CHAPTER 1

INTRODUCTION

1.1 Overview

A general layout of composite structure consists of many constituent layers of the lamina

bonded together with at least two different kinds of materials at macroscopic scale.

Lamination is being performed to unite the superior aspects of the materials present in

constituent layers and bonding material, so as to achieve a more functional material that

satisfy the design requirements. In spite of the above merits, mismatch of mechanical/

thermal properties exists at layer interfaces often made them to sustain delamination and

debonding types of failure modes, especially under high operating environments. In

addition, the cracks may appear in the layer interfaces which generally propagate into

weaker direction or lamina and leads to stress concentrations at the layer interfaces under

external loading conditions.

The aforementioned drawbacks experienced by conventional composite laminates

paves the path to discover a new kind of materials, where these problems can be addressed

in an optimum/efficient manner. As a consequence, during a space plane project in 1984 at

National Aerospace Laboratory of Japan, a group of scientists discovered a class of

advanced composite materials named as functionally graded materials (FGM). These

advance materials with engineered gradients of composition, structure and/or specific

properties in the preferred direction/orientation are superior to corresponding homogeneous

material composed of identical constituents (Koizumi 1993). Though the concept of FGMs

and their fabrication technology appears to be an engineering invention, the concept is not

new. These kinds of materials are practically accessible in plenty of forms such as Bamboo

tree, Human skin and Human bone. Although several numbers of spatial and chemical

configurations are possible in FGMs, in common, they involve a blend of two distinct

isotropic material phases. The resultant material is microscopically heterogeneous and

hence introduces the gradients by means of spatial variation of material properties along the

specified co-ordinates. Meanwhile, proper care has to be exercised while tailoring these

property gradients so as to achieve the benefit of two constituents. In most of the research

studies performed so far, FGM structure constituted by the combination of metal and

ceramic are accepted for their excellent outputs. Usually, large concentrations of ceramic

material are deposited at high temperature locations, while large concentrations of metal are

deposited at regions where mechanical strength is of great concern. The region in between

 

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these large concentrations of ceramic and metal thus results in smooth and continuous

variation of desired properties (mechanical/thermal/electrical) in the chosen direction.

The thermal conductivity of ceramics is very low compared with that of

corresponding metal part. As a result uneven stress distribution and expansion may occur in

a structure leading to fracture. In such situations, to overcome the brittle character of the

ceramic component, metals are combined with ceramics. This mixture of two distinct

isotropic materials is probable with two alternative options. The first choice is the

introduction of metal layers into ceramic structure, but it introduces interfaces and thus

results in disparity of material properties. Again this result into large concentration of

stresses, which is the undesirable effect often encountered in conventional composites. As

an alternative approach, a mixture of ceramic with the metal that exhibit a smooth and

continuously varying proportion of volume content seems to be better configuration at this

point. Thus the resulting structural component (i.e., FGM) may have gradual variation of

material properties without any interfaces and able to withstand ultra high temperature in

addition to high fracture toughness.

A general FGM structure usually refers to particulate composites where the volume

fraction of constituents varies in one or more directions (Birman and Byrd 2007). FGM may

also be formed incorporating fiber-reinforced layers with the volume fraction of constituent

phases as coordinate dependent instead of being constant and thus producing the optimal set

of structural response (Birman 1995; Birman 1997). While particulate type FGMs may be

isotropic at local domain, they may also exhibit heterogeneous properties due to the spatial

distribution of volume fraction of the constituents. Besides the particulate type of FGMs,

skeletal/orthotropic microstructure may be also manufactured using plasma spray and

electron beam physical vapor deposition techniques. In some situations, FGM may include

ceramics and two different metallic phases where the gradual variation of material

properties is achieved in the thickness direction according to power law distribution. Such a

configuration has proven to be very effective in reducing thermal stresses when compared

to conventional two-phase materials (Nemat Alla 2003).

With the potential to reduce the in-plane and transverse stresses, to improve residual

stress distribution, to enhance thermal properties, to give high fracture toughness, and

reduced stress intensity factors, FGM components registered wide applications in many

engineering and other allied fields. Supersonic flight jets, rocket components, heat

exchanger tubes, biomedical implants, flywheels and plasma facings are some of the notable

 

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fields that dominate in the list of applications. With superior thermal properties, FGM

materials are proven to be capable of resisting a temperature of 2000K with a temperature

gradient of 1000K across a section of 10mm thickness (Jha et al. 2013b). Over a course of

time, these materials have expanded their applications into chemical plants, solar energy

generators, heat exchangers, nuclear and chemical reactors, and high efficiency combustion

systems. To improve thermal, oxidation and corrosion properties they further extend their

application as coatings in thermal barriers systems. In thermo-electric field, the concept of

functional grading has been successfully implemented in sensors and thermo generators

having metal-semiconductor transition with improved efficiency. A brief chart showing the

applications of FGMs in diverse areas has been demonstrated in Figure 1.1.

Figure.1.1 Chart showing various potential areas of application of FGMs (Miyamoto 1999)

The primary step in any analysis involves the assumption of suitable kinematics

field to accurately predict the response of the system under various operating environments.

In this context, to obtain the realistic response of the structure, many displacement based

theories are developed and proposed in the literatures. Classical plate theory (CPT)

proposed by Kirchhoff (1850) is the first theory which was implemented by many

researchers for the analysis of thin plate/shell structures. But CPT neglects the effect of

shear deformation and further assumes that a normal to the mid-plane remains straight and

normal to the mid-surface after deformation. As a consequence, the CPT usually

FGM

Nuclear Projects

(Fuel pellets, Plasma wall of fusion reactor)

Space Projects

(Rocket components, Space plane frames)

Medical field

(Artificial bones, Skins, Dentistry)

Energy Sector

(Thermoelectric generators, Solar cells, Sensors)

Miscellaneous

(Building materials, Sport goods, Window glasses)

Communication field

(Optical fibers, Lenses, Semiconductors generators,

Solar cells) Sensors)

 

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underestimates the deflection parameter and overestimates the natural frequencies and

buckling loads especially for thick plates (Reddy 2004). Also, this theory may be highly

unsuitable for structures made of FGM due to the phenomenon of continuous and gradual

volume distribution of two materials observed from point to point in the specific direction.

