Crack parameter estimation in structures using finite element modeling

This paper addresses the problem of linear crack quantification, crack depth estimation and localization, in structures. An optimization technique based on a finite element model for cracked structural elements is employed in the estimation of crack parameters for beam, truss and two-dimensional frame structures. The modal data for the cracked structures are obtained by solving the corresponding eigenvalue problem. The error in the modal data is simulated by an additive noise that follows the normal distribution. The simulated reduced modal data is expanded using the eigenvector projection method. Numerical examples showed that this technique gives good results for cracks with high depth ratio. The accuracy of the estimated crack parameters depends on (1) the number of modes used, (2) the error level in the cracked structure modal data and (3) the number of measured degrees of freedom in the case of reduced modal data.     

表面裂纹蠕变分析的蠕变线弹簧模型方法

鉴于用通常的数值方法分析三维蠕变裂纹问题的困难,提出了一个三维表面裂纹蠕变断裂力学参量分析的蠕变线弹簧模型方法,并在非稳态蠕变条件下的位移、裂纹尖端J积分和C积分的工程估算公式及弹塑性线弹簧模型的基础上,建立了蠕变线弹簧模型方法的有关基本方程．具体分析计算了受均匀拉伸表面裂纹平板的J积分和C积分,并与三维有限元解进行了比较,其结果吻合良好．研究结果为进一步研究三维表面裂纹的蠕变扩展及寿命预报提供了基础．     

Three-dimensional mixed-mode stress intensity factors for cracks in functionally graded materials using enriched finite elements

Three-dimensional enriched finite elements are used to compute mixed-mode stress intensity factors (SIFs) for three-dimensional cracks in elastic functionally graded materials (FGMs) that are subject to general mixed-mode loading and constraint conditions. The method, which advantageously does not require special mesh configuration/modifications and post-processing of finite element results, is an enhancement of previous developments applied so far on isotropic homogeneous and isotropic interface cracks. The spatial variation of FGM material properties is taken into account at the level of element integration points. To validate the developed method, two- and three-dimensional mixed-mode fracture problems are selected from the literature for comparison. Two-dimensional cases include: inclined central crack in a large FGM medium under uniform tensile strain and stress loadings, a slanted crack in a finite-size FGM plate under exponentially varying tensile stress loading and an edge crack in a finite-size plate under shear traction load. The three-dimensional example models a deflected surface crack in a finite-size FGM plate under uniform tensile stress loading. Comparisons between current results and those from analytical and other numerical methods yield good agreement. Thus, it is concluded that the developed three-dimensional enriched finite elements are capable of accurately computing mixed-mode fracture parameters for cracks in FGMs.     

A computational study of local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens

In this paper, the local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens are investigated by finite element analyses. Based on the experimental observations of kinked crack growth mechanisms in lap-shear specimens under cyclic loading conditions, three-dimensional and two-dimensional plane-strain finite element models are established to investigate the local stress intensity factor solutions for kinked cracks emanating from the main crack. Semi-elliptical cracks with various kink depths are assumed in the three-dimensional finite element analysis. The local stress intensity factor solutions at the critical locations or at the maximum depths of the kinked cracks are obtained. The computational local stress intensity factor solutions at the critical locations of the kinked cracks of finite depths are expressed in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The three-dimensional finite element computational results show that the critical local mode I stress intensity factor solution increases and then decreases as the kink depth increases. When the kink depth approaches to 0, the critical local mode I stress intensity factor solution appears to approach to that for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The two-dimensional plane-strain computational results indicate that the critical local mode I stress intensity factor solution increases monotonically and increases substantially more than that based on the three-dimensional computational results as the kink depth increases. The local stress intensity factor solutions of the kinked cracks of finite depths are also presented in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. Finally, the implications of the local stress intensity factor solutions for kinked cracks on fatigue life prediction are discussed.     

Detection of crack location using cracked beam element method for structural analysis

The local flexibility introduced by cracks changes the dynamic behavior of the structure and, by examining this change, crack position and magnitude can be identified. In order to model the structure for FEM analysis, a special finite element for a cracked Timoshenko beam is developed. Shape functions for rotational and translational displacements are used to obtain the consistent mass matrix for the cracked beam element. Effect of the crack on the stiffness matrix and consistent mass matrix is investigated. Proposed is a procedure for identifying cracks in structures using modal test data.     

Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading

This paper describes the development and application of a general domain integral method to obtain J-values along crack fronts in three-dimensional configurations of isotropic, functionally graded materials (FGMs). The present work considers mode-I, linear-elastic response of cracked specimens subjected to thermomechanical loading, although the domain integral formulation accommodates elastic–plastic behavior in FGMs. Finite element solutions and domain integral J-values for a two-dimensional edge crack show good agreement with available analytical solutions for both tension loading and temperature gradients. A displacement correlation technique provides pointwise stress-intensity values along semi-elliptical surface cracks in FGMs for comparison with values derived from the proposed domain integral. Numerical implementation and mesh refinement issues to maintain path independent J-values are explored. The paper concludes with a parametric study that provides a set of stress-intensity factors for semi-elliptical surface cracks covering a practical range of crack sizes, aspect ratios and material property gradations under tension, bending and spatially-varying temperature loads.     

准脆性材料中强不连续问题的数值方法若干应用及其研究进展

综述了模拟准脆性材料开裂过程的数值计算方法的研究进展和工程应用,比较了表征强不连续问题的显式非连续模型和隐式非连续模型的优缺点.结合混凝土粘结裂纹, 重点讨论了嵌入非连续模型,扩展有限元方法和富集有限元技术等非连续方法的构造特征和本质区别.从各种富集方法的理论完备性考察,以假定发展应变为基础的嵌入非连续方法虽然可以解决混凝土开裂过程中的应力锁死,满足内部边界的静力平衡条件以及反映开裂后的位移不连续问题,但嵌入非连续所采用的富集函数在开裂单元中并不能满足协调条件,使非连续两侧的应变不独立. 其局限性是由于富集自由度在单元的水平上引入,而以单位分解为基础的扩展有限元和富集有限元的富集函数以节点自由度的方式引入,除具有嵌入非连续的优点, 还可以有效消除嵌入非连续引起裂纹两侧应变的相互影响.文中同时指出了网格重构技术,弥散裂纹模型的局限性以及扩展有限元和富集有限元技术在构造方式上的细微差别.对于节点自由度方式引入的富集函数, 其操作困难性在文中也作了说明.      

Fatigue cracking and its influence on dynamic response characteristics of spot welded specimens

X-ray imaging has been used to determine the fatigue crack growth behavior and failure mechanisms of spot welded specimens. Cracks critical to final failure of the tensileshear specimens studied are through-thickness plate cracks, which are usually initiated about 0.2–1.0 mm away from the edge of the nugget. In addition, frequency response functions (FRFs), obtained by impact hammer-accelerometer experiments throughout the fatigue process, show that the natural frequencies of these joints nonlinearly decrease with the growth of fatigue cracks. The three-dimensional finite-element analysis results for FRFs of uncracked and cracked spot welded joints are shown to be in good agreement with the experimental data. It is also shown that the fatigue cracks have different degrees of influence on different natural frequencies because of the location of cracks and vibrating modes. The results by both experiment and finite element analysis indicate that analysis of the variation of natural frequencies and vibrating modes may be used to study the fatigue crack propagating shape and the location of the fatigue crack.     

SINGLE AND DOUBLE EDGE INTERFACE CRACK SOLUTIONS FOR ARBITRARY FORMS OF MATERIAL COMBINATION

In this paper interfacial edge crack problems are considered by the application of the finite element method. The stress intensity factors are accurately determined from the ratio of crack-tip-stress value between the target given unknown and reference problems. The reference problem is chosen to produce the singular stress fields proportional to those of the given unknown problem. Here the original proportional method is improved through utilizing very refined meshes and post-processing technique of linear extrapolation. The results for a double-edge interface crack in a bonded strip are newly obtained and compared with those of a single-edge interface crack for different forms of combination of material. It is found that the stress intensity factors should be compared in the three different zones of relative crack lengths. Different from the case of a cracked homogeneous strip, the results for the double edge interface cracks are found to possibly be bigger than those for a single edge interface crack under the same relative crack length.     

