Three-dimensional Crack Problems Research Articles

Analysis of three-dimensional cracks horizontally placed in transversely isotropic FGMs under interior compressive stresses

This paper develops the dual boundary element method (DBEM) using a new fundamental singular solution. This new solution is the elastic fields of transversely isotropic multilayered materials of infinite extent under point concentrated loads. The discretization technique is utilized to approximate arbitrary variations of functionally graded materials (FGMs) with transverse isotropy in a graded direction. The proposed DBEM is used to analyze the three-dimensional crack problems in FGM halfspaces. The penny-shaped and elliptical cracks are horizontally placed on the interface between the FGM layer and homogeneous halfspace and are subjected to uniform compressive stresses on crack surfaces. The stress intensity factors are provided to understand the fracture behaviors of transversely isotropic FGMs. An influential factor is defined to describe the influence of the anisotropy of the FGMs on stress intensity factors. Results show that the anisotropy and heterogeneity of materials exert an obvious influence on the fracture properties of crack problems.

The Multi-Virtual Crack Closure Technique for three-dimensional interface crack problems

The Virtual Crack Closure Technique (VCCT) is a simple method that was found in the past to produce good results for determining stress intensity factors of cracks in homogeneous bodies and poor results for interface cracks. In recent papers, the VCCT was found to produce excellent results for two-dimensional interface cracks where many elements were used in the virtual crack extension (VCE). This approach is called here the Multi-VCCT. A criterion for the optimal number of elements used in the VCE was also found. Here, the M-VCCT is extended to three-dimensional interface crack problems. Results for three-dimensional problems containing a straight through finite length interface crack and a penny-shaped interface crack are presented. The interfaces include that between two homogeneous and isotropic dissimilar materials, as well as two transversely isotropic materials.

A novel approach to compute the spatial gradients of enriching functions in the X-FEM with a hybrid representation of cracks

The eXtended Finite Element Method (X-FEM) is a versatile technique to model discontinuities by enriching the trial functions with a prior solution. In the X-FEM, a crack can be explicitly represented by a set of triangles or implicit signed distances, i.e., level set functions, of the points of interest from the crack surface and the crack front. In the explicit representations, it is crucial to accurately evaluate surface normal, conormal, and tangent vectors along crack fronts for computations of the gradients of the enriching functions. The solution is very sensitive to these directional vectors, especially for non-planar 3D cracks. We here propose a novel approach to compute these gradients without evaluating the directional vectors. Level set functions are first set up in a hexahedral grid independent of the background mesh in the X-FEM. We can thus implement the Weight Essentially Non-Oscillatory (WENO) scheme to compute the gradients of level sets. The gradients of the enriching functions at any integration point can therefore be computed by interpolations and chain rules. We compare the implementation procedures of the explicit representation and the proposed hybrid representation in detail. A three-dimensional lens crack problem is studied to demonstrate the accuracy of the proposed method, especially for coarse meshes.

Theoretical and computational aspects of configurational forces in three-dimensional crack problems

Mechanics in material space yields configurational forces (CFs) as an appropriate means to describe driving forces acting at defects or inhomogeneities in magnitude and direction. A thermodynamically based approach to obtain a weak formulation of CFs with virtual defect displacements as test functions is introduced first, starting from a global variational energy balance instead of from the currently employed local configurational balance equation. A theoretical basis and issues of three-dimensional implementation are presented to calculate vectors of CFs along a crack front, providing information on local loading and deflection angles. Spurious nodal CFs are separated from physical ones, inter alia requiring surface integrals along the crack faces. Within a linear elastic framework, these are efficiently replaced by an algebraic expression, providing CFs at crack front nodes in a very good approximation, which is eventually demonstrated by means of analytical and numerical examples.

Effective elastic properties of three-dimensional multiple crack problems with the isogeometric boundary element parallel fast direct solver

We propose an isogeometric boundary element (IGABEM) parallel fast direct solver based on OpenMP to solve three-dimensional multiple crack problems. In this paper, the orderly and randomly distributed cracks are studied to observe the influence of the multiple crack interaction on the effective elastic properties. This paper designs the parallel implementation of the matrix low rank approximation and the matrix assembly for the hierarchical off-diagonal low rank (HODLR) matrix. In the matrix low rank approximation, we propose the m-n tree to improve the accelerated hybrid algorithm to balance the CPU performance and memory usage. The numerical results show that the effective elastic properties can respond to the strong interaction to a certain extent, but the maximum stress intensity factors (SIFs) of multiple cracks still need to be calculated to diagnose the possible crack initiation points. On the other hand, the numerical results also show that the IGABEM parallel fast direct solver can greatly improve the computational efficiency on the premise of ensuring the computational accuracy.

