to experimentally observed scatter. [39][40][41][42] Via replica and acoustic microscopy observations, eg, previous studies, [31][32][33]35,38,42,43 the interaction and evolution of short fatigue cracks and their coalescence were clarified. However, the number of static images, typically = 7 to 15 for each specimen, is too small to describe all the kinematic events possibly taken place over the specimen surface throughout its life. ...

... Such observations are common in relevant published experimental works. 34,35,60,61 The fatigue limit of a material is defined as the maximum stress range below which there is no fracture after the application of an infinite number of loading cycles. According to this definition, Δσ F = 2σ F = 332 MPa for the present material. ...

This work is an extension of applying a previously developed fracture mechanics cracking damage model to predict the fatigue lifetime of un-notched round specimens made of a ferrite-pearlite 0.4C-70/30 carbon steel in some cases of variable amplitude loading VAL. The model simulates the collective behavior of growing short fatigue cracks originating from the specimen surface roughness. Material grains of different phases, sizes and strengths are randomly distributed over the minimum circumference. Possible activities of surface cracks are predicted against loading cycles. Relevant published experimental data were utilized for comparison. The present predictions are in agreement with the corresponding experimental results.

... Bhardwaj et al. [6] presented an approach to predict the fatigue life of an interfacial cracked plate in the presence of flaws by a combination method of homogenized XFEM and isogeometric analysis. Recently, O'Hara et al. [41] discussed a two-scale G/XFEM approach to solve fracture mechanics and fatigue crack propagation problems. In their model, both macro-and micro-crack surfaces were modeled only in the local problem using the discontinuous and singular enrichment functions. ...

... As it was discussed before, the work of Kim et al. [27] and O'Hara et al. [41] used a two-scale G/XFEM method, so-called global-local G/XFEM (G/XFEM gl in order to analyze the interaction of macrocrack with the multiple microcracks. Only the effects of microcracks were studied over the macrocrack behavior and they were chosen in a fixed orientation with respect to the macrocrack position. ...

... In addition, the two-scale G/XFEM have showed its impact by delivering better approximate solutions in terms of both stress intensity factors and stresses for main crack/micro-defects interactions. -The presence of micro cracks/holes/inhomogeneities have quite clear impact on the global-local enrichment function, since they influenced the intensity factor results, similar to the results from the literature in terms of the total behavior [27,31,41]. The main difference between current work and the work of Kim et al. [27] and O'Hara et al. [41] is that the microcracks in their work were chosen in a fixed orientation with respect to the macrocrack position. ...

Generalized or extended finite element method (G/XFEM) models the crack by enriching functions of partition of unity type with discontinuous functions that represent well the physical behavior of the problem. However, this enrichment functions are not available for all problem types. Thus, one can use numerically-built (global-local) enrichment functions to have a better approximate procedure. This paper investigates the effects of micro-defects/inhomogeneities on a main crack behavior by modeling the micro-defects/inhomogeneities in the local problem using a two-scale G/XFEM. The global-local enrichment functions are influenced by the micro-defects/inhomogeneities from the local problem and thus change the approximate solution of the global problem with the main crack. This approach is presented in detail by solving three different linear elastic fracture mechanics problems for different cases: two plane stress and a Reissner-Mindlin plate problems. The numerical results obtained with the two-scale G/XFEM are compared with the reference solutions from the analytical, numerical solution using standard G/XFEM method and ABAQUS as well, and from the literature.

... FEM is come up with a symmetric and positive definite system [11]. The singular stress field is created by refining mesh at the crack tip or using special types of elements such as quarter point elements [12]. A suitable mesh is handled by advanced re-meshing algorithms. ...

... Each component of the energy release rate is represented by a subscript at G , whereby the sum of II I G G , and III G produces Total G . The energy release rate can be changed to the SIF, as shown in equation (11), (12) and (13). For further details of derivation of element arrangement at crack front can be referred to Okada, Higashi [21]. ...

Stress intensity factor (SIF) is one of the most fundamental and useful parameters in all of fracture mechanics. The SIF describes the stress state at a crack tip, is related to the rate of crack growth, and used to establish failure criteria due to fracture. The SIF is determined to define whether the crack will grow or not. The aims of this paper is to examine the best sampling statistical distributions in SIF analysis along the crack front of a structure. Box-Muller transformation is used to generate the statistical distributions which is in normal and lognormal distributions. This method transformed from the random number of the variables within range zero and one. The SIFs are computed using the virtual crack-closure method (VCCM) in bootstrap S-version finite element model (BootstrapS-FEM). The normal and lognormal distributions are represented in 95% of confidence bounds from the one hundred of random samples. The prediction of SIFs are verified with Newman-Raju solution and deterministic S-FEM in 95% of confidence bounds. The prediction of SIFs by BootstrapS-FEM in different statistical distribution are accepted because of the Newman-Raju solution is located in between the 95% confidence bounds. Thus, the lognormal distribution for SIFs prediction is more acceptable between normal distributions.

... Significant efforts have been made to develop less labor-intensive but qualified mesh systems for 3D crack analysis, e.g., XFEM [29] and GFEM [30] . These methods do not depend on the mesh for modeling explicit cracks. ...

... A new crack mesh, called mixture mesh or template mesh, has been gradually applied to introduce cracks [30] . Examples of this type of mesh include the crack-block used in ZENCRACK software [31] and the tube-like domain mesh proposed by Barenberg and Dhondt [32] . ...

... Over the last two decades unfitted finite element methods (UFEM), that allow the use of relatively simple background meshes, have proved to be useful tools for solving partial differential equations (PDE) on domains that may be highly complex and may evolve with time. Under the umbrella of unfitted finite element methods, a range of approaches and techniques have been developed including the generalised finite element method (GFEM) [1][2][3][4][5][6][7][8][9][10][11], extended finite element method (XFEM) [12][13][14][15][16][17][18] and cut finite element method (CutFEM) [19][20][21][22][23][24][25]. ...

In this work a multi-point constraint unfitted finite element method for the solution of the Poisson equation is presented. Key features of the approach are the strong enforcement of essential boundary, and interface conditions. This, along with the stability of the method, is achieved through the use of multi-point constraints that are applied to the so-called ghost nodes that lie outside of the physical domain. Another key benefit of the approach lies in the fact that, as the degrees of freedom associated with ghost nodes are constrained, they can be removed from the system of equations. This enables the method to capture both strong and weak discontinuities with no additional degrees of freedom. In addition, the method does not require penalty parameters and can capture discontinuities using only the standard finite element basis functions. Finally, numerical results show that the method converges optimally with mesh refinement and remains well conditioned.

