Investigation of anti-plane shear behavior of a Griffith permeable crack in functionally graded piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading for the permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of dual-integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allows us to using the maximum stress as a fracture criterion. The finite hoop stresses at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Finite element simulations of crack propagation in functionally graded materials under flexural loading

Cracks in stepped and continuously graded material specimens under flexural loading were investigated via finite element analysis. Calculation of mechanical energy release rates and propagation angles with crack-opening displacement correlation and the local symmetry (K_II = 0) criterion, respectively, provided results most efficiently and accurately, as compared with compliance and J-integral approaches and other deflection criteria. A routine was developed for automatic crack extension and remeshing, enabling simulation of incremental crack propagation. Effects of gradient profile and crack geometry on crack-tip stresses and crack propagation path are examined, and implications of these for optimal design of graded components against failure by fast fracture are discussed.     

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.     

Mixed-mode dynamic crack propagation in graded materials under thermo-mechanical loading

Mixed-mode dynamic crack growth behavior in functionally graded materials (FGMs) under thermo-mechanical loading is studied. Asymptotic analysis in conjunction with displacement potentials has been used to develop thermo-mechanical stress fields for a mixed mode propagating crack in a FGM. The shear modulus, mass density, thermal conductivity and coefficient of thermal expansion of the FGM are assumed to vary exponentially along the gradation direction. First, asymptotic temperature fields are derived for an exponential variation of thermal conductivity and later these temperature fields are used in deriving stress fields. Using asymptotic thermo-mechanical stress fields the variation of maximum shear stress, circumferential stress and strain-energy density as a function of temperature around the crack tip are generated. Finally, utilizing the minimum strain-energy density criterion and the maximum circumferential stress criterion, the crack growth direction for various crack-tip speeds, non-homogeneity coefficients and temperature fields are determined.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

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.     

Simulation of crack paths in functionally graded materials

One of the main interests of fracture mechanics in functionally graded materials is the influence of such an inhomogeneity on crack propagation processes. Using the Griffith’ energy principle, the change of energy has to be calculated, if the crack starts to propagate. In homogeneous linear-elastic structures (asymptotically precise) formulas for the energy release rate are known, but a direct transfer of these methods to functionally graded materials can lead to very inaccurate results. Moreover, the influence of the inhomogeneity on the crack path cannot be seen. Here, a simple model for functionally graded materials is introduced. For this model, a formula for the change of potential energy is derived, giving detailed information on the effect of the gradation on crack propagation.     

Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory

Dynamic stability of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and Timoshenko beam theory. This non-classical Timoshenko beam model contains a material length scale parameter and can interpret the size effect. The material properties of FGM microbeams are assumed to vary in the thickness direction and are estimated though Mori–Tanaka homogenization technique. The higher-order governing equations and boundary conditions are derived by using the Hamilton’s principle. The differential quadrature (DQ) method is employed to convert the governing differential equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotin’s method. Free vibration and static buckling are also discussed as subset problems. A parametric study is conducted to investigate the influences of the length scale parameter, gradient index and length-to-thickness ratio on the dynamic stability characteristics of FGM microbeams with hinged–hinged and clamped–clamped end supports. Results show that the size effect on the dynamic stability characteristics is significant only when the thickness of beam has a similar value to the material length scale parameter.     

Experimental investigation of the mixed-mode crack propagation in ZrO2/NiCr functionally graded materials

Mixed-mode fracture response of ZrO₂/NiCr functionally graded materials (FGMs) is investigated using fracture test, digital image correlation technique and extended finite element method. It is found that: (1) prior to crack initiation, the increasing elastic modulus ahead of crack-tip can enhance the mode mixity of crack-tip; (2) the crack with the increasing elastic modulus ahead of crack-tip kinks less than the one with the decreasing elastic modulus, which is caused by the elastic gradient and the local fracture toughness; (3) the heterogeneity of microstructure can cause the local perturbation but has no obvious effect on the overall crack propagation path.     

Cracks and re-entrant corners in functionally graded materials

Functionally graded materials (FGMs) are special composites in which the volume fractions of constituent materials vary gradually, giving continuously graded mechanical properties. The aim of this paper is the evaluation of the strength of structures composed by FGMs incorporating re-entrant corners - tending to the more common crack for vanishing corner angle. The end result is useful in engineering applications predicting the strength of the element corresponding to the unstable brittle crack propagation in such innovative materials. To show the general validity of the method, heterogeneous plates under tension and beam under bending containing re-entrant corners and by varying corner angle, depth and grading of the FGM are considered. Ad hoc performed numerical finite element simulations, by using the FRANC2D code, agree with the theoretical predictions.     

