3D Crack Problems Research Articles

Crack propagation arrest by the Joule heating in micro/nano-sized structures

The Joule heating technique is utilized to arrest the crack propagation so to protect some important structures being further damaged. In this approach there are created compression stresses at the crack tip vicinity due to a specific temperature distribution induced by the electric current. The Joule heating is amulti-physical problem which is applied here to micro/nano-sized structures. The size-effects are considered for description of both the elastic and the thermal fields. The heat conduction can’t be well described by classical Fourier’s law in nano-sized structures. A novel gradient theory is developed in such structures adopting the size effect of heat conduction. Besides, the strain gradients and double stresses are introduced into the constitutive laws. Then, the governing equations are given by partial differential equations with order of derivatives being higher than in classical theory. The principle of virtual work is the base of weak formulation and derivation of the discretized finite element equations for ageneral 2d boundary value problem with cracks. The collocation mixed FEM with large strain gradients is developed for numerical solution, where the C0 continuous approximation is applied independently to displacements and strains and also to temperature and temperature gradients. Kinematic constraints between strains and displacements are satisfied by the collocation method at selected interior points. The collocation mixed FEM is applied to solve 2D crack problems with Joule heating.

A review of three‐dimensional weight function methods for the analysis of various surface/corner crack problems

AbstractThe stress intensity factor (SIF) is the foundation of fracture mechanics analysis. Accurate determination of SIFs is at the very heart of damage tolerance design and fatigue crack growth life prediction. The weight function method (WFM) is a powerful method for SIF‐determination involving complex load conditions. This article presents a review of the historical development over the past five decades and the current state‐of‐the‐art in three‐dimensional (3D) WFMs. The discussions are focused on the slice synthesis weight function method (SSWFM) and the point weight function method (PWFM). The powerfulness of the 3D WFMs is shown by a variety of examples with complex part‐through crack configurations under uni‐ and bi‐variant loadings. Solution accuracy is verified by comparisons of SIFs with various numerical methods. Use of the substitute geometry concept to expand the capability of 3D WFMs for solving real‐world engineering 3D crack problems is demonstrated. Some remaining challenges are briefly discussed.

A novel SAM/X-FEM coupling approach for the simulation of 3D fatigue crack growth under rolling contact loading

The present work proposes a novel efficient numerical approach for the simulation of three-dimensional propagation of non-planar frictional crack under Rolling Contact Fatigue (RCF). The developed model relies on a global–local strategy involving the Semi-Analytical Method (SAM), dedicated to the resolution of 3D contact problems, and the eXtended-Finite Element Method (X-FEM), dedicated to the solving of 3D crack problems. Similar to a submodeling technique, it consists in performing a local X-FEM analysis of the 3D RCF crack problem by means of boundary conditions extracted from a prior global SAM analysis which solves the contact problem between two semi-infinite bodies without considering the crack. A powerful procedure for the transfer of relevant mechanical quantities between the SAM and X-FEM models is developed. It allows the use of non-matching and non-conforming discretization scheme for the global SAM model and the local X-FEM model, particularly worthwhile for the pre-processing stage. An initial arbitrary stress profile can be considered to investigate the influence of residual stresses on crack behavior. The non-intrusive SAM/X-FEM coupling is integrated into a fully automatic crack propagation algorithm. This provides a powerful and user-friendly tool suitable for industrial applications, which easily handles 3D long non-planar frictional crack growth in the region of interest. After introducing some details concerning the implementation of the model, a validation of the innovative SAM/X-FEM coupling is performed by using results from the literature. Furthermore, a practical example demonstrates the great potential of this novel numerical technique to simulate in a fast, robust and accurate way the 3D complex behavior of fatigue crack under moving contact. • Simulation of the 3D frictional crack propagation under rolling contact conditions. • Non-intrusive global–local approach based on the coupling of SAM and X-FEM. • Field transfer procedure allowing independent meshes for the SAM and X-FEM models. • Fast, efficient and robust resolution coupled with a pre-processing time saving. • Influence of residual stresses on crack behavior highlighted.

