The effect of crack faces contact interaction on the critical strain in composites under compressive loading

Aviation and aerospace structural components made of composite laminates due to their internal structure and manufacturing methods often contain a number of inter- and intra-component defects which size, dispersion and interaction alter significantly the critical compression strain level [1]. The current study investigates the effect of the cracks interaction and crack faces contact interaction on the critical strain in laminar transversally isotropic material (cross-ply) compressed in a static manner along interlaminar defects. The frictionless Hertzian contact and the shear and extensional mode of stability loss are considered for the interacting crack faces. The statement of the problem is based on the most accurate approach, the model of piecewise-homogenous medium and the 3-D stability theory [2]. The moment of stability loss in the microstructure of material is treated as the onset of the fracture process. The complex non-classical fracture mechanics problem is solved utilizing the finite elements analysis. The results are obtained for the typical dispositions of cracks. It was found that the crack faces contact interaction alter significantly the critical strain level of the composite. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

夹层压电材料中垂直于界面的共线双裂纹动力学问题分析

采用Schmidt方法分析了在简谐反平面剪切波作用下,两个半空间夹层压电材料中的共线裂纹的动力学行为．压电材料层内裂纹垂直于界面,电边界条件假设为可导通．通过Fourier变换,使问题的求解转换为两对三重积分对偶方程．通过数值计算,给出了裂纹的几何尺寸、压电材料常数、入射波频率等对于应力强度因子的影响．结果表明,在不同的入射波频率范围,动力场将阻碍或促使压电材料内裂纹的扩展．与不可导通电边界条件相比,导通裂纹表面的电位移强度因子比不可导通裂纹的电位移强度因子要小许多．     

含裂纹的圆柱体的弯曲

含裂纹的圆柱体弯曲的研究有十分重要的意义。文献[1]～[4]研究过含径向裂纹或裂纹系的情形,[5]研究过有同心圆弧裂纹的圆柱体的弯曲。本文继续[6]对内部出现在任意位置的直线裂纹的圆柱体在力与裂纹垂直时的弯曲问题,用弹性理论复变函数方法进行了讨论;得到了位移、应力和应力强度因子用级数表示的表达式;对Ah小的这种弯曲问题的应力强度因子给出了好的近似式,分析了它们随裂纹中心的变化规律。最后对裂纹的一个尖端在原点的径向裂纹圆柱体的扭转率和弯曲中心进行了计算,其结果与[1]几乎完全相同。     

压电材料中两平行对称可导通裂纹断裂性能分析

采用Schmidt研究了压电材料中对称平行的双可导通裂纹的断裂性能,利用富里叶变换使问题的求解转换为求解两对以裂纹面位移之差为未知变量的对偶积分方程,并采用Schmidt方法来对这两对对偶积分程进行数值求解。结果表明应力强度因子和电位移强度因子与裂纹的几何尺寸有关。与不可导通裂纹有关结果相比,可导通裂纹的电位移强度因子远小于相应问题不可导通裂纹的电位移强度因子。     

沿两条内部平行裂纹受压时材料的断裂问题

顺着平面内的平行裂纹受压材料的断裂问题并不能在线性断裂力学的框架内进行描述，Grif-fith－Irvin型或COC型的断裂判据，虽然可以用来处理经典的线性断裂力学，但对本题则完全不适用，这是因为这些压力对应力强度系数没有影响，与裂缝孔隙值也没有关系[1，2]，这一类问题只能采用新的方法，本文的第一作者曾建议过一种新方法，在这一方法中仍然使用了线性关系，但这种线性关系是从变形固体力学中的非线性方程导出的[3，4，5]．这里必须指出，这种方法曾在变形体稳定性问题中广泛地采用过。作为断裂开始的判据，我们采用了裂缝缺陷附近的局部失稳的判据，在这类情况下，我们认为是从失稳过程引发断裂过程的。     

Interaction between three moving Griffith cracks at the interface of two dissimilar elastic media

The paper deals with the interaction between three Griffith cracks propagating under antiplane shear stress at the interface of two dissimilar infinite elastic half-spaces. The Fourier transform technique is used to reduce the elastodynamic problem to the solution of a set of integral equations which has been solved by using the finite Hilbert transform technique and Cooke’s result. The analytical expressions for the stress intensity factors at the crack tips are obtained. Numerical values of the interaction effect have been computed for and results show that interaction effects are either shielding or amplification depending on the location of each crack with respect to other and crack tip spacing.     

