基于Kirchhoff板中裂纹尖端辛本征解的有限元应力恢复方法

对于含穿透裂纹的板结构,裂纹尖端应力场及应力强度因子的计算精度对评估板的安全性具有非常重要的影响。基于含裂纹Kirchhoff板弯曲问题中裂纹尖端场的辛本征解析解,该文提出了一个提高裂纹尖端应力场计算精度的有限元应力恢复方法。首先利用常规有限元程序对含裂纹板弯曲问题进行分析,得到裂纹尖端附近的单元节点位移;然后根据节点位移确定辛本征解中的待定系数,得到裂纹尖端附近应力场的显式表达式。数值结果表明,该方法给出的应力分析精度得到较大提高,并具有良好的数值稳定性。     

Numerical thermo-elastic analysis of singularities in two-dimensions

Linear elastic two-dimensional problems with singular points subjected to steady-state temperature distribution are considered. The stress tensor in the vicinity of the singular points exhibits singular behavior characterized by the strength of the singularity and the associated thermal stress intensity factors (TSIFs). It is shown that the TSIFs and the strength of the stress singularity can be obtained using the principle of complementary energy together with the modified Steklov method and the p-version of the finite element method. Importantly, the proposed method is applicable not only to singularities associated with crack tips, but also to multi-material interfaces and non-homogeneous materials. Numerical results of crack-tip singularities in a rectangular plate and singular points associated with a two-material inclusion are presented.Research performed while the author served as a visiting assistant professor at the Center for Computational Mechanics, Washington University, St. Louis, Missouri 63130, USA.     

Dynamic analysis of fixed cracks in composites by the extended finite element method

This paper is dedicated to simulation of dynamic analysis of fixed cracks in orthotropic media using an extended finite element method. This work is in fact an extension to dynamic problems of the recently developed orthotropic extended finite element method for fracture analysis of composites. In this method, the Heaviside and near-tip enrichment functions are used in the framework of the partition of unity for modeling crack discontinuity and crack-tip singularities within the classical finite element method. In this procedure, elements that include a crack are not required to conform to crack edges. Therefore, mesh generation can be performed without any need to comply to crack edges and the method is capable of modeling the crack propagation without any remeshing. To determine the fracture properties, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated by means of domain separation integral (J^′-integral) method. Results of the proposed method are compared with other available analytical and computational results.     

A finite element method for calculating stress intensity factors and its application to composites

A concept which allows for the development of efficient finite element techniques in the analysis of plane elastic structures containing cracks is discussed. It consists of combining a specially defined finite element in the region surrounding each crack tip with conventional CST elements describing the remaining portion of the geometry considered. For the special element a pair of displacement functions is chosen which adequately represents the singular character of the elastic solution at the crack tip. The application of this concept is illustrated through a specific numerical method developed by W. K. Wilson for the calculation of mode I stress intensity factors.

Wilson's method was coded and used to analyze an infinitely long strip under tension with a line crack perpendicular to its axis of symmetry. Circular inclusions of different material properties were assumed to be present near the tips of the crack and their effect on the mode I stress intensity factor was investigated.

It was found that more flexible inclusions increase the intensity factor while more rigid inclusions decrease it. These results are quite similar to those obtained by analytical methods in an analogous problem involving an infinite sheet, but in the case of a strip, the influence of inclusions on the intensity factor was found to be more pronounced.     

Stress intensity factors by enriched finite elements

The finite element method is extended to direct calculation of combined modes I and II stress intensity factors for axisymmetric and planar structures of arbitrary geometry and loading. High order conventional isoparametric elements are combined with a fracture mechanics enrichment of the same element so that a corner node is made to correspond to a crack-tip. The development is explained in detail and necessary modifications to a standard finite element computer program are identified. Stress intensity factors as well as a complete stress analysis are obtained directly from the computer printout. Few elements are generally required, minimizing engineering costs and eliminating the need for data generators. Example problems demonstrate the ease of using the method, the high accuracy to be expected from the method, and the fact that accuracy is relatively insensitive to variations of the element mesh in the vicinity of crack tips. Important applications include complex geometries for which other methods fail and situations wherein multiple crack tips may interfere with one another.     