Stable generalized finite element method (SGFEM) for three-dimensional crack problems

This paper proposes a stable generalized finite element method (SGFEM) for the linear 3D elasticity problem with cracked domains. Conventional material-independent branch functions serve as singular enrichments. We prove that the proposed SGFEM with the geometric enrichment scheme yields the optimal order of convergence in the energy norm, O(h), for fully 3D elasticity planar crack problems; h is the mesh parameter. To improve the conditioning of SGFEM, two stability techniques have been employed, namely, (a) a cubic polynomial has been used as the PU (partition of unity), instead of the standard FE hat-functions, to address the possible almost linear dependence between the PU functions and the enrichments, and (b) a local principal component analysis (LPCA) has been implemented to address the local bad conditioning produced by multi-fold enrichments at a node. The scaled condition number for the proposed SGFEM is shown to be O(h^{-2}) (same as that of a standard Finite Element Method), for various relative positions of crack surface and mesh. The robustness of the scaled condition number for the proposed SGFEM, with respect to the relative positions of the crack-surface and the element boundaries, has been observed numerically. The numerical experiments for both the planar and fully 3D planar crack problems are presented to show the efficiency of the proposed SGFEM.

A Precise Time-Domain Expanding Crack-Tip Element for 3D Planar and Axial Dynamic Cracks

Unlike two-dimensional (2D) crack problems of which the analytical symplectic eigensolutions are available, the analytical symplectic solution of three-dimensional (3D) planar and axial dynamic cracks has not been derived yet. We thus propose a trial displacement field for the 3D planar and axial crack problem using the existing solutions of the 2D crack problems and construct a new crack-tip element on that basis. The recently developed precise integration scheme for 3D planar and axial cracks under dynamic loading (Hu, X. F., Chen, W. H., Zhang, P. and Yao, W. A. [2021] “An explicit crack-tip element for stationary dynamic cracks,” Theor. Appl. Fract. Mech. 112, 102886) is further enhanced. A precise time-domain expanding algorithm is employed for the temporal discretization. All the field variables are expanded using a series expansion scheme, and the dynamic problem is then transformed into a series of quasi-static elastic problems. All the sub-elastic problems are related in a recursive manner. Taking advantage of a smart automatic truncating approach, the number of expanding terms can be determined automatically subjected to a prescribed error tolerance. A few numerical examples are investigated to demonstrate the developed method, numerical solutions indicate that the time dependent stress intensity factors (SIFs) can be calculated in a simple way and the present method has higher efficiency than the recently developed time integration scheme while having comparable accuracy.

Effect of fiber orientation angle on patch repaired composite plates

Abstract Stress intensity factors numerically investigated the Mode I loading of composite plates with an edge crack and repaired with a patch on a single side. The effect of the fiber orientation angle for both composite plate and the patch were analyzed with regard to crack length, adhesive properties, and plate thickness. The stress intensity factors were calculated by using the quarter point element that can be applied to 3D crack problems of homogeneous anisotropic materials.It was observed in this study that the fiber orientation angle affects the stress intensity factors significantly.

A general algorithm for numerical integration of three-dimensional crack singularities in PU-based numerical methods

With the development of PU-based numerical methods for crack problems, the evaluation of various orders of vertex/edge singularity has been one of the most critical issues, which restrains the computational efficiency of PU-based methods, especially for 3D crack problems. In this paper, based on the conventional Duffy transformation, a general algorithm for numerical integration of three-dimensional crack singularities is proposed for the vertex/edge singularity problems, which takes the integration cell shape into full consideration. Besides, the corresponding 3D conformal preconditioning strategy is constructed to fully eliminate the shape influence of tetrahedron elements. Extensive numerical examples, including ill-shaped integration cells and crack-front tetrahedron elements with parallel/nonparallel crack front, are given to validate the feasibility and accuracy of the proposed method. As a result, for each crack-front element, several hundreds of Gauss points are sufficient to achieve the precision of 10−6 for both kernels 1∕r and 1∕r, in sharp contrast with around ten thousands of Gauss points using the conventional Duffy transformation.

Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part I: Theoretical formulations

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.

Analysis of 3D planar crack problems of one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part II: Numerical approach

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are carried out systematically. Part II of the work aims to develop a general numerical approach to study 3D planar crack problems in 1D hexagonal piezoelectric QC media. Based on the theoretical formulations presented in Part I, a direct and effective method is proposed to derive the Green's functions for point extended displacement discontinuities (EDDs). These Green's functions are presented explicitly by a series of potential functions in a compact form. Using the superposition principle, the Green's functions for uniformly distributed EDDs over the crack elements are obtained. Related element solutions are used to construct the numerical approach, known as EDD boundary element method, for 1D hexagonal piezoelectric QCs. The proposed numerical method can be applicative to many complicated planar crack problems, such as multiple cracks, and cracks with non-uniform loadings, for 3D media composed of 1D hexagonal piezoelectric QCs with thermal effect. A comparison of the results obtained from the theoretical solutions given in Part I of the work with those obtained from the numerical method proposed here validates of the present investigation.

Development of an Automated System for Three-Dimensional J-Integral Analysis

By integrating the 3D solid modeler with the universal FEM code, an automatic nonlinear analysis system for 3D crack problems has been developed. A geometry model, that is, a solid containing one or several 3D cracks is defined. Multiple distributions of local node density are selected and automatically overlapped with each other over the geometry model using fuzzy knowledge processing. Nodes are created by the bucketing method, and tetrahedral square tetrahedral solid elements are generated by the Delaunay triangulation technique. A complete finite element (FE) model is created and stress analysis is performed. In this system, analysts can significantly reduce the burden of introducing 3D cracks into FE models and estimating fracture mechanics parameters. This paper describes a methodology for realizing such a function and demonstrates the validity of this system.

A review and verification of analytical weight function methods in fracture mechanics

AbstractThe weight function method is a powerful method for the determination of key parameters of stress intensity factors and crack opening displacements that are fundamental for crack analyses in fracture mechanics. The method has a remarkable computational efficiency without compromising solution accuracy and is especially suited for cracks in real‐world components with complex stress gradients. A critical issue for successful development and reliable application of the weight function method is how to achieve verifiable and high solution accuracy. This paper provides several reflections on the current state of the art in weight function methods, including a brief historical perspective and description of several analytical and numerical weight methods, with main focus on the two‐dimensional (2D) analytical approaches. A variety of typical application examples are presented. A platform for point‐by‐point accuracy assessment by using the Green's functions is established on a completely independent numerical approach with validated accuracy. Rigorous and systematic verification of the accuracy levels of different 2D analytical weight function approaches are conducted using several benchmark crack geometries. The sources of inaccuracy of different analytical weight function methods are analysed. A brief account on weight function methods for analysing three‐dimensional (3D) crack problems with arbitrary univariant and bivariant stress distributions is provided.

A family of non-conforming crack front elements of quadrilateral and triangular types for 3D crack problems using the boundary element method

This study focuses on establishing nonconforming crack front elements of quadrilateral and triangular types for 3D crack problems when the dual boundary element method is applied. The asymptotic behavior of the physical variables in the area near the crack front is fully considered in the construction of the shape function. In the developed quadrilateral and triangular crack front elements, the asymptotic term, which captures the asymptotic behavior of the physical variable, is multiplied directly by the conventional Lagrange shape function to form a new crack front shape function. Several benchmark numerical examples that consider penny-shaped cracks and straight-edge crack problems are presented to illustrate the validity and efficiency of the developed crack front elements.