含裂纹弹性结构对声散射作用研究

研究了含裂纹的弹性结构对声的散射作用,应用分配形有限元和边界元相结合的方法于含裂纹的结构声相互作用问题,利用二级分形有限元方法对含裂纹结构进行离散,这将使得自由度大为减少;使用边界元方法计算外域散射声场,这将自动满足无限远辐射边界条件,数值结果初步表明：(1)随着裂纹深度的增加,结构声耦合系统的共振频率将下降;(2)裂纹附近的声场所受的影响更为明显。     

平面幂律塑性Ⅰ型裂纹尖端应力场全解

在航空航天、船舶、石油管道和核电等领域,服役结构或部件在长期极端条件下运行,不可避免地会产生裂纹,因此,为研究含裂纹结构的准静态断裂行为,必须了解裂纹尖端附近区域的应力应变场特点.对于幂律材料裂纹构元,研究平面应变和平面应力条件下Ⅰ型裂纹尖端应力场的解析分布.基于能量密度等效和量纲分析,推导了能量密度中值点代表性体积单元(representative volume element, RVE)的等效应力解析方程,并定义其为应力因子,进而针对有限平面应变和平面应力紧凑拉伸(compact tension, CT)试样和单边裂纹弯曲(single edge bend, SEB)试样,以应力因子作为应力特征量,并构造用于表征裂尖等效应力等值线的蝶翅轮廓式和扇贝轮廓式三角特殊函数,提出描述幂律塑性条件下平面I型裂纹尖端应力场的半解析模型.该半解析模型形式简单,对CT和SEB试样的裂尖应力场的预测结果与有限元分析的结果比较表明,两者之间均密切吻合,模型公式可直接用于预测Ⅰ型裂纹尖端应力分布,方便于断裂安全评价和理论发展.     

Transient dynamic responses of a cracked solid subjected to in-plane loadings

Transient dynamic responses of an elastic cracked solid subjected to in-plane surface loadings are investigated in this study. Two vertical cracks, a surface-breaking crack and a sub-surface crack, are considered. The frequency responses of the plane strain problem are calculated by the computational mechanics combining the finite element method with the boundary integral equation. The finite element method is used for the near-field enclosing the crack, while the boundary integral equation is applied for the far-field to satisfy the Sommerfeld radiation condition. The transient responses are then obtained using fast Fourier transform. Surface displacements, crack opening displacements, and dynamic stress intensity factors are presented to show the significant effects of the cracks. The interaction between the elastic waves and the cracks as well as the mode conversion phenomena can be observed and understood through the numerical simulations.     

裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型

比例边界有限元法SBFEM(Scaled Boundary Finite Element Method)是一种半解析数值方法,在裂缝分析特别是强度因子计算上具有相当高的精度。本文提出了一种用于裂缝分析的基于虚拟结构面的SBFEM与常规FEM的耦合分析方法。首先选取裂缝周边一定范围的计算域,并将结构分成不含裂缝区域和含裂缝区域两部分。然后,对不含裂缝区域,采用FEM进行网格离散;对含裂缝区域,采用SBFEM进行网格离散;两者相互独立,在这两个域内,分别采用各自相应的位移模式。最后通过在SBFEM网格的外边界设置虚拟耦合结构面的模式,实现有限元网格和比例边界有限元网格的耦合。通过两个经典的含裂缝平板的算例研究,探讨了本文方法在I型开裂和混合型开裂分析中,影响应力强度因子精度的因素。算例表明,SBFEM具有的降维和半解析性质,使本文方法在裂缝分析中的前处理简单易行,且计算结果具有相当高的计算精度。     

含双裂纹圆截面悬臂梁的损伤识别研究

裂纹的萌生和扩展直接影响构件的振动响应，对构件的安全可靠性具有重要影响．本文以圆截面悬臂梁为对象，结合转角模态振型和模态频率等高线，研究了一种双裂纹识别技术．首先，基于应力强度因子和卡氏定理推导了无裂纹梁单元和含裂纹梁单元的刚度矩阵；在此基础上，建立了含裂纹圆截面悬臂梁的有限元动力学方程；然后，结合裂纹对梁转角模态振型和模态频率的影响，提出了双裂纹识别策略．最后，通过算例讨论了双裂纹识别策略的可行性．结果表明，圆截面悬臂梁的模态转角在裂纹位置出现突变，裂纹深度越大转角突变值越大；将识别出的裂纹位置作为已知参数，通过模态频率等高线法，可以准确地识别出双裂纹的深度．     