Investigation of anti-plane shear behavior of a Griffith permeable crack in functionally graded piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading for the permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of dual-integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allows us to using the maximum stress as a fracture criterion. The finite hoop stresses at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Non-local theory solution for the anti-plane shear of two collinear permeable cracks in functionally graded piezoelectric materials

In this paper, the non-local theory of elasticity is firstly applied to obtain the behavior of two collinear cracks in functionally graded piezoelectric materials under anti-plane shear loading for permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield finite stresses at the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses at the crack tips depend on the distance between two collinear cracks, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Cracked Elastic Bi-Material Layer Under Compressive Loads

The boundary value problem of an elastic bi-material layer containing a finite length crack under compressive mechanical loadings has been studied. The crack is located on the bi-material interface and the contact between crack surfaces is frictionless. Based on Fourier integral transformation techniques the solution of the formulated problem is reduced to the solution of singular integral equation, then, with Chebyshev`s orthogonal polynomials, to infinite system of linear algebraic equations. The expressions for contact stresses in the elastic compound layer are presented. Based on the analytical solution it is found that in the case of frictionless contact the shear and normal stresses have inverse square root singularities at the crack tips. Numerical solutions have been obtained for a series of examples. The results of these examples are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mixture of mode I and mode II type singularities. The numerical solutions show that an interfacial crack under compressive forces can become open in certain parts of the contacting crack surfaces, depending on the applied forces, material properties and geometry of the layers.     

HT有限元在Ⅰ、Ⅱ与Ⅲ型复合弹性断裂问题中的应用

探讨了HT有限元应用于Ⅰ、Ⅱ和Ⅲ型复合裂纹的弹性断裂问题。分析了Ⅲ型弹性断裂问题的HT有限元方法及高阶奇异性应力强度因子KΙΙΙ,同时,对Ⅰ和Ⅱ型断裂问题的HT有限元原理及断裂强度因子KΙ和KΙΙ的计算也进行了阐述。特别地,在计算三个强度因子时,引入了一种新的方法——附加试函数法,它主要用于满足裂尖特殊的边界条件,提高了三个奇异应力强度因子的精确性与可靠性。最后,根据HT有限元计算结果,讨论了奇异应力强度因子无量纲化系数K/Kc随裂纹单元特殊T函数项数、细划单元数、单元高斯点数及裂尖不同附加试函数的变化规律;获得了应力强度因子精确度和可靠度,并与其它有限元结果进行了比较,阐述了此方法的优越性。     

Strength of sandwich beams with mid-plane debondings in the core

Sandwich beams containing cracks in the mid-plane of the core are investigated. The cracked part is subjected to a constant remote shear stress field. Beams with different cross-section geometries and materials were analyzed by the finite element method (FEM) in order to compute the stress intensity factors at the crack tips. An analytical approach for estimation of the energy release rate, based on a potential energy calculation, is presented that agrees well with results from the FE analyses. Results from four-point bending tests with cracked beams show that the fracture load can be accurately predicted. The simplicity of the analytical model makes it possible to compute critical crack lengths and safety factors for various types of sandwich beams.     

Dynamic propagation of a finite crack in a micropolar elastic solid

Summary The dynamic propagation of a finite crack under mode-I loading in a micropolar elastic solid is investigated. By using an integral transform method, a pair of two-dimensional singular integral equations governing stress and couple stress is formulated in terms of displacement transverse to the crack, macro and micro rotations, and microinertia. These equations are solved numerically, and solutions for dynamic stress intensity and couple stress intensity factors are obtained by utilizing the values of the strengths of the square root singularities in macrorotation and the gradient of microrotation at the crack tips.     

Non-Local Theory Solution for an Anti-Plane Shear Permeable Crack in Functionally Graded Piezoelectric Materials

In this paper, the non-local theory solution of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading is obtained for the permeable electric boundary conditions, in which the material properties vary exponentially with coordinate parallel to the crack. The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variable is the jump of displacement across the crack surfaces, not the dislocation density function. To solve the dual integral equations, the jump of the displacement across the crack surfaces is directly expanded in a series of Jacobi polynomials. From the solution of the present paper, it is found that no stress and electric displacement singularities are present near the crack tips. The stress fields are finite near the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses and the electric displacements at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively. On the other hand, the angular variations of the strain energy density function are examined to associate their stationary value with locations of possible fracture initiation.     

分析不同材料界面应力奇异性的一维杂交有限元方法

该文提出了一种计算效率较高的分析不同材料界面应力奇异性的一维杂交有限元方法。为了推导该方法,首先列出了用于求解不同材料界面裂纹奇异应力场特征解的基本方程和边界条件,然后利用加权残量方法(weighted residual method),得到上述基本方程和边界条件的弱形式,该弱形式的基本变量为位移和应力。运用Galerkin有限元方法的思想及上述弱形式,最后得到了一个一维杂交有限元方法,该一维杂交有限元方法只需对扇形区域在角度方向上离散,其总体方程为一个二次特征矩阵方程。数值算例表明:该方法可以准确而高效地计算不同材料界面奇异应力场的特征解。     