... In contrast, the mode I fracture parameter K I,A of point A increases more rapidly than K II,B of point B until the mode II fracture parameter K II,A becomes negative. It is reasonably believed that the interaction due to crack tip stress [81][82][83]. This phenomenon vanishes rapidly, however, once the overlapping of two crack tips A takes place due to stress relaxation, resulting in a decrease of the SIF ranges. ...

By introducing the shape functions of Virtual node Polygonal (VP) elements into the standard extended finite element method (XFEM), a conforming elemental mesh can be created for the cracking process. Moreover, an adaptively refined meshing with the quadtree structure only at a growing crack tip is proposed without inserting hanging nodes into the transition region. A novel dynamic crack growth method termed as VP-XFEM is thus formulated in the framework of fracture mechanics. To verify the newly proposed VP-XFEM, both quasi-static and dynamic cracked problems are investigated in terms of computational accuracy, convergence, and efficiency. The research results show that the present VP-XFEM can achieve good agreement in stress intensity factor and crack growth path with the exact solutions or experiments. Furthermore, better accuracy, convergence, and efficiency of different models can be acquired, in contrast to standard XFEM and mesh-free methods. Therefore, VP-XFEM provides a suitable alternative to XFEM for engineering applications.

... Methodologies based on the partition of unity have been applied to three-dimensional crack/crack propagation analyses. These methodologies, which include the extended finite element method (X-FEM) [34][35][36] and the generalized finite element method (G-FEM) and its variations [37,38], do not require explicit crack modeling with the finite element mesh. Hence, it is quite feasible to update the geometry of the crack during the crack propagation analysis. ...

... Methodologies based on the partition of unity have been applied to three-dimensional crack/crack propagation analyses. These methodologies, which include the extended finite element method (X-FEM) [34][35][36] and the generalized finite element method (G-FEM) and its variations [37,38], do not require explicit crack modeling with the finite element mesh. Hence, it is quite feasible to update the geometry of the crack during the crack propagation analysis. ...

A software system to perform SCC crack propagation analyses for complex and realistically shaped structures consisting of dissimilar materials, such as weld metal, base metal, etc., under weld residual stresses is presented in this paper. The system consists of programs to perform automatic generation of the finite element mesh, weld residual stress mapping to the finite element model with weld residual stresses to carry out fracture analysis and finite element analysis, and evaluating the stress intensity factors and updating crack geometries for crack propagation analysis. The results of stationary crack and crack propagation analyses elucidate the influences of both the residual stress distribution and the crack geometry on the distribution of the stress intensity factors. Their influences on the crack propagation behavior are also clarified.

... Based on their analyses, an elasto-plastic analysis of the interaction problem is almost necessary in order to have accurate properties of the coalescence process. O'Hara et al. [22] discussed a two-scale generalized finite element method (FEM) approach to solve fracture mechanics and fatigue crack propagation problems. In their model, both macroand micro-crack surfaces modeled only in the local problem using the discontinuous and singular enrichment functions. ...

Fatigue is a process in engineering materials in which damage accumulates due to the fluctuating loading. One solution for a component under the fatigue process is to arrest the crack propagation before the final failure using different available retardation methods, such as drilling/stop-hole technique. In addition, structural components may also suffer from the existence of micro-cracks or voids due to their forming process or service lives. These micro-cracks/voids are very critical to study, since they can effectively play an important role in the behavior of the existing main crack in a component. This paper aims to investigate the effect of the stop-hole retardation technique and multiple micro-cracks/voids with different characteristic lengths and geometries on the fatigue crack propagation in a compact tension specimen. A modified Forman equation, the so-called NASGRO equation is used to define the transition between crack initiation and crack growth period. Also, the extended finite element method is adapted in the crack
propagation phase in order to model crack path in the geometry eliminating the need for remeshing procedure. The whole analyses are conducted in a commercial package through a user-written codes that handles all fatigue crack growth analysis. The reference solutions from the literature are used to compare and to validate results obtained from current work.

... In XFEM/GFEM, a level set technique with enrichment functions is used to represent the crack in the domain, hence avoiding the requirement of remeshing during the crack propagation. This method has since been extended for interfacial crack (Sukumar et al., 2004;Pathak et al., 2013a;Kumar et al., 2015b;Hu et al., 2016), fatigue crack growth Singh et al., 2012;Pathak et al., 2015b;Hara et al., 2016a;Pant and Bhattacharya, 2017), elasto-plastic crack growth (Elguedj et al., 2006;Kumar et al., 2014;Kumar et al., 2015c;Kumar et al., 2016), three dimensional crack growth (Areias and Belytschko, 2005;Rabczuk et al., 2010;Pathak et al., 2013b, Pathak et al., 2013c, dynamic crack growth (Zi et al., 2005;Réthoré et al., 2005;Kumar et al., 2015d) fatigue crack growth in functionally graded materials (Singh et al., 2011;Bhattacharya and Sharma, 2014) and interaction of multiple cracks (Hara et al., 2016b). Despite its success in many types of problems, there exist some limitations: (1) it introduces an error during the mapping of discontinuities from the physical space to the natural space (Fries and Belytschko, 2010); (2) the implementation in FEM can be complicated as blending elements are generally required for connecting the enriched elements to standard elements; (3) the numerical solution is sensitive to the numerical integration scheme used for the enriched elements (Rabczuk, 2013); and (4) different enrichment functions are usually required to tackle different material problems. ...

The Floating Node Method (FNM), first developed for modeling the fracture behavior of laminate composites, is here combined with a domain-based interaction integral approach for the generic fracture modeling of quasi-brittle materials from crack nucleation, propagation to final failure. In this framework, FNM is used to represent the kinematics of cracks, crack tips and material interfaces in the mesh. The values of stress intensity factor are obtained from the FNM solution using domain-based interaction integral approach. To demonstrate the accuracy and effectiveness of the proposed method, four benchmark examples of fracture mechanics are considered. Predictions obtained with the current numerical framework compare well against literature/theoretical results.

... While the interaction between multiple stationary cracks has attracted a large number of investigations, e.g., [1][2][3][4][5][6], there has been a limited amount of research involving growing cracks. The interaction between multiple growing cracks under quasi-static loading conditions has been investigated using different numerical methods such as finite element method, e.g., [7,8]; boundary element method (BEM), e.g., [9,10]; moving-least-squares-based numerical manifold method, e.g., [11]. ...

