The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

改进型扩展比例边界有限元法

结合了扩展有限元法(extended finite elementmethods,XFEM)和比例边界有限元法(scaled boundary finite elementmethods,SBFEM)的主要优点,提出了一种改进型扩展比例边界有限元法(improvedextended scaled boundary finite elementmethods,$i$XSBFEM),为断裂问题模拟提供了一条新的途径.类似XFEM,采用两个正交的水平集函数表征材料内部裂纹面,并基于水平集函数判断单元切割类型;将被裂纹切割的单元作为SBFE的子域处理,采用SBFEM求解单元刚度矩阵,从而避免了XFEM中求解不连续单元刚度矩阵需要进一步进行单元子划分的缺陷;同时,借助XFEM的主要思想,将裂纹与单元边界交点的真实位移作为单元结点的附加自由度考虑,赋予了单元结点附加自由度明确的物理意义,可以直接根据位移求解结果得出裂纹与单元边界交点的位移;对于含有裂尖的单元,选取围绕裂尖单元一圈的若干层单元作为超级单元,并将此超级单元作为SBFE的一个子域求解刚度矩阵,超级单元内部的结点位移可通过SBFE的位移模式求解得到,应力强度因子可基于裂尖处的奇异位移(应力)直接获得,无需借助其他的数值方法.最后,通过若干数值算例验证了建议的$i$XSBFEM的有效性,相比于常规XFEM,$i$XSBFEM的基于位移范数的相对误差收敛性较好;采用$i$XSBFEM通过应力法和位移法直接计算得到的裂尖应力强度因子均与解析解吻合\较好.      

Analysis on the cohesive stress at half infinite crack tip

The nonlinear fracture behavior of quasi-brittle materials is closely related with the cohesive force distribution of fracture process zone at crack tip. Based on fracture character of quasi-brittle materials, a mechanical analysis model of half infinite crack with cohesive stress is presented. A pair of integral equations is established according to the superposition principle of crack opening displacement in solids, and the fictitious adhesive stress is unknown function . The properties of integral equations are analyzed, and the series function expression of cohesive stress is certified. By means of the data of actual crack opening displacement, two approaches to gain the cohesive stress distribution are proposed through resolving algebra equation. They are the integral transformation method for continuous displacement of actual crack opening, and the least square method for the discrete data of crack opening displacement. The calculation examples of two approaches and associated discussions are give     

The vibration of a half-space due to a buried mode I crack opening

The solution of a dynamic problem for calculation of a displacement field on a half-space surface caused by an internal mode I crack opening is presented. The problem is reduced to the system of boundary integral equations (BIEs). The equations of motion are solved with the use of Helmholtz potentials and applying Fourier integral transform. The effects of the crack size, the crack depth and the distance from the crack epicenter to the observation point on the parameters of elastic waves are investigated. It is established that the increasing of the defect size leads to narrowing bandwidth of elastic waves and to lowering of center frequency. The analysis given here can be used for identification of the crack growth during technical diagnostic of an industry objects and structural elements by AE method.     

压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向刚性压电材料中受拉伸和电载荷作用的平片裂纹Ⅰ型断裂力学问题．使用自限部分概念，从二维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意．     

基于整体位移模式的平面裂纹结构数值模拟方法

对于平面裂纹问题,针对扩展有限元法和无网格伽辽金法的不足,从结构的整体位移模式出发,提出了一种新的数值模拟方法。在整个求解域内构造其试探函数,并引入裂纹修正项描述裂尖处的奇异性和裂纹面的强间断特性;同时,提出了一种新的强制边界条件施加方法,通过引入位移边界水平集函数,将位移边界条件包含在近似位移场的表达式中,有效地解决了位移边界条件问题,减小了刚度矩阵的阶数,非常方便地消除了刚度矩阵的奇异性,降低了线性方程组的求解难度。含裂纹矩形平板结构的数值算例验证了该方法的有效性。     

扩展数字图像相关方法中裂尖位移函数的表征研究

探寻适用于扩展数字图像相关方法(X-DIC)的裂纹尖端位移函数,对于提高该方法在裂尖测试精度方面具有重要意义。本文基于断裂力学裂纹尖端位移场函数,进行主导项影响分析,并探究裂尖位移函数中各项及其组合项对位移场表征的贡献程度及对测试精度的影响。通过对I型、II型及混合型裂纹进行测试,发现各项及其组合项在不同裂纹类型测试中的影响有着较大区别,从而得出在X-DIC测试计算中影响裂尖位移场表征的主导项。最后,应用上述研究成果对某航空超硬铝合金的单边裂纹进行拉伸测试。     

Analysis method of planar cracks of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium

The hyper-singular boundary integral equation method of crack analysis in three-dimensional transversely isotropic magnetoelectroelastic media is proposed. Based on the fundamental solutions or Green’s functions of three-dimensional transversely isotropic magnetoelectroelastic media and the corresponding Somigliana identity, the boundary integral equations for a planar crack of arbitrary shape in the plane of isotropy are obtained in terms of the extended displacement discontinuities across crack faces. The extended displacement discontinuities include the displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, and correspondingly the extended tractions on crack face represent the conventional tractions, the electric displacement and the magnetic induction boundary values. The near crack tip fields and the intensity factors in terms of the extended displacement discontinuities are derived by boundary integral equation approach. A solution method is proposed by use of the analogy between the boundary integral equations of the magnetoelectroelastic media and the purely elastic materials. The influence of different electric and magnetic boundary conditions, i.e., electrically and magnetically impermeable and permeable conditions, electrically impermeable and magnetically permeable condition, and electrically permeable and magnetically impermeable condition, on the solutions is studied. The crack opening model is proposed to consider the real crack opening and the electric and magnetic fields in the crack cavity under combined mechanical-electric-magnetic loadings. An iteration approach is presented for the solution of the non-linear model. The exact solution is obtained for the case of uniformly applied loadings on the crack faces. Numerical results for a square crack under different electric and magnetic boundary conditions are displayed to demonstrate the proposed method.     

Anti-plane shear cracks in ideally plastic crystals

Cracks in ductile single crystals are analyzed here for geometries and orientations such that two-dimensional states of anti-plane shear constitute possible deformation fields. The crystals are modelled as ideally plastic and yield according to critical resolved shear stresses on their slip systems. Restrictions on the asymptotic forms of stress and deformation fields at crack tips are established for anti-plane loading of stationary and quasistatically growing cracks, and solutions are presented for several specific orientations in f.c.c. and b.c.c. crystals. The asymptotic solutions are complemented by complete elastic-plastic solutions for stationary and growing cracks under small scale yielding, based on previous work by Rice (1967, 1984) and Freund (1979). Remarkably, the plastic zone at a stationary crack tip collapses into discrete planes of displacement and stress discontinuity emanating from the tip; plastic flow consists of concentrated shear on the displacement discontinuities. For the growing crack these same planes, if not coincident with the crack plane, constitute collapsed plastic zones in which velocity and plastic strain discontinuities occur, but across which the stresses and anti-plane displacement are fully continuous. The planes of discontinuity are in several cases coincident with crystal slip planes but it is shown that this need not be the case, e.g., for orientations in which anti-plane yielding occurs by multi-slip, or for special orientations in which the crack tip and the discontinuity planes are perpendicular to the activated slip plane.     

An improved atomistic simulation based mixed-mode cohesive zone law considering non-planar crack growth

A novel and improved atomistic simulation based cohesive zone law characterizing interfacial debonding is developed which explicitly accounts for the non-planarity of the crack propagation. Group of atoms in the simulation constituting cohesive zones which are used to obtain local stress and crack opening displacement data are determined dynamically during the non-planar crack growth as they cannot be determined apriori. The methodology is used to study the debonding of Σ5 (2 1 0)/[0 0 1] symmetric tilt grain boundary interface in a Cu bicrystal under several mixed mode loading conditions. Simulations show that such bicrystalline specimen exhibits three types of energy dissipative mechanisms – shear coupled GB migration (SCM) away from the crack-tips, change in spacial orientation of GB structural units rendering highly disordered grain boundary near the crack tips and brittle intergranular fracture. Which combination of these three deformation mechanism will be active influencing the degree of non-planarity of the crack propagation at various stages of loading depends on the loading mode-mixity. As the ratio of shear component of the loading parallel to the GB plane and normal to the tilt axis with respect to the normal loading increases (thereby increasing the mode-mixity), overall strain-to-failure also increases and SCM tends to become the dominant deformation mechanism. Through this framework, analytical functional forms and parameters describing cohesive laws for both normal and shear traction as a function of the mode-mixity of the loading and crack opening displacement are predicted.     

