Full Terms & Conditions of access and use can be found athttp://www.tandfonline.com/action/journalInformation?journalCode=upst20

Particulate Science and TechnologyAn International Journal

ISSN: 0272-6351 (Print) 1548-0046 (Online) Journal homepage: http://www.tandfonline.com/loi/upst20

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

Nand Jee Kanu, Umesh Kumar Vates, Gyanendra Kumar Singh & SachinChavan

To cite this article: Nand Jee Kanu, Umesh Kumar Vates, Gyanendra Kumar Singh & SachinChavan (2018): Fracture problems, vibration, buckling, and bending analyses of functionally gradedmaterials: A state-of-the-art review including smart FGMS, Particulate Science and Technology,DOI: 10.1080/02726351.2017.1410265

To link to this article: https://doi.org/10.1080/02726351.2017.1410265

Published online: 08 Mar 2018.

Submit your article to this journal

View related articles

View Crossmark data

PARTICULATE SCIENCE AND TECHNOLOGY https://doi.org/10.1080/02726351.2017.1410265

Fracture problems, vibration, buckling, and bending analyses of functionally graded materials: A state-of-the-art review including smart FGMS Nand Jee Kanua , Umesh Kumar Vatesb , Gyanendra Kumar Singhb , and Sachin Chavanc

aS. V. National Institute of Technology, Surat, Gujrat, India; bAmity School of Engineering and Technology, Amity University, Noida, Uttar Pradesh, India; cCollege of Engineering, Bharati Vidyapeeth Deemed University, Pune, Maharashtra, India

ABSTRACT 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.

KEYWORDS ABAQUS; buckling and bending analysis; carbon nanotubes (CNT); extended finite element method (XFEM); extended isogeometric analysis (XIGA); FG nanobeam; finite element solution; fracture; functionally graded material (FGM); functionally graded piezoelectric material (FGPM); multiscale models; vibration

1. Introduction

1.1. Functionally graded material

A functionally graded material (FGM) is a (microscopically heterogeneous) composite material containing at least two dif-ferent phases whose volume fraction changes gradually along at least a dimension of the solid. As shown in Figure 1, FGM has different phases such as ceramic and metal. It has been cleared from paper that functionally graded composite materials (FGCMs) are inhomogeneous (not homogeneous or uniform) materials and these are made up of two (or more) different materials, which are engineered to have continuously varying spatial composition profile (Udupa, Rao, and Gangadharan 2014). FGM is in fact a material in which both its structure and composition would be gradually changing over volume, thereby changing the properties of material to perform specific functions. And thus material properties are designed or controlled for desired mechanical, thermal, electrical, and chemical properties (Marin and Fuentes PPT).

FGMs are of many kinds, based upon their participating components (e.g., ceramic–metal, metal–metal, etc.).

Furthermore, modeling of FGMs can be discussed using Max-well model (Functionally Graded Materials, PPT by University of Victoria, British Columbia, Canada). Furthermore, FGM can be defined as an anisotropic material whose physical properties are going to vary throughout the volume (either strategically or randomly) to achieve desired qualities (features) (Modal Analysis of Rectangular Simply-Supported Functionally Graded Plates, PPT by Wes Saunders). Since these FGMs possess number of advantages that attract attention of researchers. Advantages including reduced in- plane and transverse through-the-thickness stresses, enhanced residual stress distribution, improved thermal properties, greater fracture toughness, and decreased stress intensity factors (SIFs) (Birman and Byrd 2007). Due to tedious distri-bution, FGM overcomes the limitations of stress discontinuity and delamination problem as we have in case of composite. Fiber reinforcement can also be done in FGMs, but volume fraction of that must not be constant as it should be coordinate dependent (Birman 1995, Birman 1997).

Figure 2 shows the unrefined FGM where volume fractions of constituent phases graded in one direction (Yin, Sun, and Paulino 2004). Nemat-Alla has analyzed one such material

none defined

CONTACT Nand Jee Kanu nandssm@gmail.com S. V. National Institute of Technology, Surat 395007, Gujrat, India. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/upst. Supplemental data for this article can be accessed on the publisher’s website. © 2017 Taylor & Francis

which is consisting of ceramic and two different metallic phases whose volume fraction varied in direction of thickness following a power law (Nemat-Alla 2003). After citing number of papers, it has been concluded that ceramic volume fraction will be the function of thickness coordinate z:

vc ¼2z þ h

2h

� �n� h2� z �

h2

Here, h is the thickness of the structure and n is the volume fraction.

1.2. Examples of FGM

Tungsten–copper FGM is having hybrid properties as shown in Figure 3. We can have advantages of both materials.

1.3. Classifications of FGMs

Functionally graded material may be compositionally or micro-structurally graded. In a FGM, compositions or functions are varying continuously or step wisely from one

side to the other. As shown in Figure 4, FGM is having continuous or, stepped wise graded structures. Example of continuous graded structure is bone. However, spark plug comes in category of stepped wise graded structures.

If we compare between traditional composite structures and FGMs, we find typical variation in properties in case of later as shown in Figure 5.

As showing above, we can observe straight away differences in properties in case of FGM which would be having added advantages over tradition composite structures.

Here, it is very important to note that in case of FGM we are replacing those sharp interfaces with gradient interfaces which produce smooth transition from one material to the next. And thus one can avoid delamination failure in case of FGM.

1.4. Advantages and challenges of FGMs

Advantages of FGMs are as follows: .� Provide multi-functionality. .� Provide ability to control deformation, dynamic response,

wear, corrosion, etc., and ability to design for different complex environments.

.� Provide ability to remove stress concentrations.

.� Provide opportunities to take the benefits (pros) of different material systems (e.g., ceramics and metals such as resistance to oxidation (rust), toughness, machinability, and bonding capability). Challenges of FGMs include the following:

.� Mass production,

.� quality control, and

.� cost.

1.5. Applications of FGMs

Current applications of FGMs include the following: .� Structural walls that combine two or more functions

including thermal and sound insulation as shown in Figure 6.

.� Enhanced sports equipment such as golf clubs, tennis rackets, and skis with added graded combinations of flexibility, elasticity, or rigidity as shown in Figure 6.

.� Enhanced body coatings for cars including graded coatings with particles such as dioxide/mica as shown in Figure 6.

.� Aerospace applications ceramic–metal FGMs are particularly suited for thermal barriers in space vehicles.

.� Fuel cell technology creating a porosity gradient in the electrodes, the efficiency of the reaction can be maximized.

Figure 1. Schematic representation of functionally graded material.

Figure 2. Constituent phases graded in vertical direction (Yin, Sun, and Paulino 2004).

Figure 3. Tungsten–copper FGM (Jedamzik, Neubrand, and Rodel 2000). Note: FGM, functionally graded material.

2 N. J. KANU ET AL.

2. Fractures in functionally graded materials

Functionally graded materials are advanced composite materials having two or more constituent phases and a continuously variable composition. These FGMs acquire a number of advantages which make them unique in potential applications, including a potential reduction of in-plane and transverse through-the-thickness stresses, enhanced thermal properties, an improved residual stress distribution, higher fracture toughness, and of course reduced SIFs (Pindera et al. 1994, Pindera et al. 1995; Birman and Byrd 2007).

The issues of fracture and fatigue of FGM are of great significance and they have been intentionally discussed in depth to such extent that a distinct review is required to perfectly define the state of art and to address the most favorable directions and essential requirements in this field (Birman and Byrd 2007).

2.1. Fracture problems in functionally graded material

Many recent papers are considered below to highlight the range and implications of fracture problems. Many other latest publications are listed in Table 1 where each paper is reviewed with comment on its major headline. Kim and Paulino can be referred as an illustration of a finite element answer of fracture problems in FGM (Kim and Paulino 2005a, 2005b). A finite element approach combined with a remeshing algorithm reflecting changes in the path of the crack has been used to study the propagation of mixed-mode cracks in FGM. Also, the SIFs have been derived from the interaction integral method previously developed by the same authors. And there-after these factors have been implemented in a fracture cri-terion to allocate the direction of crack propagation and the mesh used here in the finite element method (FEM) has been accordingly modified reflecting the new trajectory of the crack.

Figure 4. (a) Continuous or, (b) and (c) stepped wise graded structures (Nemat-Alla 2003).

Figure 5. Comparison between traditional composite and FGM (Cherradi, Kawasaki, and Gasik 1994). Note: FGM, functionally graded material.

Figure 6. FGM applications. Note: FGM, functionally graded material.

PARTICULATE SCIENCE AND TECHNOLOGY 3

The generation of cracks in a quasi-brittle FGM has been studied by Comi and Mariani considering Finite Element Analysis (FEA) (Comi and Mariani 2005). Here, numerical results have been shown in the paper, which are expressing the stabilization of the crack propagating into an increasingly tougher material as well as a delay in the onset of initial crack-ing in tougher zones. The influence of residual thermal stresses on fracture has gained a powerful attention. This actually has impact on the SIFs for cracks developing from the surface of stepwise functionally graded alumina/zirconia materials which have been considered by Vena, Gastaldi, and Contro (2005). Other aspect of fatigue, for example, the subcritical crack advancement as a consequence of mechanical or thermal cyc-ling loading which was applied to a FGM coating bonded to a homogeneous substrate, has been taken into consideration using a three-dimensional FEM (Inan et al. 2005).

2.1.1. Transient and dynamic fractures These have been comprehensively reviewed in recent years. Kokini and Rangaraj have done experimental and numerical investigations of functionally graded yttria stabilized zirconia bond coat alloy thermal barrier coatings (TBCs) subject to transient thermal loading and accounting of visco-plastic effects after using laser thermal shock tests (Kokini and Rangaraj 2005). Due to gradation of the coating, the character of surface cracking, that is, single versus multiple cracks, has been influenced by the tendency to multiple cracking increasing with increased gradation. Paulino and Zhang have focused on dynamic fracture in FGM using graded intrinsic cohesive elements in the fracture zone and graded finite elements in the intact material (Paulino and Zhang 2005). Other latest publications are listed in Table 2 after reviewing each paper carefully.

2.1.2. Experimental investigation of fracture Due to the complexity of the process in FGM and difficulties in its modeling and characterization, such investigation is particularly needed. Using a four-point bending test under both monotonic and cyclic loading, the onset and propagation of cracks in alumina/epoxy FGM has been monitored (Tilbrook et al. 2005). Here, both the crack growth rates and their trajectories have been registered. In this paper, finite element forecasting of the anticipated crack path being generated by ANSYS for linear elastic FGM could be quite authentic. It has been found that the deflection angles of the cracks generally increased in FGM with a larger grading gradient and obviously when the crack onset occurred close to the compliant position. In this context, latest publications are listed in Table 3 after reviewing each paper comprehensively.

3. Fracture analyses of functionally graded materials

3.1. Finite element method fracture analysis

3.1.1. Responses of probabilistic fractures in case of thermo-mechanical loading and vibrational responses Kim and Paulino have presented finite element solution of fracture problems in FGM (Kim and Paulino 2005a, 2005b). In their research, they have studied about propagation of mixed-mode cracks in FGM by finite element approach and found SIF by interaction integral method. It is a well-known fact that SIF is used in fracture criterion to establish the direc-tion of crack propagation.

Kokini and Rangaraj have applied transient thermal loading on TBCs and thus suggested that the surface cracking has been

Table 1. Representative papers on fracture and fatigue of FGM (Birman and Byrd 2007). References Especial highlight

T-stress issue in FGM Paulino and Kim (2004) Evaluation of T stresses in mixed-mode fracture mode by the application of interaction integral Becker, Cannon, and Ritchie (2001) Here, the direction vector of the in-plane mixed-mode crack in functionally graded material and with that the effect of T

stresses have been considered Application of interaction integral method during the analysis of fracture in functionally graded material Kim and Paulino (2004a, 2004b) To check out T stresses in a mixed-mode fracture problem for an orthotropic FGM with straight or curved cracks, the

interaction integral is used Kim and Paulino (2003c, 2003d) Application of interaction integral to analyze an extensive variety of crack problems in FGM Kim and Paulino (2003c, 2003d) Again, application of interaction integral to find mixed-mode stress intensity factors for arbitrary-oriented cracks in FGM Kim and Paulino (2005a, 2005b) Application of formulations of constant-constitutive tensor which are applicable to the fracture analysis of FGM by the

interaction integral method and finally compared Thermal effects on fracture in FGM Afsar and Sekine (2002) Fracture toughness analysis of FGM coating around a circular hole in an infinite elastic solid which has been subject to

thermal stresses Nemat-Alla and Noda (2000) Review of a crack in FGM plate having bidirectional variation of the coefficient of thermal expansion and subjected to

steady thermal loading El-Borgi, Erdogan, and Hidri (2004) Consideration of thermo-mechanical stresses at the tip of a partially insulated crack Xiong et al. (2005) Here, experimental examination of fracture in FGM TBCs which has been subjected to high heat flux and of course thermal

cycles Mixed-mode cracks in functionally graded material Kim and Paulino (2003c, 2003d) Extension of path-independent J-integral method to orthotropic FGM eventuating in the stress intensity factors and

energy release rates in Mode I and mixed-mode fracture problems Tvergaard (2002) Self-similar crack growth within a functionally graded material interface layer between unlike elastoplastic solids including

the impact of mode mixity on the steady-state fracture Guo, Wu, and Ma (2004) Strain energy release rates and mixed-mode stress intensity factors are achieved for a crack in a FGM coating Kim and Paulino (2003c, 2003d) Crack initiation angle, mixed-mode stress intensity factors, and T stress in FGM are obtained by the interaction integral

method united with FEA Kim and Paulino (2004a, 2004b) Finite element method solution taken with an automatic remeshing is used to keep eye on mixed-mode crack

propagation in FGM

FGM, functionally graded material.

4 N. J. KANU ET AL.

affected by gradation of coating. And so while increasing gradation, tendency to multiple cracking increases (Kokini and Rangaraj 2005).

Tilbrook et al. have recorded crack growth rate and trajectories and they have accepted that crack path generated after using ANSYS is in agreement with the reality (Tilbrook et al. 2005).

3.1.2. Application of FE to analyze 3D curved non-planar cracks in FGMs Shaghaghi and colleagues have proposed FEM to analyze 3D curved non-planar cracks in FGMs subjected to thermal load-ing and further they have discussed the effect of gradation (graded thermal and mechanical properties) of material properties on SIFs of curved non-planar cracks (Shaghaghi et al. 2015).

3.1.3. Simulations of thermal shock cracking using virtual crack closure technique in FGM plate Burlayenko et al. have studied 2D thermal shock problem in FGM plate having crack and have solved numerically after using ABAQUS (the finite element software) (Burlayenko

et al. 2015). They have used user subroutines to model variation in thermo-mechanical properties and considered steady state as well as transient responses and then in FGM plate which is under thermal shock, crack growth is analyzed using virtual crack closure technique.

3.1.4. Analysis of development of tissue phenotypes: Bone fracture healing using FGM bone plates Mehboob and Chang (2014) have examined the effect of bending stiffness on bone fracture healing while maintaining an identical bone plate modulus after having various configurations of bone plates obtained from FGCMs. They have used mechano-regulation theory with deviatoric strain to estimate healing performance and development of tissue phenotypes. After all, the analysis is done in ABAQUS and in user’s subroutine program. Finally, most effective configurations of FGM layers have been decided based on healing performance.

3.2. Extended isogeometric analysis of functionally graded materials

3.2.1. Extended isogeometric analysis application in assessing the fatigue life of FGMs Bhardwaj et al. have successfully implemented extended isogeometric analysis (XIGA) to assess fatigue life of interfacial cracked bi-layered FGMs due to flaws and further they have compared both fatigue lives (obtained after using scatter in input parameters and deterministic values (Bhardwaj, Singh, and Mishra 2015). Finally, they have concluded that Paris law exponent for alloy has more impact on fatigue life as compared to that of Paris law exponent for ceramic and also stated that if there would be holes, inclusions, and minor cracks in FGMs, then these will have significant influence on fatigue life of interfacial cracked bi-layered FGM plate.

Table 2. Representative papers on fracture and fatigue of FGM (Udupa, Rao, and Gangadharan 2014). References Especial highlight

Dynamic and transient effects Shul and Lee (2002) Under an antiplane impact, dynamic stress intensity factors have been found for a FGM coating crack Li, Weng, and Duan (2001) Dynamic stress intensity factor and transient stresses around a penny-shaped crack in a transversely isotropic FGM strip

produced by torsional impact have been discussed Feng and Zou (2003) Dynamic Mode I and mixed-mode fracture in FGM have been taken into account using the cohesive zone model which has

been implemented in a graded finite element method Zhang and Paulino (2005) Evaluation of transient thermal stresses and stress intensity factors for an edge crack in a FGM strip with changing thermal

properties and however, constant elastic properties Jin and Paulino (2001) Solution of Mode I fracture problem in a FGM viscoelastic strip is given by the correspondence principle Jin and Paulino (2002) Using weight function method, crack growth in FGM is studied, and failure of a thermal barrier under cyclic thermal

loading is considered Zhou, Wang, and Sun (2004) Consideration of dynamic fracture problem for a crack in FGM which is subjected to harmonic stress waves and thereafter

dynamic stress intensity factors are found Chen, Liu, and Zou (2002) Determination of dynamic stress intensity factors for Mode I fracture for a crack which is almost perpendicular to the

surfaces of an orthotropic FGM strip and subject to internal impact stresses Kirugulige, Kitey, and Tippur (2005) Influence of a functionally graded core on fracture resistance of sandwich structures which have been subjected to impact

loading is finally studied experimentally Jain and Shukla (2004) Deformations and stresses analysis for a crack propagating with a constant velocity along the direction of material

property gradation Guo et al. (2005) For a crack in a FGM coating that propagates in the direction perpendicular to a homogeneous substrate dynamic stress

intensity factors are found Ueda (2006) For a functionally graded piezoelectric strip with a center crack, dynamic stress intensity factor and electric displacements

are obtained Shin and Kim (2016) Using integral transform techniques, transient response analysis of a Mode III interface crack between a piezoelectric layer

and a functionally graded orthotropic material layer has been done

FGM, functionally graded material.

