The interaction of multiple rows of periodical cracks

In this paper, the interaction of multiple rows of periodical cracks contained in an infinite elastic plate with far-field stress loaded is studied. An extremely accurate and efficient numerical method for solving the problem is presented. The method is mainly by means of the crack isolating analysis technique, stress superposition principle, the Chebyshev polynomial expansion of the pseudo-traction as well as the segmental average collocation technique. This method can be used to compute the stress intensity factors of multiple cracks, periodical cracks, and multiple rows of periodical cracks. In the process of dealing with the superposition of interaction of infinite number of periodic cracks, a key series summation technique is used, which aims at numerical results with extremely high accuracy but with less computation work. Many complex computing examples are given in this paper, and, for some typical examples, numerical results are compared with analytic solutions and with previous numerical solutions. For the problem of the one periodical collinear cracks, the accuracy given by this method reaches to 6 significant digits if a/d 0.9 (where a is the half crack length, and d is the half crack spacing). And even if a/d=0.99, the error is still less than 0.5%. The computer results for multiple rows of periodic collinear and echelon cracks show that the interaction effect between two rows rapidly decline with exponential law as the array pitch increases. This method has filled the gaps in the research field on the interaction of multiple rows of general periodical cracks.     

An extended Layerwise method for composite laminated beams with multiple delaminations and matrix cracks

For the delamination and matrix crack prediction of composite laminated structures, the methods based on the damage mechanics and fracture mechanics are most commonly used. However, there are very few methods that can accurately simulate the delaminations together with matrix cracks, although the in‐plane matrix cracks always exist alongside the delaminations under impact loading. In this work, an extended layerwise method is developed to model the composite laminated beam with multiple delaminations and matrix cracks. In the displacement field, the nodes in the thickness direction are located at the middle surface of each single layer, the top surface and the bottom surface of the composite beams. The displacement field contains the linear Lagrange interpolation functions, the one‐dimensional weak discontinuous function and strong discontinuous function. The strong and weak discontinuous function are applied to model the displacement discontinuity induced by delaminations and the strain discontinuity induced by the interface between the layers, respectively. Because the nodes in the thickness direction are located at the middle surface of each single layer, the extended layerwise method can be conveniently employed to deal with the in‐plane matrix cracks combined with the extend FEM. Copyright © 2014 John Wiley & Sons, Ltd.     

Modeling of coalescence of dispersed surface cracks. Part 2. Simulation model for multiple fracture

Abstarct A simulation model for multiple fracture has been developed that reproduces random processes of initiation, growth, and coalescence of dispersed surface cracks. The model is based on the method of statistical simulation (Monte Carlo method) and on the fracture regularities determined experimentally. The main factor responsible for fracture is found to be the coalescence of dispersed cracks, especially at the final stage, which accounts for about 30% of the total life. The ultimate state of a structure is defined by the condition according to which the length of the largest of the available damages is bigger than the calculated value of the maximum crack length.Translated from Problemy Prochnosti, No. 1, pp. 108–117, January–February, 2005.     

A probabilistic simulation of fluid leakage in multiple cracks situation

ABSTRACT In pressure vessels and piping the leak‐before‐break (LBB) assessment method is employed to avoid any catastrophic failure prior to a detectable leakage. One of the most important parameter, the leak rate, is investigated in the present paper by means of the Monte Carlo method. A brief review is carried out with emphasis on aspects such as crack growth, crack size with detectable leakage, crack opening area and leak rate. Issues concerning the property and behaviour of multiple cracks are also covered along with a review of the characteristics of leak rate through distributed multiple cracks using a statistical simulation method. The simulation results show that the effect of multiple cracks is quite significant to the LBB concept. Both the relationship between leak rate and crack length (or time) and the statistical characteristics of the leak rate are considerably different for different initial crack conditions.     