The major limitation of this theory has been realized during the analysis of thick plates,

where the contribution of shear deformation cannot be neglected. To propose an alternative

solution, a theory which considers the variation of shear deformation in linear sense is

introduced by Reissner-Mindlin (Reissner 1945b; Mindlin 1951) as first order shear

deformation theory (FSDT). But, the linear assumption of shear deformation variation leads

to the use of shear correction factor in order to account for the realistic parabolic variation

of transverse shear strain. As cited in most of the literatures, these factors are very sensitive

to the geometric properties of plates, loading and boundary conditions. To address the

issues related with CPT and FSDT, many higher order shear deformation theories were

proposed (Reddy 1984b; Lo et al., 1967) to accomplish the realistic parabolic variation of

transverse stresses through the thickness. Generally HSDT involve the higher order term in

the Taylor’s expansion of the displacement component along the thickness direction.

Recently, different forms of HSDT models were proposed in the literature incorporating

sinusoidal, cubic, hyperbolic and exponential variations in the in-plane fields with constant

variation of transverse displacement (Thai and Choi 2013b) through the thickness, while

many others adopted the quadratic and cubic variation of transverse component in addition

to cubic variation of in-plane part for the analysis of FGM structures (Jha et al. 2012b; Jha

et al. 2013c; Talha and Singh 2010).

Skew plates belongs to the quadrilateral plate family are often identified as

parallelogram plate, swept plate, rhombic plate and oblique plate and serve as major load

bearing components in many structures. FGM skew plates have wide applications in areas,

where some constraints on straight alignment of structures are encounter/necessary with the

demand of resisting high temperature environments without losing their structural integrity.

They offer potential benefits in construction industry in the form of reinforced slabs/plates,

stiffened fiber-reinforced plastic super-structures, floors in bridges, ship hulls,

parallelllogram slabs in buildings and deck/skew grid of beams and girders. In aerospace

industry, swept wings of airplanes are idealized as substitute structures in the form of skew

plates. Also complex alignment problems in bridge panels are solved by the use of plates

with skew boundary due to functional, aesthetic or structural requirements. In FGM skew

plates/shells, the skew angle is a prime key factor influencing the static, dynamic, and

 

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buckling response of the structure. However, there exists strong singularity at the obtuse

vertex and hence the computation effort required for analysis of such structures increases

with increasing skew angle.

Over a period of time, the concept of functional grading is employed in sandwich

type construction, owing to their gradual and continuous variation of mechanical/thermal

properties at layer interfaces, which is not achievable in traditional sandwich arrangement.

If FGM technique is implemented in a sandwich layers, the core material could be designed

such that stiffness vary gradually from the high value at the interfaces to a lower value at

the centre, thus eliminating the large jump in material properties and hence avoids the stress

concentrations (Woodward and Kashtalyan 2011). In FGMs with sandwich layers, modeling

of plate/shell structure can be achieved in two alternative ways. In former case, the

homogenous ceramic core is introduced between the two layers of metal; thereby the

interface layer utilizes the concept of FGM. Alternatively, top and bottom skins are

occupied by the ceramic and metal isotropic materials, respectively; thereby the material in

the core portion obeys the rule of graded distribution of the constituent materials. In both

models, the gradation of material properties is governed by the volume fraction of the

individual constituent materials (Zenkour 2006). The definition of FGM could also be

employed effectively in smart structures where actuators and sensors are placed in the face

layers by appropriately selecting the core part (Xiang et al. 2010; Alibeigloo 2010; Loja et

al. 2013).

Thin walled structural members like plates and shells, used in reactor vessels,

turbines and other machine parts can experience large elastic deformations and finite

rotations and hence susceptible to failure due to excessive stresses induced by thermal or

combined thermo mechanical loading conditions. In such situations, analysis has to be

performed by considering geometric nonlinearity to predict the large deformation responses.

While doing so, non-linear strain part has to be taken care of for describing the strain

components. For non-linear analyses, strain part includes Green-Lagrange strain relation by

incorporating quadratic terms of in-plane and transverse displacement components. The

conditions of derivatives of in-plane displacement components with respect to Cartesian co-

ordinates are small and transverse displacement is independent of thickness co-ordinate (z)

are imposed on the Green-Lagrange relation to arrive for von Kármán form of strains (Fung

1965). Even though, some quadratic and cubic terms are present in von Kármán strain part,

the inadequacy is observed in the form of not capable of defining the case of moderate

rotations. When large rotations are encountered, von Kármán strains are proven to be

 

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unsuitable to describe the actual boundary conditions, since it is based on the undeformed

co-ordinate system (Pai 2007). Hence considering the full geometric nonlinearity in terms

of presence of quadratic terms of displacement components seems to be vital to describe the

structural response under large amplitudes. Due to the presence of all non-linear terms, the

formulation involves mathematical complexities in the form of large matrix sizes.

During their service life, plate and shell elements are exposed to various types of

transverse and in-plane mechanical and thermal loadings. Hence stability analysis of these

structures under such loading conditions is one of the major issues associated with the safe

and optimum design. The plate/shell structure exhibit reserve strength after the critical load

is reached, which is generally described as post-buckling strength. It is well known

observation that the plate/panel structures are capable of carrying additional load to a large

extent after buckling without any signs of failure. In order to fully exploit the strength of

FGM plates/shells in carrying in-plane loads, an accurate prediction of their load resisting

capacity in post buckling region forms essential topic in this research area.

By considering the aforementioned aspects regarding analysis and behavior of

graded structures, development of an efficient and ingenious model based on accurate

numerical tool seems to be an imperative task for researchers engaged in this field. In this

connection, suitable kinematics fields that incorporate realistic variation of transverse

displacement with the inclusion of bending and shear terms is necessary to accurately

predict the static and dynamic response of FGM structures, which generally exhibit the

stretching-bending phenomenon. Further, exploiting suitable non-linear terms in the strain

part is necessary to obtain the solutions close to the practical situation for the cases

involving finite strains and moderate rotations or large deformations. Keeping the above

imperative aspects in mind, application of graded concept in sandwich plates/shells

considering linear/non-linear analyses becomes very useful for designers and researches to

arrive optimum design. An accurate modeling, simple analysis and effective design of FGM

structures with the features of skew boundary and sandwich layers based on the above

criteria would certainly serve as milestone in the field of material research.

1.2 Objectives and Scope of the Present Research

The objective of present investigation is to develop an efficient and simple 2D

model for the analysis of single/layered FGM structure using displacement based finite

element method. The proposed numerical approach should able to incorporate the

mechanical and thermal analyses considering the thermal-dependent properties in both

 

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linear and non-linear sense. A FGM plate/shell with skew geometry is also to be modeled

by suitable transformation of boundary conditions from global co-ordinates into local

domain. FGM sandwich plates/shells are also to be modeled with either graded core or

homogeneous core in order to utilize the optimum material properties in an effective way.

In addition, to accurately predict the large deformation behaviour of such structures,

geometric non-linear analysis and buckling analysis beyond critical load range (i.e., post

buckling behavior) are to be performed.

