Weight function,stress intensity factor and crack opening displacement solutions to periodic collinear edge hole cracks

Periodic collinear edge hole cracks and arbitrary small cracks emanating from collinear holes, which are two typical multiple site damages occurred in the aircraft structures, are studied by using the weigh function method. An explicit closed form weight function for periodic edge hole cracks in an infinite sheet is obtained and further used to calculate the stress intensity factor and crack opening displacement for various loading cases. Compared to finite element method, the present weight function is accurate and highly efficient. The interactions of the holes and cracks on the stress intensity factor and crack opening displacement are quantitatively determined by using the present weight function. An approximate weight function method is also proposed for arbitrary small cracks emanating from multiple collinear holes. This method is very useful for calculating the stress intensity factor for arbitrary small cracks.     

Growth prediction of two interacting surface cracks of dissimilar sizes

When multiple cracks approach one another, the stress intensity factor changes due to the interaction of the stress field. This causes variation in the crack growth rate and shape of cracks. In particular, when cracks are parallel to the loading direction, their shape becomes non-planar due to the mixed mode stress intensity factor. In this study, the growth of interacting surface cracks was simulated by using the S-version finite element method, in which a local detailed finite element mesh (local model) is superposed on a coarse finite element model (global model) representing the global structure. First, simulations were performed for fatigue crack growth experiments and the method validity was shown. Second, simulations were conducted for various relative sizes and spacings of twin cracks. It was shown that the offset distance and the relative size were both important parameters to determine the interaction between two surface cracks; the smaller crack stopped growing when the difference in size was large. It was possible to judge whether the effect of interaction should be considered based on the correlation between the relative spacing and relative size.     

Analysis of the crack growth propagation process under mixed-mode loading

In the present paper, a computational model for crack growth analysis under Mode I/II conditions is formulated. The focus is on two issues – crack path simulation and fatigue life estimation. The finite element method is used together with the maximum principal stress criterion and the crack growth rate equation based on the equivalent stress intensity factor. To determine the mixed-mode stress intensity factors, quarter-point (Q-P) singular finite elements are employed. For verification purposes, a plate with crack emanating from the edge of a hole is examined. The crack path of the plate made of 2024 T3 Al Alloy is investigated experimentally and simulated by using the finite element method with the maximum tangential stress criterion. Then, the validation of the procedure is illustrated by applying the numerical evaluation of the curvilinear crack propagation in the polymethyl methacrylate (PMMA) beam and the Arcan specimen made of Al Alloy for which experimental results are available in the literature. In order to estimate fatigue life up to failure of the plate with crack emanating from the edge of a hole, the polynomial expression is evaluated for the equivalent stress intensity factor using values of stress intensity factors obtained from the finite element analysis. Additionally, the fatigue life up to failure of the Arcan specimen is analyzed for different loading angles and compared with experimental data. Excellent correlations between the computed and experimental results are obtained.     

Numerical Modeling of the Surface Fatigue Crack Propagation Including the Closure Effect

Presently modeling of surface fatigue crack growth for residual life assessment of structural elements is almost entirely based on application of the Linear Elastic Fracture Mechanics (LEFM). Generally, it is assumed that the crack front does not essentially change its shape, although it is not always confirmed by experiment. Furthermore, LEFM approach cannot be applied when the stress singularity vanishes due to material plasticity, one of the leading factors associated with the material degradation and fracture. Also, evaluation of stress intensity factors meets difficulties associated with changes in the stress state along the crack front circumference. An approach proposed for simulation the evolution of surface cracks based on application of the Strain-life criterion for fatigue failure and of the finite element modeling of damage accumulation. It takes into account the crack closure effect, the nonlinear behavior of damage accumulation and material compliance increasing due to the damage advance. The damage accumulation technique was applied to model the semi-elliptical crack growth from the initial defect in the steel compact specimen. The results of simulation are in good agreement with the published experimental data.     

