Singular stress analysis of an anisotropic elastic medium containing polygonal holes using a novel hybrid finite element method

A novel hybrid finite element method based on a numerical procedure is proposed to compute singular field near V-shaped notch corners in an anisotropic material containing polygonal holes. The finite element method is established by the following three steps: (1) an ad hoc one-dimensional finite element formulation is employed to determined numerical eigensolutions of the singular field near an V-shaped notch corner; (2) a super corner tip element is constructed to determine the strength of the singular field, in which the independent assumed stress fields are extracted from the eigensolutions; (3) a novel hybrid finite element equation is obtained by coupling the super corner tip element with the conventional hybrid stress elements. In numerical examples, generalized stress intensity factors for interactions between two polygonal holes with various geometry, space position and material property are mainly discussed. All the numerical results show that present method yields satisfactory singular stress field solutions with fewer elements. Compared with the conventional finite element methods and integral equation methods, the present method is more suitable for dealing with micromechanics of anisotropic materials.     

用切口尖端单元分析各向异性材料中多边形孔奇异性应力场

该文提出了一种基于全数值方法的新型杂交元方法, 用于研究各向异性复合材料中多边形孔奇异性应力场干涉问题。该方法的建立分3 个步骤:首先, 用一维有限元方法求解各向异性材料切口尖端奇异性应力场数值特征解;然后, 采用杂交有限元列式构造一种超级切口尖端单元, 其中, 假设应力场和位移场是利用上述奇异性场数值特征解推导出来的;最后, 将上述超级切口尖端单元与传统4 结点杂交应力元组装, 得到新型杂交元方法。算例中, 将裂纹问题作为考核例, 并进一步考察双菱形孔和双矩形孔的奇异性应力干涉问题。算例表明:当前模型能降低单元数, 且精度好;与传统有限元法和积分方程方法相比, 该模型更具有通用性和高效性, 为各向异性材料的细观力学分析打下了基础。     

Stress intensity factors of a surface crack near an interface end

Due to the singular behavior of the stress field near the interface edge of bonded dissimilar materials, fracture generally initiates near the interface edge, or just from the interface edge point. In this paper, an edge crack near the interface, which can be considered as being induced by the edge singularity and satisfying two conditions, is analyzed theoretically, based on the singular stress field near the interface edge and the superposition principle. It is found that the stress intensity factor can be expressed by the stress intensity coefficient of the edge singular stress field, the crack length, the distance between the interface and the crack, as well as the material combination. Boundary element method analysis is also carried out. It is found that the theoretical result coincides well with the numerical result when the crack length is small. Therefore, the theoretical representation obtained by this study can be used to simply evaluate the stress intensity factor of an edge singularity induced crack for this case. However, when the crack length becomes larger than a certain value, a significant difference appears, especially for the case with large edge singularity.     

A finite element analysis of the singular stress fields in anisotropic materials loaded in antiplane shear

A finite element formulation is developed to determine the order and angular variation of singular stress states at material and geometric discontinuities in anisotropic materials subject to antiplane shear loading. The displacement field of the sectorial element is quadratic in the angular co-ordinate direction and asymptotic in the radial direction measured from the singular point. The formulation of Yamada and Okumura¹⁴ for in-plane problems is adapted for this purpose. The simplicity and accuracy of the formulation are demonstrated by comparison to several analytical antiplane shear solutions for both isotropic and anisotropic multi-material wedges and junctions with and without disbonds. The nature and speed of convergence of the eigensolution suggests that the solution presented here could be used in developing enriched elements for accurate and computationally efficient evaluation of stress intensity factors in problems having complex global geometries.     

Singular stress analyses of V-notched anisotropic plates based on a novel finite element method

A super singular wedge tip element whose stiffness matrix is based on numerical eigensolutions is incorporated into standard hybrid-stress finite elements to study singular stress fields around the vertex of anisotropic multi-material wedge. The numerical eigensolutions are obtained by an ad hoc finite element eigenanalysis method. To demonstrate the validity of the method, singular stresses for some typical anisotropic single-material/bimaterial wedges are investigated. All numerical results show present finite element method converges rapidly to available solutions with few elements. The present method is applicable to dealing with the problems with more complex geometries.     

