A boundary layer framework considering material gradation effects

This paper describes the development and application of a novel modified boundary layer (MBL) model for graded nonhomogeneous materials, e.g. functionally graded materials (FGMs). The proposed model is based on a middle-crack tension, M(T), specimen with traction boundary conditions applied to the top and lateral edges of the model. Finite element analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates elements with graded elastic and plastic properties. Elastic crack-tip fields obtained from the proposed MBL model show excellent agreement with those obtained from full models of the cracked component for homogeneous and graded nonhomogeneous materials. The K-T dominance of FGMs is investigated by comparing the actual stress fields with the asymptotic stress fields (the Williams’ solution). The examples investigated in the present study consider a crack parallel to the material gradient. Results of the present study provide insight into the K-T dominance of FGMs and also show the range of applicability of the proposed MBL model. The MBL model is applied to analyze the elastic-plastic crack-tip response of a Ti/TiB FGM SE(T) specimen. The numerical results demonstrate that the proposed MBL model captures the effect of T-stress on plastic zone size and shape, constraint effects, etc. for such configurations.     

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.     

Thermal fracture analysis of orthotropic functionally graded materials using an equivalent domain integral approach

A new computational method based on the equivalent domain integral (EDI) is developed for mode I fracture analysis of orthotropic functionally graded materials (FGMs) subjected to thermal stresses. By using the constitutive relations of plane orthotropic thermoelasticity, generalized definition of the J-integral is converted to an equivalent domain integral to calculate the thermal stress intensity factor. In the formulation of the EDI approach, all the required thermomechanical properties are assumed to have continuous spatial variations through the functionally graded medium. Developed methodology is integrated into a fracture mechanics research finite element code FRAC2D using graded finite elements that possess cubic interpolation. Steady-state and transient temperature distribution profiles in orthotropic FGMs are computed using the finite elements based heat transfer analysis software HEAT2D. EDI method is validated and domain independence is demonstrated by comparing the numerical results obtained using EDI to those calculated by an enriched finite element method and to those available in the literature. Single and periodic edge crack problems in orthotropic FGMs are examined in order to study the influences of principal thermal expansion coefficient and thermal conductivity components, relative crack length and crack periodicity on the thermal stress intensity factors. Numerical results show that among the three principal thermal expansion coefficient components, the in-plane component perpendicular to the crack axis has the most significant influence on the mode I stress intensity factor. Gradation profile of the thermal expansion coefficient parallel to the crack axis is shown to have no effect on the outcome of the fracture analysis.     

Determination of thermal stress intensity factors for an interface crack in a graded orthotropic coating-substrate structure

The thermal fracture problem of an interface crack between a graded orthotropic coating and the homogeneous substrate is investigated by two different approaches. For the case that most of the material properties in the graded orthotropic coating are assumed to vary as an exponential function, the integral transform and singular integral equation technique is used to obtain some analytical results. In order to analyze the case with more complex material distribution, an interaction integral is presented to evaluate the thermal stress intensity factors of cracked functionally graded materials (FGMs), and then the element-free Galerkin method (EFGM) is developed to obtain the final numerical results. The good agreement is obtained between the numerical results and the analytical ones. In addition, the influence of material gradient parameters and material distribution on the thermal fracture behavior is also presented.     

Approximate J estimates for circumferential cracked pipes under primary and secondary stresses

This paper presents finite element solutions for elastic-plastic J for circumferentially cracked pipes under combined mechanical and thermal loads in terms of the V/V_o factor used within the failure assessment diagram approach. Systematic analyses suggest that the two major variables affecting V/V_o are the relative magnitude of the secondary stress and the primary load magnitude. It has been found that the variations with these parameters obtained from the FE results agree well with current R6 predictions for modest thermal loads and low primary loads. For larger thermal stresses, the R6 predictions are overly conservative and more accurate predictions are suggested.     

Nonlinear thermal bending response of FGM plates due to heat conduction

Nonlinear thermal bending analysis is presented for a simply supported, shear deformable functionally graded plate without or with piezoelectric actuators subjected to the combined action of thermal and electrical loads. Heat conduction and temperature-dependent material properties are both taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations of an FGM plate are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. A two step perturbation technique is employed to determine the thermal load–deflection and thermal load–bending moment curves. The numerical illustrations concern nonlinear bending response of FGM plates without or with surface bonded piezoelectric actuators due to heat conduction and under different sets of electric loading conditions. The results reveal that for the case of heat conduction the nonlinear thermal bending responses are quite different to those of FGM plates subjected to transverse mechanical loads, and the temperature-dependency of FGMs could not be neglected in the thermal bending analysis.     

A solution method for frictional contact problems of a crack in FGMs under mechanical and thermal loads

This article provides a numerical treatment of a finite crack in an interfacial layer with spatially varying elastic properties under in-plane mechanical and thermal loading conditions. The variation of stress intensity factors and energy release rates with the functions which are governing the material properties of the interfacial layer is studied. Transient and steady-state response of a central crack in FGMs subjected to the mechanical and thermal loads are investigated. Unlike earlier studies which consider the cracks encountered as open, the current investigation studies cracks in an essentially compressive environment in which the crack faces are in contact and frictional effects play an important role. To solve this contact problem, a simple and efficient, iterative finite element method developed by authors is used. Numerical examples are provided to verify the technique and the results are compared with those of the published papers. This revised version was published online in July 2006 with corrections to the Cover Date.     

