Stress intensity factor solutions for cracks in finite-width three layer laminates with and without residual stress effects

Approximate stress intensity factor solutions for cracks in finite-width three layer laminates, with the crack located in the middle layer, were derived on the basis of force-balance between the applied stress and the modified Westergaard form of normal stress distribution ahead of the crack tip. This yielded a simple and closed form equation for the stress intensity factor that included the effects of the ratio of the moduli of the layers and the relative layer thicknesses. A comparison of the stress intensity factor values from this equation and with finite element data indicated that the difference between these two data sets was small for most of the crack lengths and the modulus ratio of the layers. The maximum difference occurred at crack lengths approaching the interface and at high moduli ratios, but was less than 10%, in general. The equations were also modified to incorporate the effects of residual stresses that arise during cooling after laminate processing, on the stress intensity factor. A comparison of the analytical data with the finite element data obtained by imposing thermal and mechanical boundary loads on the laminate specimens indicated a good agreement. The present closed form approximate solutions may be useful in fracture analyses of finite-width laminates containing cracks.     

Mesh-free analysis of cracks in isotropic functionally graded materials

This paper presents a Galerkin-based meshless method for calculating stress-intensity factors (SIFs) for a stationary crack in two-dimensional functionally graded materials of arbitrary geometry. The method involves an element-free Galerkin method (EFGM), where the material properties are smooth functions of spatial coordinates and two newly developed interaction integrals for mixed-mode fracture analysis. These integrals can also be implemented in conjunction with other numerical methods, such as the finite element method (FEM). Five numerical examples including both mode-I and mixed-mode problems are presented to evaluate the accuracy of SIFs calculated by the proposed EFGM. Comparisons have been made between the SIFs predicted by EFGM and available reference solutions in the literature, generated either analytically or by FEM using various other fracture integrals or analyses. A good agreement is obtained between the results of the proposed meshless method and the reference solutions.     

Finite element evaluation of mixed mode stress intensity factors in functionally graded materials

This paper is directed towards finite element computation of fracture parameters in functionally graded material (FGM) assemblages of arbitrary geometry with stationary cracks. Graded finite elements are developed where the elastic moduli are smooth functions of spatial co‐ordinates which are integrated into the element stiffness matrix. In particular, stress intensity factors for mode I and mixed‐mode two‐dimensional problems are evaluated and compared through three different approaches tailored for FGMs: path‐independent J^*_k‐integral, modified crack‐closure integral method, and displacement correlation technique. The accuracy of these methods is discussed based on comparison with available theoretical, experimental or numerical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Determining the mixed mode stress intensity factors of surface cracks in functionally graded hollow cylinders

This work reports about an investigation on mixed mode stress intensity factors (SIFs) of three-dimensional (3D) surface cracks in hollow cylinders made-up of functionally graded material (FGM). A finite element implementation of the interaction energy integral in domain form is employed to extract the SIFs. In turn, surface cracks located at the inner and outer wall of the cylinders are considered, and the influence of exponentially varying Poisson’s ratio and Young’s modulus in radial direction on the SIFs is studied in detail. The computational results reported herein show that graded materials properties can significantly affect the magnitude and the distribution of SIFs along 3D crack fronts in FGM hollow cylinders.     