Table 3. Representative papers on fracture and fatigue of FGM (Udupa, Rao, and Gangadharan 2014).

References Especial highlight

Experimental FGM fracture studies Kawasaki and

Watanabe (2002) For functionally graded TBCs subject to thermal shock or

thermal cycling reflecting on the effect of grading on the spallation life, experimental study of fracture has been done

Forth et al. (2003) For effects of heat treatment promoting the formation of β-phase of titanium during aging on fatigue of a functionally graded Ti-6Al-4 V, further experimental study has been done

FGM, functionally graded material.

PARTICULATE SCIENCE AND TECHNOLOGY 5

3.2.2. Crack analysis in orthotropic FGM using extended isogeometric analysis Shojaee et al. have used XIGA to analyze crack in orthotropic FGM after numerical modeling of stationary cracks into it (Shojaee and Daneshmand 2015). They have used three formulations of interaction integrals for FGM fields. While modeling such FGM, they have applied smooth functions and thereafter, good accuracy obtained as compared to other methods in the literature.

3.2.3. Thermo-mechanical fracture review of inhomoge-neous cracked solids using extended isogeometric analysis method The extended isogeometric analysis method is used here to review the fracture of homogeneous and inhomogeneous mate-rials under mechanical and thermo-mechanical loadings. After discretizing the domain of the problem by the knot spans of the isogeometric analysis (IGA), the same basis functions have been used for constructing the geometry to discretize the solution. The exact form of the interaction integral method in inhomoge-neous materials and thermal conditions has been used to review the mixed-mode SIFs (Bayesteh, Afshar, and Mohammdi 2015).

3.3. Extended finite element method fracture analysis

3.3.1. Application of extended finite element method and interaction integral method to assess SIF Goli and Kazemi have discussed about behavior of FGCMs (advanced functionally graded composite materials) in cracked conditions and for this they have used extended finite element as tool to model discontinuity in the computational fracture mechanics area with interaction integral method to evaluate the SIFs particularly in case of transversely isotropic FGMs (Figure 7) (Goli and Kazemi 2014). In this way, computational efforts can be significantly reduced after using extended finite element method (XFEM).

3.3.2. Application of extended finite element method under thermal cyclic load indifferent materials Bhattacharya and Sharma have focused on thermal fatigue failure in FGMs after subjecting it to cyclic thermal loads. And in this regard they have carried out numerical simulation of fatigue crack growth using XFEM under thermal cyclic load in three different materials (a unidirectional FGM made of aluminum alloy and alumina (ceramic), an equivalent composite consisting of the same volume fractions of alloy and the ceramic as the FGM, and the aluminum alloy, Figure 8) (Bhattacharya and Sharma 2014).

They have concluded with quiet interesting facts that if there would be discontinuities (combination of holes/voids, inclusions, and minor cracks), then fatigue life of the materials will be reduced significantly. With that they have also con-cluded that if crack will be present on the ceramic-rich side with respect to the alloy-rich side, then FGM would fail earlier (as ceramic is weaker in fracture as compared to alloy).

3.3.3. Study of transient thermal dynamic analysis of stationary cracks in FGPMs Liu et al. have studied transient thermal dynamic analysis of stationary cracks in functionally graded piezoelectric materials (FGPMs) using XFEM (Liu et al. 2013). They have shown that behaviors of transient thermal fracture in FGPM are not same in cooling and heating. Their investigation shows that influences of polarizations, material gradation, crack length, etc. and shock loading on dynamic intensity factors (DIFs) are significantly indeed.

3.3.4. Fracture analysis of orthotropic functionally graded materials using XFEM Bayesteh and Mohammadi have done fracture analysis of orthotropic FGMs using XFEM (Bayesteh and Mohammadi 2013). They have used interaction integral method to determine SIFs. The proposed method has been further evaluated using numerical examples and quality results have been determined by far fewer degrees of freedoms.

3.3.5. Analysis of transient dynamic crack in FGPMs using XFEM Liu et al. have studied stationary cracked FGPMs, subjected to impact loading, about their transient dynamic fracture behaviors (Liu et al. 2013). In this context, they have used XFEM after assuming that material properties are varying exponentially in one direction. Dynamic XFEM model with implicit integration technique has been used along with contour interaction integral technique to assess applicable dynamic fracture parameters of FGPMs. Furthermore, they have examined DIFs determined after using XFEM. With that they have verified the model after comparing calculated dynamic responses with solutions obtained from the meshless local Petrov–Galerkin method and the FEM which agrees well.

3.4. Multiscale models for fracture analysis of functionally graded materials

3.4.1. Fracture analysis of functionally graded materials using stochastic multiscale models Three multiscale models, including sequential, invasive, and concurrent models, for fracture analysis of a crack in a two- phase, functionally graded composite, have been presented in this article. In models, stochastic description of the particle volume fractions, particle locations, and constituent material properties; a two-scale algorithm including microscale and macroscale analyses for determining crack-driving forces; and two stochastic methods for fracture reliability analysis have been involved. The multiscale model is accurate, and gives probabilistic solutions, which is very close to that generated from the microscale model, and thus can minimize

Figure 7. Cracked transversely isotropic FGMs (Goli and Kazemi 2014). Note: FGMs, functionally graded materials.

6 N. J. KANU ET AL.

the computational effort of the latter model. Apart from that, multiscale model also helps in forecasting crack trajectory very accurately as that in case of microscale model (Chakraborty and Rahman 2008).

3.4.2. Stochastic multiscale fracture analysis of 3D functionally graded composites The moment-modified polynomial dimensional decompo-sition (PDD) method has been used for stochastic multiscale fracture analysis of three-dimensional, particle-matrix, FGMs subject to arbitrary boundary conditions. This method includes Fourier-polynomial expansions of component functions by orthonormal polynomial bases, an additive control variate in conjunction with Monte Carlo simulation for the expansion coefficients, and a moment-modified random output for the effects of particle locations and geometry. The numerical conclusions from a three-dimensional, edge-cracked, FGM specimen under a mixed-mode deformation state that the stat-istical moments or probability distributions of crack-driving forces and the conditional probability of fracture initiation can be significantly produced by the univariate PDD method. The results which have been found are not sensitive to the subdomain size from concurrent multiscale analysis, which, if selected wisely, results to computationally efficient estimates of the probabilistic solutions (Rahman and Chakraborty 2011).

3.5. Miscellaneous methods on the fracture analysis of functionally graded materials

3.5.1. Influence of FGM coating thickness on apparent fracture toughness Afsar et al. have discussed the influence of FGM coating thick-ness on apparent fracture toughness (AFT) of a thick-walled cylinder where they have considered incompatible Eigen strain which has been developed in the cylinder because of non- uniform coefficient of thermal expansion (Afsar and Song 2010). Finally, they have developed approach to calculate AFT based on a generalized method of evaluating SIF.

3.5.2. Application of peridynamic model to analyze dynamic fracture in FGMs Cheng et al. have proposed peridynamic model in case of FGMs with monotonically varying volume fraction of

reinforcement and then analyze their dynamic fracture after elastic wave propagation in FGMs and comparing with analytical results for the classical model (Cheng et al. 2015). In the research work, they have done vigorous studies of impact of material gradients, elastic waves, contact time, and magnitude of impact loading on the fracture behavior in terms of crack path geometry and crack propagation speed. At the end, they have concluded with the fact that peridynamic model would help in a much better way while understanding crack propagation in FGMs. This model will also help in terms of understanding factors that could restrict crack path and its velocity in FGMs.

3.5.3. Study of fracture criteria after loading system with tensile load and heat flux Fracture criteria are studied to predict the extension of crack growth direction and internal crack in a bimaterial having a homogeneous and a FGM with internal defects (Petrova and Schmauder 2014). Here, the material is subjected to tensile load and heat flux. Petrova et al. have obtained asymptotic analytical formulas for the SIFs at the interface crack tips and further they have used SIFs to determine possible direction of crack propagation and then have discussed impact of geometry of location and orientation of cracks as well as parameters of FGM, on chief fracture characteristics.

3.5.4. Analysis of propagating crack tip in OFGMs Lee has analyzed propagating crack tip in orthotropic functionally graded materials (OFGMs) (Lee 2016). He has developed crack-tip stress and displacement fields for a propagating crack which is propagating along a gradient with constant velocity and with an exponential variation of the shear modulus and density. He has determined crack-tip fields using wave potentials and the Airy stress function by an asymptotic analysis.

3.5.5. Application of analytical model for collinear cracks in FGMs Pan et al. have developed analytical model for collinear cracks in FGMs having general mechanical properties instead of solving analytically (Pan, Song, and Wang 2015). At the end, the crack problems are resolved by reducing into singular integral equations which can be further solved by numerical method.

Figure 8. Plate with an edge crack on the alloy-rich side under thermal loading, and plate with an edge crack on the ceramic-rich side under thermal loading (moving from left) (Bhattacharya and Sharma 2014).

PARTICULATE SCIENCE AND TECHNOLOGY 7

3.5.6. Application of viscoelastic fracture mechanics model to examine crack in FGMs Wang et al. have proposed a viscoelastic fracture mechanics model to examine crack problem in viscoelastic FGMs where they have considered interface crack problem (Wang, Zhang, and Guo 2014). Study shows that SIF is significantly influenced by various mechanical properties.

3.5.7. Dynamic stress intensity factors computation in cracked FGMs using scaled boundary polygons Chiong et al. have developed scaled boundary polygons formulation for computation of dynamic SIFs in cracked FGMs and have computed accurate dynamic SIFs (Chiong et al. 2014). Five different cases of FGMs having cracks and under dynamic loadings are being modeled using the formulation which shows its versatility.

3.5.8. Analysis of interfacial fracture of bonded dissimilar strips Choi has focused on solution to the problem of interfacial behavior of unlike and homogeneous semi-infinite strips bonded through a functionally graded interlayer and weakened due to an embedded or edge interfacial crack (Choi 2015). He has presented SIFs against the geometric and material parameters. It has been seen that interlayer thickness effect is depending on the material combination; however, the effect of thickness of homogeneous strips is managed by external conditions.

3.5.9. Thermal SIFs for FG cylinders having internal circumferential cracks Eshraghi and Soltani have derived mathematical expressions for thermal SIF of cracked functionally graded cylinders (Eshraghi and Soltani 2015). They have used weight function approach for analysis of thermal fracture. To calculate SIFs, they have introduced fitting function, and studied effects of internal cooling with FGM index on thermal SIFs. Finally, they have predicted SIFs by the developed mathematical expression (weight function) which has agreed well with direct finite element SIF results.

3.5.10. Elastodynamic analysis of mode III multiple cracks in FGM Haghiri et al. have used displacement discontinuous method to model dynamic crack problems (Haghiri, Fotuhi, and Shafiei 2015). It has been seen that the overshoot of dynamic stress intensity factors (DSIFs) was greatly influenced by FGM constants. They have found that Durbin method for oscillatory loadings had greater accuracy than Stehfest method and have concluded that orthotropic materials having greater ratios would result into better behaviors in terms of DSIFs.

3.5.11. Modeling of fracture of isotropic FGMs by numerical manifold method Zhang and Ma have studied two-dimensional stationary cracks in FGMs using numerical manifold method (Zhang and Ma 2014). They have computed SIFs by interaction integral using non-equilibrium auxiliary fields. Accuracy of the proposed method has been further verified by examples of single- and

multi-branched crack. The concerned problems are handled using uniform mathematical cover system which is inde-pendent of physical boundaries. Finally, they have shown that obtained SIFs are matching well with existing reference solutions.

3.5.12. Evaluation of critical fracture load for FGSs under mixed-mode loading using new expression Salavati et al. have examined functionally graded steels (FGSs) subjected to mixed-mode loading (I + II) and weakened by notches using averaged value of strain energy density over a well-defined volume (Salavati et al. 2015). For fracture assessment, they have used local energy criterion. And this developed criterion has been used with artificial neural network. Finally, it has been seen that both experimental and theoretical results are matching with each other.

3.5.13. Application of local approach model for FGMs cleavage fracture and crack extension direction in FGMs Bezensek and Banerjee (2010) have developed local approach model for structural assessment of FGMs where yield strength and the fracture toughness vary spatially (Yamanouchi et al. 1990). This model is supposed to find the direction of crack extension and further failure probabilities of cleavage failure in case of stationary pre-crack in FGM. To validate this model, influence of independent variation in yield strength and toughness has been discussed after coupling these two through temperature. This model is reported to be agreed with experimental observations of cleavage fracture tests on mild steel under controlled temperature gradient normal to the crack.

3.5.14. Application of stochastic model to dynamically analyze crack in FGMs layer Zhang et al. have used stochastic model to dynamically analyze crack in FGMs (material properties vary randomly in the thickness direction and that crack is parallel to the materials faces) layer for plane problem under dynamic loadings (Zhang Zhao, and Su 2012). Further after dividing FGM layer into various sub-layers, the problem is then treated as analysis of composites having crack. They have analytically derived the SIF. Finally, they have concluded with the fact that mathemat-ical expectation and standard deviation of normalized stress intensity would increase with increase in crack length, random field parameter β, and crack location ratio h2/h.

3.5.15. Random dynamic response and reliability of a crack in a FGM layer Zhang et al. have proposed an analytical approach for the random dynamic analysis of a FGM (material properties varied randomly in thickness direction) layer having crack between two unlike elastic half-planes (Zhang Zhao, and Su 2012). Further after dividing FGM layer into various sub- layers, the problem is then treated as analysis of composites having crack. They have analytically derived both SIF history with its statistics and dynamic reliability. At the end, they have shown influences of related parameters through numerical calculations.

8 N. J. KANU ET AL.

3.6. XFEM fracture analysis

Further knowledge of crack analysis is required followed by analytical solutions for composites with numerical techniques. Finally, it has been concluded from literature survey that XFEM is the most efficient tool for fracture analysis in case of FGMs (Pipes and Pagano 1970; Pipes and Pagano 1974; Sih and Chen 1980; Delale and Erdogan 1983; Eischen 1983; Batakis and Vogan 1985; Houck 1987; De Masi, Sheffler, and Ortiz 1989; Niino and Maeda 1990; Yamanouchi et al. 1990; Meier et al. 1991; Holt et al. 1993; Takahashi et al. 1993; Brindley 1995; Ilschner and Cherradi 1995; Kaysser and Ilschner 1995; Lee and Erdogan 1995; Sampath et al. 1995; Chen and Erdogan 1996; Kasmalkar 1996; Gu and Asaro 1997; Gu, Dao, and Asaro 1999; Nadeau and Ferrari 1999; Kim and Paulino 2002; Kim and Paulino 2003a, 2003b).

3.6.1. Fracture analysis of functionally graded materials Due to variation in material properties, fracture mechanics is affected in asymptotic solution for near crack-tip displacement and stress fields and SIFs.

Some FGMs can be tailored as per requirement, such as resistance to fracture and failure patterns and so it is often needed to compute fracture parameters and simulation of crack growth in FGMs.

Due to manufacturing of FGMs using various processing techniques, these (FGMs) have anisotropic nature and so non-orthotropic elastic continuum formulation is required to study FGMs.

3.6.1.1. Mode I near tip fields in FGM composites. (Ozturk and Erdogan 1997). Figure 9 illustrates Mode I crack in FGM (Delale and Erdogan 1983; Erdogan and Wu 1993). Stress–displacement relations can be given as

rxx x; yð Þ ¼E x; yð Þ

1 � n2@ux x; yð Þ

@xþ n

@uy x; yð Þ

@y

� �

ryy x; yð Þ ¼E x; yð Þ

1 � n2@uy x; yð Þ

@yþ n

@ux x; yð Þ

@x

� �

rxx x; yð Þ ¼E x; yð Þ

2ðj0 þ nÞ

@ux x; yð Þ

@yþ@uy x; yð Þ

@x

� �

where j0, dimensionless constant for the power hardening law

E x; yð Þ ¼ Eðx1; x2Þ

The final solutions of the displacement field of the crack face u2 and the SIFs KI at the two crack tips (Ozturk and Erdogan 1997)

u2 x1; 0þð Þ

x0¼ �

ffiffiffiffiffiffiffiffiffiffiffiffi1 � r2p X1

n¼1

An

nUn� 1 rð Þ

" #

KIðaÞK0¼ � eaIa

X1

n¼1An

KIð� aÞK0

¼ � e� aIaX1

n¼1ð� 1ÞnAn:

3.6.1.2. Stress and displacement field. Figure 10 shows the global and local crack-tip coordinate systems in Cartesian and polar forms (Sih, Paris, and Irwin 1965). Displacement fields u1 and u2, in the x- and y-directions, respectively, are

u1 ¼ KI

ffiffiffiffiffi2rp

r

Re1

stip1 � stip

2stip

1 ptip2 g2ðhÞ � stip

2 ptip1 g1ðhÞ

h i( )

þ KII

ffiffiffiffiffi2rp

r

Re1

stip1 � stip

2ptip

2 g2ðhÞ � ptip1 g1ðhÞ

h i( )

u2 ¼ KI

ffiffiffiffiffi2rp

r

Re1

stip1 � stip

2stip

1 qtip2 g2ðhÞ � stip

2 qtip1 g1ðhÞ

h i( )

þ KII

ffiffiffiffiffi2rp

r

Re1

stip1 � stip

2qtip

2 g2ðhÞ � qtip1 g1ðhÞ

h i( )

where KI, KII, Mode I and II SIFs.

