Crack Tip Enrichment Research Articles

Modeling quasi-static crack propagation using preconditioned numerical manifold method

The numerical manifold method (NMM) is robust and effective in modeling complex fracture problems as it allows the fracture paths to intersect meshes and uses enrichments with Williams expansions to capture the tip singularities. However, mesh cutting and crack-tip enrichment lead to small cut ratios and linear dependence, respectively, for complicated fracture problems. These two situations constantly cause serious ill-conditioning issues. Therefore, ill-conditioning is extremely challenging for crack propagation modeling. A preconditioning strategy was developed previously for NMM to deal with the ill-conditioning for static crack problems (Liu et al., 2023). In this study, we employ the preconditioned numerical manifold method (PNMM) for complicated crack propagation problems. Furthermore, we use a load factor to ensure that the mechanical mechanism and fracture criterion are satisfied simultaneously for multiple crack propagation problems. Numerical examples demonstrate the remarkable stability and efficiency of PNMM for multiple crack propagation problems.

Improved XFEM for 3D interfacial crack modeling

The conventional extended finite element method (XFEM) usually suffers from the crack tip enrichment for removing the blending elements. This may be inconvenient when treating the three dimensional (3D) and curved crack modeling since the crack tip enrichment is usually defined in the local coordinate system depending on the crack front. To avoid these shortcomings of the XFEM, an improved enrichment scheme is proposed for the 3D interfacial crack modeling. The proposed enrichment scheme removes the crack tip enrichment and two weight functions are introduced to capture the strong and weak discontinuities in the element including the crack front. The new enrichment scheme has advantages for the 3D curved interfacial cracks since all the enrichment functions are defined in the global system and it is unnecessary to introduce additional coordinate systems. Several benchmark problems including the mixed mode stress intensity factors (SIFs), anti-plane load and curved cracks are analyzed by the proposed method with the local refinement technology. Numerical results demonstrate that the proposed method can obtain accurate 3D energy release rate and SIFs. In addition, the proposed method enhances the applications of XFEM and has the potential abilities to the multiple-fields coupling and interfacial crack growth phenomena due to the fact that compatible enrichment functions are utilized.

Enriched physics-informed neural networks for 2D in-plane crack analysis: Theory and MATLAB code

In this paper, a method based on the physics-informed neural networks (PINNs) is presented to analyze 2D in-plane crack problems in the linear elastic fracture mechanics. Instead of using a discretizing mesh, the PINNs-based method is meshless and can be trained on batches of discretely sampled collocation points. In order to capture the asymptotic behavior of the near-tip displacement and stress fields, the standard PINNs formulation is enriched here by including the crack-tip asymptotic functions such that the local behavior in the crack-tip region can be modeled accurately without a nodal refinement. The trainable parameters of the enriched PINNs are learned to satisfy the governing partial differential equations of the cracked elastic solid and the corresponding boundary conditions. It is found that the incorporation of the crack-tip enrichment functions in the PINNs is substantially simpler and more feasible than in the conventional finite element method (FEM) or boundary element method (BEM). The present PINNs-based algorithm is tested and verified by several representative benchmark examples for different loading and crack-mode types. Numerical results show that the present PINNs-based method can provide highly accurate stress intensity factors (SIFs) with only few degrees of freedom. For interested readers, a self-contained MATLAB code and data-sets supplementing this paper are also provided in the Supplementary material.

Preconditioned numerical manifold method for linear elastic fractures

Fractures have attracted the attention of computational scientists for several decades. The modeling and simulation of fractures have been a major motivation for developing enriched finite element methods (FEMs), such as the numerical manifold method (NMM). However, ill-conditioning has always haunted NMM and other enriched FEMs when they are utilized for linear elastic fracture problems. Generally, ill-conditioning for a fracture problem is caused by two main issues: the arbitrary cut of the mesh by the fracture path and linear dependence related to the crack-tip enrichments. It is significantly challenging to overcome these two types of ill-conditioning using a single technique. In this study, we employ a preconditioner based on global normalization and local Gram–Schmidt orthogonalization of bases to eliminate these two ill-conditioning issues in NMM entirely and simultaneously. Various numerical examples have demonstrated that the proposed preconditioning strategy is highly effective in reducing the condition number and iteration counts of a iterative solver. It is highly robust, stable, and efficient and can be incorporated into enriched FEM programs to significantly facilitate the analyses of linear elastic fractures.