Accurate stress intensity factor solutions for corner cracks at a hole

Laboratory test and in-service experience shows fatigue cracks at holes exhibit unsymmetric growth; thus, the need for the new solutions is paramount. Stress intensity factor, K, solutions for symmetric and unsymmetric corner cracks at a hole subject to general loading were determined using a hp-version of the finite element method (FEM) in conjunction with a mathematical splitting scheme to enable efficient, accurate calculations. In traditional applications of the FEM, mesh generation is labor intensive; however, using the splitting scheme, stress intensity functions are obtained without explicitly including the crack in the FE mesh of the global structure. By using the hp-version of FEM, a set of K-solutions converging exponentially fast to the exact solution is obtained. The crack is analyzed in the local domain with easily generated FE meshes. All structurally significant crack shapes were considered; specifically, crack depth to crack length ratios (a/c) of 0.1–10.0, crack depth to sheet thickness ratios (a/t) of 0.10–0.99, and hole radius to sheet thickness ratios (r/t)=1.0. The loading conditions were remote tension, remote bending, and pin loading (bearing). In addition, all combinations of a/c and a/t are analyzed at each side of the hole; thus 226,875 solutions were developed with control of the error in the computed K solutions. Calculated relative error is generally much smaller than 1% along the entire crack front including the vertex regions. Comparisons are made to solutions in the open literature. The new K solutions show the literature solutions are, in general, accurate for all three load conditions; however, for the extreme cases of a/c, a/t, and r/t; the literature solutions differ by as much as 26%.     

Interaction of parallel dielectric cracks in functionally graded piezoelectric materials

In this paper, the problem of two interacting parallel cracks in functionally graded piezoelectric materials under in-plane electromechanical loads is studied. The formulation is based on using Fourier transforms and modeling the cracks as distributed dislocations, and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model considering the crack filling effect is adopted to describe the electric boundary conditions along crack surfaces. Numerical simulations are made to show the effect of material gradient, the geometry of interacting cracks, and crack position upon fracture parameters such as stress intensity factors, electric displacement intensity factor, and COD intensity factor. By considering the effect of a dielectric medium inside the crack and crack deformation, the results obtained from the dielectric crack model are always between those from the traditional crack models with physical limitation.     

Dynamic stress intensity factor of a functionally graded material under antiplane shear loading

Summary The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.     

Transient thermo-mechanical analysis of dynamic curving cracks in functionally graded materials

Mixed-mode dynamic crack growth behavior along an arbitrarily smoothly varying path in functionally graded materials (FGMs) under transient thermo-mechanical loading is studied. An asymptotic analysis in conjunction with displacement potentials is used to develop transient thermo-mechanical stress fields around the propagating crack-tip. Asymptotic temperature field equations are derived for exponentially varying thermal properties, and later, these equations are used to derive transient thermo-mechanical stress fields for a curving crack in FGMs. The effect of the transient parameters (loading rate, crack-tip acceleration, and temperature change) and temperature gradient on the maximum principal stress and circumferential stress associated with the propagating crack-tip is discussed. Finally, using the minimum strain energy density criterion, the effect of temperature gradient, crack-tip speeds, and T-stress on crack growth directions is determined and discussed.     

Mixed-mode crack-tip stress fields for orthotropic functionally graded materials

Quasi-static mixed mode stress fields for a crack in orthotropic inhomogeneous medium are developed using asymptotic analysis coupled with Westergaard stress function approach. In the problem formulation, the elastic constants E ₁₁, E ₂₂, G ₁₂, ν ₁₂ are replaced by an effective stiffness ${E=\sqrt {E_{11} E_{22}}}$ , a stiffness ratio ${\delta =\left({{E_{11}}\mathord{\left/ {\vphantom {{E_{11}} {E_{22}}}}\right. \kern-0em} {E_{22}}} \right)}$ , an effective Poisson’s ratio ${\nu =\sqrt {\nu_{12}\nu _{21}} }$ and a shear parameter ${k=\left({E \mathord{\left/ {\vphantom {E {2G_{12}}}}\right. \kern-0em} {2G_{12}}}\right)-\nu }$ . An assumption is made to vary the effective stiffness exponentially along one of the principal axes of orthotropy. The mode-mixity due to the crack orientation with respect to the property gradient is accommodated in the analysis through superposition of opening and shear modes. The expansion of stress fields consisting of the first four terms are derived to explicitly bring out the influence of nonhomogeneity on the structure of the mixed-mode stress field equations. Using the derived mixed-mode stress field equations, the isochromatic fringe contours are developed to understand the variation of stress field around the crack tip as a function of both orthotropic stiffness ratio and non-homogeneous coefficient.     