A method for eliminating the effect of 3-D bi-material interface corner geometries on stress singularity

The stress singularities at three-dimensional (3-D) interface corners and edges have been investigated numerically using common finite element methods. The effects of variation of edge angle and vertex angle between the two free side surfaces on the order of stress singularity at bi-material interface corners are determined. It is found that the order of stress singularity at interface corner depends not only on the combination of materials and edge angles but also on the vertex angle between the two free side surfaces. The effect of the vertex angle on the order of stress singularity at interface corner can be eliminated by smoothing the intersection of interface edges, which can be achieved simply by generating a circular-arc fillet at the intersection of the two free side surfaces. The numerical results show that, after smoothing the intersection of interface edges, the order of stress singularity along the interface edges become continuous, i.e. the order of the corner singularity can be reduced to the level of the edge singularity.     

On the nature of the free edge stress singularity in composite laminated plates

The variational-asymptotic formulation for the edge layer problem is implemented by mixed-hybrid finite elements. This approach is used to analyse the stress singularity at the free edge in composite laminates. Results indicate that the stress singularity (as r → 0) is very closely approximated by log r instead of r^?α.     

Modified crack closure integral using six-noded isoparametric quadrilateral singular elements

Six-noded, isoparametric serendipity type quadrilateral regular/singular elements are used for the estimation of stress intensity factors (SIF) in linear elastic fracture mechanics (LEFM) problems involving cracks in two-dimensional structural components. The square root singularity is achieved in the six-noded elements by moving the in-side nodes to the quarter point position. The modified crack closure integral (MCCI) method is adopted which could generate accurate estimates of SIF for a relatively coarse mesh. The equations for strain energy release rate and SIF are derived for mixed mode situations using six-noded quadrilateral elements at the crack tip. The model is validated by numerical studies for a centre crack in a finite plate under uniaxial tension, a single edge notched specimen under uniaxial tension, an inclined crack in a finite rectangular plate and cracks emanating from a pin-loaded lug (or lug attachment). The results compare very well with reference solutions available in the literature.     

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.     

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

This paper proposes a definition of generalized stress intensity factors that includes classical definitions for crack problems as special cases. Based on the semi-analytical solution obtained from the scaled boundary finite-element method, the singular stress field is expressed as a matrix power function with its dimension equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the scaled boundary finite-element solution for the singular stress field by following standard stress recovery procedures in the finite element method. The definition and evaluation procedure are valid to multi-material wedges composed of any number of isotropic and anisotropic materials. Numerical examples, including a cracked homogeneous plate, a bimaterial plate with an interfacial crack, a V-notched bimaterial plate and a crack terminating at a material interface, are analyzed. Features of this unified definition are discussed.     

Fracture mechanics of delamination in ARALL laminates

Fracture mechanics of delamination in ARALL laminates is examined by using a finite element method employing special singular elements. Since these special elements contain the exact stress singularity, the delamination stress intensity factors and energy release rates can be evaluated conveniently. Solution convergence is studied to demonstrate the efficiency of this method. To ensure the validity of the result, the numerical prediction is compared with experimental results. Very good agreement is obtained.     

Variations of stress intensity factor of a semi-elliptical surface crack subjected to mixed mode loading

Maximum stress intensity factors of a surface crack usually appear at the deepest point of the crack, or a certain point along crack front near the free surface depending on the aspect ratio of the crack. However, generally it has been difficult to obtain smooth distributions of stress intensity factors along the crack front accurately due to the effect of corner point singularity. It is known that the stress singularity at a corner point where the front of 3 D cracks intersect free surface is depend on Poisson's ratio and different from the one of ordinary crack. In this paper, a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3-D semi-elliptical surface crack in a semi-infinite body under mixed mode loading. The body force method is used to formulate the problem as a system of singular integral equations with singularities of the form r ⁻³ using the stress field induced by a force doublet in a semi-infinite body as fundamental solution. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of mixed modes stress intensity factors along the crack front accurately. Distributions of stress intensity factors are indicated in tables and figures with varying the elliptical shape and Poisson's ratio.     