The use of the symmetric Galerkin boundary element method (SGBEM) for studying the quasi-static interaction between multiple growing micro-cracks is presented in this work. The micro-cracks can conveniently be modeled in infinite domains, and this type of analysis can be handled by the SGBEM in a straightforward manner. In fact, it reduces the size of the analysis due to the absence of a physical boundary. A quasi-static multi-crack growth model based upon the maximum hoop stress criterion (MHSC), and the SGBEM was developed in this work. An improved quarter-point crack-tip element and adjusted maximum crack increments were employed to enhance the accuracy and effectiveness of the crack growth prediction. The improved quarter-point element has been known for producing accurate stress intensity factors required by the MHSC, while the technique used to adjust the maximum crack increment at each iteration of crack growth simulations allows to achieve converged (accurate) crack extension paths even if a relatively large maximum crack increment is selected at the onset. Several numerical examples were presented to show the effectiveness of the proposed multi-crack growth model.

... Extending the two-scale G/XFEM approach to shell problems is a potential research for future. Another interesting work that can be done based on this research is to apply the two-scale G/XFEM method for A c c e p t e d m a n u s c r i p t fatigue analysis of plate structures similar to the work of O'Hara et al. [41], using an easy-to-implement approach presented in [29]. ...

Generalized or extended finite element method (G/XFEM) uses enrichment functions that holds a priori knowledge about the problem solution. These enrichment functions are mostly limited to two-dimensional problems. A well-established solution for problems without any specific types of analytically derived enrichment functions is to use numerically-build functions in which they are called global-local enrichment functions. These functions are extracted from the solution of boundary value problems defined around the region of interest discretized by a fine mesh. Such solution is used to enrich the global solution space through the partition of unity framework of the G/XFEM. Here it is presented a two-scale/global-local G/XFEM approach to model crack propagation in plane stress/strain and Reissner-Mindlin plate problems. Discontinuous functions along with the asymptotic crack-tip displacement fields are used to represent the crack without explicitly represent its geometries. Under the linear elastic fracture mechanics approach, the stress intensity factor (obtained from a domain-based interaction energy integral) can be used to either determine the crack propagation direction or propagation status, i.e., the crack can start to propagate or not. The proposed approach is presented in detail and validated by solving several linear elastic fracture mechanics problems for both plane stress/strain and Reissner-Mindlin plate cases to demonstrate its the robustness and accuracy.

... The global error is controlled by the quality of local error. If the local error is presented in high enrichment function, it is expected good global accuracy as well [6]. The meshing problems and difficulties of embedded crack shape is resolved with some improvements in FEM. ...

Prediction of fatigue crack growth is one of the vital issues to prevent catastrophic failure from damage tolerance perspective. The surface of crack shape usually in semi-elliptical that maintained during the whole propagation. The investigation of this paper is to illustrate the surface crack growth that subjected to fatigue loading. The four-point bending and three-point bending have been simulated by using the S-version Finite Element Model (S-FEM). The simulation is conducted for aluminium alloys A7075-T6 and A2017-T3 with all of the parameters based on the previous experiment. The semi-elliptical crack shape is applied during the simulation process to represent with the reality of crack growth phenomena. Paris' Law model approach is presented in fatigue crack growth simulation. The S-FEM produced the surface crack growth and fatigue life prediction. The results of the S-FEM prediction then compared with the previous experiments. The results presented in a graph for comparison between S-FEM prediction with the experimental results. The S-FEM results obtained is good agreement with the experiment results.

... Fracture growth methodologies which advance fractures in discrete steps, referred to in this work as "quasi-static growth schemes" [e.g. 41,19,[42][43][44][45], will have similar properties due to the requirement to advance fracture tips in discrete steps. Quasi-static growth is a reasonable assumption for subsurface fracture networks, where growth occurs locally in short ruptures, and by chemically assisted processes such as stress corrosion [46], tectonic stresses [6] and thermal deformation. ...

Concurrent growth of multiple fractures in brittle rock is a complex process due to mechanical interaction effects. Fractures can amplify or shield stress on other fracture tips, and stress field perturbations change continuously during fracture growth. A three-dimensional, finite-element based, quasi-static growth algorithm is validated for mixed mode fracture growth in linear elastic media, and is used to investigate concurrent fracture growth in arrays and networks. Growth is governed by fracture tip stress intensity factors, which quantify the energy contributing to fracture extension, and are validated against analytical solutions for fractures under compression and tension, demonstrating that growth is accurate even in coarsely meshed domains. Isolated fracture geometries are compared to wing cracks grown in experiments on brittle media. A novel formulation of a Paris-type extension criterion is introduced to handle concurrent fracture growth. Fracture and volume-based growth rate exponents are shown to modify fracture interaction patterns. A geomechanical discrete fracture network is generated and examined during its growth, whose properties are the direct result of the imposed anisotropic stress field and mutual fracture interaction. Two-dimensional cut-plane views of the network demonstrate how fractures would appear in outcrops, and show the variability in fracture traces arising during interaction and growth.

... In order to circumvent this deficiency, without the need of creating a complex and fine mesh close to crack, the option of building custom and local enrichment functions for each problem analyzed arises [18][19][20]. Developments of the global-local technique incorporated into the G/XFEM were studied for three-dimensional problems in Pereira et al. [21], O'Hara et al. [22], Li and Duarte [23], demonstrating its applicability and efficiency in several engineering scenarios. ...

The Generalized/eXtended Finite Element Method (G/XFEM) is applied as a very useful tool in the resolution of complex structural models using an effective approach to represent the existence of cracks and other micro-defects. This is an unconventional formulation of the Finite Element Method (FEM), in that there is an expansion of the approximate solution field from the use of enrichment functions associated with the nodes. Enrichment functions can be singular functions derived from analytic deductions, polynomial functions or even functions resulting from other solution processes, such as the global–local strategy. The Stable Generalized Finite Method (SGFEM) is a variation of the G/XFEM with a simple modification in its enrichment functions, reducing the condition number of the stiffness matrix as well as the approximate error in the so-called blending elements. Considering this under the global–local strategy, the solution quality and the conditioning of the stiffness matrix in 3D linear fracture problems are investigated here. The crack surface is described by the Heaviside discontinuous functions and singular/crack front functions on a local scale. The solution of the local problem is used to enrich the approximation in the global domain, which may cause bad conditioning of the resulting system of equations and poor approximation errors in the solution. In order to overcome this problem, its projection into the linear polynomial space, according to the SGFEM strategy, is subtracted from the enrichment. Numerical examples, with different load and crack configurations, of linear elastic fracture mechanics are employed. Differently from other works, the meshes of the two scales are kept constant. Only the number of nodes associated with the local enrichment functions is changing. The impact on the accuracy and conditioning of the analysis is assessed, and the importance of using the SGFEM strategy is highlighted.