Mapped finite element methods: High‐order approximations of problems on domains with cracks and corners

Linear elasticity problems posed on cracked domains, or domains with re‐entrant corners, yield singular solutions that deteriorate the optimality of convergence of finite element methods. In this work, we propose an optimally convergent finite element method for this class of problems. The method is based on approximating a much smoother function obtained by locally reparameterizing the solution around the singularities. This reparameterized solution can be approximated using standard finite element procedures yielding optimal convergence rates for any order of interpolating polynomials, without additional degrees of freedom or special shape functions. Hence, the method provides optimally convergent solutions for the same computational complexity of standard finite element methods. Furthermore, the sparsity and the conditioning of the resulting system is preserved. The method handles body forces and crack‐face tractions, as well as multiple crack tips and re‐entrant corners. The advantages of the method are showcased for four different problems: a straight crack with loaded faces, a circular arc crack, an L‐shaped domain undergoing anti‐plane deformation, and lastly a crack along a bimaterial interface. Optimality in convergence is observed for all the examples. A proof of optimal convergence is accomplished mainly by proving the regularity of the reparameterized solution. Copyright © 2016 John Wiley & Sons, Ltd.     

A wide range weight function for a single edge cracked geometry with clamped ends

A single edge cracked geometry with clamped ends is well suited for fracture toughness and fatigue crack growth testing of composites and thin materials. Stress intensity factors may be determined by the weight function method. A weight function for the single edge cracked geometry with clamped ends is developed and verified in this paper. It is based on analytical forms for the reference stress intensity factor and crack mouth opening displacement. The analytical forms are shown to be valid, by comparison with finite element results, over a wide range of crack depths and plate aspect ratios. Use of the analytical form enables the weight function to be calculated for any plate aspect ratio without the need for preliminary finite element analysis. Stress intensity factors and crack mouth opening displacements, predicted using this weight function, correlated well with finite element results for non-uniform crack surface stress distributions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Methods for modelling the ultimate strength of orthotropic plate with a central hole under uniaxial tension

The stress state in plates with circular holes made of orthotropic homogeneous material has no singularities and it can be exactly determined. The numerical stress distribution calculation by the finite element method will be compared with those obtained by the analytical equations developed by several authors. The goal of this work is to validate the finite element method, in conjunction with in-plane and out of plane failure criteria, in order to calculate not only the stress distribution for orthotropic plates with circular holes but also to determine their ultimate strength.The tool used has been a user subroutine (UMAT) specially developed for this work that implements the features of the commercial FE program (ABAQUS). The code performs an implicit analysis of the stress-state with progressive damage modelling.Finally, both of them, numerical and analytical method, will be checked with experimental tests by means of strain gauges.     

The effect of grain-size on the stress and velocity fields ahead of a crack in a material which deforms by Coble creep

The stress and velocity fields ahead of a grain-boundary crack in a material which deforms by Coble creep are investigated for a series of grain sizes using the numerical analysis technique established by Cocks [1–3]. The numerical results are compared with the linear viscous fields obtained by the boundary element method and the finite element method. It is demonstrated that a detailed knowledge of the micromechanical processes of material transport and deformation are required to determine the stress and displacement fields for the grains which immediately surround the crack tip. But the response of the surrounding material can be adequately described from a conventional linear viscous analysis of the problem.     

New crack‐tip elements for XFEM and applications to cohesive cracks

An extended finite element method scheme for a static cohesive crack is developed with a new formulation for elements containing crack tips. This method can treat arbitrary cracks independent of the mesh and crack growth without remeshing. All cracked elements are enriched by the sign function so that no blending of the local partition of unity is required. This method is able to treat the entire crack with only one type of enrichment function, including the elements containing the crack tip. This scheme is applied to linear 3‐node triangular elements and quadratic 6‐node triangular elements. To ensure smooth crack closing of the cohesive crack, the stress projection normal to the crack tip is imposed to be equal to the material strength. The equilibrium equation and the traction condition are solved by the Newton–Raphson method to obtain the nodal displacements and the external load simultaneously. The results obtained by the new extended finite element method are compared to reference solutions and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips

In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials.     

A novel singular finite element on mixed-mode bimaterial interfacial cracks with arbitrary crack surface tractions

This paper investigates interfacial cracks with arbitrary crack surface tractions. A novel singular finite element which is constructed with the analytical solution around interfacial cracks is presented. Interfacial crack problems can be analyzed numerically using the singular finite element, and Mode I and/or Mode II stress intensity factors can be obtained directly. Unlike other enriched elements for cracks, neither extra unknowns nor transition elements are required. Numerical examples are given to illustrate the validity of present method.