Analyzing interaction between coplanar square cracks using an efficient boundary element‐free method

This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Near‐crack contour behaviour and extraction of log‐singular stress terms of the self‐regular traction boundary integral equation

This work contains an analytical study of the asymptotic near‐crack contour behaviour of stresses obtained from the self‐regular traction‐boundary integral equation (BIE), both in two and in three dimensions, and for various crack displacement modes. The flat crack case is chosen for detailed analysis of the singular stress for points approaching the crack contour. By imposing a condition of bounded stresses on the crack surface, the work shows that the boundary stresses on the crack are in fact zero for an unloaded crack, and the interior stresses reproduce the known inverse square root behaviour when the distance from the interior point to the crack contour approaches zero. The correct order of the stress singularity is obtained after the integrals for the self‐regular traction‐BIE formulation are evaluated analytically for the assumed displacement discontinuity model. Based on the analytic results, a new near‐crack contour self‐regular traction‐BIE is proposed for collocation points near the crack contour. In this new formulation, the asymptotic log‐singular stresses are identified and extracted from the BIE. Log‐singular stress terms are revealed for the free integrals written as contour integrals and for the self‐regularized integral with the integration region divided into sub‐regions. These terms are shown to cancel each other exactly when combined and can therefore be eliminated from the final BIE formulation. This work separates mathematical and physical singularities in a unique manner. Mathematical singularities are identified, and the singular information is all contained in the region near the crack contour. Copyright © 2003 John Wiley & Sons, Ltd.     

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A new unified and integrated method for the numerical‐analytical calculation of J_k‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J₁ and stress intensity factors (SIFs) exists, a similar relation for J₂ in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J₂ in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed J_k values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of J_k for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Analyse the role of the non‐singular stress in brittle fracture by BEM coupled with eigen‐analysis

A coupled model resulting from the boundary element method and eigen‐analysis is proposed in this paper to analyse the stress field at crack tip. This new combine method can yield several terms of the non‐singular stress in the Williams asymptotic expansion. Then the maximum circumferential stress (MCS) criterion taken the non‐singular stress into account is introduced to predict the brittle fracture of cracked structures. Two earlier experiments are re‐examined by the present numerical method and the role of the non‐singular stress in the brittle fracture is investigated. Results show that if more terms of non‐singular stress are taken into account, the predicted crack propagation direction and the critical loading by MCS criterion are much closer to the existing experimental results, especially for dominating mode II loading conditions. Moreover, numerical results manifest that Williams series expansion can describe the stress field further from the crack tip if more non‐singular stress terms are adopted.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Weakly singular stress‐BEM for 2D elastostatics

A weakly singular stress‐BEM is presented in which the linear state regularizing field is extended over the entire surface. The algorithm employs standard conforming C⁰ elements with Lagrangian interpolations and exclusively uses Gaussian integration without any transformation of the integrands other than the usual mapping into the intrinsic space. The linear state stress‐BIE on which the algorithm is based has no free term so that the BEM treatment of external corners requires no special consideration other than to admit traction discontinuities. The self‐regularizing nature of the Somigliana stress identity is demonstrated to produce a very simple and effective method for computing stresses which gives excellent numerical results for all points in the body including boundary points and interior points which may be arbitrarily close to a boundary. A key observation is the relation between BIE density functions and successful interpolation orders. Numerical results for two dimensions show that the use of quartic interpolations is required for algorithms employing regularization over an entire surface to show comparable accuracy to algorithms using local regularization and quadratic interpolations. Additionally, the numerical results show that there is no general correlation between discontinuities in elemental displacement gradients and solution accuracy either in terms of unknown boundary data or interior solutions near element junctions. Copyright © 1999 John Wiley & Sons, Ltd.     