Based on the aforementioned objectives, the scope of the present investigation

encompasses the following salient features.

i. To perform the linear/non-linear analysis of single/three layer (sandwich) FGM

structure with/without skew boundary considering a kinematics field that

incorporates constant/quadratic variation of thickness terms in defining the

transverse displacement (w). In-plane displacement fields (u and v) are assumed to

have cubic variations across the thickness. Such a model should be able to

incorporate the effect of normal strain and realistic transverse deformation in

efficient manner.

ii. In case of spatial variations of mechanical/thermal properties as in FGM, it is not

wise to ignore the heterogeneous nature of RVE (representative volume element). In

such cases, methods which consider the grading concept at both microscopic and

macroscopic level seem to be appropriate to accurately define the effective

properties. To include this aspect of FGM, Mori-Tanaka Scheme and rule of

mixture methods of homogenization are employed in the present study.

iii. Based on the proposed C0 finite element formulation and homogenization scheme, a

numerical code is initially developed in FORTRAN 90. To utilize a more versatile

numerical platform for solving different problems (e.g. sandwich FGM plates/shells)

especially where geometric non-linearity or post bucking analyses are done,

MATLAB (R2013b) tool is widely used for different analyses.

iv. Thermal analysis is done for linear static, free vibration and buckling problems by

incorporating temperature-dependent properties of the constituents. Non-linear

through-the thickness thermal distribution is assumed by virtue of graded thermal

properties of FGMs.

v. To ensure suitable assumption of core thickness having graded or isotropic material

with respect to total thickness for FGM sandwich plate/shell structures, various

 

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schemes have been modeled that incorporate different core layer thickness with

respect to total/face sheet thickness.

vi. The developed computer codes based on FORTRAN 90 and MATLAB (R2013b)

are successfully applied to solve the static (linear/ geometric non-linear), dynamic

(free/forced response) and buckling/post buckling responses of FGM plate/shell

structures having single/sandwich layers with/without skew boundary. While

solving the problems, different choices of ceramic and metal constituents are

considered. Several parameters such as aspect ratio, side-thickness ratio, radius-

thickness ratio, skew angle, boundary conditions, shell curvatures, and volume

fraction index are considered to show their influences.

vii. Comprehensive numerical results are presented in the form of tables and graphs to

show different responses of single/multi layer FGM structures, which should be

useful for researchers/engineers working in the field. 

 

1.3 Organization of thesis

A brief overview and various issues that serve as motivation for the present investigation

are summarized in Chapter 1. The objectives and scope of the present research work are

also described in the end of Chapter 1.

Chapter 2 presents the brief overview of the existing literatures that serve as

background for the present research. In the first part, various shear deformation theories

available for the static, free/forced vibration and buckling analyses are discussed along with

the merits of each theory. Subsequently, the research works related to the analysis of

composite/FGM skew plates under mechanical and thermal loadings are discussed. In

addition, the works related to analysis of FGM sandwich plates are elaborated along with

their interesting findings. In the end, non-linear bending and post buckling studies

performed on laminated composites/FGM structures are outlined.

Chapter 3 explains the brief mathematical formulation based on constant and

quadratic variation of transverse displacement in the kinematics field. The formulation that

assumes constant transverse displacement is based on Reddy’s higher order shear

deformation theory with the implementation of C0 isoparametric formulation. The first part

covers the governing equations for linear static, vibration, dynamic and buckling analysis

followed by non-linear formulation for bending and post buckling analyses. Solution

techniques adopted for different analyses combined with computer coding are included at

the end of the chapter.

 

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Chapter 4 demonstrates the application of the present formulation in solving

various numerical problems related to functionally graded plates/shells under different

loading conditions. In each case, the applicability of the developed coding based on C0

finite element formulation has been ensured by comparing the present results with the

results published in literatures by performing the convergence study. Numerical examples

of FGM skew plates/shells are presented for different linear analyses by varying different

parameters such as aspect ratio, thickness ratio, curvature ratio, boundary conditions, skew

angle and volume fraction parameter. In case of sandwich arrangement, the influence of

material grading on thickness range of bottom/core/top layers of plate/shell are studied

under two different types of modeling. Finally, non-linear analysis is performed for bending

and post buckling of FGM plates/shells. Several new results are presented for linear and

non-linear analyses of FGM skew plates/shells considering single/sandwich configurations.

Chapter 5 summarizes different conclusions and observations of the present

research work in a concise form. Lastly, the scope of the future works in context to the

present research is described followed by the list of publications in Journals and

conferences accomplished from the present research work.

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CHAPTER 2 LITERATURE REVIEW

2.1 Introduction FGM structures are highly inhomogeneous, by virtue of choice of material constituents.

In addition if skew alignment is encountered in such structures, the analysis becomes more

tedious and hence proper attention has to be paid to ensure the reliable and optimum design. In

this context, an efficient and simple 2D modeling of FGM structures based on accurate

analytical/numerical technique is essential which should also be able to predict the results as

accurate as 3D elasticity solutions. Further, if sandwich layers are incorporated in FGM

configuration proper modeling of core and face sheet layers relies on suitable choice of material

constituent. More often these structures undergo large amplitudes with moderate rotations. In

such cases, sufficient knowledge should be acquired regarding the effect of geometric non

linearity and buckling response beyond critical load point. To solve the aforementioned issues,

different analytical/semi analytical/numerical solution strategies are proposed by many

investigators in the past, by considering appropriate displacement field and homogenization

scheme.

Since this research area is relatively new and gaining more attention from researches, a

vast body of research summary exists in the literature for static, dynamic and buckling

analyses. Hence by keeping the aim and objective of the present investigation in mind, only the

literatures that are related to the current research topic are reviewed in this chapter. Meanwhile,

care has been taken to include the recent literatures that are related to this topic. Since limited

number of literatures are available regarding FGM skew/sandwich plates/shells under large

deformation analysis, some literatures related to composite skew/sandwich plates/shells are

also appraised. Before proceeding to the present research problem, a critical survey and

assessment of the existing literatures have been performed which are categorized into the

following sections.

Brief review of various linear/non-linear shear deformation theories of plates/shells

Static, free vibration, buckling and dynamic response of FGM plates/shells

Static, free vibration, buckling and dynamic response of

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o Laminated composite skew plates/shells

o FGM skew plates/shells

Static, free vibration and buckling response of

o Laminated composites sandwich plates/shells

o FGM sandwich plates/shells

Geometric non-linear analysis of

o Laminated composite plates/shells

o FGM plates/shells

Post buckling responses of

o Laminated composite plates/shells

o FGM plates/shells

In addition to above sections, a section incorporating the summary and conclusions of different

important and relevant literatures is also included at the end.