3.6.2. Stress intensity factor 3.6.2.1. J integral. It is used in calculating SIFs in various mixed-mode fracture problems. For inhomogeneous materials, it can be written as (Kim and Paulino 2003a)

Figure 9. Mode I crack in an FGM (Erdogan and Wu 1993). Note: FGM, functionally graded material.

Figure 10. Global and local crack-tip coordinate systems in Cartesian and polar forms for an arbitrary orthotropic body (Sih, Paris, and Irwin 1965).

PARTICULATE SCIENCE AND TECHNOLOGY 9

J ¼Z

A

rijui;1 � wsd1j� �

q; ddAþZ

A

rijui;1 � wsd1j� �

q; j dA

3.6.2.2. Interaction integral. It is used to compute mode I and mode II SIFs. It is based on superposition of auxiliary and actual fields. The energy release rate in elastic media is calculated as (Yau 1979; Yau, Wang, and Corten 1980)

G ¼ J ¼ t11K2I þ t12KIKII þ t22K2

II

For calculating actual mode I and II SIFs from the local interaction integral Ml, following equations are used

Ml KauxI ¼ 1;Kaux

II ¼ 0� �

¼ 2t11KI þ t12KII

Ml KauxI ¼ 0;Kaux

II ¼ 1� �

¼ t12KI þ 2t22KII

where tij, material function.

3.6.3. Inhomogeneous XFEM The general methodology of the orthotropic XFEM can be referred for inhomogeneous problems (Ye and Ayari 1994; Ayari and Ye 1995).

3.6.3.1. XFEM approximation. The displacement for a point x located within the domain.

uh xð Þ ¼ u xð Þ þ uH xð Þ þ utip xð Þ þ utra xð Þ

Orthotropic crack-tip enrichment functions (Dolbow and Gosz 2002).

Fl r; hð Þf g4l¼1

¼ffiffirp

cosh

2;ffiffirp

sinh

2;ffiffirp

sin h cosh

2;ffiffirp

sin h sinh

2

� �

Transition domain (Szabo and Babuska 1991; Tarancon et al. 2009). For nodes t1 and t2 in Figure 11, the hierarchical shape functions in terms of the isoperimetric coordinates ξ,η are given as

NP1 n; gð Þ ¼ �12

ffiffiffi32

r

1 � n2� � 1 � g

2

NP2 n; gð Þ ¼ �12

ffiffiffi32

r

1 � g2� � 1 � n

2

XFEM discretization. After discretization using the XFEM approximation, following equation results

Kuh ¼ f

The global stiffness matrix K is assembled from the stiffness of each element Ke

ij

Krsij ¼

Z

Xe

Bri

� �TDBs

j dX r; s ¼ u; a; b; c; dð Þ

where Krsij , components of stiffness matrix.

Figure 11. Added hierarchical nodes in the blending elements (Tarancon et al. 2009).

Figure 12. XFEM static and dynamic fracture analyses (Mohammadi 2012). Note: XFEM, extended finite element method.

10 N. J. KANU ET AL.

And, nodal force vector can be given by

f ei ¼ f u

i f ai f ba

i f ci f d

i

n oT

3.6.4. Static and dynamic analyses using XFEM To carry out static and even dynamic analyses of fractures in FGMs using XFEM, below steps need to be followed as shown in Figure 12 (Mohammadi 2012).

4. Vibration, buckling, and bending analyses of functionally graded materials

The review is carried out with an emphasis to present vibration and buckling and bending characteristics of FGM plates estimated using different theories proposed by researchers without involving mathematical implication of various methodologies. The main aim of this research paper is to serve the interests of researchers and engineers who are already involved in the analysis and design of FGM structures (Swaminathan et al. 2015).

4.1. Vibration analysis

There are many methods which have been employed so far for the vibration analysis of FGM plates and also FGM sandwich plates. Those are reviewed in this section under two classified headings as analytical methods and numerical methods (Swaminathan et al. 2015).

4.1.1. Analytical methods Vibration analysis of FGM includes the solution of Eigen value problems. Solution of Eigen value problems which is often based on 3D elasticity theories would be actually difficult to achieve particularly in the case when the material properties are graded after using the power-law parameter. Finally, 2D plate theories, which are based on displacement field and stress function, are developed and efficiently being used for the analysis of FGM plates (Swaminathan et al. 2015).

4.1.1.1. Three-dimensional (3D) elasticity theory. Many other latest publications are listed in Table 4 where each paper is reviewed with comment on its major headline (Swami-nathan et al. 2015).

4.1.1.2. Two-dimensional (2D) plate theories. Many other latest publications are listed in Table 5 where each paper is reviewed with comment on its major headline (Swaminathan et al. 2015).

4.1.2. Numerical methods To avoid complexities in analytical solution for those FGMs with geometrical and material complexities, numerical methods have been preferred (Swaminathan et al. 2015).

4.1.2.1. Finite element method. Many other latest publica-tions are listed in Table 6 where each paper is reviewed with comment on its major headline (Swaminathan et al. 2015).

4.1.2.2. Meshless methods. Many other latest publications are listed in Table 7 where each paper is reviewed with comment on its major headline (Swaminathan et al. 2015).

4.1.3. Recent papers on vibration analysis of FGMs Many other latest publications are listed in Table 8 where each paper is reviewed with comment on its major headline.

Simple refined shear deformation theory was used for vibration and buckling of FGM sandwich plate which has been kept on elastic foundations. Based on nonlinear variations in the in-plane displacements through the thickness, displace-ment field was chosen. It was seen that the used theory has been more accurate as compared to existing higher-order shear deformation theories having more number of unknowns (Meziane, Abdelaziz, and Tounsi 2014).

Using a nonlocal quasi-3D theory which was having both shear deformation and thickness stretching effects, size- dependent bending and free flexural vibration behaviors of FG nanobeams were investigated herewith. Nonlocal elastic behavior was described using differential constitutive model of Eringen which has further enhanced the effectiveness of analysis and design of nanostructures. Examination of effect of material gradient index, the nonlocal parameter, and the beam aspect ratio on the global response of the FG nanobeam has opened the path of mechanical design considerations in devices having carbon nanotubes (CNT) (Bouafia et al. 2017).

For the vibration analysis of temperature-dependent FG plates, four variable refined plate theories were presented. Parabolic, hyperbolic, sinusoidal, and exponential distribu-tions of the transverse shear strains were being accounted with these theories. These theories were satisfying the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Based on Fourier series, analytical solutions for the free vibration analysis were obtained. At the end, it has been concluded that presented theories are accurate and simple in estimating the free vibration responses of temperature-dependent FG plates (Attia et al. 2015).

For the purpose of analyses of bending and free vibration of FG plates, an accurate and efficient simple higher-order shear deformation theory (HSDT) was developed. The theory has three unknowns as the classical plate theory instead of five as in the well-known first shear deformation theory (FSDT) and HSDT apart from sinusoidal variation of transverse shear strains through the thickness of the plate (Houari et al. 2016).

Authors have done vibration analysis of FGMs after con-sidering chances of porosities inside FGMs during fabrication, after implementing simple displacement field based on HSDT. These theories have few unknowns and equations of motion as compared to FSDT; however, these theories have transverse shear deformation effects. The interesting part of this theory was that it accounted for a quadratic variation of the trans-verse shear strains across the thickness, and satisfied the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors (Atmane et al. 2015).

The number of unknowns and governing equations for an efficient and simple higher-order shear and normal deformation theory were reduced after dividing transverse

PARTICULATE SCIENCE AND TECHNOLOGY 11

displacement into bending, shear, and thickness stretching parts and thus the theory was efficient and simple higher- order shear and normal deformation one of its kind. Both shear deformation and thickness stretching effects by a

hyperbolic variation of all displacements across the thickness were accounted in this theory. And this theory was found to be simple in predicting the bending and free vibration responses of plates (Belabed et al. 2014).

Table 4. Representative papers on three-dimensional (3D) elasticity theory. References Especial highlight

Harmonic vibration problem of FGM plates Reddy and Cheng (2003) 3D asymptotic theory has been formulated. Refinement of asymptotic formulation by expanding the frequency parameter

which enables the asymptotic approach to find any higher-order solution. Free and forced vibration behavior of FGMs Vel and Batra (2004) Presentation of an exact solution after using the power series expansion method. Finally, results were compared with classical

plate theory, FSDT and TSDT. It has been concluded that the FSDT performs better than TSDT. Nie and Zhong (2007) 3D analysis has been carried out under many boundary conditions using a semi-analytical approach which enables use of

state space method and 1D differential quadrature method. The semi-analytical method requires less computational effort. Nie and Zhong (2008) Extension of same procedure for analysis as in Nie and Zhong (2007) Free vibration behavior of FGMs Dong (2008) Using the Chebyshev–Ritz method, the 3D free vibration behavior of FGMs has been studied Li, Iu, and Kou (2008) Using the Ritz method, 3D elasticity solution was developed by Li et al. Tsai and Wu (2008) Using an asymptotic expansion method, there was a presentation of 3D free vibration of simply supported doubly curved FGM

magneto-electro-elastic shells. Hosseini-Hashemi et al. (2013) Presentation of an exact closed-form solutions after using the 3D elasticity theory to review both in-plane and out-of-plane

free vibration of thick simply supported FGMs.

FGMs, functionally graded materials; FSDT, first-order shear deformation theory; TSDT, third-order shear deformation theory.

Table 5. Representative papers on two-dimensional (2D) plate theories. References Especial highlight

Free vibration analysis of FGMs Cinefra et al. (2010) Extension of the Reissner mixed variational theorem with the Carrera’s unified formulation. Accuracy of theories depends

on geometrical parameters and dynamic modes. Liu, Wang, and Chen (2010) Analysis of FGM plates based on the CPT particularly when material is graded along the in-plane direction. In-plane

material gradient parameter would have a certain effect on the natural frequencies of the plate. Hosseini-Hashemi, Fadaee, and

Es’haghi (2010) Presentation of an exact closed-form frequency equation for analysis plates based on the Mindlin’s FSDT

Hosseini-Hashemi, Fadaee, and Atashipour (2011a, 2011b)

Application of Levi-type solution for analysis of plates based on the FSDT, when two opposite edges were being simply supported and other two edges were being under various boundary conditions

Hosseini-Hashemi, Fadaee, and Atashipour (2011a, 2011b)

The same method of solution as in Hosseini-Hashemi, Fadaee, and Atashipour (2011a, 2011b) for the analysis of thick rectangular plates has been used based on the Reddy’s TSDT

Uymaz, Aydogdu, and Filiz (2012) Displacement has been assumed as functions in the form of the Chebyshev polynomials for the analysis of plates. Ritz method has been used here

Zhang, Chen, and Zhang (2013) Application of Ritz energy method while reviewing the nonlinear post-buckling, nonlinear bending, and vibration of plates based on physical neutral surface and Reddy’s TSDT

Thai, Park, and Choi (2012) Shear deformation theory having four unknowns has been proposed here for analysis of plates. The theory was found for the quadratic variation of the transverse shear strains across the thickness and also justified the zero traction boundary conditions on the top and bottom surfaces of the plate even without using shear correction factors

(Su et al. 2014) Unified solution method for the free vibrations of FGMs with general boundary conditions based on the FSDT has been proposed here. Displacements and rotations have been expressed as a modified Fourier series which would be capable of representing any function including the exact solutions

Thermo-elastic vibration response of FGMs Ungbhakorn and

Wattanasakulpong (2013) Investigations of the thermo-elastic vibration response of FGM plates have been done. The observed frequencies were

found to be minimum when the distributed patch mass was placed at the center and the corresponding frequencies were found to be maximum when the patch mass was moved closer to the edge support.

FGMs, functionally graded materials; CPT, classical plate theory; FSDT, first-order shear deformation theory; TSDT, third-order shear deformation theory.

Table 6. Representative papers on finite element method. References Especial highlight

Large amplitude free flexural vibration behavior of FGMs Sundararajan, Prakash, and

Ganapathi (2005) They have used an eight-noded shear flexible quadrilateral plate element based on consistency approach for the

concerned analysis based on the FSDT Talha and Singh (2011a, 2011b) They did analysis of shear deformable FGM plates based on the higher-order shear deformation theory using a C0

continuous element with 13 DOF at each node Free vibration analysis of FGMs Pradyumna and

Bandyopadhyay (2008) They have used an eight-noded C0 continuity element for FGM panels based on the higher-order formulation

Kant and Khare (1997) Finally, they have concluded that although higher-order shear deformation theory is quiet computationally expensive, it shows good performance for thin and thick panels. So, it can be proposed for the free vibration analysis of both thin and thick FGM plates and shell panels

Malekzadeh and Shojaee (2013) They have used an eight-noded solid element along with the Newmark’s time integration scheme to study the response of plates based on the FSDT subjected to moving heat source

FGMs, functionally graded materials; CPT, classical plate theory; FSDT, first-order shear deformation theory; DOF, degrees of freedom.

12 N. J. KANU ET AL.

For the purpose of analyses of bending and vibration of FG plates which was resting on two-parameter elastic foundation, efficient and simple quasi-3D hyperbolic shear deformation theory was developed. Apart from the thickness stretching effect, the theory dealt with five unknowns as the FSDT. The theory was found to be accurate and simple in serving its purposes of analyses (Benahmed et al. 2017).

Using a five-variable refined plate theory, free vibration analysis of functionally graded sandwich plates was discussed. The number of unknowns and governing equations for the present theory was reduced after diving transverse displacement into bending, shear, and thickness stretching parts and thus the theory was simple to use. The theory has five unknowns as compared to six or more in the case of other shear and normal deformation theories. The interesting part of this theory was that it accounted for hyperbolic distribution of the transverse shear strains, and satisfied the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. The theory was found to be accurate and efficient in predicting the bending and free vibration responses of functionally graded sandwich plates (Bennoun, Houari, and Tounsi 2016).

For the purpose of analyses of bending and free vibration of FG sandwich plates having isotropic face sheets, accurate and

efficient new higher-order shear and normal deformation theory was developed. The theory has five unknowns as compared to six or more in the case of other shear and normal deformation theories. The interesting part of this theory was that it accounted for hyperbolic distribution of the transverse shear strains, and satisfied the zero traction boundary conditions on the surfaces of the plate without using shear correction factor (Bessaim et al. 2013).

Using a novel nonlocal refined trigonometric shear deformation theory for the first time, free vibration analysis of size-dependent functionally graded (FG) nanoplates resting on two-parameter elastic foundation was discussed to investigate the influence of power-law index, temperature difference, elastic foundation parameters, plate aspect ratio, and side-to-thickness ratio on the non-dimensional frequency of plates (Besseghier et al. 2017).

A zeroth-order shear deformation theory was used for free vibration analysis of FG nanoscale plates resting on elastic foundation. The interesting part of this theory was that it accounted for the influences of small scale and the parabolic variation of the transverse shear strains across the thickness of the nanoscale plate, and thus has used shear correction factors to present numerical results. In this way, influences of small scale, Winkler modulus parameter, gradient index, shear deformation, and Pasternak shear modulus parameter

Table 7. Representative papers on meshless methods. References Especial highlight

Free vibration analysis of FGMs Yas and Aragh (2011) They have proposed an elasticity solution after using the generalized differential quadrature method for analysis of a four-

parameter cylindrical panel Wu, Chiu, and Wang (2011) They have used a meshless collocation and an element-free Galerkin method for the 3D free vibration of FGM sandwich plates Zhu and Liew (2011a, 2011b) A local Kriging meshless technique has been used to construct shape functions which have Kronecker delta function property.

The FSDT and local Petrov Galerkin formulation have been used here Alijani, Amabili, and

Bakhtiari-Nejad (2011) They have reviewed influence of thermal loads on the nonlinear vibration of doubly curved FGM shells based on two different

higher-order theories after using the multi-model energy approximation Neves et al. (2013) They have used the Carrera’s unified formulation and radial basis functions collocation method for the free vibration analysis

of FGM shells based on a higher-order shear deformation theory Zhu and Liew (2011a, 2011b) They have formulated a meshless method using the Kriging interpolation method for the analysis of FGMs based on the FSDT

and von-Karman nonlinearity

FGMs, functionally graded materials; FSDT, first-order shear deformation theory.

Table 8. Representative papers on vibration analysis. References Especial highlight Vibration characteristics of functionally graded material plate with various boundary constraints using higher-order shear deformation theory Gupta, Talha, and Singh (2016) Vibration characteristics of shear deformable FGM plates have been discussed here. C0 continuous isoparametric finite

element formulation has been implemented here. Natural frequencies of the FGM plates with certain constraints along with parametric study have been performed here

Thermo-mechanical induced vibration characteristics of shear deformable functionally graded ceramic–metal plates using the finite element method Talha and Singh (2010a, 2010b) Thermo-mechanical-induced vibration characteristics of shear deformable FGM plates have been discussed here where

formulations are based on higher-order shear deformation theory. To accomplish the results C0 continuous isoparametric Lagrangian finite-element with 13 DOF per node has been implemented. It has been found that the temperature field and the gradient in the material properties have significant effect on the vibration of the plates

Large amplitude free flexural vibration analysis of shear deformable FGM plates using nonlinear finite element method Talha and Singh (2011a, 2011b) Investigation has been done here for, large amplitude free flexural vibration analysis of shear deformable FGM plates. They

have obtained results after employing C0 finite element with 13 DOFs per node. To establish the efficacy of the present model convergence tests and comparison studies have been done

Stochastic nonlinear free vibration analysis of elastically supported FGMs plate with system randomness in thermal environment Jagtap, Lal, and Singh (2011) The stochastic nonlinear free vibration response of elastically supported FGMs plate has been presented here. Higher-

order shear deformation theory with von-Karman nonlinear strains using modified C0 continuity has been used here as basic formulation. To compute the second-order statistics (mean and coefficient of variation) of the nonlinear fundamental frequency for FGM plate, direct iterative-based nonlinear finite element method in conjunction with first- order perturbation technique has been implemented and respective approach has been validated with results available in the collected literature and Monte Carlo simulation

FGMs, functionally graded materials; DOFs, degrees of freedom.