A new approach for displacement functions of a curved Timoshenko beam element in motions normal to its initial plane

In existing literature, either analytical methods or numerical methods, the formulations for free vibration analysis of circularly curved beams normal to its initial plane are somewhat complicated, particularly if the effects of both shear deformation (SD) and rotary inertia (RI) are considered. It is hoped that the simple approach presented in this paper may improve the above‐mentioned drawback of the existing techniques. First, the three functions for axial (or normal to plane) displacement and rotational angles about radial and circumferential (or tangential) axes of a curved beam element were assumed. Since each function consists of six integration constants, one has 18 unknown constants for the three assumed displacement functions. Next, from the last three displacement functions, the three force–displacement differential equations and the three static equilibrium equations for the arc element, one obtained three polynomial expressions. Equating to zero the coefficients of the terms in each of the last three expressions, respectively, one obtained 17 simultaneous equations as functions of the 18 unknown constants. Excluding the five dependent ones among the last 17 equations, one obtained 12 independent simultaneous equations. Solving the last 12 independent equations, one obtained a unique solution in terms of six unknown constants. Finally, imposing the six boundary conditions at the two ends of an arc element, one determined the last six unknown constants and completely defined the three displacement functions. By means of the last displacement functions, one may calculate the shape functions, stiffness matrix, mass matrix and external loading vector for each arc element and then perform the free and forced vibration analyses of the entire curved beam. Good agreement between the results of this paper and those of the existing literature confirms the reliability of the presented theory. Copyright © 2005 John Wiley & Sons, Ltd.     

Singular boundary elements for the analysis of cracks in plane strain

Straight and curved cracks are modelled by direct formulation boundary elements, of geometry defined by Hermitian cubic shape functions. Displacement and traction are interpolated by the Hermitian functions, supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance r from the crack tip, and subdominant modes in which it is proportional to r^1·5. The singular functions extend over many boundary elements on each crack face. A nodal collocation scheme is used, in which additional boundary integral equations are obtained by differentiation of the equation obtained from Betti's theorem. The hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack. Examples are shown of the analysis of buried and edge cracks, to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements.     

Analysis of stresses and displacements in the vicinity of blunt V‐notches by considering higher order terms

Mode I asymptotic stress and displacement fields of blunt V‐notches are calculated based on the Muskhelishvili's approach. First, by using appropriate complex potential functions, the stress and displacement components in the polar coordinate system are determined. By applying proper mapping, the notch profile is built up, and then by utilizing the near‐field boundary conditions, free parameters are specified. Afterwards, the solution procedure available for the singular term is extended for higher order terms leading to asymptotic relations for stress and displacement fields. Subsequently, the coefficients of the asymptotic stress fields are computed by using a numerical procedure named “overdeterministic” technique. It is shown that the proposed method that considers the higher order terms yields very good results, whereas the methods based only on the singular term have significant errors. Finally, some parameters affecting the convergence of the numerical results are studied and discussed.     

Boundary element‐free method (BEFM) for two‐dimensional elastodynamic analysis using Laplace transform

In this paper, we present a direct meshless method of boundary integral equation (BIE), known as the boundary element‐free method (BEFM), for two‐dimensional (2D) elastodynamic problems that combines the BIE method for 2D elastodynamics in the Laplace‐transformed domain and the improved moving least‐squares (IMLS) approximation. The formulae for the BEFM for 2D elastodynamic problems are given, and the numerical procedures are also shown. The BEFM is a direct numerical method, in which the basic unknown quantities are the real solutions of the nodal variables, and the boundary conditions can be implemented directly and easily that leads to a greater computational precision. For the purpose of demonstration, some selected numerical examples are solved using the BEFM. Copyright © 2005 John Wiley & Sons, Ltd.     

A super‐element for crack analysis in the time domain

A super‐element for the dynamic analysis of two‐dimensional crack problems is developed based on the scaled boundary finite‐element method. The boundary of the super‐element containing a crack tip is discretized with line elements. The governing partial differential equations formulated in the scaled boundary co‐ordinates are transformed to ordinary differential equations in the frequency domain by applying the Galerkin's weighted residual technique. The displacements in the radial direction from the crack tip to a point on the boundary are solved analytically without any a priori assumption. The scaled boundary finite‐element formulation leads to symmetric static stiffness and mass matrices. The super‐element can be coupled seamlessly with standard finite elements. The transient response is evaluated directly in the time domain using a standard time‐integration scheme. The stress field, including the singularity around the crack tip, is expressed semi‐analytically. The stress intensity factors are evaluated without directly addressing singular functions, as the limit in their definitions is performed analytically. Copyright © 2004 John Wiley & Sons, Ltd.     