 

2.2 Brief review of various shear deformation theories of plates/shells   This section is separated into four sub sections to provide a brief outline about the

various shear deformation theories developed for laminated composite plates/shells. At first,

various shear deformation theories developed for the analysis of laminated composite plate are

discussed followed by the implementation of such theories for linear/non-linear analysis of

functionally graded plates in section two. While third section comprises a detailed discussion

on linear/non-linear shell theories developed for analysis of laminated composites and the

section ends by providing necessary information about application of shell theories to analyze

functionally graded shells. The reference order is not intended to imply priority of any

particular theory.

A solution accuracy of any analysis problem largely depends on the assumed

displacement field based on which the strain equations are formulated. In order to capture the

accurate profile of shear deformation, many theories were proposed and implemented to

analyze the static and dynamic response of structures under complex loading conditions.

Initially, two plate theories are discussed which are widely adopted in most of the literatures to

model the plate geometry. The first theory does not consider the effect of transverse shear

deformation; while the second theory accounts for it. In both the theories, normal stress in the

15  

thickness direction was assumed as zero. In 1850, the first theory is emerged and named as

thin-plate theory or Kirchoff’s theory. The second theory is popularly known as Mindlin,

Reissner-Mindlin and Mindlin-Reissner theory (1944; 1945a; 1947). The initial credit goes to

Ashton and Whitney (1970) for implementing CPT for the analysis of composite structures.

Linear and constant variation of in-plane and shear strain, respectively, is assumed in the

investigation. The main drawback of the theory comes in the form of neglecting the transverse

shear contribution in the deformation process. As a further development, displacement theory

that includes the effect of rotary inertia and shear is formulated by Mindlin (1951) for elastic

isotropic plates. Later on, the Mindlin theory (1951) was extended to laminated anisotropic

plates (Yang et al. 1966; Whitney and Leissa 1969; Whitney and Pagano 1970), named as first

order shear deformation theory (FSDT) and provides a balance between computational

efficiency and accuracy at global domain and suitable for thin and moderately thick laminates

plates. But the theory fails to predict the responses at local domain, for example, the

interlaminar stress distribution between layers, delaminations, and etc. Since the theory

consider the transverse shear strain in constant sense, use of arbitrary number known as shear

correction factor is mandatory in the analysis (Pandit 2009).

It has been observed in many literatures that the performance of the FSDT is primarily

dependent on these shear correction and which again fluctuate according to geometry,

boundary and loading conditions (Liu et al. 2007; Reddy 2007). To offer the solution for the

various difficulties encountered in the above theories (CPT and FSDT), higher power of

thickness term is incorporated in in-plane displacement fields to describe the actual warping of

cross sections, accurately. The benefit of elimination of shear correction factor recommended

by HSDT by incorporating realistic variation of transverse shear deformation made them to be

employed in various analyses, thereafter. In developing HSDT for analysis of plate structures, a

significant contribution was recorded by Hildebrand et al. (1949) and Basset (1890). As further

improvement in HSDT, Lo et al. (1977a, b) proposed higher order plate theory incorporating

non-linear distribution of in-plane displacement with respect to thickness co-ordinate for

homogenous and laminated plates. The theory incorporates four terms in in-plane part in

addition to three terms for transverse displacement and thus finally leads to eleven unknowns in

the kinematics model.

As a continued effort, different higher order models were proposed in the literature to

account for realistic variation of transverse shear stresses and strains (Nelson and Larch 1974;

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Cho et al 2007; Makhecha et al. 2001) and highlighted the importance of higher order terms in

predicting the accurate structural responses. To further refine HSDT, cubic and linear variation

of in-plane and transverse displacement, respectively, is considered in the Taylor’s expression

by Kant et al. (1982) to analyze the thin and thick composite plates. But the disadvantage lies

with the higher number of unknowns exists in the theory compared to FSDT. To solve this

issue, a simple higher order theory is proposed by Reddy (1984b) for analysis of laminated

plates. The proposed theory has cubic variation for in-plane part, while constant variation was

assumed for transverse displacement component. The various higher order unknowns exists in

the kinematics field are solved by the satisfaction of free boundary condition of transverse

stresses on the upper and lower surfaces. While doing so, the number of unknowns are

considerably reduced which obviously reduce the computational effort required for the

analysis. In addition to above theories, the 3D continuum-based theory is also employed to

predict the interlaminar stresses in a composite laminates, but the computational cost is a major

concern of this theory.

To evaluate the transverse stresses in composite and sandwich laminates, a set of higher

order theories were proposed by Kant and Manjunatha (1994) by employing C0 iso-parametric

finite elements. For analysis of hybrid/smart graded plates, a higher order shear and normal

deformation theory is employed by Shiyekar and Kant (2010). The electric field is

approximated as layer wise linear model through the thickness of the piezoelectric fibre

reinforced composites, while displacement function is approximated by Taylor’s series

expansion. Further, a refined higher order model is proposed by Swaminathan and Patil (2008)

to solve the natural frequency of simply supported anti-symmetric angle-ply composites and

sandwich plates. Some of the other higher order theories that identify seven unknowns (Kant

and Pandya 1988), nine unknowns (Ren 1986; Pandya and Kant 1988b; Pradyumna and

Bandyopadhyay 2008), (twelve unknowns Jha et al. 2013c) in the assumed kinematic model

can also be located in the literature. While few other available higher-order theories propose

equivalent number of unknowns as in FSDT e.g., third-order shear deformation theory (Reddy

1984b; Reddy 2000), sinusoidal shear deformation theory (Touratier 1991; Ferreira et al 2005b;

Zenkour 2006), hyperbolic shear deformation theory (Soldatos 1992; Xiang et al. 2009; Akavsi

2010; Grover et al. 2013), exponential shear deformation theory (Karama et al. 2003), and

trigonometric shear deformation theory (Mantari et al. 2012a), but their equations of motion are

more complicated compared to FSDT.

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A comprehensive document that lists the different shear deformation theories with

emphasis on estimation of transverse/inters laminar stresses in laminated composites was

reviewed by Kant and Swaminathan (2000). As a parallel effort, a review of refined

displacement and stress-based shear deformation theories for analysis of isotropic and

anisotropic laminated plate was compiled by Ghugal and Shimpi (2002). In addition, a

collection of different laminated theories based on displacement hypothesis was submitted by

Liu and Li (1996), which include shear deformation theories, layer wise theories, Zigzag

theories, and the global-local double-superposition theories. Since FGM layers does not

introduce any interfaces in their geometry, due to their smooth spatial variation of effective

properties, the further discussion about zig-zag theory seems to be inappropriate at this point,

and hence subsequent discussion has been continued to discuss implementation of shear

deformation theories for analysis of FGM plates.