... Global-local enrichment functions are numerically constructed from the solution of local boundary value problems, using boundary conditions from a global problem defined on a coarse mesh. This numerical strategy is identified here as GFEM gl , and was recently expanded for the interaction and coalescence of multiple crack surfaces [28], fatigue crack simulations [29] and evaluation of stress intensity factors at spot welds [30]. ...

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.

... In such cases, a multi-scale analysis can be used under the G/XFEM approach, as proposed by Duarte and Kim (2008). Recent applications of the global-local modelling method to G/XFEM can be found in OHara et al. (2016a), OHara et al. (2016b), Plews and Duarte (2016), Li and Duarte (2018), Gerasimov et al. (2018) and Geelen et al. (2020). Another way of using both fine and coarse-scale meshes is considering a local mesh refinement associated with variable-node elements for the transition from fine to coarse-scale mesh. ...

Purpose
The 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/approach
Numerical 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.
Findings
Known 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/value
This 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.

... Further, coalescence of adjacent cracks is the main reason of having the sudden increases in shown in Fig. 6(a, b). Such observations are common in relevant published experimental works [1,65,69,70]. The fatigue limit of a material is defined as the maximum stress range below which there is no fracture after the application of an infinite number of loading cycles. ...

... On the other hand, the continuum smeared crack models require assumptions on the initial size and distribution of micro cracks, and they cannot fully describe the growth of dominant cracks leading to the macroscopic failure [8]. Other methods such as the cohesive zone method (CZM) [9][10][11], generalized/extended finite element method (G/XFEM) [12][13][14][15][16][17][18][19][20][21], meshless method [22,23], and augmented finite element method (A-FEM) [24] are currently being used to overcome the difficulties of sharp and smeared crack models. Most of these methods utilize a predefined crack path and need remeshing to predict crack propagation and fracture behavior in polycrystalline materials, which is challenging and often computationally expensive. ...

A phase-field model based on a modified form of the regularized formulation of Griffith's fracture theory is presented to investigate intergranular and transgranular crack propagations in polycrystalline brittle materials. Grains and grain boundaries are incorporated in the crack initiation and propagation model based on a phase-field model for grain growth, in which the elastic anisotropy varies based on the grain orientation angle, and the grain boundary energy is related to the misorientation angle of the adjacent grains. Correction parameters are utilized in the total free energy functional and mechanical equilibrium equations to consider the effect of material strength on crack nucleation and propagation independent of the regularization parameter. This allows controlling the strength and crack surface energy along the grain boundaries as a function of the misorientation angle in order to mediate intergranular and/or transgranular crack propagation. To demonstrate the capability of the proposed model, intergranular and transgranular crack propagation in ZrB 2 bicrystal systems under tensile loading are studied in detail. The effects of grain boundary misorientation angle, grain boundary inclination with respect to initial crack direction, and grain boundary strength (and/or crack surface energy) on the crack propagation path are investigated. Intergranular crack propagation can be promoted by specific combinations of grain boundary strength and crack surface energy, which can contribute to the fracture toughness of polycrystalline materials.

... It shows that appearance of the vice crack reduces the J-integral of the main crack significantly. This tendency is consistent with the crack shielding effect described by Oʹ Hara et al. [28]. With the decrease of a/c of the vice crack, the J-integral of the main crack decreases correspondingly. ...

For a positive displacement motor (PDM), the threaded joint connecting drive shaft shell (DSS) and universal shaft shell that is close to the bit is inclined to fracture. In this paper, elastic-plastic fracture performance of the threaded connection is simulated under both make-up torque and bending moment. Firstly, an FE model, which includes a cracked external thread of the DSS and an engaging internal thread of the universal shaft shell, is established and validated. Secondly, influences of both plastic deformation and the helix angle on fracture properties of the cracked thread are evaluated quantitatively. Meanwhile interactions between two cracks are also discussed. Finally, under the two kinds of loading conditions, i.e. loaded by pre-load only and loaded by both pre-load and bending moment, explicit relationships between characteristic J-integrals and the crack depth are obtained for the DSS.

... Based on their analyses, an elastoplastic analysis of the interaction problem is almost necessary in order to have accurate properties of the coalescence process. O'Hara et al. [19] discussed a two-scale generalized finite element method (FEM) approach to solve fracture mechanics and fatigue crack propagation problems. In their model, both macroand micro-crack surfaces modeled only in the local problem using the discontinuous and singular enrichment functions. ...

From the engineering point of view there are traditionally two ways to approach the fatigue design problem: safe-life and damage-tolerant. The damage-tolerant methodology focus on predictions of the fatigue crack growth rate and the remaining fatigue life whereas the safe-life design methodology focuses on estimating the total life. Thus, estimations of the remaining fatigue life of a flawed structure is only possible through use of the damage-tolerant approach. Fatigue is a process in engineering materials in which damage accumulates due to the fluctuating loading. One solution for a component under the fatigue process is to arrest the crack growth before the final failure using different available retardation methods. This paper aims to investigate the effect of the stop-hole retardation technique and multiple micro-cracks/voids on the fatigue crack growth (FCG) in a compact tension specimen. Considering the linear elastic fracture mechanics concept, the Paris law will be used to define the transition between crack initiation and crack growth period. Also, the extended finite element method will be adapted in the crack propagation phase in order to exclude the remeshing procedure. The whole analyses are conducted in a commercial package along with some user-written codes to make the FCG process easier.

A novel approach for mesoscale modelling of concrete composites is proposed by combining enriched scaled boundary finite element methods with quad-tree mesh. The concrete meso-structures are comprised of randomly distributed aggregates, mortar matrix, and interface transition zone. An improved random aggregate generation technique is developed to construct digital images of mesoscale concrete models. Based on the quadtree decomposition algorithm, meshes can be generated automatically from the digital images of concrete mesostructure. The whole mesh generation process is highly efficient without any artificial interference and eliminates the issue of hanging nodes faced by standard finite elements. Additionally, local remeshing is unnecessary as crack propagates. Three numerical examples are modelled to demonstrate the performance of the proposed approach, and the effects of aggregate area fraction on the mechanical properties of concrete composites are also discussed.

This paper presents a global–local strategy with the generalized finite element framework to simulate structural failure through nonlinear continuum damage models. The global problem is the scale of the structure which is discretized with relative coarse meshes while the local problem is a subdomain of interest of the global problem where refined meshes can be used to simulate localized crack growth. The proposed method uses the converged nonlinear local problem solution as enrichment functions for the global problem which is considered linear with no need for an iterative procedure to solve it. The strategy is validated and compared with reference solutions and experimental results with crack propagation in mode I and mixed-mode conditions in monotonic tests of quasi-brittle materials. The results show that the method has the capability to transfer the kinematic effects and the damage state variable that occurs in the local problem to accurately predict the global structural behavior under the damage process. The proposed strategy demonstrated the capability to accurately predict the experimental crack paths and load–displacement curves with a reduced number of iterations and degrees of freedom in relation to the conventional finite element and generalized finite element methods.

