Computation of stress intensity factors in a plane homogeneous anisotropic solid

The accurate computation of stress intensity factors (SIFs) plays a decisive role in the determination of crack paths. The calculation of SIFs with the help of singular weight functions leads to pure Neumann problem for anisotropic elasticity in a plane domain with a crack. We present a method to overcome the specific numerical difficulties which arises while calculating these solutions with Finite Element methods. the accuracy and advantage of this method are shown by a numerical example, the calculation of SIFs of a compact tension specimen. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

幂函数型曲线裂纹平面问题的一般解

通过构造一个新的、精确的和通用的保角映射,利用Muskhelishvili复势法研究了任意自然数次幂的幂函数型曲线裂纹的平面弹性问题,给出了远处受单向拉伸载荷下裂纹尖端Ⅰ型和Ⅱ型应力强度因子的一般解．当幂次取不同的自然数时,可以退化为若干已有的结果．通过数值算例,讨论了幂函数型曲线裂纹的系数、幂次及在x轴上的投影长度对Ⅰ型和Ⅱ型应力强度因子的影响规律．     

点群6一维六方准晶椭圆孔口与半无限裂纹反平面弹性问题的解析解

导出了点群6-维六方准晶反平面弹性问题的控制方程.利用复变方法,给出了点群6-维六方准晶在周期平面内的反平面弹性问题的应力分量以及边界条件的复变表示,通过引入适当的保角变换,研究了点群6-维六方准晶中带有椭圆孔口与半无限裂纹的反平面弹性问题,得到了椭圆孔口问题应力场的解析解,给出了半无限裂纹问题在裂纹尖端处的应力强度因子的解析解.在极限情形下,椭圆孔口转化为Griffith裂纹,并得到该裂纹在裂尖处的应力强度因子的解析解.当点群6-维六方准晶体的对称性增加时,其椭圆孔口与半无限裂纹的反平面弹性问题的解退化为点群6mm-维六方准晶带有椭圆孔口与半无限裂纹的反平面弹性问题的解。     

Simulation of 2D linear crack growth under constant load using GFVM and two-point displacement extrapolation method

A new approach to model two-dimensional linear crack propagation, based on the Galerkin Finite Volume Method (GFVM), is proposed. The displacement field is calculated using the GFVM method by solving two-dimensional equilibrium equations on an unstructured triangular mesh. An essential feature of this method is that it does not require matrix operations; hence, it obviously reduces computation time. The Two-Point Displacement Extrapolation (TPDE) technique is employed to calculate Stress Intensity Factors (SIFs). The accuracy of the structural solver that has been developed is evaluated using five test cases. In the first example, a Timoshenko cantilever beam, carrying an end point load, is analyzed. In the second and third examples, stress intensity factors are computed for edge and inner crack development in plates under transient loading. The GFVM results are then compared with their counterparts that resulted from the Explicit Finite Element Method (E-FEM). The comparison indicates that the FVM has an accuracy close to E-FEM, whereas the FVM drastically reduces the computational time. A case study is conducted to simulate the gradual propagation of crack. The results computed by the numerical simulation presented are in excellent agreement with the corresponding results from the analytical solution as well as experimental measurements.     

一维正交准晶中具有四条裂纹的椭圆孔口问题的解析解

运用广义复变函数方法,通过构造适当的广义保角映射,研究了一维正交准晶中具有四条裂纹的椭圆孔口的平面弹性问题.通过引入应力函数,把平面弹性问题的基本方程简化为一个四阶偏微分方程,从而给出了各个应力分量在像平面的复表示,求得了裂纹尖端的应力强度因子的解析解.当描述缺陷的各参数发生变化时,该文的结果不仅可以还原已有文献中的结论,还可给出多种常见缺陷构型的应力强度因子,为工程力学分析提供了理论依据.     

Efficient crack growth analyzes by combining fast methods for BEM

The combination of fast methods for the boundary element method (BEM) for efficient crack growth analyzes is presented. Due to the nonlinearity of fatigue crack growth an incremental procedure has to be applied. Within each increment a stress analysis is needed. Based on the asymptotic stress field the stress intensity factors (SIFs) are calculated by an extrapolation method. Then, a new crack front is determined by a reliable 3D crack growth criterion. Finally, the numerical model has to be updated for the next increment. The time dominant factor in each increment is the computation of the stress field. Due to the stress concentration problem the BEM is utilized. To speed-up the calculation several independent fast methods are exploited. An algebraic technique is the adaptive cross approximation (ACA) method which is acting on the system matrix itself. The application of the substructure technique leads to a blockwise band matrix and therefore to reduced memory requirements. Further savings in memory and computation time are reached by modelling cracks with the dual discontinuity method (DDM) and using the ACA method in each substructure. The efficiency of the combined methods is shown by a complex industrial example. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic crack propagation in a 2D elastic body. The out-of plane case