PARTICULATE SCIENCE AND TECHNOLOGY 13

on the vibration responses of the FG nanoscale plates have been investigated (Bounouara et al. 2016).

Using a simple and refined trigonometric higher-order beam theory, bending and vibration of functionally graded beams were discussed to investigate to show the effect of the inclusion of transverse normal strain on the deflections and stresses. The interesting part of this theory was inclusion of the thickness stretching effect with the displacement field hav-ing only three unknowns as in Timoshenko beam theory (Bourada et al. 2015).

The number of unknowns and governing equations for a newly developed quasi-three-dimensional (3D) hyperbolic shear deformation theory for the bending and free vibration analysis of functionally graded plates (FGPs) were reduced after dividing transverse displacement into bending, shear, and thickness stretching parts and thus the theory was one of its simple kind. Both transverse shear and normal deformations were accounted in this theory. And this theory was found to satisfy zero traction boundary conditions on the surfaces of the plate without using shear correction factor (Hebali et al. 2014).

Using a new hyperbolic shear deformation theory, free vibration analysis of functionally graded, sandwich, and laminated composite plates were discussed. The present theory has five degrees of freedom. The interesting part of this theory was that it accounted for parabolic transverse shear strains across the thickness direction, and satisfied the zero traction

boundary conditions on the surfaces of the plate without using shear correction factor. The theory was found to be accurate and efficient in predicting the bending and free vibration responses plates (Mahi and Tounsi 2015).

4.2. Buckling analysis

Like in case of vibration analysis, the research work has been classified under two headings, and those are analytical meth-ods and numerical methods (Swaminathan et al. 2015).

4.2.1. Analytical methods Buckling analysis of FGM includes extensive investigations of the critical buckling loads under various boundaries and load-ing conditions (Swaminathan et al. 2015).

4.2.1.1. Three-dimensional (3D) elasticity theory. To find critical buckling load, Eigen value problem needs to be solved. However, buckling analysis of FGM plates using three-dimensional elasticity theory (a very accurate method of analysis) so far not yet reported in any research paper (Swaminathan et al. 2015).

4.2.1.2. Two-dimensional (2D) plate theories. Many other latest publications are listed in Table 9 where each paper is reviewed with comment on its major headline (Swaminathan et al. 2015).

Table 9. Representative papers on two-dimensional (2D) plate theories. References Especial highlight Mechanical load case Najafizadeh and Eslami (2002) They have done buckling analysis of FGM plates based on the Love–Kirchhoff hypothesis and the Sander’s nonlinear

strain–displacement relation with either simply supported or clamped edges subjected to uniform radial compression. Mechanical instability of FGM plates was found to be lower than fully ceramic plates

Yanga and Shen (2003) To investigate the post-buckling behavior of fully clamped FGM rectangular plates based on the CPT under transverse and in-plane loads, perturbation technique along with one-dimensional differential quadrature approximation and Galerkin procedure has been introduced by authors. They found that though the mechanical performance of FGM plates has been quite similar to homogeneous isotropic ones, they do exhibit unique and interesting characteristics due to the grading of FGM composition

Yanga and Shen (2003) They have extended same work to review the influence of initial geometric imperfection on the post-buckling behavior of FGM plates based on the Reddy’s TSDT under boundary conditions. They found that the effect of local imperfection became much less

Prakash and Ganapathi (2006) For the asymmetric flexural vibration analysis of FGM plates based on the FSDT, a three-noded shear flexible plate element based on the field-consistency principles has been proposed here. They found that nonlinear temperature variation through the thickness results into higher critical buckling loads compared to constant through the thickness variation of temperature.

Shariat, Javaheri, and Eslami (2005) They have reviewed the buckling behavior of geometrically imperfect FGM plates which is actually based on the CPT Najafizadeh and Heydari (2008) Closed-form solution has been proposed here for the buckling of FGM circular plates which is actually based on the

Reddy’s TSDT finally subjected to uniform radial compression. The results have been compared with CPT and FSDT. They have observed that the buckling values which have been estimated by TSDT were the lowest

Thermo-mechanical load case Shariat and Eslami (2006) They have reviewed the influence of geometrical imperfections on the thermal buckling of FGM plates based on the FSDT.

The plate has been subjected to three types of thermal loadings namely uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise

Wu, Shukla, and Huang (2007) To study the post-buckling response of FGM plates under different kinds of boundary conditions subjected to uniaxial compression or uniform temperature rise based on the FSDT, a finite double Chebyshev polynomial which is capable of solving plates with non-Levy-type boundary conditions has been used. As such, no significant difference exists between the post-buckling response of FGM plates with aspect ratio equal to 3 and 4

Shariat and Eslami (2007) They have proposed a closed-form solution for the buckling analysis of rectangular thick FGM plates based on the TSDT under mechanical and thermal loads.

Bouazza et al. (2010) The Navier’s solution technique has been proposed here for the stability analysis of simply supported FGM plates which has been subjected to uniform and linear temperature rise through the thickness based on the FSDT

Bodaghi and Saidi (2010) They have proposed Levy-type solution after using a boundary layer function for the buckling analysis of thick FGMs under various boundary conditions based on the Reddy’s TSDT

FGMs, functionally graded materials; CPT, classical plate theory; FSDT, first-order shear deformation theory; TSDT, third-order shear deformation theory.

14 N. J. KANU ET AL.

4.2.2. Numerical methods To avoid complexities in analytical solution for those FGMs with geometrical and material complexities, numerical meth-ods have been preferred. That is the reason why numerical methods such as FEMs and meshless methods have been widely used for complex engineering problems (Swaminathan et al. 2015).

4.2.2.1. Finite element method. Many other latest publica-tions are listed in Table 10 where each paper is reviewed with comment on its major headline (Swaminathan et al. 2015).

4.2.2.2. Meshless methods. Many other latest publications are listed in Table 11 where each paper is reviewed with comment on its major headline (Swaminathan et al. 2015).

4.2.3. Recent papers on buckling analysis of FGMs Using a simple FSDT, thermal buckling response of functionally graded sandwich plates with various boundary conditions was dis-cussed. The theory having four unknowns was found to achieve same accuracy as we have in the existing conventional FSDT which has more number of unknowns (Bouderba et al. 2016).

Based on a four-variable refined plate theory, the buckling analysis of FGP subjected to uniform, linear, and nonlinear temperature rises across the thickness direction were discussed. The interesting part of this theory was that it accounted for parabolic distribution of the transverse shear strains, and satisfied the zero traction boundary conditions on the surfaces of the plate without using shear correction factor to investigate plate parameters on buckling temperature difference such as ratio of aspect ratio, thermal expansion, side-to-thickness ratio, and gradient index (Bousahla et al. 2016).

Table 10. Representative papers on finite element method. References Especial highlight

Mechanical load case Bateni, Kiani, and Eslami (2013) Investigation of existence of bifurcation buckling under various thermal and mechanical loads has been done here. They

have used a four-variable refined plate theory to derive the governing equations of equilibrium. Investigation has been done here for influence of temperature dependency of material properties on the critical buckling load for both uniform temperature rise and heat conduction cases. Here, in both cases, temperature dependency concluded in underestimation of the critical buckling temperature

Naei, Masoumi, and Shamekhi (2007) They have used energy method which was based on Love–Kirchhoff hypothesis to review the buckling analysis of a radially loaded circular FGM plates having variable thickness

Lee and Kim (2013) They have studied the post-buckling nature of FGM plates in hygrothermal environments based on the FSDT and von- Karman nonlinearity. Finally, they have realized that the influence of moisture on the post-buckling nature appreciably increases with the increase in the value of power-law parameter

Thermo-mechanical load case Prakash, Singha, and Ganapathi

(2008) They have used an eight-noded C0 shear flexible quadrilateral plate element to review the nonlinear bending/pseudo-

post-buckling nature of FGMs based on the Mindlin formulation under thermo-mechanical load and then realized that temperature-dependent material belongings overestimated the thermal post-buckling resistance

Prakash, Singha, and Ganapathi (2009)

They have actually extended the same work to review the effect of the position of the neutral surface on the stability nature of FGM plates

Sohn and Kim (2008) Here, a nine-noded rectangular element has been used to review the static and dynamic stability of FGMs based on the FSDT and those FGMs were actually subjected to thermal and aerodynamic loads simultaneously

FGMs, functionally graded materials; FSDT, first-order shear deformation theory.

Table 11. Representative papers on meshless methods. References Especial highlight

Mechanical load case Chen and Lie (2004) Investigation is done for the buckling behavior using the radial basis function in case of two-dimensional elastic plane stress

problem of FGMs based on the Mindlin’s plate assumption and subjected to nonlinearly distributed in-plane edge loads Zhang, Chen, and Zhang (2013) Investigation is done for stability and local bifurcation nature for a simply supported FGMs subjected to the transversal and in-

plane excitations in the uniform thermal environment using both analytical and numerical methods Mahdavian (2009) They have used the Airy stress field approach and Galerkin’s approach for the stability analysis of FGMs and while subjecting

to non-uniform in-plane compressive loads based on the CPT and von-Karman nonlinearity. Here, four types of loadings have been considered, those were concentrated load, triangular load, uniform load, reverse triangular load, and sinusoidal load. The critical buckling load coefficient has been found to be maximum in case of sinusoidal load and minimum for concentrated load

Thermo-mechanical load case Park and Kim (2006) They have studied the thermal post-buckling and vibrations in FGM plates with temperature-dependent material properties in

the pre- and post-buckled regions based on the FSDT and incremental strain–displacement relationship Li, Zhang, and Zhao (2007) To study the post-buckling of an imperfect FGM plates subjected to mechanical load and transverse non-uniform temperature

rise as well, shooting method along with the von-Karman’s plate theory has been used here. They concluded with the fact that influence of mechanical load on the deflection decreases with the increase in the value of thermal load. Also, they had realized that mechanical load became dominant in contributing on the central deflection when the effect of thermal load was very small

Liew, Zhao, and Lee (2012) To investigate the post-buckling response of FGM cylindrical shells under axial compression and thermal loads based on the FSDT and von-Karman nonlinearity, they have used the element-free kp-Ritz method. Finally, they evaluated system bending stiffness after using a stabilized conforming nodal integration method. And the membrane and shear terms have been found using direct nodal integration to remove shear locking and of course then reducing the computational cost

FGMs, functionally graded materials; CPT, classical plate theory; FSDT, first-order shear deformation theory.

PARTICULATE SCIENCE AND TECHNOLOGY 15

Many other latest publications are listed in Table 12 where each paper is reviewed with comment on its major headline.

4.3. Bending analysis

Using a four-variable refined plate theory, hygro-thermo- mechanical bending behavior of sigmoid functionally graded material (S-FGM) which has been resting on variable two-parameter elastic foundations was discussed. Four independent variables were there in formulation, as against five in other shear deformation models to investigate the influence of power-law index, temperature difference, elastic foundation parameters, plate aspect ratio, and side-to-

thickness ratio on the static behavior of S-FGM plates (Beldjelili, Tounsi, and Mahmoud 2016).

For the purpose of analyses of bending and dynamic behaviors of FG plates, a new FSDT was developed. It has been seen that there was no stretching–bending coupling effect in the formulation of neutral surface-based, and thus boundary conditions and those governing equations of plates have simple forms as in isotropic plates. The theory was found to be accurate and simple in serving its purposes of analyses (Bellifa et al. 2016).

Based on a recently developed refined trigonometric shear deformation theory, the thermo-mechanical bending response of FGPs resting on Winkler–Pasternak elastic

Table 12. Recent papers on buckling of FGMs. References Especial highlight

Post-buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties Jagtap et al. (2011) Examination has been done for the second-order statistics of post-buckling response of FGM plate subjected to mechanical and

thermal loading with non-uniform temperature changes subjected to temperature-independent and -dependent material properties. The theme has been validated with the research in literatures and independent Monte Carlo simulation

Thermo-mechanical elastic post-buckling of functionally graded materials plate with random system properties Jagtap, Achchhe, and

Singh (2013) Stochastic post-buckling response of elastically supported FGM plate, supported with two parameters of Pasternak foundation with

Winkler cubic nonlinearity, has been presented with random system properties subjected to temperature change with material properties. Higher-order shear deformation theory with von-Karman nonlinearity has been used as basic formulation using modified C0 continuity. The second-order statistics (mean and coefficient of variation) of post-buckling response of FGM plates have been computed using direct iterative-based nonlinear finite element method combined with first-order perturbation technique

FGMs, functionally graded materials.

Table 13. Representative papers on bending analysis. References Especial highlight

Static response and free vibration analysis of FGM plates using higher-order shear deformation theory Talha and Singh (2010a,

2010b) In this paper, free vibration and static analysis of FGM plates have been discussed using higher-order shear deformation theory

with particular change in the transverse displacement in conjunction with finite element models. Here, mechanical properties of the plate are assumed to vary continuously in the thickness direction according to simple power-law distribution in terms of the volume fractions of the constituents. Gradient in the material properties has significant role to play while determining the response of the FGMs

Nonlinear mechanical bending of functionally graded material plates under transverse loads with various boundary conditions Talha and Singh (2011a,

2011b) In this paper, nonlinear mechanical bending of FGM plates under transverse loads and various boundary conditions have been

discussed where nonlinear finite element formulations are based on the higher-order shear deformation theory. They have obtained results after employing C0 continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. To establish the efficacy of the present model, convergence tests and comparison studies have been done

Spline finite strip bending analysis of FGM Chi and Chung (2006) They have investigated spline finite strip bending analysis of FGM after using the power-law technique and classical plate Theory.

They have finally compared with results of the closed-form solution. Parvathy and Beena (2013) Spline finite strip is having numerous merits of classical finite strip apart from additional merits. Here, it has been realized that the

trigonometric series which has been used in classical finite strip method would not be the right approximation to model the bending behavior

Static response of FGMs Zenkour (2006) Investigation of the static response of FG plates has been done using generalized shear deformation theory Daouadji, Tounsi, and Bedia

(2013) They have formulated Navier solutions of plates based on a new higher-order shear deformation model for the static response of

functionally graded plates. The mechanical properties of the plate are supposed to vary continuously in the thickness direction as per simple power-law distribution. They have verified results with available results in the literature

Quasi-static bending response of FGMs Zenkour, Mashat, and

Elsibai (2009) For a simply supported functionally graded rectangular plate which has been subjected to a through-the-thickness temperature

field under the effect of various theories of generalized thermo-elasticity, the quasi-static bending response is proposed here. Material properties of the plate are supposed to be graded in the direction of thickness as per simple exponential law distribution in terms of the volume fractions of the constituents

Benatta et al. (2009) and Sallai et al. (2009)

For simply supported FGMs which have been subjected to uniformly distributed transverse loads using a higher-order shear deformation theory, they have analytically solved static bending deformations and gave numerical results for the deflection, and the transverse normal and the transverse shear stresses

Axisymmetric bending deflection of FGMs Ma and Wang (2004) They have considered TSDT to describe axisymmetric bending deflection and buckling loads of FGMs Critical buckling load in Timoshenko beams (FGMs) Li and Batra (2013) The importance of this research paper lies in the fact that for the clamped–clamped (C–C), simply supported–simply supported

(S–S) and clamped–free (C–F) FGM Timoshenko beams, the critical buckling load can be easily calculated from that of the corresponding homogeneous Euler–Bernoulli beam and two constants whose values depend upon the through-the-thickness variations of E and m. It is interesting to note that for C–S FGM Timoshenko beam, the transcendental equation (for the determination of the critical buckling load) is quiet similar to that for the corresponding homogeneous Euler–Bernoulli beam

FGMs, functionally graded materials; TSDT, third-order shear deformation theory.

16 N. J. KANU ET AL.

Table 14. Representative papers on wave propagation in functionally graded plates. References Especial highlight

Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories Yahia et al. (2015) For wave propagation in functionally graded plates, higher-order shear deformation plate theories were developed. As there have

been chances of porosities inside FGMs during fabrication, so authors have thought about the wave propagation in plates. These theories have few unknowns and equations of motion as compared to first-order shear deformation theory; however, these theories have transverse shear deformation effects. Whichever results have been discussed herewith could be used further in ultrasonic inspection techniques and thus in structural health monitoring

An efficient shear deformation theory for wave propagation of functionally graded material plates Boukhari et al. (2016) For wave propagation in an infinite functionally graded plate in the thermal environments, an efficient shear deformation theory was

developed. Here, both thermal effects and temperature-dependent material properties were considered. The number of unknowns and governing equations for the present theory was reduced after diving transverse displacement into bending, shear and thus the theory was simple to use. Whichever results have been discussed herewith could be used further in ultrasonic inspection techniques and thus in structural health monitoring

FGMs, functionally graded materials.

Table 15. Representative papers on CNT-reinforced functionally graded materials, functionally graded nanocomposites, functionally graded single-walled carbon nanotubes, and FG nanobeam.