The generalized finite element method (GFEM) is versatile and powerful in the numerical analysis of various engineering problems. However, its application to large deformation analysis of elastoplastic solids is rare, especially in three dimensions (3D), since it suffers from singular system matrix and awkward implementation. This issue is caused by the extra degrees of freedom (DOFs) of GFEM. In this work, a nonlinear GFEM for large deformation analysis of elastic and elastoplastic solids in 3D is developed by using the extra-DOF-free enrichments proposed by Tian (2013) Updated Lagrangian (UL) formulation incorporating both geometric and material nonlinearities is employed and hyperelastic and hypoelastic–plastic constitutive models are considered. As a result of the elimination of the extra DOFs, implementation of the developed method is much more convenient than the standard GFEM and singular system matrix does not present. The capability of the proposed nonlinear GFEM in modeling three-dimensional large elastoplastic deformations is investigated by several typical examples. Numerical results demonstrate that, in comparison to the traditional finite element method, the proposed nonlinear GFEM is remarkably more accurate and stable. Particularly, in analysis of extreme deformations, the commercial software ABAQUS fails even when sophisticated elements are used, however the proposed nonlinear extra-DOF-free GFEM is still stable and converged results can be obtained.

Cracking releases the strain energy and decreases the stiffness of structures. From the force formulation standpoint, it is equivalent to an increase in flexibility. This fact, which is called compliance concept, is the base of many special cracked finite elements. In this paper, a new element of this type is proposed. The suggested element is applicable to the plane problems with internal or edge non-propagating cracks. General quadrilateral geometry and consideration of the inclined cracks are the two major improvement of the presented formulation in comparison to the existing ones. Authors’ element is applicable to the static and dynamic analysis of cracked problems. In addition, it is possible to compute stress intensity factors of the cracked structures using this element. Various numerical examples prove capabilities of the presented formulation.

A critical issue in the structural design of glazed surfaces is the evaluation of the strain consequent to temperature variations due to environmental actions such as solar radiation, which represents one of the main causes of breakage. In the practice, approximate solutions are used, where the temperature profile across the glass thickness is constant or linear, but the consequent thermal stress cannot be adequately estimated from these. On the other hand, sophisticated thermal software is available only for important tasks.
Here, we propose a semi-analytical approach, easily implementable in a simple FEM code, to evaluate the time-dependent temperature profile through the thickness of layered glazing, which is based on the variational method proposed by Biot in the Fifties. A prompt evaluation not only of the temperature field, but also of the heat flux, can be obtained. Compared to other numerical approaches, this method rigorously accounts for energy conservation and, since it does not involve temperature gradients in the formulation, it is particularly efficient for problems with steep temperature variations. Temperature profiles that are not necessarily linear can be approximated by Hermite splines, for a precise evaluation of the thermally-induced stress. Comparisons with a direct numerical solution of the heat-conduction differential equations confirm the accuracy and the effectiveness of the proposed approach.

The modelling of a crack propagation through a finite element mesh is of prime importance in fracture mechanics. We propose here a solution based on an advanced remeshing technique. A fully automatic remesher enables us to deal with multiple boundaries and multiple materials. The propagation of the crack is achieved with both remeshing and nodal relaxation. A maximal normal stress criterion is used to compute the crack direction. Several tools are developed and presented to obtain accurate results at the crack tip: evolving mesh refinement, crack tip finite elements, ring of elements surrounding the crack. Finally, several applications are presented to show the robustness of this technique.

This paper presents a novel numerical framework based on the generalized finite element method with global-local enrichments (GFEMgl) for two-scale simulations of propagating fractures in three dimensions. A non-linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters. The cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity (POU) concept provided by the generalized/extended finite element method (GFEM/XFEM), and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface. The proposed method employs fine-scale solutions of non-linear local boundary value problems extracted from the original global problem in order to not only construct scale-bridging enrichment functions but also to identify damaged states in the global problem, thus enabling accurate global solutions on coarse meshes. This is in contrast with available GFEMgl in which the local solution field contributes only to the kinematic description of global solutions. The robustness, efficiency, and accuracy of this approach are demonstrated by results obtained from representative numerical examples.

When two planar penny-shape cracks propagate and become sufficiently close to interact, the local stress intensity factor becomes no more constant along the fronts so that the cracks shape gradually deforms. The aim of this paper is to quantify these crack front deformations and their implication on the loading, up to their coalescence. The method used is based on numerical iterations of Bueckner-Rice weight functions perturbation approach which gives the variation of the stress intensity factor when the crack fronts are slightly perturbed in their plane. It is extended here to the case of several cracks. The advantage of this method in comparison to more standard finite element based methods is that the sole crack front lines have to be meshed and that the calculation of the mechanical fields is avoided. In fatigue, we show that for the most common materials, the deformations of the cracks are small and that the number of cycles leading to coalescence is smaller of a few percent than the one predicted for two isolated cracks. In brittle fracture, we notice, as soon as the size of the cracks becomes comparable to the distance between them, large deformations and considerable decrease of the threshold loading corresponding to the onset of crack propagation.

This paper presents an extension of a two-scale generalized finite element method (GFEM) to three-dimensional fracture problems involving confined plasticity. This two-scale procedure, also known as the generalized finite element method with global-local enrichments (GFEMgl), involves the solution of a fine-scale boundary value problem defined around a region undergoing plastic deformations and the enrichment of the coarse-scale solution space with the resulting nonlinear fine-scale solution through the partition-of-unity framework. The approach provides accurate nonlinear solutions with reduced computational costs compared to standard finite element methods, since the nonlinear iterations are performed on much smaller problems. The efficacy of the method is demonstrated with the help of numerical examples, which are three-dimensional fracture problems with nonlinear material properties and considering small-strain, rate-independent J2 plasticity.

This paper presents improvements to three-dimensional crack propagation simulation capabilities of the generalized finite element method. In particular, it presents new update algorithms suitable for explicit crack surface representations and simulations in which the initial crack surfaces grow significantly in size (one order of magnitude or more). These simulations pose problems in regard to robust crack surface/front representation throughout the propagation analysis. The proposed techniques are appropriate for propagation of highly non-convex crack fronts and simulations involving significantly different crack front speeds. Furthermore, the algorithms are able to handle computational difficulties arising from the coalescence of non-planar crack surfaces and their interactions with domain boundaries. An approach based on moving least squares approximations is developed to handle highly non-convex crack fronts after crack surface coalescence. Several numerical examples are provided, which illustrate the robustness and capabilities of the proposed approaches and some of its potential engineering applications. Copyright © 2013 John Wiley & Sons, Ltd.