We discuss the propagation of a running crack under shear waves in a rigorous mathematical way for a simplified model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u (y, t) and the one-dimensional crack tip trajectory h = h (t). We handle both equations separately, assuming at first that the crack position is known. Existence and uniqueness of strong solutions of the wave equation are studied and the crack-tip singularities are derived under the assumption that the crack is straight and moves tangentially. Using an energy balance law and the crack tip behaviour of the displacement fields we finally arrive at an ordinary differential equation for h (t), called equation of motion for the crack tip. We demonstrate the crack-tip motion with corresponding nonuniformly crack speed by numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

3D discrete X-ray transform

The analysis of 3D discrete volumetric data becomes increasingly important as computation power increases. 3D analysis and visualization applications are expected to be especially relevant in areas like medical imaging and nondestructive testing, where elaborated continuous theory exists. However, this theory is not directly applicable to discrete datasets. Therefore, we have to establish theoretical foundations that will replace the existing inexact discretizations, which have been based on the continuous regime. We want to preserve the concepts, properties, and main results of the continuous theory in the discrete case. In this paper, we present a discretization of the continuous X-ray transform for discrete 3D images. Our definition of the discrete X-ray transform is shown to be exact and geometrically faithful as it uses summation along straight geometric lines without arbitrary interpolation schemes. We derive a discrete Fourier slice theorem, which relates our discrete X-ray transform with the Fourier transform of the underlying image, and then use this Fourier slice theorem to derive an algorithm that computes the discrete X-ray transform in O(n⁴logn) operations. Finally, we show that our discrete X-ray transform is invertible.     

Accurate calculation of stress intensity factors for strongly curved cracks applying path-independent integrals

Path-independent contour integrals like the J_k-integral or the I_k-integral are beneficial to calculate accurate stress intensity factors (SIF). In order to avoid special requirements concerning crack tip meshing and with regard to an efficient calculation of contour integrals including internal boundaries and interfaces, it is beneficial to apply large contours far from the crack tip. In the case of mixed-mode loading and strongly curved cracks, however, this usually leads to highly inaccurate results. In this paper, several methods are presented considerably improving the calculation of SIF based on J_k- and I_k-integrals. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transverse cracks in cross-ply laminates. 1. Stress analysis

During service loading of cross-ply laminates, transverse cracks occur in plies. The cracks parallel to the fiber direction are extended over the full thickness of transverse plies and often cross the entire test specimen width. It is widely recognized that the changes of laminate thermomechanical constants, caused by the transverse cracking of composite laminates, can be significant. Theoretical stress analysis in the cross-ply laminates in the vicinity of cracks is performed using numerical (FE) and analytical methods. The effect of transverse cracks on the degradation of elastic properties will be discussed in Part 2 [1]. Approximate analytical micromechanical models based on shear lag predictions, variational analysis, and numerical 2D finite element calculations were verified in their predictive abilities. The three variational models used are based on the principle of minimum complementary energy and have different degrees of accuracy with respect to the stress assumptions used (Hashin's, 2D 0° and 2D 0°/90° models). Using FEM, the plane stress and strain state were analyzed. The effect of material properties and layer thickness on the stress distribution in a 90° layer was evaluated by varying the crack spacing. The crack opening displacement (COD), normalized with respect to the far field strain, is proposed as a measure of reduction of the mechanical properties. Since the CODs are rather insensitive to the crack spacing (crack density) in a wide region, they will be used in modeling the stiffness reduction in these laminates [1].Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 796–820, November–December, 1997.     

Stress intensity factors for the Brazilian disc with a short central crack: Opening versus closing cracks

Closed form expressions are obtained for the stress intensity factors (SIFs) in case of a Brazilian disc with a short central crack, the length of which does not exceed one fifth of the disc radius. The disc is loaded by uniform radial pressure along two finite symmetric arcs of its periphery. The solution is achieved using the method of complex potentials introduced by Kolosov and Muskhelishvili. The advantage of the expressions obtained is that they are valid both for cracks under opening mode as well as for closing cracks. For the first case (opening cracks) the results of the present study are compared with existing approximate solutions and it is concluded that the agreement is excellent as long as the length of the crack remains relatively small compared to the radius of the disc. Regarding the case of a closing mode crack the procedure proposed here (based on a recent alternative approach of the cracked Brazilian disc) leads to a physically acceptable deformed crack shape instead to an unnatural crack with overlapped lips. At the same moment the dependence of the SIFs on the properties of the material is eliminated.     

Influence of inclusion on the stress intensity factors under thermomechanical loads in a PM superalloy

The interaction between a round inclusion and a crack under thermomechanical loading is analyzed based on a modified body force method. The traction-free condition on the crack line is mended by adding the resultant force induced by thermal stress to the force equilibrium equations, so that the coupling of mechanical and thermal loads could be taken into account. The series of integral equations can be discretized to a set of linear equations. Stress intensity factors (SIFs) are obtained through solving the linear equations. The calculated results in this paper are compared to those in open references to validate the method and code. The method is applied to a case of FGH95 PM superalloy containing Al₂O₃ inclusions under mechanical and thermal loads. The results show that the thermal load has little effect on SIF, while the mechanical load is the dominant factor.     