An interaction integral method for analysis of cracks in orthotropic functionally graded materials

This paper presents two new interaction integrals for calculating stress-intensity factors (SIFs) for a stationary crack in two-dimensional orthotropic functionally graded materials of arbitrary geometry. The method involves the finite element discretization, where the material properties are smooth functions of spatial co-ordinates and two newly developed interaction integrals for mixed-mode fracture analysis. These integrals can also be implemented in conjunction with other numerical methods, such as meshless method, boundary element method, and others. Three numerical examples including both mode-I and mixed-mode problems are presented to evaluate the accuracy of SIFs calculated by the proposed interaction integrals. Comparisons have been made between the SIFs predicted by the proposed interaction integrals and available reference solutions in the literature, generated either analytically or by finite element method using various other fracture integrals or analyses. An excellent agreement is obtained between the results of the proposed interaction integrals and the reference solutions. The authors would like to acknowledge the financial support of the U.S. National Science Foundation (NSF) under Award No. CMS-9900196. The NSF program director was Dr. Ken Chong.     

Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

Stress intensity factor solutions for cracks in railway axles

The aim of this paper is a collection of stress intensity factor solutions for cracks in railway axle geometries which the authors of the present special issue developed and/or used for damage tolerance analyses. These solutions comprise closed form analytical as well as tabled geometry functions and they refer to solid as well as hollow axles and various crack sites such as the T- and V-notch and the axle body.     

The fracture behavior of functionally graded piezoelectric materials with dielectric cracks

This article provides a comprehensive investigation on the fracture behavior of cracked functionally graded piezoelectric materials (FGPMs). To account for the effect of dielectric medium inside the crack upon the fracture behavior of FGPMs, a dielectric crack model is used in this work, in which the electric boundary condition along crack surfaces is deformation-dependent and is nonlinear. The analytical formulations are developed using Fourier transform technique and solving the nonlinear singular equations using Chebyshev polynomials. A solution technique is developed to determine the desired deformation mode of the crack. Numerical simulations are given to show the effects of material gradient and the dielectric medium filling the crack upon the fracture behavior of FGPMs. The results obtained from this dielectric crack model clearly demonstrate how the transition between electrically impermeable and permeable crack models occurs with the change of crack opening displacement in response to the applied electromechanical loads. It is also observed that a critical state for the applied electromechanical loading exists for FGPMs that determines whether the impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric crack model considering the effect of dielectric medium filling the crack.     

XFEM simulation of cracks, holes and inclusions in functionally graded materials

The present work aims at the numerical simulation of inhomogeneities/discontinuities (cracks, holes and inclusions) in functionally graded materials (FGMs) using extended finite element method (XFEM). A FGM with unidirectional gradation in material properties is modeled under plane strain condition. The domain contains a major crack either at the center or at the edge of the domain along with multiple minor discontinuities/flaws such as minor cracks and/or voids/inclusions distributed all over the domain. The effect of the variation in stress intensity factor (SIF) of the major crack due to the presence of the minor cracks and voids/inclusions is studied in detail. The simulations show that the presence of minor discontinuities significantly affects the values of SIFs.     

Determination of dynamic effective properties in functionally graded materials

Summary This work is dedicated to the investigation of the dynamic effective properties in functionally graded materials resulting from an anti-plane shear wave. A micromechanics-based elastodynamic model is developed to predict the dynamic behavior of two-phase functionally graded materials, and the distribution of dynamic effective properties in the gradation direction is presented. Generally speaking, in functionally graded materials there exist two microstructurally distinct zones: a fiber-matrix zone and a transition zone. In the fiber-matrix zone, the dispersion relation for the effective wave number is derived using the effective medium method, and the dynamic effective properties for any macroscopic material points are determined in the corresponding microstructural representative volume element (RVE). In the transition zone, a transition function is introduced to make the wave fields continuous and differentiable. Numerical examples of the dynamic effective properties in the gradation direction under different parameters are presented graphically. The obtained results reveal that the distribution of dynamic effective properties in the gradation direction is dependent on the material properties of each phase, the incident frequency, and the gradation parameter of the materials. Comparisons between numerical solutions and experimental data are also made. At last, the results are discussed in detail.     