References Especial highlight Kumar and Harsha (2016) Here, influence of carbon nanotubes on CNT-reinforced FGM nanoplate under thermo-mechanical loading has been discussed. It has

been confirmed that CNT increases significant strength in both FGM and composite. It has been cleared that in case of mechanical loading, CNT would result into more strength in composite as compared to FGM; however, in thermo-mechanical loading, FGM would be having more strength than composite

Bruck and Doherty (2007) Functionally graded nanocomposites have potential for nanotechnology to bring innovations from the bench to the Fleet, and these can be tailored for reducing costs.

Florczyk and Saha (2007) Nanotechnology can be implemented into biomedical implants after using functionally graded materials (FGM). Tailoring is possible in FGMs to alter the mechanical strength or the resorb ability of the material

El-Hadad et al. (2010) The centrifugal mixed-powder method (CMPM) has been proposed to fabricate FGMs containing nanoparticles selectively dispersed in the outer surface of the fabricated parts. To have control on particles morphology, compound formulas or sizes, alternative CM technique is being favored. Finally, advanced CMPM as reaction centrifugal mixed-powder method (RCMPM) has been presented here. It has been proposed here that after using RCMPM, Al-Al3Ti/Ti3Al FGMs having good surface properties and temperature controlled compositional gradient could be fabricated

Janghorban and Zare (2012) Bending analysis of functionally graded single-walled carbon nanotubes has been presented in research paper. They have modeled carbon nanotubes using Euler–Bernoulli beam theory. To have accuracy in work, the results are compared with other existing results. At the end, influences of different parameters such as power-law index, inner and outer radii of nanotubes, and length nanotubes have been studied

Sahmani, Aghdam, and Bahrami (2015)

Investigation is done for the free vibration response of third-order shear deformable nanobeams made of FGMs around the post- buckling domain after considering the effects of surface free energy on the basis of an efficient numerical solution. They have agreed to the fact that in post-buckling domain, the natural frequency of FGM nanobeam reduces after rise in the value of material property gradient index

Shiri et al. (2015) Here, Cu/NbC FGM [Cu–15%NbC (volume fraction)] has high electrical conductivity with the same hardness and wear properties as those of the composite sample on the composite surface. Therefore, Cu/NbC FGM having good mechanical and electrical properties would be ideal for electrical contact applications

Rahmani and Pedram (2014) Here, they have discussed Timoshenko beam theory which is responsible for the size-dependent effects in FGM beam. They have verified the model by comparing the available results with benchmark results available in the literature. A parametric study has been followed to cross check influence of the gradient index, length scale parameter, and length-to-thickness ratio on the vibration of FGM nanobeams. It has been seen that these parameters are finally essential in examination of the free vibration of a FG nanobeam

Lin and Xiang (2014) Here, investigation is done for linear free vibration of nanocomposite beams which are reinforced by single-walled carbon nanotubes. They have considered two types of CNT-reinforced beams, uniformly distributed CNT-reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams. Furthermore, comparative studies have been done for UD-CNT and FG- CNT beams based on the first-order and the third-order beam theories and finally, differences in vibration frequencies between these two theories have been highlighted

Roque et al. (2016) To review the influence of a scale parameter in the free vibration of a Timoshenko functionally graded beam, modified couple stress theory has been used here. Differential evolution optimization has been used to solve the optimization problem to minimize the free vibration frequency of the beam

Nami and Janghorban (2014) They have reviewed for the first time the resonance behaviors of functionally graded micro/nanoplates using Kirchhoff plate theory Lajevardi, Shahrabi, and

Szpunar (2013) Synthesis of functionally graded nickel–nano-Al2O3 composite coatings is done using pulse deposition where the amount of the

embedded nano-alumina particles changes in the cross section of the composite. To have functionally graded nanocomposite coatings by pulse electro-deposition under ultrasonic agitation, frequency and duty cycle changes could be applied and here in this paper, the effect of both parameters has been studied. They have mentioned the optimum condition for production of functionally graded nano-Al2O3–Ni coating which has been changing of the duty cycle from 90 to 10% at fixed frequency of 10 Hz

Khorshidi, Shariati, and Emam (2016)

Analysis of shear deformable functionally graded (FG) nanobeams in post-buckling based on modified couple stress theory has been presented here. Here, they have presented exact and generalized differential quadrature solutions for the static post-buckling response of FGM nanobeams under boundary conditions. Finally, both results are compared. It is interesting to note that first- order beam theory has few errors while estimating the amplitude of buckling

Uysal (2015) To simulate the nano-indentation test of WC (tungsten carbide) substrate-coated TiB (titanium boron), finite element model has been developed. Those models are actually based on a contact problem between axisymmetric half-space of a specimen and spherical indentation. Here, film layer through the thickness has been modeled for both functionally graded TiB coating which material properties vary linearly and the thin coating layer including 85% Ti (Titanium) and 15% TiB and those coatings have been compared finally

CNT, carbon nanotubes.

PARTICULATE SCIENCE AND TECHNOLOGY 17

foundations was discussed. The interesting part of this theory was that it accounted for trigonometric distribution of the transverse shear strains, and satisfied the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. The theory was found to be accurate and efficient in serving its purposes of analyses (Bouderba, Houari, and Tounsi 2013).

For the purpose of analyses of bending response of FGM plates resting on elastic foundation and subjected to hygro-thermo-mechanical loading, a four-variable refined plate theory was developed. The present theory has four

independent unknowns as compared to five in other shear deformation theories. The interesting part of this theory was that it accounted for both a quadratic variation of the transverse shear strains across the thickness, and satisfied the zero traction boundary conditions on the surfaces of the plate without using shear correction factor to investigate plate parameters for serving the purposes of simulation of rocket launch pad structures subjected to intense thermal loading (Zidi et al. 2014).

Many other latest publications are listed in Table 13 where each paper is reviewed with comment on its major headline.

Table 16. Representative papers on functionally graded piezoelectric materials (FGPMs). References Especial highlight

Wu and Lim (2016) An overview of semi-analytical numerical methods for quasi-three-dimensional (3D) analyses of multilayered (or sandwiched) functionally graded elastic/piezoelectric materials (FGEMs/FGPMs) plates has been presented here. To estimate the effective material properties of functionally graded structures, two micromechanical schemes (i.e., the rule of mixtures and Mori– Tanaka scheme) have been used to estimate the effective material properties of functionally graded structures here

Su, Ke, and Wang (2016) They have focused on the axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space which has been subjected to three typical rigid punches, that is, flat circular punch, spherical punch, and conical punch. The punch has been further assumed as a perfect conductor with a constant electric potential. Finally, they have discussed influence of the gradient index and punch geometry on the surface electromechanical contact behaviors

Wu and Liu (2016) They have introduced finite cylindrical layer methods to find the quasi-three-dimensional (3D) dynamic responses of simply supported, two-layered FGPM film-substrate circular hollow cylinders with open- and closed-circuit surface conditions which have been based on Reissner’s mixed variational theorem

Shin and Kim (2016) Here, transient response analysis of a Mode III interface crack between a piezoelectric layer and a functionally graded orthotropic material layer has been done using integral transform techniques

(Kong, Liu, and Nie 2016) In a FGPM layer bonded on 0.71Pb (Mg1/3Nb2/3)O3–0.29PbTiO3 (PMN–0.29PT) single crystal substrate, shear horizontal waves have been investigated. The dispersion equations have been derived here for different boundary conditions on the surface of FGPM layer. It has been demonstrated here that cut orientation of the PMN–0.29PT single crystal and magnetic boundary conditions and gradient coefficient of FGMP layer have significant impacts over dispersion behaviors

Wang and Luo (2016) To analyze the electromechanical characteristics of functionally graded piezoelectric ring transducers in radial vibration, they have used an exact solution. Furthermore, to illustrate the effect of material inhomogeneity index and elastic foundation stiffness on the radial vibration characteristics, they have taken the help of numerical results which have been depicted graphically

Liu et al. (2013) Using extended finite element method (X-FEM), transient thermal dynamic analysis has been done for stationary cracks in FGPMs. They have considered both heating and cooling shocks. The effects of the crack length, poling direction, etc., on the dynamic intensity factors have been demonstrated here. Results of transient dynamic crack behaviors under the cooling shock differ from those which are under the heating shock

Jamia, El-Borgi, and Usman (2016)

Investigation is done for two collinear mixed-mode limited-permeable cracks which are embedded in an infinite medium made of a FGPM with crack surfaces subjected to electromechanical loadings. To review the influence of interaction of two cracks, material gradient parameter describing FGPMs and lattice parameter on the mechanical stress and electric displacement field near crack tips, investigation has been done

Sayyaadi and Farsangi (2014) Here, analytical solution has been given for free vibration and dynamic behavior of doubly curved laminated shell consisting of a functionally graded core layer and surface attached functionally graded piezoelectric layers. They have reviewed the influence of shell curvature on the spectra of maximax response

Salah, Amor, and Ghozlen (2015)

Investigation is done for wave propagation in a three-layer structure for both electrically open and shorted cases. Achieved results set guidelines for the design of high-performance surface acoustic wave devices as well as for the measurement of material properties in a functionally graded piezoelectric layered system after using Love waves

Liu et al. (2013) Using X-FEM, they have studied transient dynamic fracture behaviors of stationary cracked FGPMs under impact loading. They have come out with conclusion that the dynamic crack behaviors in the non-homogeneous FGPMs are quiet more complicated than those in the homogeneous materials

Ansari et al. (2016) Here, analytical solution steps for the nonlinear post-buckling analysis of piezoelectric functionally graded carbon nanotubes reinforced composite (FG-CNTRC) cylindrical shells subjected to combined electro-thermal loadings, axial compression, and lateral loads have been carried out. It is important to note that carbon nanotubes are being aligned and straighten with uniform and functionally graded distributions in the direction of thickness. They have cleared that the carrying capacity of the structure increases because shell has been integrated by the piezoelectric layers and reinforced by higher CNT volume fraction

Shen and Yang (2015) Investigation has been done for large amplitude flexural vibration of a hybrid laminated beam resting on an elastic foundation in thermal environments

Kumar, Panda, and Chakraborty (2015)

They have designed a graded fiber-reinforced composite lamina and graded laminates with an aim of reduced inter-laminar stress discontinuity in composite laminates. Finally after dynamic analysis, it has been cleared about the suitability of proposed lamination scheme for the use of graded laminated composite plate under transverse harmonic loads

Rao and Kuna (2010) Domain form of the interaction integrals based on three independent formulations have been presented here for computation of the stress intensity factors and electric displacement intensity factor for cracks in FGPMs which has been subjected to steady-state thermal loading

Lezgy-Nazargah, Vidal, and Polit (2013)

Here, static, free vibration, and dynamic response of FGPM beams have been dealt using an efficient three-noded beam element. Here, beam finite element is being based on a refined sinus model. Validation of proposed FE is done through static, free vibration, and dynamic tests for FGPM beams. Excellent agreement has been found between the results from the proposed formulation and reference results from open literature or 3D FEM for different electrical and mechanical boundary conditions

18 N. J. KANU ET AL.

4.4. Wave propagation in functionally graded plates

Two recent publications are discussed in Table 14 where papers are reviewed with comment on its major headline.

5. Smart functionally graded materials

The carbon nanotubes reinforced FGCM functionally graded nanocomposites, functionally graded single-walled CNT, FG Nanobeam as well as FGPMs are expected to be the new gener-ation material having potential applications in various technologi-cal areas such as defense, aerospace, automobile, energy, etc.

5.1. CNT reinforced functionally graded materials, functionally graded nanocomposites, functionally graded single-walled carbon nanotubes, FG nanobeam

Many other latest publications are listed in Table 15 where each paper is reviewed with comment on its major headline.

5.2. Functionally graded piezoelectric materials

Many other latest publications are listed in Table 16 where each paper is reviewed with comment on its major headline.

6. Conclusion

Functionally graded materials are still recent area of research. The example of human bones becomes important for engineering prospective as these have unique properties (ability to adapt in adverse environment and self-healing through suitable bond formation). We can now realize that challenges of FGMs are in terms of mass production, low cost, and quality control. It has been seen that FGMs can be compositionally or micro-structurally graded.

The paper outlines a review on various processes applied for manufacturing of FGMs. The techniques include powder metallurgy (PM) methods, infiltration and graded casting processes, thermal spraying, and laser-assisted process and finally vapor deposition methods. It can now be said that that the manufacturing techniques of FGMs have advanced significantly. Few aspects have been highlighted as the challenges in developing innovative technique for FGMs preparation such as the suitability for mass production, the cost-effectiveness, and the convenience level in controlling the quality. Finally, it has been concluded that the powder metallurgy as the most efficient method for the manufacturing of FGMs in the future. The issue while using the PM method is the sintering process which should be further explored to attain improvement in the mechanical properties and microstructure of the manufactured FGMs.

Possible applications of FGMs lie in fields of nuclear energy, aerospace, energy conversion, and optics. It is thus expected that scientists and engineers will go on optimizing properties of material systems and thus multi-functionality of FGMs will be in lime light. There is a need to analyze the fracture and fatigue characteristics of FGM structures. There

is a need of procedures and protocols that guarantee a reliable and of course predictable distribution of material constituent phase and properties throughout the structure. With this review, the emphasis is mainly given on fracture analysis of FGM materials.

Earlier performed numerical analyses of structures were based on the FEM. The FEM can be easily applied to complex geometries and general boundary conditions and is well advanced into almost every possible engineering application, including fracture analyses.

After introduction of IGA, a new phase has been opened in unifying computer-aided design and numerical solutions using the non-uniform rational B-splines functions. IGA has been successfully implemented in several engineering problems, including structural dynamics. Recently, a combination of XFEM and IGA methodologies has been used for general mixed-mode crack propagation problems after the introduc-tion of XIGA.

The extension of FEM into XFEM has allowed new potentiality while preserving the finite element original merits. The two main advantages of XFEM are its potentiality in reproducing the singular stress state at a crack tip, and allowing several cracks or arbitrary crack propagation paths to be simulated on an independent unaltered mesh. The mesh does not need to conform to the virtual crack path, the exact analytical solutions for singular stress and discontinuous displacement fields around the crack (tip) are recreated by inclusion of a special set of enriched shape functions which are further obtained from the asymptotic analytical solutions.

The recent computational advances in the form of multiscale simulations where the part of model which requires a more accurate theoretical basis or numerical approximation, due to lack of theoretical bases or existing inconsistencies of many conventional models, is actually simulated by a finer modeling scale, which can represent details of the material behavior and the interacting effects of material constituents in a finest way.

The review is also primarily focused on new advances in analytical and numerical methods for the stress, vibration, and buckling analyses of FGMs. Direct displacement method for 3D exact analysis is limited only for FGM plates under uniform load. 3D analysis using a numerical technique would require additional computational effort and of course larger computer memory than 2D analysis. Therefore, emphasis has been primarily on to restrict 2D analysis with sorts of compro-mise in the accuracy of results. FSDT and third-order shear deformation theory were extensively used among the various 2D plate theories. FSDT can help us in terms of getting reason-ably accurate results with less computational afford. Meshless methods are best alternative to FEMs, irrespective of the variation of plate thickness. However, meshless methods would require high computational afford in 3D analysis. The role of gradients is significant in knowing the response of FGM plates.

Quasi-static bending response of FGMs has been studied in relation with the generalized thermo-elasticity theories. Here, material properties of the plate are supposed to be graded in the thickness direction as per simple exponential law distri-bution in terms of the volume fractions of the constituents. Here, exact solution has been obtained.

PARTICULATE SCIENCE AND TECHNOLOGY 19

Spline finite strip method results after using the power-law function have been reported to be in favor of the theoretical result which has been developed for the deflection of the FGM under uniformly distributed load.

Furthermore, a new higher-order shear deformation model has been proposed to realize the static nature of FGMs.

To facilitate the flexibility in design and then having excellent performance under thermal and mechanical loading, FGMs have been seen as potential structural materials for future engineering applications. And so it is needed to improve the solution techniques and methodologies for the analysis of FGMs.

At the end, this review is focused on smart FGMs such as CNT-reinforced functionally graded composites functionally graded nanocomposites, functionally graded single-walled CNT, FG Nanobeam as well as FGPMs. Also manufacturing technique, reaction centrifugal mixed-powder method to fabricate FGMs containing nanoparticles has been included in this review.

7. Future scope

Possible applications of FGMs lie in fields of nuclear energy, aerospace, energy conversion, and optics. It is thus expected that scientists and engineers will go on optimizing properties of material systems and thus multi-functionality of FGMs will be in lime light. There is a need to analyze fabrication techniques, fracture problems, vibration, buckling, and bending characteristics of FGM structures. There is a need of procedures and protocols that guarantee a reliable and of course predictable distribution of material constituent phase and properties throughout the structure.

Attention is required in designing numerical techniques for 3D analysis of FGMs particularly while using advanced computational techniques which could effectively reduce computational afford and obviously cost. We are required to develop analytical formulation and solution methodology using 3D elasticity theory for FGM plates which would be having power-law type of variations in properties.

FGM is an excellent advanced 21st century material. Major roadblock is cost as substantial part of the cost expended on powder processing and fabrication method. Intense research needs to be conducted on improving the performance of FGPMs and CNT-reinforced FGMs.

