Cracks In Functionally Graded Materials Research Articles

Free vibration of cracked FGM Mindlin plate in fluid

The free vibration of cracked functionally graded material (FGM) plate in fluid is investigated in this paper. The material components are supposed following the power law and formulated by the Voigt model. The Mindlin plate theory (MPT) including the shear deformation effect is established and the physical neutral plane is adopted to simplify the derivation due to the in-plane displacements of this plane assumed as zero. A massless linear rotational spring model (LRSM) is established to simulate the through-width surface crack. The hydrodynamic pressure at the fluid-plate interface is derived by the potential flow theory, and is regarded as the added mass of cracked FGM plate during the analyses of fluid-plate coupled vibration. The derivation and discretization of vibrational equations for cracked FGM plates in fluid are respectively achieved though the Hamilton’s principle and differential quadrature (DQ) method. The frequencies and mode shapes are iteratively calculated with these values of the vacuum case. The influences of immersed depth, fluid density, gradient index, crack depth, crack location, length-to-width ratio, length-to-thickness ratio, and boundaries on vibration characteristics are conducted in the section of numerical results. The present vibration analyses of cracked structures can be applicable to avoid structural resonance and failure.

Galerkin finite block method in solid mechanics

In this paper, the Galerkin Finite Block Method (GFBM) is proposed for the first time to solve two-dimensional elasticity problems with functionally graded materials (FGMs). The physical variables are approximated with double layer Chebyshev polynomials with unknown coefficients for two-dimensional elasticity. The main idea is to establish a set of linear algebraic equations by the Galerkin method using Chebyshev polynomials in the weak form of the partial differential equations and the boundary conditions. By introducing the mapping technique, a block with irregular shape of the boundary is transformed from the Cartesian coordinates (x, y) to the normalized coordinates (ξ, η), |ξ|⩽1, |η|⩽1. The continuity conditions of the displacement and traction on the interface between two blocks are introduced both in the weak form and strong form respectively. Several numerical examples are presented and comparisons are made with the Finite Element Method (FEM) including a cracked FGM sheet, in order to demonstrate the accuracy and efficiency of the GFBM.

On an annular crack near an arbitrarily graded interface in FGMs

Annular cracks can be generated during the fabrication of materials and around the pre-existing defects. Previous works on the annular crack problems were most limited to homogenous materials. This paper extends the analysis and examines the axisymmetric problem of a flat mixed-mode annular crack near and parallel to an arbitrarily graded interface in functionally graded materials (FGMs). The crack is modelled as plane circular dislocation loop and an efficient solution for dislocation in FGMs is used to calculate the stress field at the crack plane. The three-parts mixed boundary value problem is solved by a series expansion method. Analytical solutions of the stress intensity factors are obtained. Compared with many other existing analytical treatments to the crack problems in FGMs with the assumption of special gradation, the present method can consider arbitrarily graded shear modulus and Poisson's ratio. It is shown that the present solution can reduce to existing solutions in literatures for annular crack in homogenous materials. Numerical study is conducted to investigate the fracture mechanics of annular crack in FGMs. A practical example of an annular crack in a real Epoxy-Glass FGM with measured data of material properties is also considered.

Extended wavelet Galerkin method for mixed-mode cracked FGM plate under static and dynamic loads

The extended wavelet Galerkin method (XWGM) is presented for fracture mechanics analysis of functionally graded material (FGM) cracked plates under static and dynamic loads. The nonhomogeneous stress distribution and stress wave transition can be approximated by the piecewise truncated polynomial (B-spline) functions. The steep stress gradients around the crack tip can be captured by superposing the higher resolution wavelet functions with the nature of multiresolution analysis (MRA) in the wavelet theory. Additionally, a trigonometric function and Heaviside function are employed to enrich the wavelet bases for the fracture modeling. The interaction integral method is utilized to evaluate the static and dynamic mixed-mode stress intensity factors taking the inertia and material nonhomogeneity into account. Applicability of XWGM to the FGM crack problems is discussed through the numerical examples.