Developments in the analysis of interacting cracks

This paper presents an overview of the finite element alternating technique for the analysis of interacting cracks. To illustrate the ease and accuracy of this method the technique is used to analyse several problems associated with both widespread fatigue and multi-site damage, a problem which is attracting worldwide attention. Whilst this paper presents an overview of the technique for both two- and three-dimensional problems attention is focused on three-dimensional problems. In particular, the interaction effects between two fully embedded elliptical flaws and between two semi-elliptical surface flaws, and the effects of crack proximity and crack aspect ratio on the stress intensity factors are presented. For semi-elliptical surface flaws these results indicate that as the cracks approach each other the position of the point on the crack front with the highest stress intensity factor shifts. This subsequently suggests that surface cracks will tend to grow preferentially towards each other. The same trend is evidenced for fully embedded cracks. However, in this case there is no shift in the position of the maximum stress intensity factor. A discussion of the results in terms of stress intensity magnification factors is also presented.     

A method for stress intensity factor calculation of infinite plate containing multiple hole-edge cracks

Multiple site damage is the occurrence of small fatigue cracks at several sites within aging aircraft structures. Focusing on this typical structure, an analytical method for calculating the stress intensity factor of an infinite plate containing multiple hole-edge cracks was introduced in this paper. The properties of complex variable functions are used to evaluate the stress function. The approximate superposition method is applied to solve stress intensity factor problems on multiple holes. The equivalent crack is introduced to modify the method. Some numerical examples of an infinite plate containing two hole-edge cracks are examined by the method. By comparing the analytical and finite element analysis results it was realized that the analytical results are accurate and reliable. This modified analytical method is easier to apply than some traditional analytical methods and can provide stress intensity factor solutions for an infinite plate containing a random distribution of multiple hole-edge cracks.     

Growth evaluation of multiple interacting surface cracks. Part II: Growth evaluation of parallel cracks

In Part I of the current work, experiments on fatigue crack growth from notches and crack growth simulation for a coalesced crack with re-entrant portion were conducted. It was revealed that the growth rate in area is the same for the same applied stress and area of crack face. The main conclusion was that a crack with a re-entrant portion can be replaced with a semi-elliptical crack of the same area for the growth prediction. In this study, the influence of the interaction on the growth of semi-elliptical/semi-circular surface cracks in the parallel position was investigated. The stress intensity factor for various relative positions and shapes was evaluated by finite element analyses, and the magnitude of the interaction was quantified. Then a crack growth simulation for parallel surface cracks was developed. It was revealed that the magnitude of the interaction increases continuously during the crack growth and that, for a growth prediction, the parallel surface cracks can be replaced with a single crack of the same size on the projected plane when the relative spacing is close enough. It was concluded that the simulation can estimate the crack growth of interacting cracks and the replacement can be carried out when the offset distance is less than the crack depth.     

Generic Overlapping Cracks in Polymers: Modeling of Interaction

This paper deals with modeling of the interaction in overlapping cracks that the authors have earlier identified to be generic to a wide range of polymeric systems (Ramasamy and Lesser, J Polym Sci B Phys, 2003). A complex stress function method is used for evaluating stress intensity factors for interacting cracks. The interaction between two parallel overlapping cracks is considered first. It is shown for this case that the stress intensity factor can fall below the threshold value when there is sufficient overlap, leading to arrest of crack growth at the overlapping tip. Then the interaction in a doubly periodic infinite array of cracks is considered. The interaction in the array is found to be non-linear. However, at a given stress level, the highest density of stable cracks is related to the threshold value for crack propagation K_th though a simple set of equations. It is also shown that in an infinite array of cracks, the energy release rate criterion for crack growth is different from the stress intensity factor criterion due to a reduced stiffness of the material.     

有限板共线多孔MSD疲劳裂纹扩展有限元模拟

介绍了二维断裂分析有限元软件FRANC2D/L在疲劳裂纹扩展模拟方面的基本步骤．利用该软件对有限板中心孔边对称裂纹的疲劳裂纹扩展进行模拟计算,对比模拟结果和试验数据,发现两者吻合良好,证明了利用该方法模拟疲劳裂纹扩展的可靠性．将FRANC2D/L应用到有限板共线多孔MSD疲劳裂纹扩展的有限元模拟上,得到了各孔边裂纹的长度和疲劳扩展寿命之间的关系曲线．模拟计算结果表明,在相同条件下,有限板中心孔边对称裂纹的裂纹扩展寿命要远远高于MSD结构中中心孔边裂纹的疲劳扩展寿命;由于MSD结构中影响各孔边裂纹的因素有所差异,各条裂纹的疲劳扩展寿命也会有所不同．另外,还给出了不含主裂纹的MSD和含主裂纹的MSD两种情况下的疲劳裂纹扩展历程,通过比较得知,含主裂纹的MSD结构更容易发生裂纹的合并和贯穿致使结构发生破坏．     

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.     