The dual boundary contour method for two-dimensional crack problems

This paper concerns the dual boundary contour method for solving two-dimensional crack problems. The formulation of the dual boundary contour method is presented. The crack surface is modeled by using continuous quadratic boundary elements. The traction boundary contour equation is applied for traction nodes on one of the crack surfaces and the displacement boundary contour equation is applied for displacement nodes on the opposite crack surface and noncrack boundaries. The direct calculation of the singular integrals arising in displacement BIEs is addressed. These singular integrals are accurately evaluated with potential functions. The singularity subtraction technique for determining the stress intensity factor K_I, K_II and the T-term are developed for mixed mode conditions. Some two-dimensional examples are presented and numerical results obtained by this approach are in very good agreement with the results of the previous papers. This revised version was published online in August 2006 with corrections to the Cover Date.     

Boundary element analysis of three‐dimensional mixed‐mode thermoelastic crack problems using the interaction and energy domain integrals

A three‐dimensional boundary element method (BEM) implementation of the interaction integral methodology for the numerical analysis of mixed‐mode three‐dimensional thermoelastic crack problems is presented in this paper. The interaction integral is evaluated from a domain representation naturally compatible with the BEM, since stresses, strains, temperatures and derivatives of displacements and temperatures at internal points can be evaluated using the appropriate boundary integral equations. Several examples are analysed and the results compared with those available in the literature to demonstrate the efficiency and accuracy of the implementation to solve straight and curved crack‐front problems. Copyright © 2007 John Wiley & Sons, Ltd.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

Rotation‐free triangular shell element using node‐based smoothed finite element method

In this paper, a triangular thin flat shell element without rotation degrees of freedom is proposed. In the Kirchhoff hypothesis, the first derivative of the displacement must be continuous because there are second‐order differential terms of the displacement in the weak form of the governing equations. The displacement is expressed as a linear function and the nodal rotation is defined using node‐based smoothed finite element method. The rotation field is approximated using the nodal rotation and linear shape functions. This rotation field is linear in an element and continuous between elements. The curvature is defined by differentiating the rotation field, and the stiffness is calculated from the curvature. A hybrid stress triangular membrane element was used to construct the shell element. The penalty technique was used to apply the rotation boundary conditions. The proposed element was verified through several numerical examples.     

A time‐dependent formulation of dual boundary element method for 3D dynamic crack problems

In this paper, the dual boundary element method in time domain is developed for three‐dimensional dynamic crack problems. The boundary integral equations for displacement and traction in time domain are presented. By using the displacement equation and traction equation on crack surfaces, the discontinuity displacement on the crack can be determined. The integral equations are solved numerically by a time‐stepping technique with quadratic boundary elements. The dynamic stress intensity factors are calculated from the crack opening displacement. Several examples are presented to demonstrate the accuracy of this method. Copyright © 1999 John Wiley & Sons, Ltd     

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

A high‐order immersed boundary discontinuous‐Galerkin method for Poisson's equation with discontinuous coefficients and singular sources

We adopt a numerical method to solve Poisson's equation on a fixed grid with embedded boundary conditions, where we put a special focus on the accurate representation of the normal gradient on the boundary. The lack of accuracy in the gradient evaluation on the boundary is a common issue with low‐order embedded boundary methods. Whereas a direct evaluation of the gradient is preferable, one typically uses post‐processing techniques to improve the quality of the gradient. Here, we adopt a new method based on the discontinuous‐Galerkin (DG) finite element method, inspired by the recent work of [A.J. Lew and G.C. Buscaglia. A discontinuous‐Galerkin‐based immersed boundary method. International Journal for Numerical Methods in Engineering, 76:427‐454, 2008]. The method has been enhanced in two aspects: firstly, we approximate the boundary shape locally by higher‐order geometric primitives. Secondly, we employ higher‐order shape functions within intersected elements. These are derived for the various geometric features of the boundary based on analytical solutions of the underlying partial differential equation. The development includes three basic geometric features in two dimensions for the solution of Poisson's equation: a straight boundary, a circular boundary, and a boundary with a discontinuity. We demonstrate the performance of the method via analytical benchmark examples with a smooth circular boundary as well as in the presence of a singularity due to a re‐entrant corner. Results are compared to a low‐order extended finite element method as well as the DG method of [1]. We report improved accuracy of the gradient on the boundary by one order of magnitude, as well as improved convergence rates in the presence of a singular source. In principle, the method can be extended to three dimensions, more complicated boundary shapes, and other partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.