This work presents a new multiscale technique to investigate advancing cracks in three dimensional space. This fully adaptive multiscale technique is designed to take into account cracks of different length scales efficiently, by enabling fine scale domains locally in regions of interest, i.e. where stress concentrations and high stress gradients occur. Due to crack propagation, these regions change during the simulation process. Cracks are modeled using the extended finite element method, such that an accurate and powerful numerical tool is achieved. Restricting ourselves to linear elastic fracture mechanics, the
$J$
J
-integral yields an accurate solution of the stress intensity factors, and with the criterion of maximum hoop stress, a precise direction of growth. If necessary, the on the finest scale computed crack surface is finally transferred to the corresponding scale. In a final step, the model is applied to a quadrature point of a gas turbine blade, to compute crack growth on the microscale of a real structure.

The main feature of partition of unity methods such as the generalized or extended finite element method is their ability of utilizing a priori knowledge about the solution of a problem in the form of enrichment functions. However, analytical derivation of enrichment functions with good approximation properties is mostly limited to two-dimensional linear problems. This paper presents a procedure to numerically generate proper enrichment functions for three-dimensional problems with confined plasticity where plastic evolution is gradual. This procedure involves the solution of boundary value problems around local regions exhibiting nonlinear behavior and the enrichment of the global solution space with the local solutions through the partition of unity method framework. This approach can produce accurate nonlinear solutions with a reduced computational cost compared to standard finite element methods since computationally intensive nonlinear iterations can be performed on coarse global meshes after the creation of enrichment functions properly describing localized nonlinear behavior. Several three-dimensional nonlinear problems based on the rate-independent J
2 plasticity theory with isotropic hardening are solved using the proposed procedure to demonstrate its robustness, accuracy and computational efficiency.

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.

A high-order generalized finite element method (GFEM) for non-planar three-dimensional crack surfaces is presented. Discontinuous p-hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element mesh and thus the proposed method allows fully automated fracture analysis using an existing finite element discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non-planar, non-smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non-planar curved crack fronts and the robustness of the method.

A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a posteriori error estimation for this new method are also proved. © 1997 by John Wiley & Sons, Ltd.

A new quadrature scheme and a family of hierarchical assumed strain elements have been developed to enhance the performance of the displacement-based hierarchical shell elements. Various linear iterative procedures have been examined for their suitability to solve system of equations resulting from hierarchic shell formulations.

The energy-based growth formulation and accompanying simulation technique introduced in Part I of this series is generalized in this work to predict arbitrary, mixed-mode, non-planar crack evolution. The implementation uses a novel basis-function approach to generate a crack extension expression, rather than relying on the local, point-by-point approach described in Part I. The basis-function expression dampens the effect of numerical noise on crack growth predictions that could produce numerically unstable simulation results. Two simulations are presented to demonstrate the technique’s ability to capture both general non-planar behavior, as well as local mixed-mode phenomena, e.g. “factory-roof” formation, along the crack front.

This paper describes a fracture simulation system, the FRacture ANalysis SYStem (FRANSYS), which will be capable of modeling general, three-dimensional crack propagation in an arbitrary body. This system is currently under development at the Program of Computer Graphics at Cornell University. The system design, philosophy, and some preliminary results are presented here. The purpose of FRANSYS is to allow a person knowledgeable about fracture mechanics and numerical solutions to partial differential equations to perform simulations of the behavior of structures containing cracks. In addition, FRANSYS provides a test bed in which new algorithms, theories, and ideas in numerical fracture mechanics can be quickly prototyped and checked with both simple and complex three-dimensional models. Current plans are to support both boundary elements and displacement formulated finite elements. The system, however, is not limited to these two computational techniques.

Keywords: Fracture propagation, stress intensity factors, generalized finite element method, extended finite element method, contour integral method. Abstract. 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 formulations are tailored for the Generalized Finite Element Method and mixed-mode 3-D propagating cracks. The case of crack faces loaded by prescribed tractions is also considered. Another contribution of this paper is a procedure to control the noise of extracted SIFs based on the Moving Least Squares Method. The proposed approach provides a continuous and smooth approximation of 3-D SIF functions for each fracture mode. Numerical experiments demonstrating the accuracy and robustness of the proposed methodology are presented. They include 3-D mixed-mode fatigue crack growth simulations and the case of a pressurized crack.

In this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered. Problems of the type considered arise in the analysis of unidirectional composites, where the coefficients represent the properties of the material. Several methods for this class of problems are presented, and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients. The methods are referred to as special finite element methods because they are of finite element type but employ special shape functions, chosen to accurately model the unknown solution.

This paper investigates a coupled computational analysis framework that uses reduced-order models and the generalized finite element method to model vibratory induced stress near local defects. The application area of interest is the life prediction of thin gauge structural components exhibiting nonlinear, path-dependent dynamic response. Full-order finite element models of these structural components can require prohibitively large amounts of processor time. Recent developments in nonlinear reduced-order models have demonstrated efficient computation of the dynamic response. These models are relatively insensitive to small imperfections. Conversely, the generalized finite element method provides the ability to model local defects without geometric dependency on the mesh. A more robust version of the method, with numerically built enrichment functions, provides a multiple-scale modeling capability through direct coupling of global and local finite element models. Replacing the component finite element model with a reduced-order model allows for efficient computation of dynamic response while providing the necessary information to drive local, solid analyses which can zoom in on regions containing stress risers or cracks. This paper describes the coupling of these approaches to enable fatigue and crack propagation predictions. Numerical/experimental examples are provided.

A new methodology to build discrete models of boundary-value problems is presented. The h-p cloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. This new method uses radial basis functions of varying size of supports and with polynomial-reproducing properties of arbitrary order. The approximating properties of the h-p cloud functions are investigated in this article and several theorems concerning these properties are presented. Moving least squares interpolants are used to build a partition of unity on the domain of interest. These functions are then used to construct, at a very low cost, trial and test functions for Galerkin approximations. The method exhibits a very high rate of convergence and has a greater flexibility than traditional h-p finite element methods. Several numerical experiments in 1-D and 2-D are also presented.

