Cracks In Functionally Graded Materials Research Articles

Mode shape analysis of multiple cracked functionally graded beam-like structures by using dynamic stiffness method

Mode shapes of multiple cracked beam-like structures made of Functionally Graded Material (FGM) are analyzed by using the dynamic stiffness method. Governing equations in vibration theory of multiple cracked FGM beam are derived on the base of Timoshenko beam theory; power law variation of material; coupled spring model of crack and taking into account the actual position of neutral axis. A general solution of vibration in frequency domain is obtained and used for constructing dynamic stiffness matrix of the multiple cracked FGM Timoshenko beam element that provides an efficient method for modal analysis of multiple cracked FGM frame structures. The theoretical development is illustrated by numerical analysis of crack-induced change in mode shapes of multi-span continuous FGM beam.

Determination of stress intensity factors of 3D curved non-planar cracks in FGMs subjected to thermal loading

A finite element analysis is presented about 3D curved non-planar cracks in functionally graded materials (FGMs) subjected to steady state temperature gradients. The interaction energy integral is employed in conjunction with a computational strategy that does not require a priori information about crack front and crack surface curvatures. The influence of graded thermal and mechanical properties on mixed mode stress intensity factors (SIFs) is explored in detail. Results highlighted the robustness of the proposed computational framework, which therefore can be effectively deployed in the analysis of thermal fracture in FGMs containing curved non-planar cracks.

An analytical model for collinear cracks in functionally graded materials with general mechanical properties

The crack problems of functionally graded materials (FGMs) with general mechanical properties are usually to solve analytically. In this paper, an analytical model for collinear cracks in FGMs with arbitrary mechanical properties is developed. In the model, the general mechanical properties of the FGM are approached by dividing it into some layers with continuous properties along the gradient direction. The problem is finally reduced to a group of singular integral equations which are solved by numerical method. The interaction between the collinear cracks is discussed. The influences of the nonhomogeneity constants and geometric parameters on the stress intensity factors (SIFs) are analyzed. Some important conclusions are drawn.

Free Vibration Analysis for Cracked FGM Beams by Means of a Continuous Beam Model

Based on Euler-Bernoulli beam theory and a continuous stiffness beam model, the free vibration of rectangular-section beams made of functionally graded materials (FGMs) containing open edge cracks is studied. Assuming the material gradients follow exponential distribution along beam thickness direction, the conversion relation between the vibration governing equations of a FGM beam and that of an isotropic homogenous beam is deduced. A continuous function is used to characterize the bending stiffness of an edge cracked FGM beam. Thus, the cracked FGM beam is treated as an intact beam with continuously varying bending stiffness along its longitudinal direction. The characteristic equations of beams with different boundary conditions are obtained by transfer matrix method. To verify the validity of the proposed method, natural frequencies for intact and cracked FGM beams are calculated and compared with those obtained by three-dimensional finite element method (3D FEM) and available data in the literature. After that, further discussions are carried out to analyze the influences of crack depth, crack location, material property, and slenderness ratio on the natural frequencies of the cracked FGM beams.

Stochastic fatigue crack growth simulation of interfacial crack in bi-layered FGMs using XIGA

In this work, extended isogeometric analysis (XIGA) is employed to compute the fatigue life of interfacial edge cracked functionally graded materials (FGMs). The effect of multiple defects (holes, inclusions and minor cracks) on the fatigue life of the interfacial cracked FGM plate is evaluated by XIGA. The bottom layer of the bi-layered plate is made of aluminum alloy while the upper one is made of FGM. The gradation in FGM layer is taken from left to right with 100% aluminum alloy on left and 100% alumina (ceramic) on the right. The material properties in FGM layer vary exponentially from the left (alloy side) to the right (ceramic side). For stochastic fatigue crack growth modeling, the input parameters (fracture toughness, Paris constant, Paris exponent) are generated using a log-normal distribution. Paris law is used to evaluate the fatigue life of the interfacial cracked bi-layered problems. The effect of these input parameters is studied on the fatigue life in detail. The values of stress intensity factors are computed using domain based interaction integral approach. Several interfacial crack bi-layered FGM problems are solved and analyzed by the XIGA.