Scattering of antiplane shear wave by a propagating crack at the interface of two dissimilar elastic media

An analysis of the scattering of horizontally polarized shear wave by a semi-infinite crack running with uniform velocity along the interface of two dissimilar semi-infinite elastic media has been carried out. The mixed boundary value problem has been solved completely by the Wiener-Hopf technique. The effect of different values of the material parameter, the angle of incidence of incident wave and the crack propagation velocity on the stress intensity factor have been illustrated graphically.     

Dual boundary integral formulation for 2-D crack problems

An efficient integral equation formulation for two-dimensional crack problems is proposed with the displacement equation being used on the outer boundary and the traction equation being used on one of the crack faces. Discontinuous quarter point elements are used to correctly model the displacement in the vicinity of crack tips. Using this formulation a general crack problem can be solved in a single-region formulation, and only one of the crack faces needs to be discretised. Once the relative displacements of the cracks are solved numerically, physical quantities of interest, such as crack tip stress intensity factors can be easily obtained. Numerical examples are provided to demonstrate the accuracy and efficiency of the present formulation.     

Computation of crack propagation in 3-dimensional anisotropic solids

This contribution presents ideas, how crack propagation in three-dimensional solids composed of anisotropic materials can be predicted using the Griffith energy principle. Since the work of Irwin the change of potential energy caused by a straight elongation of a crack in an isotropic two-dimensional homogeneous structure can be expressed in quadratic terms of the stress intensities at the crack tip. This result was generalized in the last decades using methods of asymptotic analysis by many authors [1] to more complicated geometries, to anisotropic and inhomogeneous materials. With the energy release rate at hand, quasi-static scenarios of crack propagation can be simulated for plane problems [2], but this is still a complicated task for three-dimensional problems [3]. We show an idea how the change of energy caused by propagation of a crack surface in a fully three-dimensional solid of nearly arbitrary shape can be computed in anisotropic materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress intensity factor for multiple cracks in bonded dissimilar materials using hypersingular integral equations

This paper deals with the multiple inclined or circular arc cracks in the upper half of bonded dissimilar materials subjected to shear stress. Using the complex variable function method, and with the help of the continuity conditions of the traction and displacement, the problem is formulated into the hypersingular integral equation (HSIE) with the crack opening displacement function as the unknown and the tractions along the crack as the right term. The obtained HSIE are solved numerically by utilising the appropriate quadrature formulas. Numerical results for multiple inclined or circular arc cracks problems in the upper half of bonded dissimilar materials are presented. It is found that the nondimensional stress intensity factors at the crack tips strongly depends on the elastic constants ratio, crack geometries, the distance between each crack and the distance between the crack and boundary.     

压电压磁复合材料中一对平行裂纹对弹性波的散射

利用Schmidt方法对压电压磁复合材料中一对平行对称裂纹对反平面简谐波的散射问题进行了分析,借助富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程．在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式,最终获得了应力强度因子、电位移强度因子、磁通量强度因子三者之间的关系．结果表明,压电压磁复合材料中平行裂纹动态反平面断裂问题的应力奇异性与一般弹性材料中的动态反平面断裂问题的应力奇异性相同,同时讨论了裂纹间的屏蔽效应．     

Phase field simulation of thermomechanical fracture

In Francfort and Marigo's variational free-discontinuity formulation of brittle fracture [1] cracking is regarded as an energy minimization process, where the total energy is minimized with respect to any admissible crack set and displacement field. No additional criterion is needed to determine crack paths, branching of cracks and crack initiations. However, a direct discretization of the model is faced with significant technical problems, as it involves minimizations in a set of possibly discontinuous functions. A regularized version of the model has been introduced by Bourdin [2] and based on this, we use the concept of a continuum phase field model to simulate cracking processes. Cracks are indicated by the order parameter of the phase field model and cracking can be regarded as a phase transition problem. Additionally, introducing the heat equation into the model, it is capable to also take account of crack propagation due to thermal stresses. In the numerical implementation, crack parameter as well as temperature are treated as additional degrees of freedom and the coupled field equations are solved using the finite element method together with an implicit time integration scheme. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