This article aims at evaluating the extent of the surface region in notched Middle Cracked Tension specimens. Firstly, a fully automatic fatigue crack growth technique is developed to obtain stable crack shapes. After that, the stress triaxiality along the crack front is evaluated for different notch shapes. Then, objective criteria are defined to quantify the extent of the surface region from the stress triaxiality data collected. Next, the extent of the surface region is related to the elastic stress concentration factor of the uncracked geometry by a linear relationship. Finally, empirical two-constant equations able to evaluate the extent of the surface region from the thickness, notch radius, notch depth and elastic stress concentration factor are formulated.

A finite-element-based simulation technique has been developed to predict arbitrary shape evolution of 3-D, geometrically explicit, planar cracks under stable growth conditions. Point-by-point extensions along a crack front are predicted using a new, energy-based growth formulation that relies on a first-order expansion of the energy release rate. The crack-growth formulation is incorporated into an incremental-iterative solution procedure that continually updates the crack configuration by re-meshing. The numerical technique allows crack shapes to evolve according to energy-based mechanics, while reducing the effects of computational artifacts, e.g. mesh bias. Three crack growth simulations are presented as verification of the new simulation technique.

This work presents a new multiscale technique for the efficient simulation of crack propagation and crack coalescence of macrocracks and microcracks. The fully adaptive multiscale method is able to capture localization effect mesh independently. By modeling macrocracks and microcracks, the extended finite element method offers an accurate solution and captures cracks and their propagation without changing the mesh topology. Propagating and coaliting cracks of different length scales can therefore be easily modeled and updated during the computation process. Hence, the presented method is an efficient and accurate option for modeling cracks of different length scales. This is demonstrated in several numerical examples showing the interaction of microcracks and macrocracks. Copyright © 2012 John Wiley & Sons, Ltd.

In this paper, heat transfer problems with sharp spatial gradients are analyzed using the Generalized Finite Element Method with global-local enrichment functions (GFEM
gl). With this approach, scale-bridging enrichment functions are generated on the fly, providing specially-tailored enrichment functions for the problem to be analyzed with no a-priori knowledge of the exact solution. In this work, a decomposition of the linear system of equations is formulated for both steady-state and transient heat transfer problems, allowing for a much more computationally efficient analysis of the problems of interest. With this algorithm, only a small portion of the global system of equations, i.e., the hierarchically added enrichments, need to be re-computed for each loading configuration or time-step. Numerical studies confirm that the condensation scheme does not impact the solution quality, while allowing for more computationally efficient simulations when large problems are considered. We also extend the GFEM
gl to allow for the use of hexahedral elements in the global domain, while still using tetrahedral elements in the local domain, to allow for automatic localized mesh refinement without the use of constrained approximations. Simulations are run with the use of linear and quadratic hexahedral and tetrahedral elements in the global domain. Convergence studies indicate that the use of a different partition of unity (PoU) in the global (hexahedral elements) and local (tetrahedral elements) domains does not adversely impact the solution quality.

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.

This paper describes a hierarchical overlay of a p-version finite element approximation on a coarse mesh and an h-approximation on a geometrically independent fine mesh. The length scales of the local problem may be some orders of magnitude below the scale of the global problem. Despite the incompatibility of the meshes used, continuity can easily be guaranteed in the proposed method. The paper shows how finite element meshes can be constructed adaptively on the local and the global scales. It is demonstrated how a block-iteration allows a simple and efficient implementation of the method. Typical fields of application and the efficiency of the method are shown in a numerical example.

A methodology to improve the quality of the finite element calculations in the regions of unacceptable errors has been developed. Unlike the existing adaptive techniques, where either the mesh is refined (h-version), or the polynomial order is increased (p-version), or a combination of both (h−p version), the s-version increases the resolution by superimposing additional mesh(es) of higher-order hierarchical elements. C0 continuity of the displacement field is maintained by imposing homogeneous boundary conditions on the superimposed field in the portion of the boundary which is not contained within the boundary of the problem. The superimposed regions can be of arbitrary shape, unlimited by the problem geometry, boundary conditions and the underlying mesh topography. Numerical experiments for linear problems involving singularities and smooth solutions as well as the shear banding problem in viscoplastic solid, are presented to validate the present formulation.

In spite of the advances in computer technology, there is still a need for more computationally efficient methods for performing stress analysis. One approach which is receiving increasing attention is global/local analysis. Such analyses can take a variety of forms. The form described herein uses two distinct meshes (one global and one local), but retains the same level of accuracy as one would obtain if one was to use a single refined global mesh. The accuracy is retained by using an iterative procedure to enforce equilibrium between the global and local regions. The procedure was tested for three configurations. The good performance observed indicates that the iterative global/local procedure warrants further examination.

This paper describes a pilot design and implementation of the generalized finite element method (GFEM), as a direct extension of the standard finite element method (SFEM, or FEM), which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the FEM. The development of the GFEM is illustrated for the Laplacian in two space dimensions in domains which may include several hundreds of voids, and/or cracks, for which the construction of meshes used by the FEM is practically impossible. The two main capabilities are: (1) It can construct the approximation using meshes which may overlap part, or all, of the domain boundary. (2) It can incorporate into the approximation handbook functions, which are known analytically, or are generated numerically, and approximate well the solution of the boundary value problem in the neighborhood of corner points, voids, cracks, etc. The main tool is a special integration algorithm, which we call the Fast Remeshing approach, which is robust and works for any domain with arbitrary complexity. The incorporation of the handbook functions into the GFEM is done by employing the partition of unity method (PUM). The presented formulations and implementations can be easily extended to the multi-material medium where the voids are replaced by inclusions of various shapes and sizes, and to the case of the elasticity problem. This work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the GFEM, which is needed before analogous implementations are attempted in the three-dmensional and nonlinear cases, which are the cases of main interest for future work.

This paper is dedicated to my friend and colleague Professor Dr Jaap Schijve, required to retire like many good aircraft components according to a «safe life» criterion but who is still young in mind, body and spirit, has not yet reached his unfactored endurance limit, and who could continue subject to a needed change in philosophy to «retirement for cause». Professor Schijve has dedicated his life to aviation safety through his outstanding continuous research into aircraft fatigue and fracture phenomena. The Federal Aviation Administration, whose primary goal is aviation safety, is indebted to Professor Schijve for his continous counsel through many contributions to the literature in his subject

Numerical analyses based on the finite element (FE) method and remeshing techniques have been employed in order to develop a damage tolerance approach to be used for the design of aeroengines shaft components. Preliminary experimental tests have permitted the calculation of fatigue crack growth parameters for the high strength alloy steel adopted in this research. Then, a robust numerical study have been carried out to understand the influence of various factors (such as: crack shape, crack closure) on non-planar crack evolution in solid and hollow shafts under mixed-mode loading. The FE analyses have displayed a satisfactory agreement compared to experimental data on compact specimens (CT) and solid shafts.