A novel triangular boundary crack front element for 3D crack problems based on 8-node serendipity element

This paper developed a novel 6-node triangular boundary crack front element for 3D crack problems. The developed element method was constructed based on an 8-node serendipity element and a distance function that defines the distance from arbitrary point inside the element to the crack front. The specially constructed shape function captures the square-root asymptotic behavior of the distribution of the displacement nearby the crack front. In order to employ the crack front element in the dual boundary element method, conversion from conforming triangular crack front elements to non-conforming triangular ones was performed through a matrix transformation. In the implementation of dual boundary element method, a displacement extrapolation method is adopted to compute the stress intensity factor at the crack front. Finally, three examples were presented to verify the efficiency of our proposed crack front element. In all the numerical examples, the presented crack front element exhibits high accuracy and efficiency in the examples.

Analysis solution method for 3D planar crack problems of two-dimensional hexagonal quasicrystals with thermal effects

• An analysis solution method is proposed to analyze 3D planar crack problems for 2D hexagonal quasicrystals. • Corresponding EDD boundary integral equations are transferred to simplified integral-differential forms. • Various kinds of loadings, including normal, tangential and thermal loadings, applied on the crack are considered. • Using the proposed solution, influences of phonon-phason coupling effect on fracture parameters are assessed. An analysis solution method (ASM) is proposed for analyzing arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystal (QC) media. The extended displacement discontinuity (EDD) boundary integral equations governing three-dimensional (3D) crack problems are transferred to simplified integral-differential forms by introducing some complex quantities. The proposed ASM is based on the analogy between these EDD boundary equations for 3D planar cracks problems of 2D hexagonal QCs and those in isotropic thermoelastic materials. Mixed model crack problems under combined normal, tangential and thermal loadings are considered in 2D hexagonal QC media. By virtue of ASM, the solutions to 3D planar crack problems under various types of loadings for 2D hexagonal QCs are formulated through comparison to the corresponding solutions of isotropic thermoelastic materials which have been studied intensively and extensively. As an application, analytical solutions of a penny-shaped crack subjected uniform distributed combined loadings are obtained. Especially, the analytical solutions to a penny-shaped crack subjected to the anti-symmetric uniform thermal loading are first derived for 2D hexagonal QCs. Numerical solutions obtained by EDD boundary element method provide a way to verify the validity of the presented formulation. The influences of phonon-phason coupling effect on fracture parameters of 2D hexagonal QCs are assessed.

Crack analysis in magneto-electro-elastic solids by gradient theory

This paper presents a computational method to analyze 2D crack problems in magneto-electro-elastic solid described by the gradient theory with the direct flexoelectric effect. The finite-element method (FEM) is developed for these problems, where the constitutive equations for electric displacement and magnetic induction contain the strain-gradients. The coefficients staying at these terms are proportional to the scaling parameters. It expresses the size effect phenomenon in the considered advanced continuum model. The governing equations are derived by applying the variational principle. The FEM formulation is subsequently developed and implemented for the gradient theory of magneto-electro-elasticity. The path-independent J-integral is derived in the framework of considered theory.

Convergence properties of the enriched Scaled Boundary Finite Element Method in fracture mechanics

AbstractThe standard Finite Element Method (FEM) yields optimal rates of convergence for smooth domains. The presence of cracks, corners or notches within the considered domain leads to low rates of convergence which are independent of the polynomial degree of the employed trial functions. In order to regain full rates of convergence, an appropriate adaptation and a skilful choice of test and trial space of the FEM is necessary. The Scaled Boundary FEM (SBFEM) has turned out to be an appropriate method for the treatment of 2D crack problems, in which the singularity is entirely located within the considered domain. However, in 3D problems, singularities are still present at the meeting points of the crack front and the discretized boundary. Then, the SBFEM suffers from the same drawbacks as the standard FEM. In order to dispel this inconvenience, an extension of the trial space with asymptotic crack tip displacement solutions is considered resulting in improved rates of convergence.