Analytical solution for bonded wedges under thermal stresses

After a change in temperature, high stresses leading to destruction may occur in bonded dissimilar materials near the point of the interface line intersection with the edge. In terms of linear elasticity, these high stresses are described by the singular terms of the stress field expansion at the corner point. In the present paper, the explicit representation of the singular terms and exact values of the stress intensity factors in the case of infinite wedge-shaped joint geometry are obtained by the Mellin transform technique. Systematic comparison with the FEM results for samples of finite size has shown that the values of stress intensity factors are in good agreement if the singularity is not too strong (the singularity orders _k<0.2). With the stronger singularity, the analytical solution is in qualitative agreement with the FEM one, such that it can be used for fast parametrical study of finite samples as well.     

Calculation of stress intensity factors by the force method

The force method is a simple and accurate technique for calculating stress intensity factors (SIFs) from finite element (FE) models, but it has been scarcely used. This paper shows three important advantages of the force method, which make it particularly attractive for designers and researchers. First, it can be employed without special singular quadratic finite elements at the crack tip. Actually, linear reduced integration elements may be used. Second, the force method can be applied to highly anisotropic materials without requiring knowledge of complicated elasticity relations for the stress field around the crack tip. Third, it can handle mixed-mode fracture problems.     

Plastic intensity factors for cracked plates

An elastic-plastic analysis is performed for two problems relevant to fracture mechanics: a semiinfinite body with an edge crack in a far out-of-plane shearing field and an infinite plate under plane stress conditions containing a finite line crack in a remote tensile field. Amplitudes of the dominant singularity in the plastic region at the crack tip, the plastic stress and strain intensity factors, are calculated for applied stress levels approaching the yield stress. A technique is developed for using the dominant singular solution in conjunction with the finite element method to make accurate calculations for the near-tip fields. Additionally, a comparative study of deformation theory with flow theory is performed for cracks in an anti-plane shear field. Elastic fracture mechanics is extended to high levels of applied stress for which the plastic zone is no longer small compared to the crack length by relating the critical stress for fracture initiation to the plastic intensity factors.     

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.     

各向异性复合材料尖劈的应力奇异性次数研究

为了确定各向异性复合材料尖劈的应力奇异性次数,提出了一种确定广义平面应变情况下各向异性弹性接合材料界面端应力奇异性次数的一维特殊有限元法。该方法基于最小势能原理,通过坐标变换和变量分离消除径向坐标,把应力奇异性次数的求解降阶为仅与环向坐标相关的一维问题。采用三节点一维等参数二次单元,对该一维线性领域做网格划分。数值计算结果与理论解比较,表明该方法具有很高的精度和效率。     

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.     

Computation of the stress intensity factors of sharp notched plates by the fractal‐like finite element method

The fractal‐like finite element method (FFEM) is an accurate and efficient method to compute the stress intensity factors (SIFs) of different crack configurations. In the FFEM, the cracked/notched body is divided into singular and regular regions; both regions are modelled using conventional finite elements. A self‐similar fractal mesh of an ‘infinite’ number of conventional finite elements is used to model the singular region. The corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co‐ordinates after performing a global transformation by using global interpolation functions. In this paper, we extend this method to analyse the singularity problems of sharp notched plates. The exact stress and displacement fields of a plate with a notch of general angle are derived for plane‐stress/strain conditions. These exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the SIFs. The use of the global interpolation functions reduces the computational cost significantly and neither post‐processing technique to extract SIFs nor special singular elements to model the singular region are needed. The numerical examples demonstrate the accuracy and efficiency of the FFEM for sharp notched problems. Copyright © 2008 John Wiley & Sons, Ltd.     

压电与导体双材料界面端的奇异性研究

提出了一种求解平面应变条件下横观各向同性压电与导体双材料界面端的应力及电位移奇异性的特征值法。基于横观各向同性压电材料的基本方程和一阶近似假设,利用分离变量形式的位移函数和电势函数,导出了关于应力和电位移奇异性指数的奇异性特征方程。求解由无网格法离散的特征方程,即可得到应力和电位移的各阶奇异性指数,同时还可得到相应的应力和电位移角函数。数值计算结果与文献中给出的结果非常吻合,表明该方法具有很高的精度和效率。