Modal analysis of multi-cracked beams with circular cross section

This paper proposes a numerical model that combines the finite element and component mode synthesis methods for the modal analysis of beams with circular cross section and containing multiple non-propagating open cracks. The model virtually divides a beam into a number of parts from the crack sections and couples them by flexibility matrices considering the interaction forces that are derived from the fracture mechanics theory. The main feature of the presented approach is that the natural frequencies and mode shapes of a beam with an arbitrary number of cracks and any kind of two end conditions can be conveniently determined with a reasonable computational time. Three numerical examples are given to investigate the effects of location and depth of cracks on the natural frequencies and mode shapes of the beams. Moreover, it is shown through these examples that the evaluation of modal data obtained by the proposed model gives valuable information about the location and size of defects in the beams.     

Dynamics of anisotropic composite cantilevers weakened by multiple transverse open cracks

In this paper an exact solution methodology, based on Laplace transform technique enabling one to analyze the bending free vibration of cantilevered laminated composite beams weakened by multiple non-propagating part-through surface cracks is presented. Toward determining the local flexibility characteristics induced by the individual cracks, the concept of the massless rotational spring is applied. The governing equations of the composite beam with open cracks as used in this paper have been derived via Hamilton's variational principle in conjunction with Timoshenko's beam model. As a result, transverse shear and rotatory inertia effects are included in the model. The effects of various parameters such as the ply-angle, fiber volume fraction, crack number, position and depth on the beam free vibration are highlighted. The extensive numerical results show that the existence of multiple cracks in anisotropic composite beams affects the free vibration response in a more complex fashion than in the case of beam counterparts weakened by a single crack. It should be mentioned that to the best of the authors' knowledge, with the exception of the present study, the problem of free vibration of composite beams weakened by multiple open cracks was not yet investigated.     

Multiple crack weight for solution of multiple interacting cracks by meshless numerical methods

We devise a multiple crack weight (MCW) method for the accurate and effective solution of strongly interacting cracks by meshless numerical methods. The MCW method constructs weight functions around cracks so that they simultaneously characterize all the cracks present in the single nodal domain of influence. This approach reduces the number of nodes necessary to achieve sufficient accuracy and consequently it decreases the computational effort. Numerical examples demonstrate that the method allows an accurate solution of multiple cracks problems. Convergence of the method is analysed and discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamic analysis of cracks using the SGBEM for elastodynamics in the Laplace-space frequency domain

Dynamic analysis of a system can be carried out either in the time or frequency domains. Time responses/histories of this system may be directly obtained using time-domain formulations. In the frequency domain, analysis can be performed in either the Fourier or Laplace spaces. The symmetric-Galerkin boundary element method (SGBEM) for 2-D elastodynamics in the Fourier-space frequency domain has been previously reported in the literature. In this paper, the SGBEM for elastodynamics in the Laplace-space frequency domain using the standard continuous quadratic element and its application to dynamic analysis of cracks is presented for the first time. The technique developed is employed together with the fast Laplace inverse transform by Durbin to obtain time-dependent results for several typical examples including both crack and non-crack problems. These results are highly accurate when compared to those obtained from other numerical techniques. It is shown in this work that the very same boundary element code can be utilized to perform frequency domain analysis in either the Fourier or Laplace spaces. However, if time responses are required, the accuracy and computational effectiveness of the analysis may depend on the type of space selected as it determines the type of transforms (inverse Fourier/Laplace transforms) needed for converting frequency solutions to the desired time responses.     

Transient analysis of multiple parallel cracks under anti-plane dynamic loading

The problem of a homogeneous linear elastic body containing multiple non-collinear cracks under anti-plane dynamic loading is considered in this work. The cracks are simulated by distributions of dislocations and an integral equation relating tractions on the crack planes and the dislocation densities is derived. The integral equation in the Laplace transform domain is solved by the Gaussian–Chebyshev integration quadrature. The dynamic stress intensity factor associated with each crack tip is calculated by a numerical inverse Laplace scheme. Numerical results are given for one crack and two or three parallel cracks under normal incidence of a plane horizontally shear stress wave.     

