Local integral equation method for potential problems in functionally graded anisotropic materials

Efficient computational techniques are developed for 2D potential problems in anisotropic media with continuously variable material coefficients. The method is based on integral relationships considered on local sub-domains and domain-type approximations of the field variable. Three different kinds of integral equations are combined with either a domain element interpolation or a meshless point interpolation. The physical background of the formulation is discussed briefly. The accuracy and the convergence of the proposed techniques are tested by several examples and compared with benchmark analytical solutions.     

Two-dimensional stress analysis of functionally graded solids using the MLPG method with radial basis functions

The meshless local Petrov–Galerkin (MLPG) method is used for analysing two-dimensional (2D) static and dynamic deformations of functionally graded materials (FGMs) with material response modelled as either linear elastic or as linear viscoelastic. The multiquadric radial basis function (RBF) is employed to approximate the trial solution. Results are computed with two different choices of test functions, namely a fourth-order spline weight function, and a Heaviside step function, each having a compact support. No background mesh is used to numerically evaluate integrals appearing in the weak formulation of the problem, thus the method is truly meshless. A benefit of using RBFs is that they possess the Kronecker delta property; thus it is easy to satisfy essential boundary conditions. For five problems, the computed results are found to match well with those either from their analytical solutions or numerical solutions of other researchers who employed different algorithms. For a dynamic problem, the Laplace-transform technique is utilised. The numerical examples illustrate that displacements and stress distributions in a structure made of an FGM differ considerably from those at the corresponding points in the same structure made of a homogeneous material. Thus, the inhomogeneity in material properties can be exploited to optimise stress distribution, minimise deflection and reduce the maximum stress.     

Domain element local integral equation method for potential problems in anisotropic and functionally graded materials

An efficient numerical method is proposed for 2-d potential problems in anisotropic media with continuously variable material coefficients. The method is based on the local integral relationships (integral form of balance equation and/or integral equations utilizing fundamental solutions) and consistent approximation of field variable using standard domain-type elements. The accuracy and convergence of the proposed method is tested by several examples and compared with benchmark analytical solutions.     

3D crack analysis in functionally graded materials

Elastostatic crack analysis in three-dimensional, continuously non-homogeneous, isotropic and linear elastic functionally graded materials and structures is presented in this paper. A boundary-domain-integral equation formulation is applied for this purpose, which uses the elastostatic fundamental solutions for homogeneous, isotropic and linear elastic materials and involves a domain-integral due to the material’s non-homogeneity. To avoid displacement gradients in the domain-integral, normalized displacements are introduced. The domain-integral is transformed into boundary-integrals over the global boundary of the cracked solids by using the radial integration method. A meshless scheme is developed, which requires only the conventional boundary discretization and additional interior nodes instead of interior cells or meshes. Numerical examples for three-dimensional crack problems in continuously non-homogeneous, isotropic and linear elastic FGMs are presented and discussed, to show the effects of the material gradation on the crack-opening-displacements and the stress intensity factors.     

Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method

Advanced computational method for transient heat conduction analysis in continuously nonhomogeneous functionally graded materials (FGM) is proposed. The method is based on the local boundary integral equations with moving least square approximation of the temperature and heat flux. The initial-boundary value problem is solved by the Laplace transform technique. Both Papoulis and Stehfest algorithms are applied for the numerical Laplace inversion to obtain the time-dependent solutions. Numerical results are presented for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

New implementation of MLBIE method for heat conduction analysis in functionally graded materials

The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

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.     

Frictional contact analysis of functionally graded materials with Lagrange finite block method

Based on the one‐dimensional differential matrix derived from the Lagrange series expansion, the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials. In this paper, the formulation of the Lagrange finite block method with boundary type in the strong form is presented and applied to non‐conforming contact problems for the functionally graded materials subjected to either static or dynamic loads. The first order partial differential matrices are only needed both in the governing equations and in the Neumann boundary condition. By introducing the mapping technique, a block of quadratic type is transformed from the Cartesian coordinate of global system to the normalized coordinate with eight seeds. Time dependent partial differential equations are analyzed in the Laplace transformed domain and the Durbin's inversion method is applied to determine all the physical values in the time domain. Conforming and non‐conforming contacts are investigated by using the iterative algorithm with full load technique. Illustrative numerical examples are given and comparisons have been made with analytical solutions. Copyright © 2015 John Wiley & Sons, Ltd.     

XFEM fracture analysis of orthotropic functionally graded materials

In the present study, the extended finite element method (XFEM) has been used for fracture analysis of orthotropic functionally graded materials. Orthotropic crack tip enrichments have been used to reproduce the singular stress field near a crack tip. Moreover, the incompatible interaction integral method has been employed to extract the stress intensity factor components. Accuracy and convergence of the proposed method have been evaluated by numerical examples and quality results have been obtained by far fewer DOFs. Also, crack propagation in isotropic and orthotropic FGMs in the presence of crack tip enrichments has been investigated and various propagation criteria have been compared, and verified, if available, by experimental and numerical data in the literature. Application of XFEM in combination of the maximum circumferential tensile stress criterion for investigation of crack propagation in orthotropic FGM problems is performed for the first time.     