The CPT which neglects the effect of shear deformation has been extended to perform

the stability analysis of functionally graded plates (Javaheri and Eslami 2002; Zhang and Zhou

2008; Mohammadi et al. 2010; Bodagi and Saidi 2011). In some research works, the FSDT has

been employed for free vibration and buckling analysis of functionally graded plates by many

researchers (Croce and Venini 2004; Ganapathi et al., 2006; Zhao and Liew 2009a; Hashemi et

al. 2010; Hashemi et al. 2011). Batra and Jin (2005) employed FSDT to analyze free vibration

problem of FGM plates in combination with finite element method. Few of the earlier works

include quadratic, cubic and higher order variation of in-plane displacements through the

thickness of the plate (Reddy 2000; Karama et al. 2003; Zenkour 2005a; Zenkour 2005b; Xiao

et al. 2007; Matsunaga 2008; Pradyumna and. Bandyopadhyay 2008; Fares et al. 2009; Talha

and Singh 2010; Benyoucef et al. 2010; Atmane et al. 2010; Talha and Singh 2011; Meiche et

al. 2011; Mantari et al. 2012b) with the combination of either constant/linear/quadratic

variation of transverse displacement component. A sinusoidal shear deformation theory (SSDT)

having four unknowns that accounts for sinusoidal variation of transverse shear stresses and

have resemblance with conventional sinusoidal shear deformation theory is proposed for

bending, vibration and buckling analysis of functionally graded plates by Thai and Vo (2013).

The similarities between the conventional and improved SSDT are observed by means of

equations of motion, boundary conditions and stress resultant expressions.

A Reissner’s mixed variation theorem (RMVT) for bending analysis of functionally

graded plates has been formulated by Brischetto and Carrera (2010) that incorporate both

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displacements and transverse normal/shear stresses as primary field variables in order to obtain

the significant enhancement over classical models based on principal of virtual displacements

(PVD), where only the displacements are assumed as primary variables. Different orders of

expansion have been considered for primary variables through the thickness, and these

unknowns are described either by single or layer wise theory. Such a theory enables an analyst

to combine the different plate cases in a unified manner. Qian et al., (2004) obtained solutions

for static, free vibration and forces response of thick FGM plates using higher order shear and

normal deformation plate theory. A generalized shear deformation theory proposed by Zenkour

for the analysis of cross ply laminated and visco elastic composite plates (2004a; 2004b; 2004c)

has been extended to static analysis of functionally graded plates (Zenkour 2006) which

enforces traction-free boundary conditions at the faces of the plate. The theory proposed by

Zenkour (2006) have similarity with the higher order theory of Reddy (2000) and have similar

unknowns as involved in FSDT, and also able to predict the transverse stresses in accurate

manner.

Different forms of shear deformation theories that include higher order terms in HSDT

for functionally graded plates (Mantari et al. 2012b), trigonometric HSDT for exponentially

graded plates (Mantari and Soares 2012 a), including thickness stretching effect in HSDT for

functionally graded plates (Mantari and Soares 2013 a), hybrid quasi 3D shear deformation

theory (Mantari and Soares 2012 b) and generalized HSDT (Mantari and Soares 2013 b) for

static analysis of advanced composite plates, has been proposed by Mantari and his associates.

An optimized sinusoidal HSDT incorporating sine and cosine terms in in-plane and transverse

displacement, respectively, is proposed by Mantari and Soares (2014), recently, for the bending

analysis of functionally graded plates and shells. The proposed SSDT include the effect of

thickness stretching by means of shear strain shape functions and related to the arbitrary

parameters m and n (these parameters are selected based upon appropriate displacements and

stresses). The theory employs no shear correction factor since it satisfies the tangential stress-

free boundary conditions on the plate boundary surface. To predict the accurate evaluation of

mechanical stresses in functionally graded plates and shells, thickness stretching effect has

been incorporated in the formulation by Carrera et al. (2011a). A quasi 3D hybrid theory also

known as polynomial and trigonometric theory is proposed by Ferreira and his co-workers

(Neves et al. 2011; Neves et al. 2012b; Ferreira et al. 2011a) for static and free vibration

analysis of functionally graded plates in the framework of mesh free methods. Having a brief

19  

overview about various shear deformation theories, the application of such theories for the non-

linear problems of functionally graded plates is discussed in the following paragraph.

The CPT is implemented to study the non-linear cylindrical bending of FGM plates

with the variation of material properties as a sigmoid function in the thickness direction (Kaci

and Bakhti, 2013). To perform the non-linear thermo-elastic response of FGM plates (Praveen

and Reddy 1998; Zhao and Liew 2009b) FSDT is incorporated with the constant variation of

transverse shear stress through the thickness. Recently, FSDT is employed to study the

geometric non-linear analysis of functionally graded plates using cell-based smoothed three-

node Mindlin element (Van et al. 2014). To deal with small strains and moderate rotations, von

Kármán assumptions are imposed by adopting C0 HSDT formulation. Third order shear

deformation beam theory has been performed to predict the size-dependent non-linear free

vibration response of micro beams made of FGM materials (Sahmani et al. 2014). In the study,

moderate strain gradient elasticity theory and von Kármán assumptions are implemented. Shen

(2002) employed Reddy’s theory for non-linear bending of FGM plates subjected to transverse

uniform and sinusoidal load. Similarly, third order plate theory of Reddy and von Kármán

assumptions are incorporated for kinematics and kinetics field to find non-linear thermo-elastic

bending response of FGM plates by Aliga and Reddy (2004). To analyze the cylindrical non-

linear bending of FGM plates under thermal and mechanical loads, a four variable refined plate

theory is proposed by Fahsi et al. (2012). The transverse displacement consists of bending and

shear component; where, bending components do not contribute toward shear forces and vice

versa. The theory account for quadratic variation of the transverse shear strains across the

thickness and eliminates the use of shear factor by incorporating zero traction conditions on the

top and bottom surfaces of the plate. The theory proposed by Fahsi et al. (2012) is extended to

study the cylindrical bending of FGM nano composite plates by Bakhti et al. (2013) in

conjunction with Von Kármán theory and potential energy principle. In a parallel track, Kaci et

al. (2013) proposed cylindrical bending of FGM nano composite plates reinforced by single

walled carbon nano tubes by incorporating Reddy’s third-order plate theory and von Kármán

geometric nonlinearity to describe kinematic and kinetic fields. A general nonlinear third-order

plate theory that accounts for (a) geometric nonlinearity, (b) microstructure-dependent size

effects, and (c) two-constituent material variation through the plate thickness (i.e., functionally

graded material plates) is presented using the principle of virtual displacements by Reddy and

20  

Kim (2012). The modified couple stress theory includes a material length scale parameter that

can capture the size effect in a functionally graded material.