A Method on Estimates of Stress Intensity Factors of Outer Circle and Inner Ellipse Submarine Pipeline Under Bending Moment

The slender circle submarine pipeline possesses both shell and beam characteristics, which are widely used in practical engineering. Unfortunately, for some reason, there will be some geometric defects in the cross-section of the pipeline (such as machining errors and seawater corrosion, etc.), resulting in the stiffness of the circular submarine pipeline being different. The cracked variable stiffness pipeline is actually a typical three-dimensional complex crack problem. In this paper, taking the outer circle and inner ellipse submarine pipeline as an example, a method based on elementary mechanics and the concept of the conservation law is proposed to determine the stress intensity factors (SIFs) of variable stiffness pipeline, and a series of closed-form expressions of SIF are derived for the cracked pipeline. The results from the present method are shown to agree well with available solutions and FEM. In particular, the present method can also be easily applied to other similar variable stiffness submarine pipelines.

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.

Boundary element method applied to three-dimensional crack analysis in exponentially graded viscoelastic materials

In the present work, three-dimensional crack problems in viscoelastic isotropic exponentially graded solids are investigated using the boundary element method (BEM). To reproduce the viscoelastic behaviour of the material, the BEM formulation is incorporated with an approach based on the differential constitutive relations for linear viscoelasticity employing Kelvin–Voigt and Boltzmann models. Moreover, the special case of material exponential gradation which is governed by a fundamental solution available in the literature is included into the boundary integral kernels. Because this methodology allows the material response to vary in time and space, it can be used to obtain the effects on the stress intensity factors, crack opening displacements, and energy release rates in practical fracture problems. Numerical examples are presented to demonstrate the applicability of the used methodology.

An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems

In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and J-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.

Mode-I stress intensity factors for Periodic Cracked Stiffened Panels under tension

Stress intensity factor is the nuclear physical parameters in fracture mechanics. Stiffened panels structure is heavily used in actual engineering field, such as aviation, marine and other bearing structures. The SIF for periotic cracked stiffened panels under tension is a representative three-dimensional crack problem. In the present article, the solution of SIFs for stiffened panels periodic cracks are derived on the basis of materials mechanics and the conservation law J 2- integral.

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.

Three-dimensional assessments of crack tip constraint

Crack tip constraint can be used as a means to relate the stresses at the crack tip to a measure of fracture toughness of a material. Current technique to determine crack tip constraint demonstrated in structural integrity standards are based on two-dimensional plane strain approach. This paper explores the possibility of extending the current two-dimensional crack tip constraint solutions to three-dimensional crack problems through an investigation of the asymptotic structure of three-dimensional crack tip stress fields under elastic-plastic non-hardening responses. Analysis was based on three-dimensional boundary layer formulations, combined with full-field solutions of single edge notched bend bars and centre cracked tension panels having a range of thicknesses. The fields along the crack front were examined as a function of load level and thickness. From the results, it is demonstrated that, at the crack tip (r = 0), a family of asymptotic fields develop which feature a constant stress sector directly at the crack tip. Within this sector the fields differ hydrostatically while being similar in respect of the maximum stress deviator. At the intersection with the free surface, an elastic perfectly-plastic corner field which differs from the plane stress field is shown to develop. It is shown that in these fields the level of constraint is parameterised by J/zσo, where z is the distance from the free surface. J/zσo unifies both the change in the maximum principal stress and the mean stress, as a function of thickness and deformation level from all sections of different geometry and thickness specimens. An explicit expression has been developed which demonstrated the constraint loss due to in-plane and out-of plane crack tip constraint effects, allowing a two-parameter fracture mechanics methodologies in a modified form to be extended to fully three-dimensional fields.

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.

Mode-I stress intensity factors for cracked special-shaped shells under bending

The slender special-shaped shells possess both shell and beam characteristics, which are widely used in practical engineering. A slender special-shaped shell under bending with some complex cracks, such as a frost flower-shaped crack and circumferential periodic cracks in a pipeline, is actually a typical three-dimensional complex crack problem with many different singular stress fields and stress intensity factors. In the present article, a very simple and practical technique based on the elementary mechanics and the concept of the conservation law is proposed to determine the stress intensity factors. The results from present method are shown to agree well with available solutions and FEM. In particular, the present method can also be easily applied to the other similar cracked structural shells.