Three-dimensional thermal buckling analysis of functionally graded materials

Three-dimensional thermal buckling analysis is performed for functionally graded materials. Material properties are assumed to be temperature dependent, and varied continuously in the thickness direction according to a simple power law distribution in terms of the volume fraction of a ceramic and metal. The finite element model is adopted by using an 18-node solid element to analyze more accurately the variation of material properties and temperature field in the thickness direction. Furthermore, the assumed strain mixed formulation is used to prevent locking as well as maintaining kinematic stability of the finite element model for thin plates and shells. The thermal buckling behavior under uniform or nonuniform temperature rise across the thickness is analyzed. Numerical results are compared with those of the previous works. In addition, the changes of critical buckling temperature due to the effects of temperature field, volume fraction distributions, and system geometric parameters are studied.     

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.     

Crack analysis in unidirectionally and bidirectionally functionally graded materials

Mixed-mode crack analysis in unidirectionally and bidirectionally functionally graded materials is performed by using a boundary integral equation method. To make the analysis tractable, the Young's modulus of the functionally graded materials is assumed to be exponentially dependent on spatial variables, while the Poisson's ratio is assumed to be constant. The corresponding boundary value problem is formulated as a set of hypersingular traction boundary integral equations, which are solved numerically by using a Galerkin method. The present method is especially suited for straight cracks in infinite FGMs. Numerical results for the elastostatic stress intensity factors are presented and discussed. Special attention of the analysis is devoted to investigate the effects of the material gradients and the crack orientation on the elastostatic stress intensity factors.     

Mixed-mode fatigue crack growth analysis of functionally graded materials by XFEM

The present study investigates the fatigue life of a functionally graded material (FGM) made of aluminum alloy and alumina (ceramic) under cyclic mixed mode loading. The fatigue lives of aluminum alloy, FGM and an equivalent composite (having the same composition as of FGM) are compared for a major edge crack in the rectangular domain. The extended finite element method is used to simulate the fatigue crack growth under plane strain conditions. Various discontinuities such as minor cracks, holes and inclusions of arbitrary sizes are randomly located in the domain along with the major edge crack. Paris law is used to evaluate the fatigue life of the aluminium alloy, FGM and equivalent composite.     

A continuum shape sensitivity method for fracture analysis of isotropic functionally graded materials

This paper presents a new continuum shape sensitivity method for calculating mixed-mode stress-intensity factors for a stationary crack in two-dimensional, linear- elastic, isotropic FGMs with arbitrary geometry. The method involves the material derivative concept taken from continuum mechanics, the mutual potential energy release rate, and direct differentiation. Since the governing variational equation is differentiated prior to discretization, resulting sensitivity equations are independent of approximate numerical techniques, such as the finite element method, boundary element method, mesh-free method, or others. The discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires multiplication of displacement vectors and stiffness sensitivity matrices. By judiciously selecting the velocity field, the method only requires displacement response in a subdomain close to the crack tip, thus making the method computationally efficient. Seven finite-element based numerical examples, which comprise mode-I and mixed-mode deformations and/or single or multiple interacting cracks, are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method. Comparisons have been made between stress-intensity factors predicted by the proposed method and available reference solutions in the literature, generated either analytically or numerically using various other fracture integrals or analyses. Excellent agreement is obtained between the results of the proposed method and previously obtained solutions. Therefore, shape sensitivity analysis provides an attractive alternative to fracture analysis of cracks in homogeneous and non-homogeneous materials.     

Geometrically nonlinear analysis of functionally graded materials based on reproducing kernel particle method

International Journal of Mechanics and Materials in Design - Using the total Lagrange formulation, the reproducing kernel particle method (RKPM) for the geometrically nonlinear problem of the...     

An extended element free Galerkin method for fracture analysis of functionally graded materials

An extended element free Galerkin method (XEFGM) has been adopted for fracture analysis of functionally graded materials (FGMs). Orthotropic enrichments functions are used along with the sub-triangle technique for enhancing the Gauss quadrature accuracy near the crack, and the incompatible interaction integral method is employed to calculate the stress intensity factors. Numerical simulations have proved that XEFGM provides more accurate results by less number of nodes (DOFs) in comparison with the unenriched EFGM and other conventional methods for several FGM problems with different crack locations and loadings. The results have been compared with the reference results, showing the reliability, stability, and efficiency of present XEFGM.

Received 9 June 2014 Accepted 17 September 2014.     

Sensitivity analysis of heat conduction for functionally graded materials

A sensitivity analysis is presented for the steady-state and transient heat conduction of functionally graded materials (FGMs). Based on the finite element method, the sensitivity equations of heat conduction are presented by using the direct method and the adjoint method. In the solution of transient problem, the precise time integration (PTI) is employed. The spatial volume fractions of materials of FGM (size problem) and the shape design parameters are considered. Detailed formulations especial for the FGMs are provided. The numerical examples are presented to demonstrate the precision and applicability of the proposed method.     

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