Vibration and wave propagation in functionally graded beams with inclined cracks

This paper proposes an FEA method to study the vibration and wave propagation in functionally graded material (FGM) beams with multiple inclined cracks by introducing the local flexibility matrix caused by the cracks. The bending and tensile stiffnesses of the inclined cracks and their interaction are taken into consideration. Two-node (three degrees of freedom per node) beam element is used. The inclined cracks are equivalent to transversal cracks to calculate the additional flexibility matrix of the inclined cracked FGM beam element. The material properties of the FGM beam are supposed to satisfy the exponential law along its thickness. The governing equations of motion for the FGM beam with multiple inclined cracks are deduced in the frame of Euler-Bernoulli theory and Lagrange function, and numerically solved by the Newmark average acceleration method. Both theoretical and numerical comparisons are presented to validate the present method for the inclined cracked FGM beam with varying boundary conditions, arbitrary inclined crack angles as well as varying inclined crack lengths. The effects of the location, length, inclined angle and number of the inclined cracks as well as material property ratio on the fundamental frequencies and wave propagation characteristics of the inclined cracked cantilever FGM beams are comprehensively investigated.

Influence of Material Uncertainty on Vibration Characteristics of Higher-Order Cracked Functionally Gradient Plates Using XFEM

In this study, the influence of material uncertainty on the vibration characteristics of the cracked functionally graded materials (FGM) plates is investigated. Extended stochastic finite element formulation is implemented to model the cracked FGM plate with material uncertainty using higher-order shear deformation theory (HSDT). The level set function is employed to track the crack in the FGM domain. The concept of partition of unity technique is implemented to enrich the primary variable with additional functions. The gradation of the material properties along the thickness direction is done using the power-law distribution. The first-order perturbation technique (FOPT) is incorporated in the methodology for stochastic vibration analysis. The convergence and validation study has been performed to verify the efficacy and accuracy of the formulation. Numerical results are obtained to show the effects of various influential parameters like crack length, gradient index, thickness ratio, and boundary condition on the covariance of the square of natural frequencies. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of cracked FGM plates with material randomness.

Computation of SIFs for cracks in FGMs and TBC under mechanical and thermal loadings

The objective of this study is to present a numerical modeling of mixed-mode fracture in isotropic functionally graded materials (FGMs), under mechanical and thermal loading conditions. In this paper, a modified displacement extrapolation technique DET was proposed to calculate the stress intensity factor (SIFs) for isotropic FGMs. Using the Ansys Parametric Design Language, the continuous variations of the material properties are incorporated by specified parameters at the centroid of each element. Four numerical examples are presented to evaluate the accuracy of SIFs calculated by the proposed method. Comparisons have been made between the SIFs predicted by the DET and the available reference solutions in the current literature. A good agreement is obtained between the results of the DET and the reference solutions.

The Homogenized Transformation Method for the Calculation of Stress Intensity Factor in Cracked FGM Structure

A convenient calculation method is proposed for the stress intensity factor (SIF) in cracked functionally graded material (FGM) structures. In this method, the complex computational problem for SIFs in cracked FGM plate and cylinder can be simplified as the calculation problem of empirical formulas of SIFs in cracked homogenous plate and cylinder with same loading conditions and the calculation problem of related transition parameters. The results show that the SIF in cracked FGM structure can be obtained accurately without using matrix and integral. The validity and usefulness of the present method are proved by comparing with the results of the conventional method.