Crack bifurcation in laminar ceramics having large compressive stress

Crack bifurcation is observed in laminar ceramics that contain large residual compressive stress. In such composites, alternating material layers have tensile and compressive residual stress, due to thermal expansion mismatch or other sources. The compressive stress ensures that crack growth leading to failure in the laminar system is mediated by threshold strength, but, in some cases, it also leads to bifurcation of the propagating flaw. The phenomenon of bifurcation takes place when the crack tip is propagating in the compressive layer, and occurs typically at a distance equal to a few laminate thicknesses below the free surface and beyond. The observation of this phenomenon is usually associated with the presence of edge cracking in the compressive layers of the laminar ceramic, although it can also occur in the absence of such edge cracks. In the few cases where bifurcation occurs without edge cracks, the residual stresses and layer thicknesses are close to the condition in which edge cracks will occur. In addition, in this case the bifurcation is confined to near the specimen free surface, and below the bifurcation plane, the cracks are straight. The energy release rates for the straight and bifurcated cracks are calculated from the results of finite element computations and compared. When edge cracking is ignored, the crack is simulated as a through-thickness crack in an infinite body, and the energy release rate is used to predict crack deviation and bifurcation. Based on this, the finite element model successfully predicts bifurcation in only one material combination that was investigated in experiments. However, the experimental bifurcation takes place in two additional material combinations. When the effect of edge cracking is incorporated into the finite element simulations, the energy release rate calculations successfully predict the phenomenon of bifurcation in three material combinations, as observed in the experiments. Since no edge cracks are present in the fourth material combination tested experimentally, its lack of bifurcations is automatically predicted by the model. The presence of edge cracking, or its incipience, is thus concluded to be critical to the occurrence of crack bifurcation in laminar ceramic composites.     

拉伸载荷作用下单边缺陷-边裂纹应力强度因子研究

利用杂交位移不连续法研究拉伸载荷作用下矩形板中单边缺陷-边裂纹(半圆孔裂纹和半方孔裂纹)问题,给出了这三种平面弹性裂纹问题的应力强度因子的详细数值解。通过半圆孔裂纹问题和半方孔裂纹问题与单边裂纹问题的应力强度因子的比较,发现半圆孔和半方孔对单边裂纹有屏蔽影响。此外,本文的研究结果表明,杂交位移不连续法用于分析平面弹性有限体中复杂裂纹问题的应力强度因子简单且又准确。     

Non-linear dynamics of a cracked cantilever beam under harmonic excitation

The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam.The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.     

Predicting the fatigue life and crack aspect ratio evolution in complex structures

This paper discusses a computationally efficient method for determining the behaviour of complex structures containing three-dimensional cracks. A simple method is presented for calculating the mode I stress intensities for semi-elliptical cracks emanating from the saddle point of two intersecting tubular members. This method, which gives results in good agreement with published values, uses the finite element technique, but does not require the crack to be modelled explicitly. The technique is then used, in conjunction with FASTRAN II, to study fatigue crack growth and the results are compared to experimental data. Good agreement is achieved between both the predicted and measured fatigue crack growth and the evolution of the crack aspect ratios.     

Damage analysis of thermopiezoelectric properties: Part I - crack tip singularities

This is Part I of the work on a two-dimensional analysis of thermal and electric fields of a thermopiezoelectric solid damaged by cracks. It deals with finding the singular crack tip behavior for the temperature, heat flow, displacements, electric potential, stresses and electric displacements. By application of Fourier transformations and the extended Stroh formalism, the problem is reduced to a pair of dual integral equations for the temperature field with the aid of an auxiliary function. The electroelastic field is governed by another pair of dual integral equations. The inverse square root singularity is found for the heat flow field while the logarithmic singularity prevailed for the electroelastic field regardless of whether the crack lies in a homogeneous piezoelectric solid or at an interface of two dissimilar piezoelectric materials. Results are given for the energy release rate and a finite length crack oriented at an arbitrarily angle with reference to the external disturbances. Part II of this paper considers the modelling of a piezoelectric material containing microcracks. A representative cracked area element is used to obtain the effective conductivity and electroelastic modulus. Numerical results are given for a peizoelectric Bati O₃ ceramic with cracks.     

内部压力作用下矩形板中源于椭圆孔的分支裂纹应力强度因子的一种数值分析

应用一种边界元方法来研究内部压力作用下矩形板中源于椭圆孔的分支裂纹。该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者最近提出的裂尖位移不连续单元构成。在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界。本数值结果进一步证实这种数值方法对计算有限大板中复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示裂纹体几何对应力强度因子的影响。