A catalog of non-linear classical theories is proposed for thin shells based on Kirchhoff-

Love hypotheses (Donnell 1934; Novozhilov 1953; Sanders 1963; Koiter 1966; and Ginsberg

1973) by including shallow shell assumptions. In Donnel’s theory (Donnell 1934), infinitesimal

in-plane displacements are considered; while transverse displacement is assumed to be in the

order of shell thickness. Since the theory discounts for the in-plane inertia, it gives accurate

results for only thin shell categories which is the quite contradictory observation found as

compared to Donnel’s linear shell theory (1933). In the theory (Donnell 1934), non-linear terms

are retained only in the transverse displacement and neglected for in-plane field; which is

analogous to the von Kármán assumptions assumed in non-linear plate theories. The classical

shell theory (CST) proposed by Sanders (1963) is considered to be a more refined form of

tensorial based shell theory. An improved form of Sander’s theory is presented by Koiter

(1966) in the name of Sander-Koiter theory to consider finite deformations with small strains

and moderate rotations. The non-linear terms appear in strain-displacement relations are

dependent on both in-plane and transverse displacement components. The consequence of

curvature changes and torsion of the middle-surface are assumed in the linear sense by both the

theories (Sanders-Koiter 1966; Donnell 1934). However, Donnell’s theory (1934) yield

accurate results for moderately thick shells and modes of high circumferential waver number,

only.

In the non-linear shell theories proposed by Novozhilov (1953) and Ginsberg (1973),

non-linear terms are added to the curvature and torsion part. However, the strain-displacement

relation identical to that of Sander-Koiter theory (Koiter 1966) is assumed. Because the shear

deformation and rotary inertia are neglected in the above classical theories, various shear

deformation theories are developed in due course of time as an alternative solution. In the

category of shear deformation theories, the dominant role is played by two theories namely;

first-order and higher-order shear deformation theories; while the first category demand the

suitable estimation of shear factor to satisfy equilibrium condition and the later theory employs

the boundary condition similar to higher order plate theory. Parisch (1995) and Sansour (1995)

proposed shell theories to introduce quadratic assumption of shell displacement over the shell

thickness. The linear shell theory that incorporate the effect of thickness stretching was

submitted by Carrera et al. (2011a) and Ferreira et al. (2011b). An enhanced form of first order

21  

shear deformation theory in the frame work of finite element formulation is presented by

Arciniega and Reddy (2007a) for the non-linear analysis of ample range of shell geometries

that include isotropic, laminated composite and FGM structures.

By incorporating Sanders-Koiter non-linear terms (Koiter 1966), Reddy and

Chandrashekhara (1985) developed non-linear FSDT, by defining the deformation process in

terms of five independent variables (three translations and two rotations). As a further

enhancement in this direction, non-linear terms are included in Reddy’s theory and

implemented in the framework of finite element method (Reddy 2004; Dennis and Palazotto

1990; Palazotto and Dennis 1992). An extended application of higher order shell theory to the

case of anisotropic sandwich shells having compressible core is evident from the work of Hohe

and Librescu (2003). They assumed the Kirchhoff-Love hypotheses for the face sheets and a

second/third-order power series expansion for the case of core displacements. As a

development, Reddy in association with Amabili developed a refined non-linear shell theory

for closed and open shells by retaining rotary inertia, shear deformation and non-linear terms in

both in-plane and transverse displacements (Amabili and Reddy 2010). The so formed new

theory (Amabili and Reddy 2010) has proved excellent performance in predicting the large-

amplitude vibrations of moderately thick laminated circular cylindrical and deep shells

(Amabili 2011) and curved panels (Alijani and Amabili 2013). As a further progress, a

modification of the theory in the form of incorporating thickness stretching effect and

geometric imperfections is performed by Amabili (2013) by means of third-order variation of

normal strain in the non-linear theory developed by Amabili and Reddy (2010). Very recently,

a theory that accounts for normal strain in the kinematics field by means of third-order

variation of thickness is executed by Amabili (2014).The benefit of retaining transverse normal

strain components is that it utilizes all the constitutive equations and such a consideration is

predominantly suitable for materials where large deformations are achieved by large thickness

reduction.

In addition to above shell theories, a huge list of tensor based geometrically non-linear

shell theories are also proposed in the literature (Eremeyev and Pietraszkiewicz 2004; Opoka

and Pietraszkiewicz 2004; Pietraszkiewicz and Szymczak 2005; Arciniega and Reddy 2007b;

Opoka and Pietraszkiewicz 2009; Berdichevsky 2010; Xiaoqin et al. 2010; Pietraszkiewicz

2012; Steigmann 2013). A widespread assemblage of various higher deformation theories is

carried out by Reddy (2004), Amabili (2008), and Carrera et al. (2011b). In addition, a

22  

profound discussion is performed on linear shear deformable and zigzag theories by Reddy and

Arciniega (2004) and Carrera (2002; 2003).More recently, an in depth review of various shell

theories is executed by Alijani and Amabili (2014) and they also made an attempt to discuss

several other related aspects of nonlinear vibration of shells, for example, fluid-structure

interaction, geometric imperfections, influence of thermal and electrical loads in a brief

manner. In what follows, a discussion regarding application of various shell theories to FGMs

is discussed.

Based on Love’s shell theory (Love 1952), Loy et al. (1999) studied the frequencies of

simply supported FGM cylindrical shells using Ritz method. This study was further extended to

incorporate the effects of various boundary conditions on natural frequencies of FGM

cylindrical shell by Pradhan et al. (2000). Based on TSDT of Loy et al. (1999), Najafizadeh and

Isvandzibaei (2007) presented the free vibration response of thin cylindrical shells with

arbitrarily fixed ring support along the shell. This study was further extended by the authors to

study the influence of various shear deformation theories (Najafizadeh, 2009) on free vibration

response of cylindrical shells. The FSDT as a special case of higher order shear deformation

theory of Reddy is assumed to represent the kinematics field. The FSDT considering rotary

inertia and transverse shear strains is used to study the effect of thermal load on free vibration,

buckling, and dynamic stability of FGM shells by Sheng (2008). Based on FSDT, free vibration

of cylindrical, conical, and annular FGM shell structures is studied by Tornabene and his co-

workers using four-parameter power law distribution ( Tornabene 2009a, Tornaence 2009 b).