Interface waves in functionally graded piezoelectric materials

We analyze the propagation of antiplane or shear-horizontal waves near the interface between two half-spaces of piezoelectric ceramics. The material properties vary in the direction perpendicular to the interface. Both electroded and unelectroded interfaces are considered. Transcendental equations that determine the dispersion relations of the waves are obtained. They reduce to a few known results in the literature as special cases. Different from similar waves in homogeneous materials, the waves obtained are dispersive. The equations for the dispersion relations are solved numerically. It is found that the wave speeds are sensitive to the variation of material properties. This suggests the possibility of manipulating the wave propagation behavior through proper design of materials.     

Interaction integrals for thermal fracture of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium.     

Thermal stress in rotating functionally graded hollow circular disks

The analysis of thermoelastic problem of a rotating functionally graded hollow circular disk is made. The hollow disk is assumed to have varying material properties along the radial direction. An analytical method is presented to investigate steady thermal stresses in a functionally graded circular annulus rotating with constant angular velocity about its central axis. The associated boundary value problem is reduced to a Fredholm integral equation. The thermal stresses and radial displacement are obtained by numerically solving the resulting equation. A comparison of the numerical results with the exact ones for material properties of special power-law profile confirms the effectiveness of the method. For generally varying material parameters, numerical results are presented graphically to show the effects of gradient parameter, temperature change, angular velocity and thickness of the disk on the distribution of thermal stresses and radial displacement.     

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.     

An extension to classical lamination theory for use with functionally graded plates

An extension to classical lamination theory is presented for the improved analysis of thin to moderately thick functionally graded plates. The method results in an explicit formulation that accommodates any through-thickness variation in the elastic, hygrothermal and piezoelectric properties of each layer. Additionally, variations in the material rotation angle, temperature, moisture content and electric field strength through each layer are taken into account. The method relies on representing with polynomial series the variation in both the properties of each ply and the hygrothermal and piezoelectric loading. Validation problems are presented that demonstrate the application and accuracy of the method.     

Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness

The interface crack problem for a composite layer that consists of a homogeneous substrate, coating and a nonhomogeneous functionally graded interphase was formulated for singular integral equations with Cauchy kernels, which were integrated using the Lobatto–Chebyshev collocation technique. Mixed-Mode Stress Intensity Factors (SIFs) and Strain Energy Release Rates were calculated. The SIFs were compared for accuracy with relevant results previously published. The parametric studies were conducted for the various thickness of each layer and for various nonhomogeneity ratios. Particular application to the Zirconia thermal barrier on steel substrate is demonstrated.     

The modified couple stress functionally graded Timoshenko beam formulation

In this paper, a size-dependent formulation is presented for Timoshenko beams made of a functionally graded material (FGM). The formulation is developed on the basis of the modified couple stress theory. The modified couple stress theory is a non-classic continuum theory capable to capture the small-scale size effects in the mechanical behavior of structures. The beam properties are assumed to vary through the thickness of the beam. The governing differential equations of motion are derived for the proposed modified couple-stress FG Timoshenko beam. The generally valid closed-form analytic expressions are obtained for the static response parameters. As case studies, the static and free vibration of the new model are respectively investigated for FG cantilever and FG simply supported beams in which properties are varying according to a power law. The results indicate that modeling beams on the basis of the couple stress theory causes more stiffness than modeling based on the classical continuum theory, such that for beams with small thickness, a significant difference between the results of these two theories is observed.     

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.     

The effects of residual stresses and degradation on the response of viscoplastic functionally graded materials

Functionally graded materials (FGMs), having ceramic and metallic constituents, are commonly used for extreme temperature applications. Under extreme temperature changes, the mismatches in the thermo-mechanical properties of the ceramics and metallic constituents could cause pronounced thermal stresses and could lead to degradation in the properties of the constituents. High stresses in the metallic constituent lead to plastic deformations and high tensile stresses in the ceramic constituent induce cracking. An elastic–viscoplastic micromechanical model is formulated for analyzing residual stresses and strains and degradation in ceramic–metal FGMs undergoing temperature changes due to conduction of heat. A degradation parameter that depends on the temperature and strain is introduced in order to determine the level of material degradation in the ceramics and metallic constituents. The Perzyna viscoplastic model is considered for the metallic constituent while the ceramic constituent is assumed linear elastic. The material parameters in these constituents change with the degradation. The problem leads to time-dependent coupled thermal, deformation, and degradation behaviors. The micromechanical model is implemented in a displacement based finite element (FE), which is used to determine the performance of the viscoplastic functionally graded structures subject to external thermo-mechanical stimuli.     

Stress intensity factors for functionally graded solid cylinders

This paper presents the mode I stress intensity factors for functionally graded solid cylinders with an embedded penny-shaped crack or an external circumferential crack. The solid cylinders are assumed under remote uniform tension. The multiple isoparametric finite element method is used. Various types of functionally graded materials and different gradient compositions for each type are investigated. The results show that the material property distribution has a quite considerable influence on the stress intensity factors. The influence for embedded cracks is quite different from that for external cracks.