Experimental investigation of crack growth direction in multiple cracks

In this paper, the interaction between multiple cracks in crack growth direction is studied in an aluminium alloy under static and fatigue loading. Self similar as well as non‐self‐similar crack growth has been observed which depends on the relative crack positions defined by crack offset distance and crack tip distance. On the basis of experimental observations, the criterion for crack coalescence and crack growth direction are expressed in terms of the crack positions defined by crack offset and crack tip distances. The criterion presented in this study can be used to determine the limiting value of crack tip and crack offset distance and to determine the mode in which cracks coalesce during their growth process. Experimental results and crack interaction criterion presented under various crack positions and size conditions could be used to derive a new evaluation method of crack growth in multiple crack geometry.     

Fatigue crack growth of multiple interacting cracks: Analytical models and experimental validation

This article deals with the fatigue propagation of multiple cracks in finite width holed panels, which are typical of aircraft structural components. Theoretical studies in the literature have been considered and critically analyzed. Some of them have been translated into analytical models and implemented in a computer code. To check the effectiveness of the used models, a fatigue testing campaign has been conducted on six different configurations of notches and cracks. The comparison between experimental results and those obtained from the implemented models has shown a good agreement.     

Frictional contact analysis of multiple cracks by incremental displacement and resultant traction boundary integral equations

Based on the merits of the dual boundary element technique, a modified dual boundary element technique is extended to deal with the frictional contact of a finite plate with arbitrarily distributed multiple cracks. Besides establishing the incremental displacement boundary integral equation on the outer boundary, the resultant traction boundary integral equation on one of the crack surfaces is also developed. Since the resultant traction instead of incremental traction on the crack surface is introduced, the computed resultant contact tractions under sliding condition satisfy the Coulomb's friction law directly. Hence, as compared with the authors' previous work, only very few computation iterations are required by this method to accurately describe the contact situations of crack surfaces. As a result, not only the linear cracks, but also other types of multiple cracks, for example, curved and kinked cracks, can be tackled. The effects of friction and interaction among cracks on the computation of stress intensity factors are also displayed.     

Matrix Crack Detection in Composite Plate with Spatially Random Material Properties using Fractal Dimension

Fractal dimension based damage detection method is investigated for a composite plate with random material properties. Composite material shows spatially varying random material properties because of complex manufacturing processes. Matrix cracks are considered as damage in the composite plate. Such cracks are often seen as the initial damage mechanism in composites under fatigue loading and also occur due to low velocity impact. Static deflection of the cantilevered composite plate with uniform loading is calculated using the finite element method. Damage detection is carried out based on sliding window fractal dimension operator using the static deflection. Two dimensional homogeneous Gaussian random field is generated using Karhunen-Loeve (KL) expansion to represent the spatial variation of composite material property. The robustness of fractal dimension based damage detection method is demonstrated considering the composite material properties as a two dimensional random field.     

Analysis of multiple cracks in laminate composites under anti-plane static loads

This paper investigates multiple cracking of a laminate composite subjected to anti‐plane loads. The laminate composite is made up of a central layer sandwiched between two layers of different properties. There is a periodic array of cracks in the central layer along the central axis of the medium. A singular integral equation is formulated in terms of the crack‐face displacement. The model developed is applied to the analysis of fibre multiple cracking in fibre‐matrix laminate composites. It is also applied to the analysis of matrix cracking in fibre‐matrix laminate composites. Numerical results have been given for the effects of crack spacing and constituent volume fraction on the crack tip field intensity factor and stress.     

Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method

A numerical technique for modeling fatigue crack propagation of multiple coplanar cracks is presented. The proposed method couples the extended finite element method (X-FEM) [Int. J. Numer. Meth. Engng. 48 (11) (2000) 1549] to the fast marching method (FMM) [Level Set Methods & Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, UK, 1999]. The entire crack geometry, including one or more cracks, is represented by a single signed distance (level set) function. Merging of distinct cracks is handled naturally by the FMM with no collision detection or mesh reconstruction required. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity [Comput. Meth. Appl. Mech. Engng. 139 (1996) 289]. This enables the domain to be modeled by a single fixed finite element mesh with no explicit meshing of the crack surfaces. In an earlier study [Engng. Fract. Mech. 70 (1) (2003) 29], the methodology, algorithm, and implementation for three-dimensional crack propagation of single cracks was introduced. In this paper, simulations for multiple planar cracks are presented, with crack merging and fatigue growth carried out without any user-intervention or remeshing.     

Modeling of interface cracks in fiber-reinforced composites with the presence of interphases using the boundary element method

An advanced boundary element method (BEM) with thin-body capabilities was developed recently for the study of interphases in fiber-reinforced composite materials (Y.J. Liu, N. Xu and J.F. Luo, Modeling of interphases in fiber-reinforced composites under transverse loading using the boundary element method, ASME J. Appl. Mech. 67 (2000) 41–49). In this BEM approach, the interphases are modeled as thin elastic layers based on the elasticity theory, as opposed to spring-like models in the previous BEM and some FEM work. In the present paper, this advanced BEM is extended to study the interface cracks at the interphases in the fiber-reinforced composites. These interface cracks are curved cracks between the fiber and matrix, with the presence of the interphases. Stress intensity factors (SIFs) for these interface cracks are evaluated based on the developed models. The BEM approach is validated first using available analytical and other numerical results for curved cracks in a single material and straight interface cracks between two materials. Then, the interface cracks at the interphases of fiber-reinforced composites are studied and the effects of the interphases (such as the thickness and materials) on the SIFs are investigated. As a special case, results of the SIFs for sub-interface cracks are also presented. It is shown that the developed BEM is very accurate and efficient for the interface crack analyses, and that the properties of the interphases have significant influences on the SIFs for interface cracks in fiber-reinforced composites.     

Analysis of the dynamic behavior of two parallel symmetric cracks using the non-local theory

In this paper, the dynamic behavior of two parallel symmetric cracks under harmonic anti-plane shear waves is studied using the non-local theory. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the problem to obtain the stress occurs near the crack tips. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the lattice parameter and the distance between two parallel cracks, respectively.     

The interaction between matrix cracks and delaminations during quasi-static impact of composites

This paper discusses the investigation of the impact process for beam-like composite specimens. The research is focused on the initiation of delaminations at matrix cracks and on the interaction of delaminations with matrix cracks. The research is based on experiments and on finite element calculations. This study improves the fundamental understanding of the impact process in composite laminates, and gives clear indications on the relative influence of different material properties (transverse strength, interlaminar fracture toughness) on the damage development during impact.     

Determination of crack surface displacements for cracks emanating from a circular hole using weight function method

Cracks emanating from a circular hole are of significant engineering importance, especially in aerospace industry. Accurate determination of key fracture mechanics parameters is essential for damage tolerance design and fatigue life predictions. The purpose of this paper is to provide an efficient and accurate closed‐form weight function approach to the calculation of crack surface displacements for radial crack(s) emanating from a circular hole in an infinite and finite‐width plate. Results were presented for two loading conditions: remote applied stress and uniform stress segment applied to crack surfaces, and extensively compared to recent studies using other methods in the literature. Both single and double radial cracks were considered, and also the effect of finite plate width on crack surface displacements has been investigated. A brief assessment was made on an engineering estimation of displacements based on a correction of stress intensity factor ratio. It has been demonstrated that the Wu‐Carlsson closed‐form weight functions are very efficient, accurate and easy‐to‐use for calculating crack surface displacements for arbitrary load conditions. The method will facilitate fatigue crack closure and other fracture mechanics analyses where accurate crack surface displacements are required.