Fracture modeling of isotropic functionally graded materials by the numerical manifold method

Abstract Two-dimensional stationary cracks in isotropic functionally graded materials (FGMs) are studied by the numerical manifold method (NMM). The near-tip behavior of a crack in FGMs is manifested by a special choice of cover functions, and the displacement jump across a crack face is naturally represented taking the benefit of the NMM. The stress intensity factors (SIFs) are computed by the equivalent domain form of the interaction integral using the nonequilibrium auxiliary fields. Typical examples involving single- and multi-branched crack are conducted to verify the accuracy of the proposed method. Problems are tackled with the uniform mathematical cover system independent of the physical boundaries and the calculated SIFs match well with the existing reference solutions.

Three-dimensional vibrations of cracked rectangular parallelepipeds of functionally graded material

A novel examination of the three-dimensional (3-D) vibrations of rectangular parallelepipeds of functionally graded material (FGM) having side cracks is summarized. Employing 3-D theory of elasticity and a variational Ritz methodology, new hybrid series of mathematically complete orthogonal polynomials and crack functions as the assumed displacement fields are proposed to enhance the convergence modeling of the stress singular behavior of the crack terminus edge front in a rectangular FGM parallelepiped. The proposed admissible hybrid series properly describe the ϑ(1/r)3-D stress singularities at the terminus edge front of the crack, allowing for displacement discontinuities across the crack sufficient to explain the most general 3-D “mixed modes” of local crack-edge deformation and stress fields typically seen in fracture mechanics. The correctness and validity of the vibration analysis are confirmed through comprehensive convergence studies and comparisons with published results for cracked rectangular FGM parallelepipeds modeled as homogeneous rectangular plates with side cracks and FGM rectangular plates with no cracks based on various plate theories. Two types of FGM parallelepipeds, Al/Al2O3 and Al/ZrO2, are included in the study. The locally effective material properties are estimated by a simple power law and the effects of the volume fraction on the frequencies are investigated. For the first time in the published literature, this work reports frequency data and nodal patterns for FGM rectangular parallelepipeds modeled as moderately thick plates with several combinations of hinged, clamped, and completely free kinematic and stress conditions along the four side faces, and having side cracks with varying crack size effects implying flaw-size influence in FGM parallelepiped vibration and fracture, including crack length ratios (d/a and d/b), crack positions (cx/a and cy/b), and crack inclination angles (α).

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.

Dynamic curving cracks in functionally graded materials under thermo-mechanical loading

Mixed-mode dynamic crack growth along an arbitrarily smoothly varying path in functionally graded materials (FGMs) under thermo-mechanical loading is studied. The property gradation in FGMs is considered by varying shear-modulus, mass density, thermal conductivity and coefficient of thermal expansion exponentially along the gradation direction. Asymptotic analysis in conjunction with displacement potentials is used to develop the stress fields around propagating cracks in FGMs. Asymptotic temperature fields are developed first for the exponential variation of thermal conductivity and later these temperature fields are used to derive thermo-mechanical stress fields for a curving crack in FGMs. Using these thermo-mechanical stress fields, various components of the stresses are developed and the effect of curvature parameters, temperature and gradation on these stresses are discussed. Finally, using the minimum strain energy density criterion, the effect of curvature parameters, crack-tip speeds, non-homogeneity values and temperature gradients on crack growth directions are determined and discussed.

Analysis of the Dynamic Behavior of a Moving Crack in FGMs Using Non-Local Theory

A theoretical and numerical study has been conducted to investigate the dynamic crack propagation in functionally graded materials (FGMs) by making use of non-local theory. The variation of the shear modulus and mass density of the FGMs are modeled by a exponential increase along the direction perpendicular to the crack surface. The Poisson’s ratio is assumed to be constant. The mixed boundary value problem is reduced to a pair dual integral equations through Fourier. In solving the dual integral equations, the crack surface displacement is expanded in a series using Jacobi’s polynomials and Schmidt’s method is used. Contrary to the classical elasticity solution, the crack-tip stress fields does not retains the inverse square root singularity. The analysis revals that the peak values of crack-tip stress increase with the the crack velocity as characteristic length is decreased.