Large deformation vibration behavior of FGM cylindrical shell of finite length embedded in

elastic medium is performed under thermal environment is presented by Shen (2012). Higher

order shear deformation theory that includes the effect of shell–foundation interaction is

incorporated in the study. General shell theory combined with finite element method is

employed to study the vibration analysis of FGM cylindrical, doubly curved, hyperbolic

paraboloid shell (Yang 2012). More recently, Ebrahimi and Najafizadeh (2014) studied the free

vibration response of two dimensional functionally graded (2D FG) cylindrical shells using

Love’s first approximation CST.

The problem of geometric non-linearity, initial geometrical imperfection and Pasternak

type elastic foundation based on CPT is solved for nonlinear axi-symmetric response of

shallow spherical FGM shells under thermal and mechanical loads by Duc et al. (2014). Based

on FSDT, finite element method has been employed to study the dynamic stability of

23  

functionally graded shallow spherical shells (Ganapathi 2007). Geometric non-linearity is

considered in von Kármán sense and Newton iteration schemes are considered to solve non-

linear iteration equations. Non-linear axi-symmetric dynamic buckling behavior of clamped

FGM spherical caps is performed by Prakash et al. (2007) based on FSDT and von Kármán

assumptions. Bisch and his co-workers performed static and dynamic non-linear analysis of

FGM spherical shells under different loading environments by considering CST and geometric

imperfections (Bich 2009; Bich et al 2010; Bich et al. 2011; Bich et al. 2012).

Hence an exact kinematic model that incorporates the realistic variation of shear

distribution through the thickness and considers the effect of normal strain in the transverse

direction seems to be very important for accurate modeling of FGM structures under

linear/non-linear responses.

2.3 Static, dynamic and buckling responses of FGM plates/shells

3D elasticity solutions (Pagano 1969; Pagano 1970; Srinivas and Rao 1970; Srinivas et

al. 1970) are generally utilized to assess the accuracy of various 2D approximate plate theories

(Pandya and Kant 1988a; Pandya and Kant 1988b; Pandya and Kant 1988c; Reddy 2004). In

this regard, several bench mark solutions are placed in the literature based on 3D theories for

simply supported laminated plates. But the solution methodology lays the limitation for FGMs,

where the material properties are generally inhomogeneous in nature. Therefore many

displacement based 2D theories are proposed in conjunction with analytical, semi-analytical

and numerical solutions. An exceptional introduction to the fundamentals of FGMs and a

comprehensive literature review in FGM technology was provided by Suresh and Mortensen

(1998). In addition, Birman and Byrd (2007) have documented an exhaustive list of research

works regarding developments in FGM research by addressing the various topics like

characterization, modeling and analysis of FGM. Important discussion include manufacturing,

design, homogenization of particulate FGM, heat transfer problems, stress, stability and

dynamic analyses, fracture studies and various application areas. Recently, Jha et al. (2012a)

have made an attempt to present the exhaustive literature survey on deformation, stress,

vibration and stability problems of FGM plates. In this section, the literature works is focused

on the research works in the field of static, dynamic and buckling analyses of FGM

plates/shells published since 1999. Although it was an unfeasible task to discuss all the works

24  

in a single document, an endeavor has been made by the author to incorporate the important

and relative works in this area.

The free vibration response of FG cylindrical shells made of stainless steel-nickel alloy

was studied by Loy et al. (1999) based on Love’s shell theory (Love 1952) and Rayleigh-Ritz

method. The effective mechanical properties of the shell are assumed to be graded in the

thickness direction and follow power law distribution in terms of volume fraction of

constituents. Based on the assumption of nickel/stainless steel on inner/outer surfaces, two

types of FGM shells (type I and type II) are analyzed in the study. For the value of

circumferential wave number greater than one, natural frequencies of type II FG cylindrical

shells (nickel on outer surface and stainless steel on inner surface) are recorded to be higher

than type I FG (nickel on inner surface and stainless steel on outer surface) cylindrical shells

and for circumferential wave number other than this value, the FGM shells exhibit quite

opposite trend. For type I and type II FG cylindrical shells, for all the values of volume fraction

indexes, frequencies in between that of stainless steel and nickel alloy are discerned.

A finite element model of cylinder was employed to study the thermo-elastic response

of FGMs by Praveen et al. (1999). Considering the fact that the inner surface of the cylinder

was subjected to a rapid increase in temperature, the solution for unsteady heat transfer

equation has been solved. The constitutive equation has been updated at each time step, with

the temperature at each time step, and which is further solved by energy equation. The inertia

terms and thermo-mechanical coupling are neglected in the equilibrium equations. When the

average volume fraction of the ceramic attains value less than 0.5, no significant change of

maximum temperature was observed in the cylinder. This reflects the statement that FGMs with

50% of volume fraction of ceramic are good enough to withstand large temperatures, and no

gain has been visualized beyond that range. Variation of radial compressive, radial tensile and

hoop stresses is plotted against average volume fraction of ceramics of FGM cylinder. When

temperature-dependent properties are considered, the maximum tensile stress at the inner

surface is independent of the average volume fraction of ceramic below 50%. Thus increasing

the volume fraction reduces the tendency of any crack growth at tensile mode. The final

observation was that, beyond certain percentage of average volume fraction of ceramic, an

asymptotic response has been observed. Again this response is a function of different field

variables chosen to perform the study.

25  

The free vibration study performed on FG cylindrical shells by Loy et al. (1999) has

been extended to study the influence of boundary conditions on free vibration response of FG

cylindrical shells by Pradhan et al. (2000). The displacement field and solution method similar

to that of Loy et al. (1999) was considered. Clamped, simply supported and free boundary

conditions are considered to generate the frequency results. Frequency characteristics similar to

that of isotropic case are discerned for FG shells also. Further, they observed that for smaller

value of volume fraction index (n=0.1), the frequency data of FG cylindrical shells are close to

isotropic metal shell (stainless steel shell). For other higher values of volume fraction index

(n>>10), the frequency response of FG cylindrical shells are close to that of isotropic ceramic

shell (zirconia shell). This observation guides to choose the appropriate value of volume

fraction to get the required frequency response of FG shells. For different L/R

(length/curvature) ratios, the frequency responses of clamped and free edge cylindrical shells

are found to be identical.

Initially stressed FGM plates are analyzed for free and forced vibration response by

Yang and Shen (2002) under thermal environment. Temperature dependent material properties

and having power law variation through-the thickness are considered. Reddy’s higher order

shear deformation theory was assumed and uniform temperature variation under thermal part

was considered. Modal superposition method has been employed to get the transient response

of the plate under lateral dynamic loads. It was observed that for the mixed material mixture,

frequency rises by elevating in-plane tension but reduces by increasing initial edge

compression. Since Young’s modulus decreases at higher temperatures, the frequency

parameter declines at higher temperatures. Also, higher bending moments compared to

isotropic plates are observed in FGM case, due to the lower thermal expansion of the plate at

top compared to bottom. It has been noticed that dynamic response of FGM plates are not

necessarily lie between isotropic cases under thermal loading conditions. Further direct

proportion of thermally induced bending moments was observed with the volume fraction

index.