Study on Thermally Induced Crack Propagation Behavior of Functionally Graded Materials Using a Modified Peridynamic Model

Peridynamic (PD) theory is used to study the thermally induced cracking behavior of functionally graded materials (FGMs). A modified thermomechanical peridynamic model is developed. The thermal crack propagation of a ceramic slab in quenching is calculated to validate the modified PD model. The results predicted by the modified PD model agree with previously published numerical and experimental ones. Compared with the original PD model, the calculation accuracy of the modified PD model for thermal cracking is improved. The thermal cracking in FGMs is also simulated. The effects of material shape, initial temperature, and ceramic fracture toughness on thermal crack propagation behaviors are studied. It can be found that the thermal cracks in FGMs are still in periodical and hierarchical forms. The metal materials in FGMs can prevent crack initiation and arrest the long cracks. The crack number tends to be increased with the increasing initial temperature, while the strengthened ceramic fracture toughness can decrease it.

Computation of SIFs for cracked FGMs under mechanical and thermal loadings

AbstractThe objective of this study is to present a numerical modeling of mixed-mode fracture in isotropic functionally graded materials (FGMs), under mechanical and thermal loading conditions. In this paper, a modified displacement extrapolation technique (DET) was proposed to calculate the stress intensity factor (SIFs) for isotropic FGMs. Using the Ansys Parametric Design Language APDL, the continuous variations of the material properties are incorporated by specified parameters at the centroid of each element. Three numerical examples are presented to evaluate the accuracy of SIFs calculated by the proposed method. Comparisons have been made between the SIFs predicted by the DET and the available reference solutions in the current literature. A good agreement is obtained between the results of the DET and the reference solutions.

T-stress evaluation for multiple cracks in FGMs by the numerical manifold method and the interaction integral

The numerical manifold method (NMM) is superior in discontinuous analysis, benefiting from its bi-cover systems, i.e., the mathematical cover and the physical cover. In this work, the NMM is devoted to compute the T-stress for two-dimensional functionally graded materials (FGMs) with multiple cracks. The displacement field for cracked FGMs is firstly solved by the NMM and then the related T-stress is evaluated in the NMM postprocessing via the interaction integral method. Two typical examples are firstly tested to verify the accuracy of the proposed method and the domain-independence of the interaction integral, and then another three cases are examined to reveal the effects of material gradients, crack geometries and boundary conditions on the T-stress.

Crack identification in multiple cracked beams made of functionally graded material by using stationary wavelet transform of mode shapes

This paper presents crack identification in multiple cracked beams made of functionally graded material (FGM) by using stationary wavelet transform (SWT) of mode shapes and taking into account influence of Gaussian noise. Mode shapes are obtained from multiple cracked FGM beam element and spring model of cracks. The theoretical development was illustrated and validated by numerical examples. The investigated results show that crack identification method by using SWT of mode shapes is efficient and realizable.

Analysis of cracked functionally graded material plates using XIGA based on generalized higher-order shear deformation theory

A generalized higher-order shear deformation theory (GHSDT) along with extended isogeometric analysis (XIGA) is presented for the analysis of cracked functionally graded material (FGM) plates. The material properties of the FGM vary exponentially along the thickness of the plate. The GHSDT ensures a traction-free boundary conditions at the surfaces of the plate without a need of shear correction factor (SCF). To model a crack, the isogeometric analysis (IGA) approximation is enriched with partition of unity (PU) enrichment functions. The C1 continuity requirement of the GHSDT is easily achieved in IGA through Non-uniform rational B-splines (NURBS). The stress intensity factor (SIF) at any location along the thickness of the plate is computed using domain based interaction integral approach. In the interaction integral, 2-D asymptotic crack-tip fields obtained from the LEFM provide a straightforward and simple solution scheme for SIF along the thickness of the FGM plate. The accuracy of the proposed approach is demonstrated by solving various problems. The effect of various parameters such as loading, boundary conditions, crack length and plate thickness is investigated on the SIF. The solutions of several cracked FGM plate problems obtained by GHSDT based XIGA (GHSDT-XIGA) are compared with FSDT based XIGA (FSDT-XIGA) and reference solutions.