Accurate SIF Analysis of a Partially Cylindrical Side Crack Opened by Far-Field Tension

In recent years, due to a remarkable progress of qualified mesh generation algorithms combinedwith computational image processing technologies, a complex shaped 3D crack analysis has been carried out more and more easily by finite element method. Generally speaking, in order to assess thereliability of numerical solution, existence of a closed-form solution or a practically exact numericalsolution is important and thank-worthy. In contrast with 2D problems, however, there are very fewnumber of closed form solutions regarding to 3D crack problems. In the present analysis, we examineda highly accurate body force method (BFM) analysis for a partially cylindrical 3D surface crackin which a suitable basic density function can be supposed reasonably by taking account the crackgeometry. The present SIF solution could be used effectively for the purpose of benchmark the finiteelement solution for general non-planar crack problems.

Analytical and numerical analyses for a penny-shaped crack embedded in an infinite transversely isotropic multi-ferroic composite medium: semi-permeable electro-magnetic boundary condition

The problem of a penny-shaped crack embedded in an infinite space of transversely isotropic multi-ferroic composite medium is investigated. The crack is assumed to be subjected to uniformly distributed mechanical, electric and magnetic loads applied symmetrically on the upper and lower crack surfaces. The semi-permeable (limited-permeable) electro-magnetic boundary condition is adopted. By virtue of the generalized method of potential theory and the general solutions, the boundary integro-differential equations governing the mode I crack problem, which are of nonlinear nature, are established and solved analytically. Exact and complete coupling magneto-electro-elastic field is obtained in terms of elementary functions. Important parameters in fracture mechanics on the crack plane, e.g., the generalized crack surface displacements, the distributions of generalized stresses at the crack tip, the generalized stress intensity factors and the energy release rate, are explicitly presented. To validate the present solutions, a numerical code by virtue of finite element method is established for 3D crack problems in the framework of magneto-electro-elasticity. To evaluate conveniently the effect of the medium inside the crack, several empirical formulae are developed, based on the numerical results.

An adaptive cracking particle method providing explicit and accurate description of 3D crack surfaces

SummaryCracks in 3‐dimensions (3D) have arbitrary shapes and therefore present difficulties for numerical modelling. A novel adaptive cracking particle method with explicit and accurate description of 3D cracks is described in this paper. In this meshless method, crack surfaces are described by a set of discontinuous segments, which are associated with particles. This group of particles are assumed all to be “cracked” and split into 2 subparticles with modified support domains. Compared to the original method where the spherical supports at particles are equally divided, the proposed method makes use of nonplanar segments to account for changes in crack face direction. The orientations of those segments and the angular changes of cracks during crack propagation steps are recorded using triangular meshes. Supports of weight functions are modified according to those changes so that quasi‐continuous crack surfaces can be obtained, avoiding the spurious cracking seen in earlier cracking particle methods. An adaptive approach in 3D is then introduced to capture stress gradients around crack fronts. Several 3D crack problems with reference results have been tested to validate the proposed method with good agreement being achieved using the new method, showing it to be potentially a significant advance for 3D fracture prediction problems.

Analysis of Planar Crack Coalescence by Mesh-Free Body Force Method

In a standard body force method analysis, a mesh division is required to define the boundary of a problem and to solve a governing equation using discretization procedure. However, in the present study, a moving least square strategy is introduced to define a weight function for the density of body force doublet and therefore a crack analysis is carried out without providing a standard mesh-division. Hence, the standard crack face elements are not required at all. A variety of 3D crack problems can be analyzed simply by providing a data that only de nes a crack front. Besides the nodal points for crack front, several internal nodes are generated on the crack face to represent a distribution of unknown function. At the internal nodes, an unknown variable is assigned which uniquely de ne a distribution of the relative crack face displacement. In the present approach, a crack problem is formulated as a singular integral equation whose unknown is a value of the weight function at the internal nodal points. A crack growth can be simulated directly by changing the shape of crack front, by means of adding a new nodal point in the vicinity of the current crack front. In the present paper, the proposed method is used to simulate a coalescence of interacting planar cracks.