A periodic array of cracks in functionally graded materials subjected to transient loading

A periodic array of cracks in an infinite functionally graded material under transient mechanical loading is investigated. In-plane normal (mode I) and shear (mode II) loading conditions are considered. For each individual loading mode, a singular integral equation is derived, in which the crack surface displacements are unknown functions. Numerical results are obtained to illustrate the variation of the stress intensity factors as a function of the crack periodicity for different values of material inhomogeneity, either at the transient state or steady state. The material inhomogeneity can increase or decrease the mode I and mode II stress intensity factors. Compared with the single crack solution, it is also shown that multiple cracking may decrease the mode I stress intensity factors, but enhance the mode II stress intensity factors significantly.     

Evaluation of the stress intensity factor for cracks in elastomers

We analyze the applicability of the method of the J-integral to the solution of problems of fracture mechanics for structures made of low-compressible elastomers. The components of the J-integral are computed by the method of equivalent three-dimensional integration. To take into account weak compressibility, we use the finite-element moment scheme. Lugansk Agricultural Institute, Lugansk, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 81–85, July–August, 1999.     

Characterisation methods for functionally graded materials

In this paper the characterisation of functionally graded materials is elucidated by several different methods. These methods described here are used for the quantitative analysis of materials with a local dependence of microstructure parameters. Using X-ray microscopy (computed tomography) for 3D-measurements and optical microscopy on polished sections for 1D and 2D measurements on the same sample, a ceramic filter consisting of sintered spherical particles, various mathematical evaluation methods are described and compared.     

Higher-order theory for functionally graded materials

This paper presents the full generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials developed by the authors during the past several years. This theory circumvents the problematic use of the standard micromechanical approach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites by explicitly coupling the local (microstructural) and global (macrostructural) responses. The theoretical framework is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composite's functionally graded microstructure. The generalization outlined herein involves extension of the theoretical framework to enable the analysis of materials characterized by spatially variable microstructures in three directions. Specialization of the generalized theoretical framework to previously published versions of the higher-order theory for materials functionally graded in one and two directions is demonstrated. In the applications part of the paper we summarize the major findings obtained with the one-directional and two-directional versions of the higher-order theory. The results illustrate both the fundamental issues related to the influence of microstructure on microscopic and macroscopic quantities governing the response of composites and the technologically important applications. A major issue addressed herein is the applicability of the classical homogenization schemes in the analysis of functionally graded materials. The technologically important applications illustrate the utility of functionally graded microstructures in tailoring the response of structural components in a variety of applications involving uniform and gradient thermomechanical loading.     

The development of mode I, linear-elastic stress intensity factor solutions for cracks in mechanically fastened joints

Accurate representation of crack tip stress intensity is an essential part of the assessment of the damage tolerance capability of aerospace structures. In typical applications, mode I stress intensity solutions are taken as factorial combinations of the fundamental form and various boundary correction factors for any given geometry and applied load. Total solutions are then developed by superimposing individual solutions for various load conditions. This technique is discussed for typical structure involving loaded fastener holes. Tabulated and plotted results are presented for cracks emanating from both open and loaded holes in finite width plates, lugs and multi-fastener joints. In the case of the open hole and lug solutions, the results are compared with published finite element values. The expressions presented herein lend themselves to programming on a digital computer. Once established, they afford the engineer the ability to develop accurate K_I solutions for a wide range of applications which are typical of the geometry/loading configurations encountered in the damage tolerance analysis of mechanically fastened joints. Equally as important, they offer accuracy comparable to that obtained with finite element modeling at a fraction of the cost.