The Analysis of Dynamic Stress Intensity Factor for Semi-Circular Surface Crack Using Time-Domain BEM Formulation

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｔｒａｎｓｉｅｎｔａｎａｌｙｓｉｓｆｏｒ 3＿Ｄｅｌａｓｔｏｄｙｎａｍｉｃｃｒａｃｋｐｒｏｂｌｅｍｓｉｓａｎｉｍｐｏｒｔａｎｔｃｏｎｔｅｎｔｉｎｆｒａｃｔｕｒｅｍｅｃｈａｎｉｃｓ.Ｂｅｃａｕｓｅｏｆｔｈｅｄｉｆｆｉｃｕｌｔｉｅｓｏｆｔｈｅｐｒｏｂｌｅｍｓ,ｔｈｅｃａｌｃｕｌａｔｉｏｎｏｆＤＳＩＦｆｏｒ 3＿Ｄｃｒａｃｋｉｓａｐｒｏｂｌｅｍｕｎｓｏｌｖｅｄｃｏｍｐｌｅｔｅｌｙｕｐｔｏｔｈｅｐｒｅｓｅｎｔ.Ｃｈｅｎ[1]ｆｉｒｓｔｌｙｕｓｅｄｔｈｅｆｉｎｉｔｅｄｉｆｆｅｒｅｎｃｅ…     

Identification of the representative crack length evolution in a multi-level interface model for quasi-brittle masonry

It is proposed here to identify the law of crack length evolution with a small number of parameters governing a recently presented model (Rekik and Lebon, submitted for publication) describing the interface behavior in damaged masonry. Studies on non-confined medium- and large-sized masonry structures have shown that it is necessary to obtain a linear increasing crack in the post-peak part of the “stress–strain or –displacement” diagram. In confined masonry structures showing softening and sliding parts, the results obtained with this crack evolution failed to match the experimental data. The crack lengths identified in the post-peak part at several points on the experimental “stress–displacement” diagram show that the representative crack length is a bilinear or trilinear function describing the increase in the crack length with respect to the decrease in the shear stress. Numerical studies on medium- and large-sized masonry structures consisting of the same materials subjected to various loads were performed to determine the ultimate crack length, and the results are relatively insensitive to the size of the masonry and the type of the load applied. The numerical local fields determined in the elementary and full-scale structures investigated were used to test the validity of the present model at the local scale, as well as to obtain an additional unilateral condition in the case of compressed masonry structures in order to prevent overlapping between the masonry components.     

Investigation of the crack problem in non-local piezoelectric materials under combined electromechanical loadings

The present investigation of the crack problem in piezoelectric materials is performed based on the non-local theory. After some manipulations, the impermeable crack, the permeable crack （the crack gap is full of NaCI solution）, and the semi-permeable crack （the crack gap is full of air or silicon oil） are reduced to a uniform formulation by assuming the normal electric displacement on the crack surfaces to be an unknown variable. Thus, a triple integral equation with the unknown normal electric displacement is established. By using the Newton iterative method and solving the triple integral equation, it is found that the normal electric displacement on the crack surfaces is no longer a constant as determined by previous studies, rather, it depends upon the remote combined electromechanical loadings. Numerical results of the stresses and electric displacement fields show that there are no singularities at the crack tips so that the stresses remain finite. It is of great significance that the concrete electric boundary condition on the crack surfaces exerts significant influence on the near-tip fields and in this way plays an important role in evaluating the crack stability in the non-local piezoelectric materials. More specifically, the impermeable crack model always overestimates the finite stresses at the crack tips, whereas the permeable crack model always underestimates them.     

CRACK PROPAGATION IN STRUCTURES SUBJECTED TO PERIODIC EXCITATION

In the present paper,a simple mechanical model is developed to predict the dynamic response of a cracked structure subjected to periodic excitation,which has been used to identify the physical mechanisms in leading the growth or arrest of cracking.The structure under consideration consists of a beam with a crack along the axis,and thus,the crack may open in Mode I and in the axial direction propagate when the beam vibrates.In this paper,the system is modeled as a cantilever beam lying on a partial elastic foundation,where the portion of the beam on the foundation represents the intact portion of the beam.Modal analysis is employed to obtain a closed form solution for the structural response.Crack propagation is studied by allowing the elastic foundation to shorten(mimicking crack growth)if a displacement criterion,based on the material toughness,is met.As the crack propagates,the structural model is updated using the new foundation length and the response continues.From this work,two mechanisms for crack arrest are identified.It is also shown that the crack propagation is strongly influenced by the transient response of the structure.     

An integral equations method for solving the problem of a plane crack of arbitrary shape

The problem of a plane crack of arbibrary shape, subjected to arbitrary loading, is studied. The displacement field is represented by two elastic potentials, the single-layer and the double-layer potential of the second kind. The equations for the crack displacement discontinuities are derived. An approximate analysis of the crack opening displacement under pressure is discussed.     