ORCID

Nand Jee Kanu http://orcid.org/0000-0003-3919-5098 Umesh Kumar Vates http://orcid.org/0000-0002-1614-5082 Gyanendra Kumar Singh http://orcid.org/0000-0003-3765-9071 Sachin Chavan http://orcid.org/0000-0003-3113-9761

References

Afsar, A. M., and H. Sekine. 2002. Inverse problems of material distributions for prescribed apparent toughness in FGM coatings around a circular hole in infinite elastic media. Composites Science and Technology 62:1063–77. doi:10.1016/s0266-3538(02)00049-0.

Afsar, A. M., and J. I. Song. 2010. Effect of FGM coating thickness on apparent fracture toughness of a thick-walled cylinder. Engineering

Fracture Mechanics 77 (14):2919–26. doi:10.1016/j.engfracmech.2010. 07.001.

Alijani, F., M. Amabili, and F. Bakhtiari-Nejad. 2011. Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory. Composite Structures 93(10):2541–53. doi:10.1016/j.compstruct.2011.04.016.

Ansari, R., T. Pourashraf, R. Gholami, and A. Shahabodini. 2016. Analytical solution for nonlinear post buckling of functionally graded carbon nanotube-reinforced composite shells with piezoelectric layers. Composites Part B: Engineering 90:267–77. doi:10.1016/j.compositesb. 2015.12.012.

Atmane, H. A., A. Tounsi, F. Bernard, and S. R. Mahmoud. 2015. A computational shear displacement model for vibrational analysis of functionally graded beams with porosities. Steel and Composite Structures 19 (2):369–84. doi:10.12989/scs.2015.19.2.369.

Attia, A., A. Tounsi, E. A. Bedia, and S. R. Mahmoud. 2015. Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories. Steel and Composite Structures 18 (1):187–212. doi:10.12989/scs.2015. 18.1.187.

Ayari, M. L., and Z. Ye. 1995. Maximum strain theory for mixed mode crack propagation in anisotropic solids. Engineering Fracture Mechanics 52 (3):389–400. doi:10.1016/0013-7944(95)00040-3.

Batakis, A. P., and W. Vogan. 1985. Rocket thrust chamber thermal barrier coatings. NASA Contractor Report 1750222.

Bateni, M., Y. Kiani, and M. R. Eslami, 2013. A comprehensive study on stability of FGM plates. International Journal of Mechanical Sciences 75:134–44. doi:10.1016/j.ijmecsci.2013.05.014.

Bayesteh, H., A. Afshar, and S. Mohammdi. 2015. Thermo-mechanical fracture study of inhomogeneous cracked solids by the extended isogeometric analysis method. European Journal of Mechanics A/Solids 51 (8):123–39. doi:10.1016/j.euromechsol.2014.12.004.

Bayesteh, H., and S. Mohammadi. 2013. XFEM fracture analysis of orthotropic functionally graded materials. Composites Part B: Engineering 44 (1):8–25.

Becker, T. L., Jr., R. M. Cannon, and R. O. Ritchie. 2001. Finite crack kinking and T-stresses in functionally graded materials. International Journal of Solids and Structures 38:5545–63. doi:10.1016/s0020-7683 (00)00379-6.

Belabed, Z., M. S. A. Houari, A. Tounsi, S. R. Mahmoud, and O. A. Bég. 2014. An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates. Composites Part B 60:274–83.

Beldjelili, Y., A. Tounsi, and S. R. Mahmoud. 2016. Hygro-thermo- mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory. Smart Structures and Systems 18 (4):755–86. doi:10.12989/sss.2016.18.4.755.

Bellifa, H., K. H. Benrahou, L. Hadji, M. S. A. Houari, and A. Tounsi. 2016. Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position. Journal of the Brazilian Society of Mechanical Sciences and Engineering 38 (1):265–75.

Benahmed, A., M. S. A. Houari, S. Benyoucef, K. Belakhdar, and A. Tounsi. 2017. A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foun-dation. Geomechanics and Engineering 12 (1):9–34. doi:10.12989/ gae.2017.12.1.009.

Benatta, M. A., A. Tounsi, I. Mechab, and M. B. Bouiadjra. 2009. Mathematical solution for bending of short hybrid composite beams with variable fibers spacing. Applied Mathematics and Computation 212:337–48. doi:10.1016/j.amc.2009.02.030.

Bennoun, M., M. S. A. Houari, and A. Tounsi. 2016. A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates. Mechanics of Advanced Materials and Structures 23 (4):423–31. doi:10.1080/15376494.2014.984088.

Bessaim, A., M. S. Houari, A. Tounsi, S. R. Mahmoud, and E. A. A. Bedia. 2013. A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. Journal of Sandwich Structures & Materials 15 (6):671–703. doi:10.1177/1099636213498888.

20 N. J. KANU ET AL.

Besseghier, A., M. S. A. Houari, A. Tounsi, and S. R. Mahmoud. 2017. Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory. Smart Structures and Systems 19 (6):601–14.

Bezensek, B., and A. Banerjee. 2010. A local approach model for cleavage fracture and crack extension direction of functionally graded materials. Engineering Fracture Mechanics 77 (17):3394–407. doi:10.1016/j. engfracmech.2010.09.002.

Bhardwaj, G., I. V. Singh, and B. K. Mishra. 2015. Stochastic fatigue crack growth simulation of interfacial crack in bi-layered FGMs using XIGA. Engineering Fracture Mechanics 284:186–229. doi:10.1016/j. cma.2014.08.015.

Bhattacharya, S., and K. Sharma. 2014. Fatigue crack growth simulations of FGM plate under cyclic thermal load by XFEM. Procedia Engineering 86:1257–62. doi:10.1016/j.proeng.2014.11.091.

Birman, V. 1995. Stability of functionally graded hybrid composite plates. Composites Engineering 5:913–21. doi:10.1016/0961-9526(95) 00036-m.

Birman, V. 1997. Stability of functionally graded shape memory alloy sandwich panels. Smart Materials and Structures 6:278–86. doi:10.1088/0964-1726/6/3/006.

Birman, V., and L. W. Byrd. 2007. Modeling and analysis of functionally graded materials and structures. Transactions of the ASME Applied Mechanics Reviews 60:195–216. doi:10.1115/1.2777164.

Bodaghi, M., and A. R. Saidi. 2010. Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Applied Mathematical Modelling 34 (11):3659–73. doi:10.1016/j.apm.2010.03.016.

Bouafia, K., A. Kaci, M. S. A. Houari, A. Benzair, and A. Tounsi. 2017. A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams. Smart Structures and Systems 19 (2):115–26. doi:10.12989/sss.2017.19.2.115.

Bouazza, M., A. Tounsi, E. A. Adda-Bedia, and A. Megueni. 2010. Thermoelastic stability analysis of functionally graded plates: An analytical approach. Computational Materials Science 49 (4):865–70. doi:10.1016/j.commatsci.2010.06.038.

Bouderba, B., M. S. A. Houari, and A. Tounsi. 2013. Thermomechanical bending response of FGM thick plates resting on Winkler–Pasternak elastic foundations. Steel and Composite Structures 14 (1):85–104. doi:10.12989/scs.2013.14.1.085.

Bouderba, B., M. S. A. Houari, A. Tounsi, and S. R. Mahmoud. 2016. Thermal stability of functionally graded sandwich plates using a simple shear deformation theory. Structural Engineering and Mechanics 58 (3):397–422. doi:10.12989/sem.2016.58.3.397.

Boukhari, A., H. A. Atmane, A. Tounsi, B. Adda, and S. R. Mahmoud. 2016. An efficient shear deformation theory for wave propagation of functionally graded material plates. Structural Engineering and Mechanics 57 (5):837–59. doi:10.12989/sem.2016.57.5.837.

Bounouara, F., K. H. Benrahou, I. Belkorissat, and A. Tounsi. 2016. A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation. Steel and Composite Structures 20 (2):227–49. doi:10.12989/scs.2016. 20.2.227.

Bourada, M., A. Kaci, M. S. A. Houari, and A. Tounsi. 2015. A new simple shear and normal deformations theory for functionally graded beams. Steel and Composite Structures 18 (2):409–23. doi:10.12989/ scs.2015.18.2.409.

Bousahla, A. A., S. Benyoucef, A. Tounsi, and S. R. Mahmoud. 2016. On thermal stability of plates with functionally graded coefficient of thermal expansion. Structural Engineering and Mechanics 60 (2):313–35. doi:10.12989/sem.2016.60.2.313.

Brindley, W. 1995. Properties of plasma sprayed bond coats. Proceedings of the Thermal Barrier Coating Workshop, NASA-Lewis, Cleveland, OR.

Bruck, H., and R. Doherty. 2007. Energetic systems and nanotechnology— A look ahead. International Journal of Energetic Materials and Chemical Propulsion 6 (1):39–48. doi:10.1615/intjenergeticmaterialschemprop. v6.i1.30.

Burlayenko, V. N., H. Altenbach, T. Sadowski, and S. D. Dimitrova. 2015. Computational simulations of thermal shock cracking by the virtual

crack closure technique in a functionally graded plate. Computational Materials Science 116:11–21. doi:10.1016/j.commatsci.2015.08.038.

Chakraborty, A., and S. Rahman. 2008. Stochastic multiscale models for fracture analysis of functionally graded materials. Engineering Fracture Mechanics 75 (8):2062–86. doi:10.1016/j.engfracmech.2007.10.013.

Chen, Y. F., and F. Erdogan. 1996. The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate. Journal of the Mechanics and Physics of Solids 44:771–87. doi:10.1016/0022-5096(96)00002-6.

Chen, X. L., and K. M. Lie. 2004. Buckling of rectangular functionally graded material plates subjected to nonlinearly distributed in-plane edge loads. Smart Materials and Structures 13 (6):1430–7. doi:10.1088/ 0964-1726/13/6/014.

Chen, J., Z. Liu, and Z. Zou. 2002. Transition internal crack problem for a nonhomogeneous orthotropic Strip_Mode I. International Journal of Engineering Science 40:1761–74. doi:10.1016/s0020-7225 (02)00038-1.

Cheng, Z., G. Zhang, Y. Wang, and F. Bobaru. 2015. A peridynamic model for dynamic fracture in functionally graded materials. Com-posite Structures 133:529–46. doi:10.1016/j.compstruct.2015.07.047.

Cherradi, N., A. Kawasaki, and M. Gasik. 1994. Worldwide trends in functional gradient materials research and development. Composites Engineering 4:883. doi:10.1016/s0961-9526(09)80012-9.

Chi, S.-H., and Y.-L. Chung. 2006. Mechanical behavior of functionally graded material plates under transverse load—Part I, Analysis. International Journal of Solids and Structures 43 (13):3657–74. doi:10.1016/j.ijsolstr.2005.04.011.

Chiong, I., E. T. Ooi, C. Song, and F. Tin-Loi. 2014. Computation of dynamic stress intensity factors in cracked functionally graded materials using scaled boundary polygons. Engineering Fracture Mechanics 131:210–31.

Choi, H. J. 2015. Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation. Mechanics Research Communications 78:93–99. doi:10.1016/j. mechrescom.2015.08.006.

Cinefra, M., S. Belouettar, M. Soave, and E. Carrera. 2010. Variable kinematic models applied to free-vibration analysis of functionally graded material shells. European Journal of Mechanics-A/Solids 29 (6):1078–87. doi:10.1016/j.euromechsol.2010.06.001.

Comi, C., and S. Mariani. 2005. Extended finite elements for fracture analysis of functionally graded materials. Proceedings of the 8th International Conference on Computational Plasticity, COMPLAS VIII, CIMNE, Barcelona (E. Onate and D. R. J. Owen, eds.).

Daouadji, T. H., A. Tounsi, and E. A. A. Bedia. 2013. Analytical solution for bending analysis of functionally graded plates. Scientia Iranica B 20 (3):516–23.

De Masi, T., K. D. Sheffler, and M. Ortiz. 1989. Thermal barrier coating life prediction model development. NASA Contractor Report 182230.

Delale, F., and F. Erdogan. 1983. The crack problem for a non- homogeneous plane. ASME Journal of Applied Mechanics 50 (3):609–14. doi:10.1115/1.3167098.

Dolbow, J. E., and M. Gosz. 2002. On the computation of mixed-mode stress intensity factors in functionally graded materials. International Journal of Solids and Structures 39:2557–74. doi:10.1016/s0020-7683 (02)00114-2.

Dong, C. Y. 2008. Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev–Ritz method. Materials & Design 29 (8):1518–25. doi:10.1016/j.matdes.2008.03.001.

Eischen, J. 1983. Fracture of nonhomogeneous materials. International Journal of Fracture 34:3–22.

El-Borgi, S., F. Erdogan, and L. Hidri. 2004. A partially insulated embedded crack in an infinite functionally graded medium under thermo-mechanical loading. International Journal of Engineering Science 42:371–93. doi:10.1016/s0020-7225(03)00287-8.

El-Hadad, S., H. Sato, E. Miura-Fujiwara, and Y. Watanabe. 2010. Fabrication of Al-Al3Ti/Ti3Al functionally graded materials under a centrifugal force. Materials 3:4639–56. doi:10.3390/ma3094639.

Erdogan, E., and B. H. Wu. 1993. Interface crack problems in layered orthotropic materials. Journal of Mechanics, Physics and Solids 41 (5):889–917. doi:10.1016/0022-5096(93)90004-y.

PARTICULATE SCIENCE AND TECHNOLOGY 21

Eshraghi, I., and N. Soltani. 2015. Thermal stress intensity factor expres-sions for functionally graded cylinders with internal circumferential cracks using the weight function method. Theoretical and Applied Fracture Mechanics 80:170–81. doi:10.1016/j.tafmec.2015.09.003.

Feng, W. J., and Z. Z. Zou. 2003. Dynamic stress field for torsional impact of a penny-shaped crack in a transversely isotropic functionally graded strip. International Journal of Engineering Science 41:1729–39. doi:10.1016/s0020-7225(03)00066-1.

Florczyk, S. J., and S. Saha. 2007. Ethical issues in nanotechnology. Journal of Long-Term Effects of Medical Implants. 17 (3):271–280. doi:10.1615/JLongTermEffMedImplants.v17.i3.90.

Forth, S. C., L. H. Favrow, W. D. Keat, and J. A. Newman. 2003. Three- dimensional mixed-mode fatigue crack growth in a functionally graded titanium alloy. Engineering fracture mechanics 70:2175–85. doi:10.1016/s0013-7944(02)00237-0.

Functionally Graded Materials. Victoria, BC, Canada: PPT by University of Victoria.

Goli, E., and M. T. Kazemi. 2014. XFEM modeling of fracture mechanics in transversely isotropic FGMs via interaction integral method. Proce-dia Materials Science 3:1257–62. doi:10.1016/j.mspro.2014.06.204.

Gu, P., and R. J. Asaro. 1997. Cracks in functionally graded materials. International Journal of Solids and Structures 34 (1):1–17. doi:10.1016/ 0020-7683(95)00289-8.

Gu, P., M. Dao, and R. Asaro. 1999. A simplified method for calculating the crack-tip field of functionally graded materials using the domain integral. Journal of Applied Mechanics 66:101–08. doi:10.1115/ 1.2789135.

Guo, L.-C., L. Wu, and L. Ma. 2004. The inverse crack problem under a concentrated load for a functionally graded coating-substrate composite system. Composite Structures 63:397–406.

Guo, L.-C., L.-Z. Wu, T. Zeng, and L. Ma. 2005. The dynamic fracture behavior of a functionally graded coating-substrate system. Composite Structures 64:433–42. doi:10.1016/j.compstruct.2003.09.044.

Gupta, A., M. Talha, and B. N. Singh. 2016. Vibration characteristics of functionally graded material plate with various boundary constraints using higher order shear deformation theory. Composites Part B 94:64–74. doi:10.1016/j.compositesb.2016.03.006..

Haghiri, H., A. R. Fotuhi, and A. R. Shafiei. 2015. Elastodynamic analysis of mode III multiple cracks in a functionally graded orthotropic half-plane. Theoretical and Applied Fracture Mechanics 80:155–169. doi:10.1016/j.tafmec.2015.09.006.

Hebali, H., A. Tounsi, M. S. A. Houari, A. Bessaim, and E. A. A. Bedia. 2014. A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. ASCE J. Engineering Mechanics 140:374–83. doi:10.1061/(asce)em.1943- 7889.0000665.

Holt, J. B., M. Koizumi, T. Hirai, and Z. A. Munir, eds. 1993. Proceedings of the second international symposium on functionally graded materials, ceramic transactions, vol. 34. Westerville, OH: American Ceramic Society.

Hosseini-Hashemi, S., M. Fadaee, and S. R. Atashipour. 2011a. A new exact analytical approach for free vibration of Reissner–Mindlin func-tionally graded rectangular plates. International Journal of Mechanical Sciences 53 (1):11–22. doi:10.1016/j.ijmecsci.2010.10.002.

Hosseini-Hashemi, S., M. Fadaee, and S. R. Atashipour. 2011b. Study on the free vibration of thick functionally graded rectangular plates according to a new exact closed form procedure. Composite Structures 93 (2):722–35. doi:10.1016/j.compstruct.2010.08.007.

Hosseini-Hashemi, S., M. Fadaee, M. Es’haghi. 2010. A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates. International Journal of Mechanical Sciences 52 (8):1025–35. doi:10.1016/j.ijmecsci.2010.04.009.

Hosseini-Hashemi, S., H. Salehipour, S. R. Atashipour, and R. Sburlati. 2013. On the exact in plane and out-of-plane free vibration analysis of thick functionally graded rectangular plates: Explicit 3D elasticity solutions. Composites Part B: Engineering 46:108–15.

Houari, M. S. A., A. Tounsi, A. Bessaim, and S. R. Mahmoud. 2016. A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates. Steel and Composite Structures 22 (2):257–76. doi:10.12989/scs.2016.22.2.257.