Numerical Modeling Technique of Damage Behavior of MaSonry-Infilled RC Frames

The damage pattern of masonry-infilled reinforced concrete (RC) frame structures in earthquake events is complicated, and understanding the detailed failure behavior of these structures and modeling it accurately has been a challenging task. In this paper, the extended finite element method (XFEM) is introduced to reproduce arbitrary cracks initiating and propagating in concrete frame and masonry units, combined with interface elements to model various behaviors of masonry-infilled RC frames. Within the finite element analysis program FEAP, a user element subroutine is adopted for the incorporation of XFEM and two types of extended finite elements with and without crack tip enrichments are built to simulate the behavior of concrete material for frame members and masonry blocks for the infill panel, respectively. In addition, a macro command is created to check the crack-propagation criterion and update crack and enrichment information. Furthermore, numerical examples are performed with existing test data, which reveal the efficiency of the implementation procedure. A comparison of the analytical and experimental results show that the proposed modeling can be used to predict the crack and failure process and the load-bearing capacity curves of the structures and reflect accurately the interaction of masonry infill and RC frames.

Three-dimensional non-planar crack growth analysis using enriched finite elements

The paper presents computational mixed mode non-planar fatigue crack growth analyses using Fracture and Crack Propagation Analysis System (FCPAS), which employs enriched crack tip finite elements to calculate the mixed mode stress intensity factors (SIFs) and utilizes from other sub-modules to predict the evolving crack growth trajectories and the resulting fatigue crack growth lives. Numerical applications on different problems selected from the literature; a hollow shaft, an H-shaped specimen and cylindrical rods containing cracks with different initial orientations and sizes and subjected to different fatigue loading types are presented. Comparisons of results from the numerical analyses with the corresponding data from the literature agree well in terms of computed SIFs, predicted evolving crack paths and crack propagation lives.

Modeling the transient dynamic fracture and quasi-static crack growth in cracked functionally graded composites by the extended four-node gradient finite elements

This paper presents numerical simulations of transient dynamic fracture behaviors and quasi-static crack growth in cracked functionally graded composites using the enhanced extended consecutive-interpolation quadrilateral element (XCQ4). The mechanical properties of functional composites are assumed to vary continuously in spatial coordinates. In terms of XCQ4 modeling, only one function of ramp type (either linear, quadratic or cubic order), alternatively to the usual four branch functions, is used as crack-tip enrichment, somewhat reducing the complexity in implementation. Another advantage of employing the ramp function, which is not derived from asymptotic solutions, lies in the less additional degrees of freedom (DOFs) required for the branch functions, saving the computational effort. The merits of the developed approach are demonstrated through our numerical experiments, which are devoted to both dynamic loading with stationary cracks and quasi-static crack growth simulation in FGMs composites. The accuracy of the developed model is verified by comparing the computed results with respect to reference solutions derived from analytical solution, other numerical methods, and experimental data.

Fracture Modelling of a Cracked Pressurized Cylindrical Structure by Using Extended Iso-Geometric Analysis (X-IGA)

In this study, a NURBS basis function-based extended iso-geometric analysis (X-IGA) has been implemented to simulate a two-dimensional crack in a pipe under uniform pressure using MATLAB code. Heaviside jump and asymptotic crack-tip enrichment functions are used to model the crack’s behaviour. The accuracy of this investigation was ensured with the stress intensity factors (SIFs) and the J-integral. The X-IGA—based SIFs of a 2-D pipe are compared using MATLAB code with the conventional finite element method available in ABAQUS FEA, and the extended finite element method is compared with a user-defined element. Therefore, the results demonstrate the possibility of using this technique as an alternative to other existing approaches to modeling cracked pipelines.