Numerical and analytical modeling of crack path for three-dimensional mixed mode crack problem using global approach

Civil engineering and mechanical structures are usually submitted to mixed mode loading. Consequently, a mixed mode crack path process occurs. In three-dimensional crack problem, the study of this problem is very important due to multiple problems that require more understanding of the corner point effect and the local effect due to the torsion mode (mode III) and also due to the necessity to take into account thickness effect. In order to study the mixed mode loading for three-dimensional crack problem, a new three-dimensional contour integral entitled M3D integral is modeled using global approach. Combining real and virtual mechanical displacement fields, this new integral is used to separate numerically mode I, mode II and mode III in the mixed mode ratio. In earlier research works, Moutou Pitti et al [1] have proposed a new specimen called Mixed Mode Crack Growth (MMCG), and El kabir et al [2] have studied numerically the stability of this specimen for various geometries in two-dimensional case. This work deals with numerical and analytical modeling to study the crack path stability in real three-dimensional case for mixed mode crack problem. Using MMCG specimen, the non-dependence of integration domain is presented, and the stability of the calculation of M3D Integral with respect to various geometries and thicknesses is shown for the opening mode (Mode I), the shear mode (Mode II), the out-of-plan shear mode (Mode III) and also for the mixed mode ratio by computing the energy release rate versus the crack length. Finally, the analytical generalization of the M3D-integral is done. That will allow to take into account the mixed-mode crack growth analysis coupling mechanical and thermal loads.

Application of the GTN model to ductile crack growth simulation in through-wall cracked pipes

This paper presents ductile fracture simulation using the GTN (Gurson-Tvergaard-Needleman) model of through-wall cracked STPT410 carbon steel pipe test under pure bending. Three issues related to practical application are addressed; (i) how to determine the parameters in the GTN model from fracture toughness data, (ii) how to incorporate the element-size effect to the GTN model and (iii) the effect of the crack-tip mesh design on determination of the GTN model parameters. It is found that determined values of the parameters depend on the crack-tip mesh design and the finite element size used in simulation. The split of the crack tip element into two makes parameter determination easier and gives better simulation results, but is difficult to apply to three-dimensional surface crack problems.

Enriched finite element method for three-dimensional viscoelastic interface crack problems

The enriched finite element method is developed for three-dimensional problems of an interface crack between elastic and viscoelastic (including dissimilar viscoelastic) materials. According to the displacement fields of elastic interface crack, the displacement fields of viscoelastic interface crack are derived through the correspondence principle. By incorporating the displacement expressions into the displacement model of regular element, the incremental formulations of enriched element are derived. The stress intensity factors and strain energy release rates can be solved based on the enriched degree of freedoms. A 3-D through interface crack at the center of jointed dissimilar viscoelastic plate subjected to remote tension and a quarter-circular viscoelastic interface corner crack subjected to uniform thermal loading was investigated using the enriched finite element method. It is shown that the present solutions are consistent with the analytical solutions, which indicates the present method is correct and efficient.

Integral equations for 2D and 3D problems of the sliding interface crack between elastic and rigid bodies

This paper revisits the sliding interface crack problem between elastic and rigid half-planes studied by Bui and Oueslati and provides an alternative method of derivation of the solution, which will then be extended to three-dimensional (3D) crack problems. Based upon the displacement continuation technique of complex potentials, an appropriate Green function for the isolated edge dislocation dipole at the interface is given. Then by considering the sliding condition along the interface crack, the field equations can be obtained for the two-dimensional (2D) problem. Furthermore, it is shown that the edge dislocation dipole in 2D appears to be a particular form of the fundamental Kupradze–Basheleishvili tensor in 3D, which provides a method for deriving the coupled nonlinear integral equations for the same frictional interface plane crack of an arbitrary shape. The present work describes how the 3D sliding interface crack is related to the same problem in 2D.

Complex variables boundary element analysis of three-dimensional crack problems

Abstract This paper presents a new boundary element-based approach for solving three-dimensional problems of an elastic medium containing multiple cracks of arbitrary shapes. The medium could be loaded by far-field stress (for infinite domains), surface tractions (including those at the cracks surfaces), or point loads. Constant body forces are also allowed. The elastic fields outside of the cracks are represented by integral identities. Triangular elements are employed to discretize the boundaries. Integration over each element is performed analytically. In-plane components of the fields are combined in various complex combinations to simplify the integration. No singular integrals are involved since the limit, as the field point approaches the boundary, is taken after the integration. The collocation method is used to set up the system of linear algebraic equations to find the unknown boundary displacements and tractions. No special procedure is required to evaluate the fields outside of the boundaries, as the integration is performed before the limit is taken. Several numerical examples are presented to demonstrate the capacity of the method.