Fatigue crack growth analyses of offshore structural tubular joints

This study investigated various aspects of a fatigue crack growth analysis, ranging from the stress intensity factor solutions to the simulation of a fatigue crack coalescence process of a tubular joint weld toe surface flaw. Fracture mechanics fatigue crack growth analyses for offshore structural tubular joints are not simple, because of the difficulty to calculate the stress intensity factors due to their geometric complexity. The fully mixed-mode stress intensity factors of nine weld toe surface cracks of an X-shaped tubular joint under tension loading were calculated by detailed three-dimensional finite element analyses. Using these stress intensity factor solutions, a fatigue crack growth study was performed for the X-joint until (the crack surface length grew to two times the tube thickness. Through this study, the crack shape change during the fatigue crack propagation was investigated in detail. Fatigue life calculations were also performed for a range of crack geometries using the stress intensity factor solutions of the nine flaws. These calculations indicate that the natural fatigue crack growing path for a crack is its quickest growing path. The study demonstrated that detailed fracture mechanics fatigue analyses of tubular joints can be practical using the finite element method.     

A method for tracking internal crack propagation in railhead

This paper presents a method for tracking two-dimensional propagation of internal cracks, in particular vertical split head (VSH) defects, due to contact loading in railhead. Generalised curved crack is assumed to have present in the railhead and its propagation is simulated by solving interaction of arbitrary shaped cracks successively. Furthermore, the finite shape of the rail section is also modelled as the continuous distribution of dislocation within an infinite plane. Crack propagation is simulated using the criterion of vanishing of Mode II stress intensity factors (SIF) at crack tips. Examples are provided for tracking the propagation of pre-existing internal cracks in railheads turning into VSH defects under centric contact loading.     

Fatigue crack initiation and propagation under cyclic contact loading

A computational model for contact fatigue damage analysis of gear teeth flanks is presented in this paper. The model considers the conditions required for the surface fatigue crack initiation and then allows for proper simulation of the fatigue crack propagation that leads to the appearance of small pits on the contact surface. The fatigue process leading to pitting is divided into crack initiation and a crack propagation period.The model for prediction of identification of critical material areas and the number of loading cycles, required for the initial fatigue crack to appear, is based on Coffin-Manson relations between deformations and loading cycles, and comprises characteristic material fatigue parameters. The computational approach is based on continuum mechanics, where a homogenous and elastic material model is assumed and results of cyclic loading conditions are obtained using the finite element method analysis.The short crack theory together with the finite element method is then used for simulation of the fatigue crack growth. The virtual crack extension (VCE) method, implemented in the finite element method, is used for simulating the fatigue crack growth from the initial crack up to the formation of the surface pit. The relationship between the stress intensity factor K and crack length a, which is needed for determination of the required number of loading cycles N_p for a crack propagation from the initial to the critical length, is shown.     

FATIGUE CRACK GROWTH UNDER MIXED-MODE I AND II LOADING

Abstract— Mixed-mode fatigue crack growth has been studied using four point bend specimens under asymmetric loads. A detailed finite element analysis provides the stress intensity factors for curved cracks under different mixed-mode load conditions. Both fatigue crack growth direction and crack growth rate are studied. The maximum tangential stress and the minimum strain energy density criteria were found to provide satisfactory predictions of the crack growth directions. An effective stress intensity factor was used to correlate the fatigue crack growth rates successfully. It is found that the use of mode I fatigue crack growth rate properties results in a conservative crack growth rate prediction for mixed-mode load conditions.     