Extended Voronoi cell finite element methods for fracture analysis of composites

The extended Voronoi cell finite element model used to solve the fracture problems of isotropic materials and particle reinforced composites has been proven to be accurate and efficient. In this paper, the extended Voronoi cell finite element method is developed to analyse cracked orthotropic composite structures. The X-VCFEM is enhanced by orthotropic crack-tip enrichment stress functions to accurately capture the crack tip singularity. In addition, the stress functions also include polynomial stress functions. For elliptical internal cracks, reciprocal stress functions associated with polynomial functions is adopted, which are obtained by conformal mapping technique. The traction-free at crack face is introduced into functionals by the Lagrange multiplier method. The proposed method is applied for numerical investigation of fracture mechanics parameters for orthotropic composites, in which the stress intensity factors (SIFs) are calculated using the least square method. The effectiveness of the proposed method is validated by comparing with the results from ABAQUS or reference solutions. This method yields accurate results with fewer elements, which is the advantage of the proposed method.

An oblique circular cylinder element for 3D interfacial cracks in composites

This study further extends a developed symplectic analytical singular element (SASE) library to include three-dimensional (3D) cracks in dissimilar materials. We construct a trial symplectic eigen solution for 3D bimaterial interfacial cracks and use it as the crack tip enrichment to define the interior displacement field of a new singular element. The symplectic approach is applied for the numerical modelling of 3D bimaterial cracks for the first time. In the derivation, a non-standard shape of the 3D crack-tip element, i.e., a cylinder with two oblique ends with arbitrary orientations, is considered to better fit the mesh of curved 3D cracks. It is proven that the proposed 3D element has improved the accuracy and is still convenient to use. The present contribution has provided a new insight for choosing effective and efficient crack tip enrichments of 3D interfacial cracks, and is also an important step forward of the SASE library.

Explicit edge-based smoothed numerical manifold method for transient dynamic modeling of two-dimensional stationary cracks

Modeling and prediction of dynamic behaviors of cracked bodies are vital to the safety evaluation of structures. The numerical manifold method (NMM) has achieved considerable success in crack analysis as it provides a powerful representation of complex discontinuities. Strain smoothing has been incorporated into enriched methods such as NMM to improve their accuracy, efficiency, and tolerance to mesh distortion. However, conventional strain smoothing cannot reproduce the large spatial variance in the strain surrounding crack tips. Consequently, the accuracy is reduced when it is applied to crack-tip enrichment bases. High-fidelity representation of crack-tip fields can be achieved by crack-tip enrichment. Therefore, it is desirable to combine compatible strain for the enriched areas with edge-based smoothed strain for the remaining areas. When explicit time integration is employed, it is beneficial to use a lumped mass matrix because it provides a larger critical time step and a straightforward solution of the linear algebraic equation. A local-approximation-based mass lumping strategy is used to obtain a block-diagonal mass matrix with small blocks. Finally, the proposed methodology is applied to various static and dynamic crack analyses, and highly accurate and stable results are obtained.

Extended isogeometric analysis for fracture in functionally graded magneto-electro-elastic material

In this work, we present an extended isogeometric analysis (XIGA) for cracked functionally graded magneto-electro-elastic (FGMEE) material. The material properties inside the FGMEE domain vary exponentially. An electrically and magnetically impermeable crack in the FGMEE domain is modeled by adding appropriate enrichment functions into the isogeometric analysis (IGA) approximation. Heaviside function is utilized to model the displacement, electric, and magnetic potential along the crack face. In contrast, conventional four-field crack tip enrichment functions are used to mimic the crack tip singularity characteristics. The generalized intensity factors (IFs), i.e., the stress intensity factors (SIFs), electric displacement intensity factor (EDIF) and magnetic induction intensity factor (MIIF), are assessed using magneto-electro-mechanical interaction integral. A series of numerical studies over the cracked homogeneous and FGMEE material are presented. The accuracy of XIGA is examined by comparing the obtained results with the reference solutions. Various parameters such as volume fraction, crack length, loading combination parameter (both electrical and magnetic), material non-homogeneity parameter and crack orientation are taken to examine their effects on the generalized IFs.