Finite element evaluation of stress intensity factors in curved non-planar cracks in FGMs

Abstract In this work the mixed mode stress intensity factors (SIFs) of three-dimensional curved non-planar cracks in functionally graded materials (FGMs) are analyzed using the interaction energy integral. In particular, the finite element method is employed to extract the SIFs along the front of a lens shaped crack in a FGM. In order to cross check the capabilities of the present implementation, material gradation is reduced to zero and the numerical results are in turn compared to the analytical solutions for SIFs available for homogeneous materials. Finally, the contribution of material gradation to the SIFs is explored considering exponential spatial variations for Poisson’s ratio as well as for Young’s modulus.

An improved numerical method for computation of stress intensity factors along 3D curved non-planar cracks in FGMs

A simplified strategy based on the interaction energy integral is implemented in the finite element framework to evaluate mixed mode Stress Intensity Factors (SIFs) in 3D non-planar cracks. The proposed approach does not require any a priori information about crack front and crack surface curvatures, therefore different arbitrary non-planar cracks can be easily investigated. In particular, both conical and lens-shaped cracks in homogeneous materials are considered as case studies in order to demonstrate the accuracy of the present approach. Finally, the computational strategy is extended to Functionally Graded Materials (FGMs) and the effect of graded material properties (Young’s modulus and Poisson’s ratio) on the SIFs is studied in detail.

Anti-Plane Moving Crack in Functionally-Graded Material

In this paper the anti-plane moving crack in a functionally-graded material is studied by the analytical method. First the governing equations for a functionally-graded material are obtained using a Fourier cosine integral transform. Then the dual integral equations for moving crack are established according to the mixed boundary value conditions. It is shown that the dual integral equations can be reduced to the Fredholm integral equation of the second kind. Numerical results shown in the present paper indicate that the non-homogeneity of material has an important influence on the dynamic stress intensity factor.

Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials

Free vibration and elastic buckling of beams made of functionally graded materials (FGMs) containing open edge cracks are studied in this paper based on Timoshenko beam theory. The crack is modeled by a massless elastic rotational spring. It is assumed that the material properties follow exponential distributions along beam thickness direction. Analytical solutions of natural frequencies and critical buckling load are obtained for cracked FGM beams with clamped-free, hinged-hinged, and clamped-clamped end supports. A detailed parametric study is conducted to study the influences of crack depth, crack location, total number of cracks, material properties, beam slenderness ratio, and end supports on the free vibration and buckling characteristics of cracked FGM beams.

ANALYSIS OF APPROXIMATED CURVED CRACKS IN HOMOGENEOUS AND GRADED MATERIALS

In this paper two stages of analysis are studied. In stage I, the influence of crack shape on thecrack-tip stresses, critical loads and subsequent propagation direction is investigated via a simpleanalytical model for cracks in homogeneous materials. This model is verified through finite elementsimulations using ANSYS. It is demonstrated that accurate predictions of mechanical energy releaserate and crack deflection angle may be obtained from a smaller number of crack shape parameters.In stage II, this concept is extended to curved cracks in functionally graded materials (FGMs).It iscommon that analytical and computational models of fracture in FGMs have focused almostextensively on straight cracks. If it can be demonstrated that straight cracks give an adequateapproximation of curved cracks in graded materials, then the existing solutions for straight cracksprovide a sufficient foundation for fracture analysis of FGMs. On the other hand, if straight cracksdo not adequately approximate curved cracks in FGMs, then the development of solutions for nonstraight cracks in graded materials is priority. Three cracks shapes approximations are performed tocompare with the actual crack in isotropic and graded materials. The crack propagation and the SIFswere simulated using finite element method. It was concluded that piecewise linear crack shapesprovide a significantly better approximation than straight crack shapes. Accordingly, analyticalsolutions for piecewise linear cracks in graded materials would be very useful, and should be afocus of future work in this area.