Discrete crack growth analysis methodology for through cracks in pressurized fuselage structures

A methodology for simulating the growth of long through cracks in the skin of pressurized aircraft fuselage structures is described. Crack trajectories are allowed to be arbitrary and are computed as part of the simulation. The interaction between the mechanical loads acting on the superstructure and the local structural response near the crack tips is accounted for by employing a hierarchical modelling strategy. The structural response for each cracked configuration is obtained using a geometrically non-linear shell finite element analysis procedure. Four stress intensity factors, two for membrane behaviour and two for bending using Kirchhoff plate theory, are computed using an extension of the modified crack closure integral method. Crack trajectories are determined by applying the maximum tangential stress criterion. Crack growth results in localized mesh deletion, and the deletion regions are remeshed automatically using a newly developed all-quadrilateral meshing algorithm. The effectiveness of the methodology, and its applicability to performing practical analyses of realistic structures, is demonstrated by simulating curvilinear crack growth in a fuselage panel that is representative of a typical narrow-body aircraft. The predicted crack trajectory and fatigue life compare well with measurements of these same quantities from a full-scale pressurized panel test.     

Growth evaluation of multiple interacting surface cracks. Part I: Experiments and simulation of coalesced crack

In order to develop a procedure for assessing the growth of interacting surface cracks, the relationship between the area of the crack face and fatigue crack growth behavior was investigated. Fatigue crack growth tests were conducted using stainless steel plate specimens with surface notches. Then, finite element analyses were performed to simulate the growth behavior obtained by the experiment. It was shown that the change in area can be predicted by assuming the extension of crack front based on evaluated stress intensity factor at each position along the front. Based on experimental and analysis results, it was revealed that the growth of interacting surface cracks as well as independent cracks can be represented well by change in area and showed good correlation with the driving force based on area. It was also shown that, in the case of parallel cracks, the area on the projected plane was dominant. It was concluded that, when the magnitude of the interaction is sufficiently large, by replacing the two cracks with a semi-elliptical crack of the same area on the projected plane, the growth in area can be predicted precisely.     

Fatigue crack growth in fibre metal laminates with multiple open holes

The fatigue crack growth behaviour of hybrid S2‐glass reinforced aluminium laminates (Glare) with multiple open holes was investigated experimentally and analytically. It was observed that the presence of multiple‐site fatigue damage would increase crack growth rates in the metal layers as two propagating cracks converged. An analytical crack growth model was established for predicting crack growth rates based on empirical Paris equation. The effective stress intensity factor at crack tips is a function of mode I far‐field stress intensity factor, crack opening stress intensity factor and effective non‐dimensional stress intensity factor that incorporated the crack‐bridging effect in fibre metal laminates. The predicted results under different applied stress can capture the trend of averaged crack growth rates in experiments, although deviation exists in the predictions.     

Orthotropic thermoelasticity problem of symmetrical heat flow disturbed by three coplanar cracks

This article presents a study on the plane thermoelasticity problem of an infinite orthotropic plate split by three coplanar cracks under the action of symmetrical heat flow. Using the technique of Fourier transforms, the related four-part mixed boundary value problems are reduced to two kinds of quadruple integral equations with cosine and sine kernels which are solved by use of finite Hilbert transformation. Closed form solutions to the temperature, thermal displacements and thermal stresses on the crack surfaces, and especially, the thermal stress intensity factors at crack tips are obtained for the case of uniform heat flow. The known solutions to the orthotropic thermoelasticity problem of uniform heat flow disturbed by a pair of coplanar cracks or a central planar crack can be deduced from the above results in a straightforward manner, including the solution of thermal stress intensity factors for the corresponding thermoelasticity problem with two collinear cracks which is another form of the solution, equivalent to that of series expressions obtained by the authors in a previous paper, but much simpler. It is found that extremely large magnitudes of stress singularity may occur as the distance between two adjacent cracks approaches zero.     

压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

Exact solutions of two semi-infinite collinear cracks in a strip

The stress intensity factors (SIFs) are calculated for an infinitely long strip of finite height containing two straight semi-infinite collinear cracks, which is a very useful model in simulating the interaction of faults in the study of tectonic earthquake. The new solutions are obtained by complex function method. It is shown that two well-known exact solutions for the crack problems are the limiting cases of the present results.