An explicit crack-tip element for stationary dynamic cracks

A crack-tip element to accurately capture dynamic response of three-dimensional (3D) cracks is proposed in the present contribution. Unlike other extended finite element methods (XFEM) which use branch functions from the analytical solution of two-dimensional cracks as the crack-tip enrichment, we construct a 3D crack-tip solution (in the form of eigen expansion) for the problem investigated. On that basis, the finite element formulation is derived strictly. It is interesting to find that the proposed crack-tip element works perfectly with an explicit precise integration method and thus gives high accuracy and efficiency. Thanks to the advantages of the precise integration method, the present method is essentially an explicit scheme and is unconditionally stable. The advantages of the combined crack-tip element and precise integration method provide a new insight of modelling 3D dynamic crack which is normally considered to be a complicated and costly task.

Dynamic crack propagation analysis combined the stable scheme and continuous-discontinuous cellular automaton

A continuous-discontinuous cellular automaton method (CDCA) is developed for dynamic crack propagation. First, a separated time integral scheme is built, in which time difference is divided into two separated time steps: one balances the change of the geometric discretization caused by crack growing, and the other balances the dynamic loading. Second, an adaptive energy conservation strategy is proposed for the CDCA that includes the change in the degree of nodal freedom for the crack tip element, and a new strategy for the inheritance of the degrees of nodal freedom is proposed. This method retains the enrichment scheme for the discontinuity, and the Shepard interpolation method is employed to transform the variables’ values, so as to keeps the energy conservation caused by crack tip location change. Then, the enriched degree of nodal freedom values are easily passed from the crack tip enrichment to the penetrated enrichment and crack tip local coordinate system changing with crack growing. Third, a dynamic crack propagation criterion is developed, which can be used for tensile fractures and compression shear fractures. Based on those theories, a new cellular automaton model is built for dynamic crack propagation. Compared to the extended finite element method (XFEM), the calculation efficiency has been greatly improved, and several numerical examples are given to illustrate accuracy and efficiency of the present method.

Extended finite volume method with enriched HPCK shape functions for dynamic crack propagation modeling

In the present work, an extended finite volume method (EFVM) is incorporated with the enriched hp-cloud approximate function which has Kronecker delta property (EHPCK) for modeling crack propagation in 2D media. In this approximation function, appropriate crack path enrichment and crack tip enrichment functions are considered for modeling the discontinuous displacement field of cracked bodies. Unlike the conventional FVM approach, in the proposed EFVM, excessive control volumes (CVs) are embedded inside the main CVs which are called here as enrichment control volumes (ECVs). The EFVM method doesn't restrict the crack propagation path to obey any predefined path or edges of control volumes. To increase the prediction accuracies, the subtriangular technique is applied for the integration on the cracked control volumes. Capability of introduced EFVM is investigated by solving different problems such as dynamic stress intensity factor calculations and dynamic crack propagation.

A polygonal finite element approach for fatigue crack growth analysis of interfacial cracks

The fatigue crack growth analysis of an interfacial cracked structure plays an essential role in the design process as well as the safety assessment for various engineering fields in real life. This paper proposes a novel and effective computational approach based on polygonal meshes for crack growth simulation of interfacial crack problems. Polygonal meshes are fully exploited to provide greater flexibility in mesh generation for complicated domains as well as obtain more accuracy in terms of the numerical results. The twelve crack tip enrichment functions to capture the near-tip oscillating singular field are first implemented for polygonal meshes to improve the obtained results. The Wachspress shape function is applied for arbitrary polygonal elements. The strong discontinuities e.g. crack face, crack tip can be modeled, respectively, via Heaviside and asymptotic crack tip enrichment functions while the material interface discontinuities are defined by signed distance functions. The stress intensity factors (SIFs) of the interfacial crack problems can obtain by using the domain based interaction integral approach while the fatigue life is estimated via the Paris law. The accuracy and efficiency of the present approach are demonstrated through several numerical investigations regarding the crack growth problems in both homogeneous materials and bi-materials. The results which are generated from the proposed approach are compared with other earlier ones as well as the conventional XFEM solutions.