压电材料中两平行不相等界面裂纹的动态特性研究

利用Schmidt方法,研究了压电材料中两个平行不相等的可导通界面裂纹对简谐反平面剪切波的散射问题．利用Fourier变换,使问题的求解转换为对两对以裂纹面张开位移为未知变量的对偶积分方程的求解．数值计算结果表明,动态应力强度因子及电位移强度因子受裂纹的几何参数、入射波频率的影响．在特殊情况下,与已有结果进行了比较分析．同时,电位移强度因子远小于不可导通电边界条件下相应问题的结果．     

HYPERSINGULAR INTEGRAL EQUATIONS FOR PERIODIC ARRAYS OF PLANAR CRACKS IN A PERIODICALLY-LAYERED ANISOTROPIC ELASTIC SPACE UNDER ANTIPLANE SHEAR STRESS

1991MRSubjectClassification75M25,45E991IntroductionDuringthelasttenyearsorsojmanyresearchersinappliedmathematicsandmechanicshaveshownasurginginterestinformulatinglinearcrackproblemsillterlllsofsystel-alsofHadamardfillite-part(hypersingular)integralequations,e.g.Ioakimidis['],Lin'kovandMogilevskaya[']andAnal'].Anadvantageofsuchaformulationisthatthe11nkllowllfllnctionsaredirectlyrelatedtothejlllxlpillthedisplacementsacrossoppositeera(:kfaces.Oncetheyaredeterlttillied,crackparaliietersofinter…     

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.     

Closed-form solutions for a rectangular layer with central crack under anti-plane shear stress

We consider the problem of determining the stress distributionin a finite rectangular elastic layer containing a Griffithcrack which is opened by internal shear stress acting alongthe length of the crack. The mode III crack is assumed to belocated in the middle plane of the rectangular layer. The followingtwo problems are considered: (A) the central crack is perpendicularto the two fixed lateral surfaces and parallel to the othertwo stress-free surfaces; (B) all the lateral surfaces of therectangular layer are clamped and the central crack is parallelto the two lateral surfaces. By using Fourier transformations,we reduce the solution of each problem to the solution of dualintegral equations with sine kernels and a weight function whichare solved exactly. Finally, we derive closed-form expressionsfor the stress intensity factor at the tip of the crack andthe numerical values for the stress intensity factor at theedges of the cracks are presented in the form of tables.     

On the elastic symmetries of growing mixed-mode cracks

Many continuum damage mechanics models for quasi-brittle materials are based on the reduction of stiffness due to elliptical crack or penny-shaped microcracks in the material. Because of this a numerical study of growing elliptical cracks in a unit cube is undertaken with the help of an FEM simulation.The propagation of the crack is governed by the principle of maximum driving force [1]. For each propagation step the tensor of elasticity is calculated and its symmetries are analyzed. It will be shown that the elastic symmetry in each step is close to orthotropy and can be approximated by an elliptical crack. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Crack propagation calculation using trial cracks

We present a method to perform finite element calculations for crack propagation problems with arbitrary crack directions in two dimensions. The crack direction (angle of propagation) is determined by inserting small “trial cracks” at the crack tip. For each trial crack, the domain is remeshed to allow crack propagation between elements. The trial cracks are then opened and the energy release rate is measured. The optimum crack direction (i.e., the crack direction with maximum energy release) is determined by an optimisation procedure. Although the method is computationally expensive due to the need to perform several calculations for each crack increment, it has the advantage that the energy release rate can be calculated even in cases where other methods fail. After explaining the method, it is applied to some test examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of a bridged crack in a discrete lattice

The paper addresses a problem of partial fracture of a latticeby a propagating fault modelling a crack bridged by elasticfibres. It is assumed that the strength of bonds within thelattice alternates periodically, so that during the dynamiccrack propagation only weaker bonds break, whereas the strongerbonds remain intact. The mathematical problem is reduced tothe functional equation of the Wiener–Hopf type, whichis solved analytically. The load–crack speed dependenceis presented, which also has implications on the stability analysisfor the bridged crack propagating within the lattice. In particular,we address the evaluation of the dissipation rate, which isfound to be strongly dependent on the crack speed. In this latticemodel, our results also cover the case of the supercriticalcrack speed.     

Mixed finite element methods for the approximation of time-dependent problems

A given elliptic boundary problem can, in general, be approximated by several different mixed finite element methods, depending on what physical unknowns are most desired. The use of certain mixed methods for time-dependent problems has been proposed by Johnson and Thomee [5]. We analyze here some additional mixed methods for the time-dependent case which can be used to obtain direct approximations to alternate physical quantities of interest.