Dynamic stress intensity factor KⅢ and dynamic crack propagation characteristics of anisotropic materials

Based on the mechanics of anisotropic materials,the dynamic propagation problem of a mode Ⅲ crack in an infinite anisotropic body is investigated.Stress,strain and displacement around the crack tip are expressed as an analytical complex function,which can be represented in power series.Constant coefficients of series are determined by boundary conditions.Expressions of dynamic stress intensity factors for a mode Ⅲ crack are obtained.Components of dynamic stress,dynamic strain and dynamic displacement around the crack tip are derived.Crack propagation characteristics are represented by the mechanical properties of the anisotropic materials,i.e.,crack propagation velocity M and the parameter α.The faster the crack velocity is,the greater the maximums of stress components and dynamic displacement components around the crack tip are.In particular,the parameter α affects stress and dynamic displacement around the crack tip.     

Damage detection with spatial wavelets

This paper discusses a structural damage detection technique based on wavelet analysis of spatially distributed structural response measurements. The premise of the technique is that damage (e.g. cracks) in a structure will cause structural response perturbations at damage sites. Such local perturbations, although they may not be apparent from the measured total response data, are often discernible from component wavelets. The viability of this new technique is demonstrated with two examples: one based on numerically simulated deflection responses of a uniform beam containing a short transverse crack under both static and dynamic loading conditions, and the other based on smooth analytical crack-tip displacement fields. In each of these examples, the deflection or displacement response is analyzed with the wavelet transform, and the presence of the crack is detected by a sudden change in the spatial variation of the transformed response. This damage detection technique may serve the purpose of structural health monitoring in situations where spatially distributed measurements of structural response in regions of critical concern can be made with, for example, networks of distributed sensors, optical fibers, computer vision and area scanning techniques. It appears that this new technique does not require any analysis of the complete structure in question, nor any knowledge of the material properties and prior stress states of the structure.     

三维冲击荷载弹塑性弯曲裂纹张开位移

主要研究冲击载荷作用下的三维弹塑性弯曲裂纹尖端的张开位移问题．综合考虑了冲击作用应力,三维塑性区域边界上正应力与剪应力,利用二阶摄动方法计算了三维弹塑性弯曲裂纹尖端的张开位移．用数值解法计算出三维弹塑性弯曲裂纹尖端张开位移,作图分析了三维弹塑性弯曲裂纹尖端张开位移与三维裂纹体几何尺寸之间的变化关系．三维弹塑性弯曲裂纹尖端张开位移随着三维裂纹体厚度的增大而减小,随着三维裂纹体厚度的均匀增大,三维弹塑性弯曲裂纹尖端张开位移尺寸不断减小,减小的幅度越来越小,最终趋于平面应变状态下的弹塑性弯曲裂纹尖端张开位移尺寸．当三维裂纹体几何尺寸相同时,三维弯曲裂纹尖端动态张开位移随外部冲击载荷的不断增大而逐渐增大,三维弯曲裂纹尖端动态张开位移随动荷系数的增大而迅速增大,建立了一个计算三维弹塑性弯曲裂纹尖端动态张开位移的崭新理论模型．     

Transient response of a magnetoelectroelastic solid with two collinear dielectric cracks under impacts

Dynamic analysis of two collinear electro-magnetically dielectric cracks in a piezoelectromagnetic material is made under in-plane magneto-electro-mechanical impacts. Generalized semi-permeable crack-face boundary conditions are proposed to simulate realistic opening cracks with dielectric. Ideal boundary conditions of a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions are several limiting cases of the semi-permeable dielectric crack. Utilizing the Laplace and Fourier transforms, the mixed initial-boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. Dynamic intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) near the inner and outer crack tips are determined in the Laplace transform domain. Numerical results for a special magnetoelectroelastic solid are calculated to show the influences of the dielectric permittivity and magnetic permeability inside the cracks on the crack-face electric displacement and magnetic induction. By means of a numerical inversion of the Laplace transform, the variations of the normalized intensity factors of stress and COD are discussed against applied magnetoelectric impact loadings and the geometry of the cracks for fully impermeable, vacuum, fully permeable cracks and shown in graphics.     

APPLICATION OF THE HYPERSINGULAR INTEGRAL EQUATIONS METHOD TO THE INTERACTION BETWEEN TWO PARALLEL PLANAR CRACKS IN 3-D ELASTICITY

By using the analytic theory of hypersingular integral equations in three-dimensional fracture mechanics, the interactions between two parallel planar cracksunder arbitrary loads are investigated. According to the concepts and method of finite-part integrals, a set of hypersingular integral equations is derived, in which theunknown functions are the displacement discontinuities of the crack surfaces. Then itsnumerical method is proposed by combining the finite-part integral method with theboundary element method. Based on the above results, the method for calculating thestress intensity factors with the displacement discontinuities of the crack surfaces ispresented. Finally, several typical examples are calculated and the numerical resultsare satisfactory.