Fracture problems,vibration, buckling,and bending analyses of functionally graded materials: A state-of-the-art review including smart FGMS

Composite materials fail under extreme working conditions, particularly at high temperature, due to delamination (separation of fibers from matrix). And therefore it is needed to switch over functionally graded materials (FGMs) which can sustain at high temperature conditions (250–2000°C). There is a need to analyze the fracture and fatigue characteristics of FGM structures and so through this review the emphasis is given on fracture analysis of FGM materials. It has been reported that a combination of extended finite element method and isogeometric analysis methodologies has been used for general mixed-mode crack propagation problems after the introduction of extended isogeometric analysis. Furthermore, recent computational advances have been in the form of multiscale simulations where the part of model is simulated by a finer modeling scale, which can represent details of the material behavior and the interacting effects of material constituents in the finest way. The review is also focused on new advances in analytical and numerical methods for the stress, vibration, and buckling analyses of FGMs. Emphasis has been primarily on to restrict 2D analysis with sorts of compromise in the accuracy of results. First shear deformation theory (FSDT) and third-order shear deformation theory have been extensively used among the various 2D plate theories. FSDT can help us in terms of getting reasonably accurate results with less computational afford. This paper also outlines review on carbon nanotubes (CNT) reinforced FGMs, functionally graded nanocomposites, functionally graded single-walled CNT, FG nanobeam as well as functionally graded piezoelectric materials. Future applications would be based on these smart materials which are supposed to serve us in adverse conditions. Of course, with rise and advent of promising nanotechnology and its potential impact on aerospace industry as well as on other areas, it becomes important to us to compile this review article.     

A thermomechanical fracture modeling and simulation for functionally graded solids using a residual-strain formulation

This paper addresses mixed-mode crack growth in two-dimensional functionally graded solids under thermomechanical loads, and investigates the effect of mechanical and thermal loads as well as the T-stress on their crack growth behavior. A novel residual strain-based formulation in the interaction integral method is developed and used for the accurate evaluation of mixed-mode stress intensity factors and/or the T-stress. Simulation of mixed-mode crack propagation in functionally graded materials including solid oxide fuel cells under thermomechanical loads is performed by means of the finite element method and the generalized interaction integrals in conjunction with a remeshing algorithm. An iterative procedure is used for crack growth simulation including the calculation of mixed-mode stress intensity factors and/or the T-stress by means of the generalized interaction integral method, determination of crack growth direction and crack initiation condition based on selected fracture criteria, and local automatic remeshing along the crack path. The present approach employs a user-defined crack increment at the beginning of the simulation. Crack trajectories and fracture parameters obtained by the present simulation for thermomechanical loads are assessed for some numerical examples in comparison with those for mechanical loads.     

Evaluation of R-curve behavior of ceramic-metal functionally graded materials by stable crack growth

This paper deals with the fracture toughness and R-curve behavior of ceramic-metal functionally graded materials (FGMs). A possibility of stable crack growth in a three-point-bending specimen is examined based on the driving force and resistance for crack growth in FGMs, and the distribution of fracture toughness or R-curve behavior is evaluated on FGMs fabricated by powder metallurgy using partially stabilized zirconia (PSZ) and stainless steel (SUS 304). The materials have a functionally graded surface layer (FGM layer) with a thickness of 1 mm or 2 mm on a SUS 304 substrate. Three-point-bending tests are carried out on a rectangular specimen with a very short crack in the ceramics surface. On the three-point-bending test, a crack is initiated from a short pre-crack in unstable manner, and then it propagates in stable manner through the FGM layer with an increase in the applied load. From the relationship between applied load and crack length during the stable crack growth in the FGM layer, the fracture toughness is evaluated. The fracture toughness increases with an increase in a volume fraction of SUS 304 phase.     

Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack

In this paper the plane thermo-mechanical behavior of a crack in a viscoelastic functionally graded materials (FGMs) coating with arbitrary material properties bonded to a homogeneous substrate is studied. In order to avoid the complex forms that describe the viscoelastic properties of FGMs, a multi-layered model for the FGMs coating is developed. The compliance and thermal conductivity in the multi-layered model linearly vary in each layer. In this mixed boundary value problem, the system is reduced to singular integral equations and solved numerically with the Lobatto-Chebyshev collocation technique. Using the correspondence principle and Laplace transform, the problem of an interface crack between a homogeneous substrate and a viscoelastic FGMs is solved. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the multi-layered model. The numerical results confirm that the fracture toughness of materials can be greatly improved by the graded variation of material parameters. It is also confirmed that the specific variation of material parameters greatly influences the fracture behavior of viscoelastic FGMs coating.     

Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress

Thermal stresses, one of the main causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. Two efficient numerical procedures in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses are presented. The virtual crack extension method and the crack closure integral method are modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIFs are obtained using these methods.     

Numerical simulations of cracked plate using XIGA under different loads and boundary conditions

This article deals with the numerical simulation of cracked plate using extended isogeometric analysis (XIGA) under different loads and boundary conditions. The plate formulation is done using first-order shear deformation theory. The crack faces are modeled by the Heaviside function, whereas the singularity in stress field at the crack tip is modeled by crack tip enrichment functions. The stress intensity factors for the cracked plate are numerically computed using a domain-based interaction integral. The results obtained by XIGA for the center and edge crack plate are compared with extended finite element method and/or literature results for different types of loads and boundary conditions.     

A new conservation integral for circular arc crack under multiple loads

In this paper, a path independent integral that represents the rate of energy flux at the tip during crack extension in a homogeneous and isotropic material has been derived from the principle of virtual work for a two-dimensional stationary circular arc crack subjected to multiple loads. This integral is an extension of the two-dimensional version of F-integral and includes the presence of the effects of thermal strains, initial strains and body forces, hitherto, unavailable in open literature, to the best of the authors’ knowledge. It has been further demonstrated that Rice’s J-integral, the -integral derived by Kishimoto et al. and the F-integral proposed by Lorentzon et al. are special cases of the generalized integral . The integral has been implemented into a finite element post-processing program for examining the path independence behavior under elastic and elastic-plastic deformation subjected to mechanical loads and thermo-elastic analyses under pure thermal loads. Within the limits of numerical accuracy, the application demonstrates that the solutions for the energy release rate on different contours preserve nearly identical values over the computational range.     

Mixed-mode J-integral formulation and implementation using graded elements for fracture analysis of nonhomogeneous orthotropic materials

The path-independent J_k^*-integral, in conjunction with the finite element method (FEM), is presented for mode I and mixed-mode crack problems in orthotropic functionally graded materials (FGMs) considering plane elasticity. A general procedure is presented where the crack is arbitrarily oriented, i.e. it does not need to be aligned with the principal orthotropy directions. Smooth spatial variations of the independent engineering material properties are incorporated into the element stiffness matrix using a “generalized isoparametric formulation”, which is natural to the FEM. Both exponential and linear variations of the material properties are considered. Stress intensity factors and energy release rates for pure mode I and mixed-mode boundary value problems are numerically evaluated by means of the equivalent domain integral especially tailored for orthotropic FGMs. Numerical results are discussed and validated against available theoretical and numerical solutions.     

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.     

On the reduce of interfacial shear stresses in fiber reinforced polymer plate retrofitted concrete beams

One major problem when using bonded fiber reinforced polymer (FRP) plate is the presence of high interfacial shear stresses near the end of the composite (edge effect) which might govern the failure of the strengthening schedule. It is known that the decrease of plate thickness reduces the magnitude of stress concentration at plate ends. Another way is to use a plate end tapering. In this paper, the analytical solution of interfacial shear stresses obtained has been extended by a numerical procedure using the modal analysis of finite element method (FEM) in a retrofitted concrete (RC) beam with the FRP plate with tapered end, which can significantly reduce the stress concentration. This approach allows taking into consideration the variation of elastic properties of adhesive and plate as well as the complicated geometrical configurations and effects of thermal loads.     

Mixed-interpolated finite elements for functionally graded cylindrical shells

In this paper cylindrical shells made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGMs shells are determined using a variational formulation arising from the Naghdi theory. Moreover a strategy to achieve an improved transverse shear factor is investigated by energy equivalence. To approximate the problem a family of mixed-interpolated finite elements is used. It is based on a suitable reduction of the shear and membrane energy. Several numerical simulations are carried out in order to show the capability of the proposed elements to capture the properties of shells of various gradings, subjected to thermo-mechanical loads.     

Steady state thermoelastic contact problem in a functionally graded material

This paper is concerned with the stationary plane contact of a functionally graded heat conducting punch and a rigid insulated half-space. The frictional heat generation inside the contact region due to sliding of the punch over the half-space surface and the heat radiation outside the contact region are taken into account. Elastic coefficient μ, thermal expansion coefficient α_t and coefficient of thermal conductivity k are assumed to vary along the normal to the plane of contact. With the help of Fourier integral transform the problem is reduced to a system of two singular integral equations. The equations are solved numerically. The effects of nonhomogeneity parameters in FGMs and thermal effect are discussed and shown graphically.     

Impact response of an anti-plane crack in a FGPM bonded to a homogeneous piezoelectric strip

A finite crack under transient anti-plane shear loads in a functionally graded piezoelectric material (FGPM) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential functions along the thickness of the strip, and that the two layered strips is under combined anti-plane shear mechanical and in-plane electrical impact loads. The analysis is conducted on the electrically unified crack boundary condition. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Fredholm integral equations of the second kind in the Laplace transform domain. Then, a numerical Laplace inversion is performed and the dynamic intensities are obtained as functions of time and geometric parameters, which are displayed graphically.