Nonlinear vibration of edge cracked functionally graded Timoshenko beams

Nonlinear vibration of beams made of functionally graded materials (FGMs) containing an open edge crack is studied in this paper based on Timoshenko beam theory and von Kármán geometric nonlinearity. The cracked section is modeled by a massless elastic rotational spring. It is assumed that material properties follow exponential distributions through beam thickness. The Ritz method is employed to derive the governing eigenvalue equation which is then solved by a direct iterative method to obtain the nonlinear vibration frequencies of cracked FGM beams with different end supports. A detailed parametric study is conducted to study the influences of crack depth, crack location, material property gradient, slenderness ratio, and end supports on the nonlinear free vibration characteristics of cracked FGM beams. It is found that unlike isotropic homogeneous beams, both intact and cracked FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of bending–extension coupling in FGM beams.

Postbuckling analysis of edge cracked functionally graded Timoshenko beams under end shortening

In this paper, the postbuckling response of beams made of functionally graded materials (FGMs) containing an open edge crack is studied based on Timoshenko beam theory and von Karman nonlinear kinematics. The cracked section is modeled by a massless elastic rotational spring. It is assumed that material properties follow exponential distributions through thickness direction. Ritz method is employed to derive the nonlinear governing equations, which are then solved by using Newton-Raphson method to obtain the postbuckling load-end shortening curves and postbuckling deflection-end shortening curves. A detailed parametric study is conducted to study the influences of crack depth, crack location, material property gradient, and slenderness ratio on the postbuckling behavior of cracked FGM beams. It is found that both intact and cracked FGM beams exhibit similar postbuckling behavior under end shortening. Unlike isotropic homogeneous beams, bifurcation buckling does not occur for both intact and cracked FGM beams due to the presence of bending-extension coupling effect.

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.

Fracture analysis of functionally graded materials by a BEM

In this paper, crack analysis in two-dimensional (2D), continuously nonhomogeneous, isotropic and linear elastic functionally graded materials (FGMs) is presented. For this purpose, a boundary element method (BEM) based on a boundary-domain integral equation formulation is developed. An exponential variation with spatial variables is assumed for Young’s modulus of the FGMs, while a constant Poisson’s ratio is considered. Fundamental solutions for homogeneous, isotropic and linear elastic solids are applied in the formulation. To avoid displacement gradients in the domain integral, normalized displacements are introduced. By using the radial integration method, the domain integral is transformed into boundary integrals over the global boundary. The normalized displacements in the domain integral are approximated by a combination of radial basis functions and polynomials in terms of global coordinates, which leads to a meshless scheme. Special attention of the analysis is devoted to the computation of the most important crack-tip characterizing parameters of cracked FGMs, namely the stress intensity factors. To show the effects of the material gradation on the stress intensity factors, numerical examples are presented and discussed.

Caustics analysis of the crack initiation and propagation of graded materials

In this paper, dynamic crack initiation and propagation of functionally graded materials (FGMs) are experimentally studied using the optical method of caustics. First, through a novel numerical iteration method, the caustics governing equations that relate to the caustics principles and the elastic solution of mode I propagating crack in FGMs are obtained in terms of the dynamic stress intensity factor, material constants, graded index, crack velocity and the characteristic size of the caustics. Second, the three-point bending beams with crack located at both the stiff and the compliant sides are experimentally studied combining dynamic caustics and high-speed photography. A series of dynamic caustics patterns surrounding the propagating crack tip are recorded at different time intervals. Some dynamic fracture parameters are extracted including dynamic stress intensity factor and crack velocity. Finally, the influences of gradient variation on dynamic fracture behaviors in FGMS are analyzed and discussed based on both experimental result and finite element modeling.