A. D. Alexandrov’s problem for non-positively curved spaces in the sense of Busemann

This paper is the last of a series devoted to the solution of Alexandrov’s problem for non-positively curved spaces. Here we study non-positively curved spaces in the sense of Busemann. We prove that isometries of a geodesically complete connected at infinity proper Busemann space X are characterized as follows: If a bijection f: X → X and its inverse f ⁻¹ preserve distance 1, then f is an isometry.     

Relaxation Time for the Discrete D/G/1 Queue

When queueing models are used for performance analysis of some stochastic system, it is usually assumed that the system is in steady-state. Whether or not this is a realistic assumption depends on the speed at which the system tends to its steady-state. A characterization of this speed is known in the queueing literature as relaxation time.The discrete D/G/1 queue has a wide range of applications. We derive relaxation time asymptotics for the discrete D/G/1 queue in a purely analytical way, mostly relying on the saddle point method. We present a simple and useful approximate upper bound which is sharp in case the load on the system is not very high. A sharpening of this upper bound, which involves the complementary error function, is then developed and this covers both the cases of low and high loads.For the discrete D/G/1 queue, the stationary waiting time distribution can be expressed in terms of infinite series that follow from Spitzer’s identity. These series involve convolutions of the probability distribution of a discrete random variable, which makes them suitable for computation. For practical purposes, though, the infinite series should be truncated. The relaxation time asymptotics can be applied to determine an appropriate truncation level based on a sharp estimate of the error caused by truncating.This revised version was published online in June 2005 with corrected coverdate     

一维六方准晶中带三条不对称裂纹的圆形孔口问题的解析解

利用复变函数方法,通过构造保角映射,研究了一维六方准晶中带不对称三裂纹的圆形孔口的反平面剪切问题,给出了Ⅲ型裂纹问题的应力强度因子,在极限情形下,不仅可以还原为已有的结果,而且求得一维六方准晶中L裂纹问题在裂纹尖端的应力强度因子.     

Vertex singularities associated with conical points for the 3D laplace equation

The solution u to the Laplace equation in the neighborhood of a vertex in a three‐dimensional domain may be described by an asymptotic series in terms of spherical coordinates $$u = \sum\nolimits_i {A_i}{\rho ^{{\nu _i}}}{f_i}(\theta ,\phi )$$ . For conical vertices, we derive explicit analytical expressions for the eigenpairs ν_i and f_i(θ, φ), which are required as benchmark solutions for the verification of numerical methods. Thereafter, we extend the modified Steklov eigen‐formulation for the computation of vertex eigenpairs using p/spectral finite element methods and demonstrate its accuracy and high efficiency by comparing the numerically computed eigenpairs to the analytical ones. Vertices at the intersection of a crack front and a free surface are also considered and numerical eigenpairs are provided. The numerical examples demonstrate the efficiency, robustness, and high accuracy of the proposed method, hence its potential extension to elasticity problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

一维六方准晶中带双裂纹的椭圆孔口问题的解析解

利用复变函数方法,通过构造保角映射,研究了一维六方准晶中带双裂纹的椭圆孔口的反平面剪切问题,给出了Ⅲ型裂纹问题的应力强度因子,在极限情形下,不仅可以还原为已有的结果,而且求得一维六方准晶中带双裂纹的圆形孔口问题、十字裂纹问题在裂纹尖端的应力强度因子．     

Curved Hexagonal Packings of Equal Disks in a Circle

For each k ≥ 1 and corresponding hexagonal number h(k) = 3k(k+1)+1, we introduce packings of h(k) equal disks inside a circle which we call the curved hexagonal packings. The curved hexagonal packing of 7 disks (k = 1, m(1)=1) is well known and one of the 19 disks (k = 2, m(2)=1) has been previously conjectured to be optimal. New curved hexagonal packings of 37, 61, and 91 disks (k = 3, 4, and 5, m(3)=1, m(4)=3, and m(5)=12) were the densest we obtained on a computer using a so-called ``billiards' simulation algorithm. A curved hexagonal packing pattern is invariant under a rotation. For , the density (covering fraction) of curved hexagonal packings tends to . The limit is smaller than the density of the known optimum disk packing in the infinite plane. We found disk configurations that are denser than curved hexagonal packings for 127, 169, and 217 disks (k = 6, 7, and 8). In addition to new packings for h(k) disks, we present the new packings we found for h(k)+1 and h(k)-1 disks for k up to 5, i.e., for 36, 38, 60, 62, 90, and 92 disks. The additional packings show the ``tightness' of the curved hexagonal pattern for k ≤ 5: deleting a disk does not change the optimum packing and its quality significantly, but adding a disk causes a substantial rearrangement in the optimum packing and substantially decreases the quality. Received May 15, 1995, and in revised form March 5, 1996.