Analysis of cracked functionally graded piezoelectric material using XIGA

In this paper, an extended isogeometric analysis (XIGA) is presented to model the crack in functionally graded piezoelectric materials (FGPMs). The material properties of functionally graded piezoelectric material (FGPM) vary in an exponential manner along the length of the domain. To model a crack in the FGPM, the isogeometric analysis (IGA) approximation is enriched with the Heaviside and crack tip enrichment functions. Conventional four-fold enrichment functions of isotropic materials are employed to capture the crack tip singularity. To determine the fracture parameters, the domain form of electromechanical interaction integral is used. The accuracy of the proposed XIGA approach is highlighted by validating the several cracked homogeneous and functionally graded piezoelectric material against the analytical/reference solution available in the literature. Several example problems of cracked FGPM domain are solved by XIGA. The influence of material non-homogeneity parameter, loading combination parameter and crack length are examined on the stress intensity factors (SIFs) and electrical displacement intensity factor (EDIF).

A semi-homogenized extended isogeometric analysis approach for fracture in functionally graded materials containing discontinuities

In this paper, we present a semi-homogenized extended isogeometric analysis approach for assessment of stress intensity factor of center cracked functionally graded domain containing discontinuities such as holes and inclusions. In the semi-homogenized approach, the presence of discontinuities is considered in the 30% region in the vicinity of crack, and the remaining part of the domain is modeled using equivalent homogenous material properties. The Heaviside and singular crack tip enrichment functions are employed to model the center crack, whereas crack topology is tracked using level set functions. The domain form of interaction integral approach is employed to estimate the stress intensity factors at the tips of the center crack. Several problems containing discontinuities in 30% portion of the domain are investigated using the semi-homogenized approach. The efficiency and accuracy of the developed method are examined by comparing the computed results with those attained by modeling discontinuities in the entire domain. The advantages of the semi-homogenized extended isogeometric analysis approach are highlighted in terms of higher computational efficiency.

Fracture mechanism simulation of inhomogeneous anisotropic rocks by extended finite element method

The vast majority of rock masses is anisotropic due to factors such as layering, unequal in-situ stresses, joint sets, and discontinuities. Meanwhile, given the frequently asymmetric distribution of pores, grain sizes or different mineralogical compounds in different locations, they are often classified as inhomogeneous materials. In such materials, stress intensity factors (SIFs) at the crack tip, which control the initiation of failure, strongly depend on mechanical properties of the material near that area. On the other hand, crack propagation trajectories highly depend on the orthotropic properties of the rock mass. In this study, the SIFs are calculated by means of anisotropic crack tip enrichments and an interaction integral are developed for inhomogeneous materials with the help of the extended finite element method (XFEM). We also use the T-stress within the crack tip fields to develop a new criterion to estimate the crack initiation angles and propagation in rock masses. To verify and validate the proposed approach, the results are compared with experimental test results and those reported in the literature. It is found that the ratio of elastic moduli, shear stiffnesses, and material orientation angles have a significant impact on the SIFs. However, the rate of change in material properties is found to have a moderate effect on these factors and a more pronounced effect on the failure force. The results highlight the potential of the proposed formulation in the estimation of SIFs and crack propagation paths in inhomogeneous anisotropic materials.

Adaptive orthotropic XIGA for fracture analysis of composites

This paper is concerned with the numerical investigation of fracture mechanics parameters for cracked orthotropic composite structures by using an efficient computational approach. The adaptive extended isogeometric analysis (XIGA) based on locally refined (LR) B-splines is thus developed. This adaptive XIGA is enhanced by both signed-distance and orthotropic crack-tip enrichment functions to represent strong discontinuity and singularity induced by crack, respectively. In addition, to improve the integration accuracy, the ‘sub-triangle’ and ‘almost polar’ techniques are adopted for the cut and crack tip elements, respectively. For adaptivity, the smoothed stress field is achieved to develop the posteriori error estimator based local refinement, according to the Zienkiewicz–Zhu estimation. Such posteriori error estimator is then used to determine refinement domains, where the local refinement takes place. The proposed approach is applied for fracture analyses of orthotropic composites, in which the stress intensity factors (SIFs) are evaluated using the contour interaction integral technique. The accuracy of the proposed adaptive orthotropic XIGA is validated through a comparison of SIFs obtained from the proposed method and the available reference solutions. Furthermore, numerical results show that the convergence rate of the adaptive local refinement is faster than that of the global refinement approach.