Houck, D. L. 1987. Proceedings of the First National Thermal Spray Conference, ASM International.

Ilschner, B., and N. Cherradi, eds. 1995. Proceedings of the 3rd international symposium on structural and functional gradient materials, Presses Polytechniques et Universitaires Romands, Lausanne, Switzerland.

Inan, O., S. Dag, and F. Erdogan. 2005. Three dimensional fracture analy-sis of, FGM Coatings. Functionally Graded Materials VIII (FGM2004), Proceedings of the 8th International Symposium on Multifunctional and Functionally Graded Materials, Materials Science Forum, vols. 492–493.

Jagtap, K. R., L. Achchhe, and B. N. Singh. 2013. Thermomechanical elastic post-buckling of functionally graded materials plate with random system properties. International Journal for Computational Methods in Engineering Science and Mechanics 14 (3):175–94. doi:10.1080/15502287.2012.711423.

Jagtap, K. K., A. Lal, and B. N. Singh. 2011. Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment. Composite Structures 93:3185–99. doi:10.1016/j.compstruct.2011.06.010.

Jain, N., and A. Shukla. 2004. Displacements, strains and stresses associated with propagating cracks in materials with continuously varying properties. Acta Mechanica 171:75–103. doi:10.1007/s00707- 004-0126-x.

Jamia, N., S. El-Borgi, and S. Usman. 2016. Non-local behavior of two collinear mixed mode limited-permeable cracks in a functionally graded piezoelectric medium. Applied Mathematical Modeling 40 (11): 5988–6005. doi:10.1016/j.apm.2016.01.036.

Janghorban, M., and A. Zare. 2012. Harmonic differential quadrature method for static analysis of functionally graded single walled carbon nanotubes based on Euler-Bernoulli beam theory. Latin American Journal of Solids and Structures 9:633–41. doi:10.1590/s1679- 78252012000600001

Jedamzik, R., A. Neubrand, and J. Rodel. 2000. Functionally graded mate-rials by electrochemical processing and infiltration: application to tungsten/copper composites. Journal of Materials Science 35:477–86.

Jin, Z.-H., and G. H. Paulino. 2001. Transient thermal stress analysis of an edge crack in a functionally graded material. International Journal of Fracture 107:73–98.

Jin, Z.-H., and G. H. Paulino. 2002. A viscoelastic functionally graded strip containing a crack subjected to in-plane loading. Engineering Fracture Mechanics 69 (14):1769–90. doi:10.1016/s0013-7944(02) 00049-8.

Kant, T., and R. K. Khare. 1997. A higher-order facet quadrilateral composite shell element. International Journal for Numerical Methods in Engineering 40 (24):4477–99. doi:10.1002/(sici)1097-0207 (19971230)40:24<4477::aid-nme229>3.0.co;2-3.

Kasmalkar, M. 1996. Surface and internal crack problems in a homogeneous substrate coated by a graded layer. PhD dissertation, Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA, USA.

Kawasaki, A., and R. Watanabe. 2002. Thermal fracture behavior of metal/ceramic functionally graded materials. Engineering Fracture Mechanics 69:1713–28. doi:10.1016/s0013-7944(02)00054-1.

Kaysser, W. A., and B. Ilschner. 1995. FGM research activities in Europe. MRS Bulletin 20 (1):22–26. doi:10.1557/s0883769400048879.

Khorshidi, M. A., M. Shariati, and S. A. Emam. 2016. Post buckling of functionally graded Nanobeams based on modified couple stress theory under general beam theory. International Journal of Mechanical Sciences 110:160–69. doi:10.1016/j.ijmecsci.2016.03.006.

Kim, J.-H., and G. H. Paulino. 2002. Mixed-mode fracture of orthotropic functionally graded materials using finite elements and the modified crack closure method. Engineering Fracture Mechanics 69:1557–86. doi:10.1016/s0013-7944(02)00057-7.

Kim, J.-H. and G. H. Paulino. 2003a. Mixed-mode J-integral formulation and implementation using graded finite elements for fracture analysis of nonhomogeneous orthotropic materials. Mechanics of Materials 35 (1–2):107–28. doi:10.1016/s0167-6636(02)00159-x.

Kim, J.-H., and G. H. Paulino. 2003b. An accurate scheme for mixed- mode fracture analysis of functionally graded materials using the

22 N. J. KANU ET AL.

interaction integral and micromechanics models. International Journal for Numerical Methods in Engineering 58:1457–97. doi:10.1002/ nme.819.

Kim, J.-H., and G. H. Paulino. 2003c. The interaction integral for fracture of orthotropic functionally graded materials: Evaluation of stress intensity factors. International Journal of Solids and Structures 40:3967–4001. doi:10.1016/s0020-7683(03)00176-8.

Kim, J.-H., and G. H. Paulino. 2003d. T-Stress, mixed-mode stress intensity factor, and crack initiation angles in functionally graded materials: A unified approach using the interaction integral method. Computer Methods in Applied Mechanics and Engineering 192:1463–94. doi:10.1016/s0045-7825(02)00652-7.

Kim, J.-H., and G. H. Paulino. 2004a. Simulation of crack propagation in functionally graded materials under mixed-mode and non- proportional loading. International Journal of Mechanics and Materials in Design 1:63–94. doi:10.1023/b:mamd.0000035457.78797.c5.

Kim, J.-H., and G. H. Paulino. 2004b. T-Stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms. International Jour-nal of Fracture 126:345–84. doi:10.1023/b:frac.0000031092.47424.f0.

Kim, J.-H., and G. H. Paulino. 2005a. Consistent formulations of the interaction integral method for fracture of functionally graded materi-als. ASME Journal of Applied Mechanics 72:351–64. doi:10.1115/ 1.1876395.

Kim, J.-H., and G. H. Paulino. 2005b. Mixed-mode crack propagation in functionally graded materials. Functionally Graded Materials VIII (FGM2004), Proceedings of the 8th International Symposium on Multifunctional and Functionally Graded Materials, Materials Science Forum, vols. 492–493.

Kirugulige, M. S., R. Kitey, and H. V. Tippur. 2005. Dynamic fracture behavior of model sandwich structures with functionally graded core: A feasibility study. Composites Science and Technology 65:1052–68. doi:10.1016/j.compscitech.2004.10.029.

Kokini, K., and S. V. Rangaraj. 2005. Time-dependent behavior and fracture of functionally graded thermal barrier coatings under thermal shock. Functionally graded materials VIII (FGM2004), Proceedings of the 8th International Symposium on Multifunctional and Functionally Graded Materials, Materials Science Forum, vols. 492–493.

Kong, Y., J. Liu, and G. Nie. 2016. Propagation characteristics of SH waves in a functionally graded piezomagnetic layer on PMN-0.29PT single crystal substrate. Mechanics Research Communications 73:107–12. doi:10.1016/j.mechrescom.2016.02.012.

Kumar, L., and S. P. Harsha. 2016. Effect of carbon nanotubes on CNT reinforced FGM nanoplate under thermo mechanical loading. Procedia Technology 23:130–37. doi:10.1016/j.protcy.2016.03.008.

Kumar, M. S. A., S. Panda, and D. Chakraborty. 2015. Design and analysis of a smart graded fiber-reinforced composite laminated plate. Composite Structures 124:176–95. doi:10.1016/j.comp.struct. 2014.11.015.

Lajevardi, S. A., T. Shahrabi, and J. A. Szpunar. 2013. Synthesis of func-tionally graded nano Al2O3–Ni composite coating by pulse electrode-position. Applied Surface Science 279:180–88. doi:10.1016/j.apsusc. 2013.04.067.

Lal, A., K. R. Jagtap, and B. N. Singh. 2013. Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties. Applied Mathematical Modeling 37:2900–20. doi:10.1016/j.apm.2012.06.013.

Lee, K. H. 2016. Analysis of a propagating crack tip in orthotropic functionally graded materials. Composites Part B: Engineering 84:83–97.

Lee, Y. D., and E. Erdogan. 1995. Residual stresses in FGM and laminated thermal barrier coatings. International Journal of Fracture 69:145–65.

Lee, C. Y., and J. H. Kim. 2013. Hygrothermal postbuckling behavior of functionally graded plates. Composite Structures 95:278–82. doi:10.1016/j.compstruct.2012.07.010.

Lezgy-Nazargah, M., P. Vidal, and O. Polit. 2013. An efficient finite element model for static and dynamic analyses of functionally graded piezoelectric beams. Composite Structures 104:71–84. doi:10.1016/j. compstruct.2013.04.010.

Li, S.-R., and R. C. Batra. 2013. Relations between buckling loads of func-tionally graded Timoshenko and homogeneous Euler–Bernoulli

beams. Composite Structures 95:5–9. doi:10.1016/j.compstruct. 2012.07.027.

Li, Q., V. P. Iu, and K. P. Kou. 2008. Three-dimensional vibration analysis of functionally graded material plates in thermal environment. Journal of Sound and Vibration 311 (1–2):498–515. doi:10.1016/j.jsv.2009.02.036.

Li, C., G. J. Weng, and Z. Duan. 2001. Dynamic behavior of a cylindrical crack in a functionally graded interlayer under torsional loading. International Journal of solids and Structures 38:7473–85. doi:10.1016/s0020-7683(01)00046-4.

Li, S. R., J. H. Zhang, and Y. G. Zhao. 2007. Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection. Thin-Walled Structures 45 (5):528–36. doi:10.1016/j.tws.2007.04.002.

Liew, K. M., X. Zhao, and Y. Y. Lee. 2012. Post buckling responses of functionally graded cylindrical shells under axial compression and thermal loads. Composites Part B: Engineering 43 (3):1621–30.

Lin, F., and Y. Xiang. 2014. Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Applied Mathematical Modeling 38 (15–16):3741–54. doi:10.1016/j. apm.2014.02.008.

Liu, D. Y., C. Y. Wang, and W. Q. Chen. 2010. Free vibration of FGM plates with in-plane material inhomogeneity. Composite Structures 92 (5):1047–51. doi:10.1016/j.compstruct.2009.10.001.

Liu, P., T. Yu, T. Q. Bui, and C. Zhang. 2013. Transient dynamic crack analysis in non-homogeneous functionally graded piezoelectric materi-als by the X-FEM. Computational Materials Science 69:542–58. doi:10.1016/j.commatsci.2012.11.009.

Ma, L.-S., and T.-J. Wang. 2004. Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory. International Journal of Solids and Structures 41:85–101. doi:10.1016/j.ijsolstr.2003. 09.008.

Mahdavian, M. 2009. Buckling analysis of simply-supported functionally graded rectangular plates under non-uniform in-plane compressive loading. Journal of Solid Mechanics 1 (3):213–25.

Mahi, A., and A. Tounsi. 2015. A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Applied Mathemat-ical Modelling 39:2489–508. doi:10.1016/j.apm.2014.10.045

Malekzadeh, P., and S. A. Shojaee. 2013. Dynamic response of functionally graded plates under moving heat source. Composites Part B: Engineering 44 (1):295–303.

Marin, A., and A. A. Fuentes. 2014. Human bone: Functionally graded material structures with complex geometry and loading. PPT by Department of Mechanical Engineering, the University of Texas-Pan American.

Mehboob, H., and S.-H. Chang. 2014. Evaluation of the development of tissue phenotypes: Bone fracture healing using functionally graded material composite bone plates. Composite Structures 117:105–13. doi:10.1016/j.compstruct.2014.06.019.

Meier, S. M., D. M. Nissley, and K. D. Sheffler. 1991. Thermal barrier coating life prediction model development. NASA Contractor Report 189111.

Meziane, M. A. A., H. H. Abdelaziz, and A. Tounsi. 2014. An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. Journal of Sandwich Structures & Materials 16 (3):293–318. doi:10.1177/ 1099636214526852.

Saunders, W. Modal analysis of rectangular simply-supported functionally graded plates, PPT.

Mohammadi, S. 2012. XFEM fracture analysis of composites. Iran: John Wiley and Sons, University of Tehran.

Nadeau, J., and M. Ferrari. 1999. Microstructural optimization of a functionally graded transversely isotropic layer. Mechanics of Materials 31:637–51. doi:10.1016/s0167-6636(99)00023-x.

Naei, M. H., A. Masoumi, and A. Shamekhi. 2007. Buckling analysis of cir-cular functionally graded material plate having variable thickness under uniform compression by finite-element method. Journal of Mechanical Engineering Science 221 (11):1241–7. doi:10.1243/09544062jmes636.

Najafizadeh, M. M., and M. R. Eslami. 2002. Buckling analysis of circular plates of functionally graded materials under uniform radial

PARTICULATE SCIENCE AND TECHNOLOGY 23

compression. International Journal of Mechanical Sciences 44 (12):2479–93. doi:10.1016/s0020-7403(02)00186-8

Najafizadeh, M. M., and H. R. Heydari. 2008. An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression. International Journal of Mechanical Sciences 50 (3):603–12. doi:10.1016/j.ijmecsci.2007.07.010.

Nami, M. R., and M. Janghorban. 2014. Resonance behavior of FG rectangular micro/nanoplate based on nonlocal elasticity theory and strain gradient theory with one gradient constant. Composite Structures 111:349–53. doi:10.1016/j.compstruct.2014.01.012

Nemat-Alla, M. 2003. Reduction of thermal stresses by developing two- dimensional functionally graded materials. International Journal of Solids and Structures 40:7339–56. doi:10.1016/j.ijsolstr.2003.08.017

Nemat-Alla, M., and N. Noda. 2000. Edge crack problem in a semi-infi-nite, FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading. Acta Mechanica 114:211–29. doi:10.1007/bf01170176

Neves, A. M. A., A. J. M. Ferreira, E. Carrera, R. M. N. Cinefra Mjorge, and C. M. M. Soares. 2013. Free vibration analysis of functionally graded shells by a higher-order shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations. European Journal of Mechanics-A/Solids 37:24–34. doi:10.1016/j.euromechsol.2012.05.005.

Nie, G. J., and Z. Zhong. 2007. Semi-analytical solution for three- dimensional vibration of functionally graded circular plates. Computer Methods in Applied Mechanics and Engineering 196 (49–52):4901–10. doi:10.1016/j.cma.2007.06.028.

Nie, G. J., and Z. Zhong. 2008. Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges. Composite Structures 84 (2):167–76. doi:10.1016/j.compstruct.2007.07.003.

Niino, M., and S. Maeda. 1990. Recent development status of functionally graded materials. The Iron and Steel Institute of Japan 30:699–703.

Ozturk, M., and F. Erdogan. 1997. Mode I crack problem in an inhomo-geneous orthotropic medium. International Journal of Engineering Science 35 (9):869–83. doi:10.1016/s0020-7225(97)80005-5.

Pan, H., T. Song, and Z. Wang. 2015. An analytical model for collinear cracks in functionally graded materials with general mechanical properties. Composite Structures 132:359–71. doi:10.1016/j.compstruct. 2015.05.055.

Park, J. S., and J. H. Kim. 2006. Thermal post buckling and vibration analyses of functionally graded plates. Journal of Sound and Vibration 289 (1–2):77–93. doi:10.1016/j.jsv.2005.01.031.

Parvathy, U., and K. P. Beena. 2013. Spline finite strip bending analysis of functionally graded plate using power-law function. International Journal of Scientific and Engineering Research 4 (5):163. ISSN 2229– 5518.

Paulino, G. H., and J.-H. Kim, 2004. A new approach to compute T-stress in functionally graded materials by means of the interaction integral method. Engineering Fracture Mechanics 71:1907–50. doi:10.1016/j. engfracmech.2003.11.005.

Paulino, G. H., and Z. Zhang. 2005. Dynamic fracture of functionally graded composites using an intrinsic cohesive zone model. Function-ally Graded Materials VIII (FGM2004), Proceedings of the 8th Inter-national Symposium on Multifunctional and Functionally Graded Materials, Materials Science Forum, vols. 492–493.

Petrova, V., and S. Schmauder. 2014. FGM/homogeneous bimaterial with systems of cracks under thermo-mechanical loading: Analysis by fracture criteria. Engineering Fracture Mechanics 130:12–20. doi:10.1016/j.engfracmech.2014.01.014.

Pindera, M.-J., J. Aboudi, S. M. Arnold, and W. F. Jones. 1995. Use of composites in multi-phased and functionally graded materials. Composites Engineering 5:743–974.

Pindera, M.-J., S. M. Arnold, J. Aboudi, and D. Hui. 1994. Use of composites in functionally graded materials. Composites Engineering 4:1–145.

Pipes, R. B., and N. J. Pagano. 1970. Interlaminar stresses in composite laminates under uniform axial extensions. Journal of Composite Materials 4:538–48. doi:10.1177/002199837000400409.

Pipes, R. B., and N. J. Pagano. 1974. Interlaminar stresses in composite laminates—An approximate elasticity solution. Journal of Applied Mechanics 41:668–72. doi:10.1115/1.3423368.

Pradyumna, S. A., and J. N. Bandyopadhyay. 2008. Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation. Journal of Sound and Vibration 318:176–92. doi:10.1016/j.jsv.2008.03.056.

Prakash, T., and M. Ganapathi. 2006. Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method. Composites Part B: Engineering 37 (7–8):642–9.

Prakash, T., M. Singha, and M. Ganapathi. 2008. Thermal post buckling analysis of FGM skew plates. Engineering Structures 30 (1):22–32. doi:10.1016/j.engstruct.2007.02.012.

Prakash, T., M. Singha, and M. Ganapathi. 2009. Influence of neutral surface position on the nonlinear stability behavior of functionally graded plates. Computational Mechanics 43 (3):341–50. doi:10.1007/ s00466-008-0316-9.