Investigation of the thermal-elastic problem in cracked semi-infinite FGM under thermal shock using hyperbolic heat conduction theory

In this paper, a thermoelastic analytical model is established for a functionally graded half-plane containing a crack under a thermal shock in the framework of hyperbolic heat conduction theory. The moduli of functionally graded materials (FGMs) are assumed to vary exponentially with the coordinates. By employing the Fourier transform and Laplace transform, coupled with singular integral equations, the governing partial differential equations under mixed, thermo-mechanical boundary conditions are solved numerically. For both the temperature distribution and transient stress intensity factors (SIFs) in FGMs, the results of hyperbolic heat conduction model are significantly different than those of Fourier’s Law, which should be considered carefully in designing FGMs.

Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method

Abstract In the present paper, the free and forced vibration of multiple cracked multi span continuous beams made of functionally graded material (FGM) is investigated by the dynamic stiffness method. First, there are constructed dynamic stiffness matrix and nodal load vector of multiple cracked FGM beam element using Timoshenko beam theory and massless double spring model of crack. Then, effect of cracks parameters on free and forced vibration of the FGM continuous beams is examined. The theoretical developments are validated by numerical examples. The obtained results provide an efficient method to analyze free and forced vibration of multiple cracked FGM framed structures and assessment of the behavior of damaged structures.

Buckling of stomatopod-dactyl-club-inspired functional gradient plates: A numerical study

The translation of natural (or biological) structures into synthetic materials offers a spectrum of feasible pathways towards enhanced or even unprecedented material properties for a wide range of applications. Inspired by excellent mechanical properties of stomatopod dactyl club, herein we propose a new functionally graded material (FGM) model that can adequately describe the characteristics of hierarchical structures in the stomatopod dactyl club. The proposed FGM model is incorporated into extended finite element method with stabilized discrete shear gap to investigate the mechanical buckling behavior of the plate made of materials similar with that in stomatopod dactyl club. The critical buckling factors for cracked FGM plates are computed and analyzed. Numerical results show that the design motif of bioinspired FGM increases the normalized buckling factor, even higher than that of homogenous plates, which suggests that the stability of plates can be enhanced by the use of bioinpsired FGM. The proposed FGM model opens a broad avenue to explore more fascinating mechanical properties of hierarchical gradient materials inspired by stomatopod dactyl club.

Multiple cylindrical interface cracks in FGM coated cylinders under torsional transient loading

In the present study, the problem of multiple cylindricalinterfacecracks between a homogeneouscircular cylinder and its functionally graded materials (FGM) coating under torsional impact loading is investigated. First, by using the Laplace and complex Fourier transforms, the solution for Somigliana-type dynamic rotational ring dislocation in a cylinder and its coating is obtained. Next, the distributed dislocation technique is employed to derive a set of Cauchy singular integral equations for multiple cylindricalinterface cracks situated between the isotropic cylinder and its FGM coating. These integral equations are solved by Erdogan’s collocation method; the dislocation densities on the faces of the cracks are obtained, and the results are used to determine dynamic stress intensity factors (DSIFs). Several examples are provided to study the influences of material nonhomogeneity constant, the FGM layer thickness and crack geometry configuration on the DSIFs at the tips of cracks and, the interaction between the cracks.

Size effects in elastic-plastic functionally graded materials

We develop a strain gradient plasticity formulation for composite materials with spatially varying volume fractions to characterize size effects in functionally graded materials (FGMs). The model is grounded on the mechanism-based strain gradient plasticity theory and effective properties are determined by means of a linear homogenization scheme. Several paradigmatic boundary value problems are numerically investigated to gain insight into the strengthening effects associated with plastic strain gradients and geometrically necessary dislocations (GNDs). The analysis of bending in micro-size functionally graded foils shows a notably stiffer response with diminishing thickness. Micro-hardness measurements from indentation reveal a significant increase with decreasing indenter size. And large dislocation densities in the vicinity of a crack substantially elevate stresses in cracked FGM components. We comprehensively assess the influence of the length scale parameter and material gradation profile to accurately characterize the micro-scale response and identify regimes of GNDs relevance in FGMs.

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.