In order to develop a procedure for assessing the growth of interacting surface cracks, the relationship between the area of the crack face and fatigue crack growth behavior was investigated. Fatigue crack growth tests were conducted using stainless steel plate specimens with surface notches. Then, finite element analyses were performed to simulate the growth behavior obtained by the experiment. It was shown that the change in area can be predicted by assuming the extension of crack front based on evaluated stress intensity factor at each position along the front. Based on experimental and analysis results, it was revealed that the growth of interacting surface cracks as well as independent cracks can be represented well by change in area and showed good correlation with the driving force based on area. It was also shown that, in the case of parallel cracks, the area on the projected plane was dominant. It was concluded that, when the magnitude of the interaction is sufficiently large, by replacing the two cracks with a semi-elliptical crack of the same area on the projected plane, the growth in area can be predicted precisely.

Short duration tensile stress pulses have been used to study the interaction of two parallel cracks in a brittle solid. The interaction of these cracks was found to agree very well with that predicted by Yokobori et al.[5]. The significance of these observations to evaluation of flaw severity and cleavage step formation is pointed out.

In this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered. Problems of the type considered arise in the analysis of unidirectional composites, where the coefficients represent the properties of the material. Several methods for this class of problems are presented, and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients. The methods are referred to as special finite element methods because they are of finite element type but employ special shape functions, chosen to accurately model the unknown solution.

A comprehensive treatment of fracture mechanics suitable as a graduate
text and as a reference for engineers and researchers is presented. The
general topics addressed include: fundamental concepts of linear elastic
and elastic-plastic fracture mechanics; dynamic and time-dependent
fracture mechanics; micromechanisms of fracture in metals and alloys;
fracture mechanisms in polymers, ceramics, and composites; applications
to fracture toughness testing of metals and nonmetals, to structures,
fatigue crack propagation, and computational fracture mechanics.
Reference materials usually found in fracture mechanics handbooks is
provided.

A minimal remeshing finite element method for crack growth is presented. Discontinuous enrichment functions are added to the finite element approximation to account for the presence of the crack. This method allows the crack to be arbitrarily aligned within the mesh. For severely curved cracks, remeshing may be needed but only away from the crack tip where remeshing is much easier. Results are presented for a wide range of two-dimensional crack problems showing excellent accuracy. Copyright © 1999 John Wiley & Sons, Ltd.

This paper presents the results of an experimental study of fatigue crack coalescence and interaction in transparent polycarbonate test specimens. The specimen geometries studied were a plate with a centered hole loaded in remote tension and a beam subjected to cyclic bending. Each specimen contained an initial pair of surface and/or corner cracks which were grown until coalescence of the two flaws occurred. Crack length vs elapsed cycles data generated from the tests were compared with predictions made by a computer algorithm. Good predictions were obtained for the period required for the two initial flaws to coalesce, but the numerical results were somewhat conservative after the cracks coalesced into a single flaw.

We present a new multiscale method for crack simulations. This approach is based on a two-scale decomposition of the displacements and a projection to the coarse scale by using coarse scale test functions. The extended finite element method (XFEM) is used to take into account macrocracks as well as microcracks accurately. The transition of the field variables between the different scales and the role of the microfield in the coarse scale formulation are emphasized. The method is designed so that the fine scale computation can be done independently of the coarse scale computation, which is very efficient and ideal for parallelization. Several examples involving microcracks and macrocracks are given. It is shown that the effect of crack shielding and amplification for crack growth analyses can be captured efficiently. Copyright © 2007 John Wiley & Sons, Ltd.

A methodology is developed to simulate adaptively and hierarchically fatigue crack growth in structural components. Cracks are modelled, by overlaying portions of the finite element mesh free of cracks with a discontinuous finite element field containing unconstrained double nodes along the discontinuity. Crack propagation is simulated by advancing the crack front in the superimposed mesh only keeping the underlying mesh fixed. Adaptivity in time and space domain together with the hierarchical nature of the method ensure both economical and reliable simulation of crack propagation. Numerical results of fatigue crack growth in the attachment lug were found to be, in good agreement with the experimental data.

An adaptive multi-level methodology is developed in this paper to create a hierarchy of computational sub-domains with varying resolution for multiple scale problems. It is intended to concurrently predict evolution of variables at the structural and microstructural scales, as well as to track the incidence and propagation of microstructural damage in composite and porous materials. The microstructural analysis is conducted with the Voronoi cell ®nite element model (VCFEM), while a conventional displacement based FEM code executes the macroscopic analysis. The model intro-duces three levels in the computational domain which include macro, macro±micro and microscopic analysis. It dif-ferentiates between non-critical and critical regions and ranges from macroscopic computations using continuum constitutive relations to zooming in ahotspots' for pure microscopic simulations. Coupling between the scales in regions of periodic microstructure is accomplished through asymptotic homogenization. An adaptive process signi®-cantly increases the eciency while retaining appropriate level of accuracy for each region. Numerical examples are conducted for composite and porous materials with a variety of microscopic architectures to demonstrate the potential of the model.

A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a posteriori error estimation for this new method are also proved. © 1997 by John Wiley & Sons, Ltd.

An extended finite element method (X-FEM) for three-dimensional crack modelling is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three-dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.

In this paper, a new multiscale–multiphysics computational methodology is devised for the analysis of coupled diffusion–deformation problems. The proposed methodology is based on the variational multiscale principles. The basic premise of the approach is accurate fine-scale representation at a small subdomain where it is known a priori that important physical phenomena are likely to occur. The response within the remainder of the problem domain is idealized on the basis of coarse-scale representation. We apply this idea to evaluate a coupled mechano-diffusion problem that idealizes the response of titanium structures subjected to a thermo–chemo–mechanical environment. The proposed methodology is used to devise a multiscale model in which the transport of oxygen into titanium is modeled as a diffusion process, whereas the mechanical response is idealized using concentration-dependent elasticity equations. A coupled solution strategy based on operator split is formulated to evaluate the coupled multiphysics and multiscale problem. Numerical experiments are conducted to assess the accuracy and computational performance of the proposed methodology. Numerical simulations indicate that the variational multiscale enrichment has reasonable accuracy and is computationally efficient in modeling the coupled mechano-diffusion response. Copyright © 2011 John Wiley & Sons, Ltd.

A combination of the extended finite element method (XFEM) and the mesh superposition method (s-version FEM) for modelling of stationary and growing cracks is presented. The near-tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity. The two displacement fields are matched through a transition region. The method can robustly deal with stationary crack and crack growth. It simplifies the numerical integration of the weak form in the Galerkin method as compared to the s-version FEM. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.