Rahman, S., and A. Chakraborty. 2011. Stochastic multiscale models for fracture analysis of functionally graded materials. Engineering Fracture Mechanics 78:27–46. doi:10.1016/j.engfracmech.2007.10.013.

Rahmani, O., and O. Pedram. 2014. Analysis and modeling the size effect on vibration of functionally graded Nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science 77:55–70. doi:10.1016/j.ijengsci.2013.12.003.

Rao, B. N., and M. Kuna. 2010. Interaction integrals for thermal fracture of functionally graded piezoelectric materials. Engineering Fracture Mechanics 77 (1):37–50. doi:10.1016/j.eng.frac.mech.2009.09.009.

Reddy, J. N., and Z. Q. Cheng. 2003. Frequency of functionally graded plates with three dimensional asymptotic approach. Journal of Engin-eering Mechanics 129 (18):896–900. doi:10.1061/(asce)0733-9399 (2003)129:8(896).

Roque, C. M. C., P. A. L. S. Martins, A. J. M. Ferreira, and R. M. N. Jorge. 2016. Differential evolution for free vibration optimization of functionally graded Nanobeams. Composite Structures 156:29–34. doi:10.1016/j.compstruct.2016.03.052.

Sahmani, S., M. M. Aghdam, and M. Bahrami. 2015. On the free vibration characteristics of post buckled third-order shear deformable FGM Nanobeams including surface effects. Composite Structures 121:377–85. doi:10.1016/j.compstruct.2014.11.033.

Salah, I. B., M. B. Amor, and M. H. B. Ghozlen. 2015. Effect of a function-ally graded soft middle layer on Love waves propagating in layered piezoelectric systems. Ultrasonics 61:145–50. doi:10.1016/j.ultras.2015. 04.011.

Salavati, H., Y. Alizadeh, A. Kazemi, and F. Berto. 2015. A new expression to evaluate the critical fracture load for bainitic functionally graded steels under mixed mode (I + II) loading. Engineering Failure Analysis 48:121–36. doi:10.1016/j.engfailanal.2014.11.005.

Sallai, B. O., A. Tounsi, I. Mechab, M. B. Bachir, M. B. Meradjah, and E. A. Adda. 2009. A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams. Computational Materials Science 44:1344–50. doi:10.1016/j.commatsci.2008.09.001.

Sampath, S., H. Herman, N. Shimoda, and T. Saito. 1995. Thermal spray processing of FGMs. MRS Bulletin 20 (1):27–31. doi:10.1557/ s0883769400048880.

Sayyaadi, H., and M. A. A. Farsangi. 2014. An analytical solution for dynamic behavior of thick doubly curved functionally graded smart panels. Composite Structures 107:88–102. doi:10.1016/j.compstruct.2013.07.039.

Shaghaghi, A. M., and M. Alfano. 2015. Determination of stress intensity factors of 3D curved non-planar cracks in FGMs subjected to thermal loading. Engineering Fracture Mechanics. doi:10.1016/j. engfracmech.2015.07.040.

Shariat, B. A. S., and M. R. Eslami. 2006. Thermal buckling of imperfect functionally graded plates. International Journal of Solids and Structures 43 (14–15):4082–96. doi:10.1016/j.ijsolstr.2005.04.005.

Shariat, B. A. S., and M. R. Eslami. 2007. Buckling of thick functionally graded plates under mechanical and thermal loads. Composite Structures 78 (3):433–9. doi:10.1016/j.compstruct.2005.11.001.

Shariat, B. A. S., R. Javaheri, and M. R. Eslami. 2005. Buckling of imper-fect functionally graded plates under in-plane compressive loading. Thin-Walled Structures 43 (7):1020–36. doi:10.1016/j.tws.2005.01.002.

24 N. J. KANU ET AL.

Shen, H.-S., and D.-Q. Yang. 2015. Nonlinear vibration of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators in thermal environments. Engin-eering Structures 90:183–92. doi:10.1016/j.engstruct.2015.02.005.

Shin, J. W., and T.-U. Kim. 2016. Transient response of a Mode III interface crack between piezoelectric layer and functionally graded orthotropic layer. International Journal of Solids and Structures 90:122–8. doi:10.1016/j.ijsolstr.2016.03.033.

Shiri, S. G., P. Abachi, K. Pourazarang, and M. M. Rahvard. 2015. Preparation of in-situ Cu/NbC nanocomposite and its functionally graded behavior for electrical contact applications. Transactions of Nonferrous Metals Society of China 25 (3):863–72. doi:10.1016/s1003- 6326(15)63675-5.

Shojaee, S., and A. Daneshmand. 2015. Crack analysis in media with orthotropic functionally graded materials using extended isogeometric analysis. Engineering Fracture Mechanics 147:203–27. doi:10.1016/j. engfracmech.2015.08.025.

Shul, C. W., and K. Y. Lee. 2002. A subsurface eccentric crack in a functionally graded coating layer on the layered half-space under an anti-plane shear impact load. International Journal of solids and Structures 39:2019–29. doi:10.1016/s0020-7683(02)00016-1.

Sih, G. C., and E. P. Chen. 1980. Normal and shear impact of layered composite with a crack: Dynamic stress intensification. Journal of Applied Mechanics 47:351–58. doi:10.1115/1.3153668.

Sih, G. C., P. Paris, and G. Irwin. 1965. On cracks in rectilinearly anisotropic bodies. International Journal of Fracture Mechanics 1 (3):189–203. doi:10.1007/bf00186854.

Sohn, K. J., and J. H. Kim. 2008. Structural stability of functionally graded panels subjected to aero-thermal loads. Composite Structures 82 (3):317–25. doi:10.1016/j.compstruct.2007.07.010.

Su, J., L.-L. Ke, and Y.-S. Wang. 2016. Axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space under a conducting punch. International Journal of Solids and Structures 90:45–59. doi:10.1016/j.ijsolstr.2016.04.011.

Su, Z., G. Jin, S. Shi, T. Ye, and X. Jia. 2014. A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions. International Journal of Mechanical Sciences 80:62–80. doi:10.1016/j.ijmecsci.2014. 01.002.

Sundararajan, N., T. Prakash, and M. Ganapathi. 2005. Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under thermal environments. Finite Elements in Analysis and Design 42 (2):152–68. doi:10.1016/j.finel.2005.06.001.

Swaminathan, K., D. T. Naveenkumar, A. M. Zenkour, and E. Carrera. 2015. Stress, vibration and buckling analyses of FGM plates—a state- of-the-art review. Composite Structures 120:10–31. doi:10.1016/j. compstruct.2014.09.070.

Szabo, B. A., and I. Babuska. 1991. Finite element analysis. New York: John Wiley and Sons.

Takahashi, H., T. Ishikawa, D. Okugawa, and T. Hashida. 1993. Laser and plasma-arc thermal shock/fatigue fracture evaluation procedure for functionally gradient materials. In Thermal shock and thermal fatigue behavior of advanced ceramics, eds. G. Schneider and G. Petzow. Dordrecht: Kluwer Academic Publishers.

Talha, M., and B. N. Singh. 2010a. Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Applied Mathematical Modeling 34:3991–4011. doi:10.1016/j.apm.2010. 03.034.

Talha, M., and B. N. Singh. 2010b. Thermo-mechanical induced vibration characteristics of shear deformable functionally graded ceramic–metal plates using the finite element method. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 225:50–65. doi:10.1243/09544062JMES2115.

Talha, M., and B. N. Singh. 2011a. Large amplitude free flexural vibration analysis of shear deformable FGM plates using nonlinear finite element method. Finite Elements in Analysis and Design 47 (4):394–401. doi:10.1016/j.finel.2010.11.006.

Talha, M., and B. N. Singh. 2011b. Nonlinear mechanical bending of functionally graded material plates under transverse loads with various boundary conditions. International Journal of Modeling, Simulation,

and Scientific Computing 2 (2):237–58. doi:10.1142/S17939623110 00451.

Tarancon, J. E., A. Vercher, E. Giner, and F. J. Fuenmayor. 2009. Enhanced blending elements for XFEM applied to linear elastic fracture mechanics. International Journal for Numerical Methods in Engineering 77:126–48. doi:10.1002/nme.2402.

Thai, H. T., T. Park, and D. H. Choi. 2012. An efficient shear deformation theory for vibration of functionally graded plates. Archive of Applied Mechanics 83 (1):137–49. doi:10.1007/s00419-012-0642-4.

Tilbrook, M., L. Rutgers, R. Moon, and M. Hoffman. 2005. Fracture and fatigue crack propagation in graded composites. Functionally Graded Materials, V. I.II (FGM2004), Proceedings of the 8th International Symposium on Multifunctional and Functionally Graded Materials, Materials Science Forum, vols. 492–493.

Tsai, Y. H., and C. P. Wu, 2008. Dynamic responses of functionally graded magneto-electroelastic shells with open-circuit surface con-ditions. International Journal of Engineering Science 46 (9):843–57. doi:10.1016/j.ijengsci.2008.03.005.

Tvergaard, V. 2002. Theoretical investigation of the effect of plasticity on crack growth along a functionally graded region between dissimilar elastic-plastic solids. Engineering Fracture Mechanics 69:1635–45. doi:10.1016/s0013-7944(02)00051-6.

Udupa, G., S. S. Rao, and K. V. Gangadharan. 2014. Functionally graded composite materials: An overview. Procedia Materials Science 5:1291–99. doi:10.1016/j.mspro.2014.07.442.

Ueda, S. 2006. Transient response of a center crack in a functionally graded piezoelectric strip under electromechanical impact. Engineering Fracture Mechanics 73:1455–71. doi:10.1016/j.engfracmech.2006. 01.025.

Ungbhakorn, V., and N. Wattanasakulpong. 2013. Thermo-elastic vibration analysis of third-order shear deformable functionally graded plates with distributed patch mass under thermal environment. Applied Acoustics 74:1045–59. doi:10.1016/j.apacoust.2013.03.010.

Uymaz, B., M. Aydogdu, and Filiz S. 2012. Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method. Compo-sites Structures 94 (4):1398–405. doi:10.1016/j.compstruct.2011.11.002.

Uysal, M. U. 2015. Numerical modeling of functional graded TiB coating in nano indentation on determination of mechanical properties. Materials Today: Proceedings 2 (1):217–23. doi:10.1016/j.matpr.2015. 04.025.

Vel, S. S., and Batra, R. C. 2004. Three-dimensional exact solution for the vibration of functionally graded rectangular plates. Journal of Sound and Vibration 272 (3–5):703–30. doi:10.1016/s0022-460x(03)00412-7.

Vena, P., D. Gastaldi, and R. Contro. 2005. Effects of the thermal residual stress field on the crack propagation in graded alumina/zirconia cer-amics. Functionally Graded Materials VIII (FGM2004), Proceedings of the 8th International Symposium on Multifunctional and Function-ally Graded Materials, Materials Science Forum, vols. 492–493.

Wang, H. M., and D. S. Luo. 2016. Exact analysis of radial vibration of functionally graded piezoelectric ring transducers resting on elastic foundation. Applied Mathematical Modeling 40 (4):2549–59.

Wang, Z.-H., L. Zhang, and L.-C. Guo. 2014. A viscoelastic fracture mechanics model for a functionally graded materials strip with general mechanical properties. European Journal of Mechanics-A/Solids 44:75–81. doi:10.1016/j.euromechsol.2013.10.008.

Wu, C. P., K. H. Chiu, and Y. M. Wang. 2011. RMVT-based meshless collocation and element free Galerkin methods for the quasi-3D free vibration analysis of multilayered composite and FGM plates. Compo-sites Structures 93 (5):1433–48. doi:10.1016/j.compstruct.2010.07.001.

Wu, C.-P., and X.-F. Lim. 2016. Coupled electro-mechanical effects and the dynamic responses of functionally graded piezoelectric film- substrate circular hollow cylinders. Thin-Walled Structures 102:1–17. doi:10.1016/j.tws.2016.01.008.

Wu, C.-P., and Y.-C. Liu. 2016. A review of semi-analytical numerical methods for laminated composite and multilayered functionally graded elastic/piezoelectric plates and shells. Composite Structures 147:1–15. doi:10.1016/j.compstruct.2016.03.031.

Wu, T. L., K. K. Shukla, and J. H. Huang. 2007. Post-buckling analysis of functionally graded rectangular plates. Composites Structures 81 (1):1–10. doi:10.1016/j.compstruct.2005.08.026.

PARTICULATE SCIENCE AND TECHNOLOGY 25

Xiong, H.-P., A. Kawasaki, Y.-S. Kang, and R. Watanabe. 2005. Experimental study of heat insulation performance of functionally graded metal/ceramic coatings and their behavior at high surface tem-perature. Surface and Coatings Technology 194:203–14. doi:10.1016/j. surfcoat.2004.07.069.

Yahia, S. A., H. A. Atmane, M. S. A. Houari, and A. Tounsi. 2015. Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories. Structural Engineering and Mechanics 53 (6):1143–65. doi:10.12989/sem.2015.53.6.1143.

Yamanouchi, M., M. Koizumi, T. Hirai, and I. Shiota, eds. 1990. Proceedings of the 1st International Symposium on Functionally Graded Materials. FGM-90, FGM Forum, Tokyo, Japan.

Yanga, J., and H. Shen. 2003. Non-linear analysis of functionally graded plates under transverse and in-plane loads. International Journal of Non-Linear Mechanics 38 (4):467–82. doi:10.1016/s0020-7462(01)00070-1.

Yas, M. H., and B. S. Aragh. 2011. Elasticity solution for free vibration analysis of four-parameter functionally graded fiber orientation cylindrical panels using differential quadrature method. European Journal of Mechanics-A/Solids 30 (5):631–8. doi:10.1016/j.euromechsol. 2010.12.009.

Yau, J. F. 1979. Mixed mode crack analysis using a conservation integral. PhD thesis, Dept. of Theoretical and Applied Mechanics, University of Illinois.

Yau, J. F., S. S. Wang, and H. T. Corten. 1980. A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. ASME Journal of Applied Mechanics 47 (2):335–41. doi:10.1115/1.3153665.

Ye, Z., and M. L. Ayari. 1994. Prediction of crack propagation in anisotropic solids. Engineering Fracture Mechanics 49 (6):797–808. doi:10.1016/0013-7944(94)90017-5.

Yin, H. M., L. Z. Sun, and G. H. Paulino. 2004. Micromechanics- based elastic model for functionally graded materials with particle interactions. Acta Materials 52:3535–43. doi:10.1016/j.actamat. 2004.04.007.

Zenkour, A. M. 2006. Generalized shear deformation theory for bending analysis of functionally graded plates. Applied Mathematical Modeling 30:67–84. doi:10.1016/j.apm.2005.03.009.

Zenkour, A. M., D. S. Mashat, and K. A. Elsibai. 2009. Bending analysis of functionally graded plates in the context of different theories of thermoelasticity. Mathematical Problems in Engineering 2009:1–15. doi:10.1155/2009/962351.

Zhang, D. G. 2013. Modeling and analysis of FGM rectangular plates based on physical neutral surface and high order shear deformation theory. International Journal of Mechanical Sciences 68:92–104. doi:10.1016/j.ijmecsci.2013.01.002.

Zhang, X., F. Chen, and H. Zhang. 2013. Stability and local bifurcation analysis of functionally graded material plate under transversal and in-plane excitations. Applied Mathematical Modelling 37 (10–11):6639–51. doi:10.1016/j.apm.2013.01.031.

Zhang, H. H., and G. W. Ma. 2014. Fracture modeling of isotropic functionally graded materials by the numerical manifold method. Engineering Analysis with Boundary Elements 38:61–71. doi:10.1016/ j.enganabound.2013.10.006.

Zhang, Z., and G. H. Paulino. 2005. Cohesive zone modeling of dynamic failure in homogeneous and functionally graded materials. International Journal of Plasticity 21 (6):1195–254. doi:10.1016/j. ijplas.2004.06.009.

Zhang, H., X. Zhao, and J. Su. 2012. Random dynamic response and reliability of a crack in a functionally graded material layer between two dissimilar elastic half-planes. Engineering Analysis with Boundary Elements 36 (11):1560–70. doi:10.1016/j.enganabound.2012.04.010.

Zhang, H., X. Zhao, J. Zhang, and Z. Zhou. 2012. Random dynamic response of crack in functionally graded materials layer for plane problem. Transactions of Nonferrous Metals society of China 22 (Suppl. 1):s198–s206. doi:10.1016/s1003-6326(12)61709-9.

Zhou, Z. G., B. Wang, and Sun, Y.-G. 2004. Investigation of the dynamic behavior of a finite crack in the functionally graded materials by the use of the Schmidt method. Wave Motion 39:213–25. doi:10.1016/j. wavemoti.2003.09.001.

Zhu, P., and K. M. Liew. 2011a. A local Kriging meshless method for free vibration analysis of functionally graded circular plates in thermal environments. Procedia Engineering 2012 (31):1089–94. doi:10.1016/j. proeng.2012.01.1147.

Zhu, P., and K. M. Liew. 2011b. Free vibration analysis of moderately thick functionally graded plates by local Kriging meshless method. Composites Structures 93 (11):2925–44. doi:10.1016/j.compstruct. 2011.05.011.

Zidi, M., A. Tounsi, M. S. A. Houari, and O. A. Bég. 2014. Bending analysis of FGM plates under hygro-thermomechanical loading using a four variable refined plate theory. Aerospace Science and Technology 34:24–34. doi:10.1016/j.ast.2014.02.001.

26 N. J. KANU ET AL.