The axi-symmetric stability of circular FGM plates was considered by Najafizadeh and

Eslami (2002) based on Love-Kirchoff hypothesis. The linear Sander’s shell theory was

employed to approximate the strain displacement field. Results are presented for simply

supported and clamped boundary conditions and material properties are assumed to be graded

26  

in the radial direction. They observed that the critical buckling load of FG plates was lower

than corresponding pure isotropic circular plates.

A three-dimensional analytical solution was proposed by Vel and Batra (2003) for

simply supported functionally graded plates subjected to time-dependent thermal loads. The

governing transient heat conduction equation has been reduced into ordinary partial differential

equations by means of Laplace transform equations and then solved by the power series

method. The micromechanical models based on Mori-Tanaka and self consistent scheme are

incorporated in the study. The important finding of the study was that, the transient longitudinal

stresses are approximately 8 times their respective steady state value, when rapid time-

dependent surface temperatures are prescribed. But the transient stresses are less than their

steady state values. As the time elapses, the pattern of stress changes from compressive to

tensile in case of both longitudinal and transverse shear stresses.

The study carried out by Vel and Batra (2003) has been extended to study the vibration

of functionally graded rectangular plates by Vel and Batra (2004). The solution method and

micromechanical models identical to that of Vel and Batra (2003) were incorporated in the

model. In addition, the transient response of the plate was performed under the sinusoidal

spatial distribution of pressure applied on the top surface. The authors manifested that, the

displacement and stresses in the functionally graded plates exhibit anti symmetric pattern with

respect to the mid-plane, by virtue of their anti symmetric properties about the mid-plane.

Under forced response, the displacements and stresses are large as the forcing frequency

approaches the natural frequency. Also, the normal and transverse stresses are more for the case

of sinusoidal normal pressure applied on the top of the plate. Even though, the transverse

normal and shear stresses are computed by integrating the 3D elasticity equation in the study, a

considerable deviation between the analytical solution and the CPT results was noticed,

especially for thick plates. The FSDT results are found to be close to analytical solution than

results by TSDT.

In addition to above discussed works, Batra and his co-workers studied the fracture

concepts in functionally graded materials (Jin and Batra 1996), stress intensity relaxation study

in the cracked functionally graded material subjected to thermal shock (Jin and Batra 1996),

and R-curve and strength behavior of functionally graded materials (1998). Since these topics

are not relevant to present research topic, not discussed briefly in this section.

27  

Najafizadeh and Heydari (2004) studied the thermal buckling of functionally graded

circular plates based on HSDT (Reddy and Khdeir 1989). The fundamental partial differential

equations are established by variational approach and mechanical properties are assumed to be

graded in the thickness direction in proportion with their volume fraction index. They

compared the various numerical results with the FSDT and CPT results and concluded that the

CPT and FSDT over estimate the buckling temperature parameter. Under uniform temperature

rise, buckling temperature was found to be lower at volume fraction index equals 2.7, while

maximum at volume fraction index equals 10, and thus it was recommended to opt for the

value between 4.0 and 10.0 for the parameter. Further, the critical buckling temperature of FGP

was observed to decline by reducing the value of volume fraction index.

Free/forced vibration and static study of thick functionally graded plates was performed

by Qian et al. (2004). The meshless Petrov-Galerkin method in the framework of higher order

shear and normal deformable plate theory was utilized to perform the analysis. In addition, the

response of the plate under impulse load was considered by employing different values of

volume fraction index. To account for the interaction between the adjacent inclusions, Mori-

Tanaka approach was used to derive the effective elastic constants. The top surface of the plate

was loaded with sinusoidal form of traction represented as Fourier series to obtain the static

solution and the solution was obtained by the superposition method. A time dependent

harmonic normal pressure of uniformly distributed was applied for the time period 0 t 5 ms

and suddenly removed. The through-the-thickness variation of deflection for both isotropic and

FGM plate exhibit un symmetric variation about the mid-surface, due to the non symmetric

pattern of applied load. Also, the tensile stresses observed at the bottom segment of the plate

are reduced towards top by the addition of ceramic content and compressive stresses at the top

are increased with the addition of ceramic part. As far as the effect of volume fraction was

concerned, the deflection variation with volume fraction index was noticeable, while it is not

for the case of axial stress variation of FG plates. Also for linear volume fraction value, the

natural frequencies of pure ceramic and pure metal are the upper and lower bounds of the

frequencies of the FG plate. Regarding dynamic response, the oscillation time period of pure

ceramic plates is found to be less than that of pure metallic plate.

Ferreira et al. (2005a) presented the static problem of functionally graded plates using

third-order shear deformation theory in conjunction with meshless method. Two kinds of

homogenization schemes namely, Mori-Tanaka approach and rule of mixture are adopted to

28  

estimate the effective properties of the FG plate. Two types of material combinations are

considered; the one with equal Poisson’s ratio value of constituents and the other combination

with a wide variation of Poisson’s ratio. An interesting observation regarding the influence of

Poisson’s ratio of two constituents on displacement has been noticed i.e., when Poisson’s ratio

becomes equal, both the models produce quite close results and have large variation for

different values of Poisson’s ratio. While plotting the axial stress profile for FG plates, the

lower and higher values of volume fraction index have sharp gradient change near the bottom

and top, respectively, due to sharp variation of material properties at the corresponding points.

The micro mechanical model based on Mori-Tanaka scheme, global collocation

method, the FSDT and HSDT are employed by Ferreira et al. (2006a) to study the natural

frequencies of FGPs. The solution proposed in the work does not require any nodal

connectivity procedure and evaluation of integral was performed over a sub domain. Different

parameters such as boundary conditions, thickness ratio, and volume fraction index are

considered to present the natural frequencies of FGM plate. It was noticed that frequency

depends on number of collocation points, their corresponding locations, and the parameter c

present in the multi quadratic basis functions.

The proportionality constant that exists between the homogenous and FGM plates was

derived and proposed by Abrate (2006) for static, free vibration and buckling problems. The

author observed that the natural frequencies, deflection and buckling load of FGM plates are

proportional to those of corresponding homogeneous isotropic plate, while the other parameters

are kept constant. Different examples are selected from the literature to show the correlation

between the homogenous and FGM plates for different analyses. Numerical expressions are

derived that will supply the suitable proportionality constant to predict the static, vibration and

buckling responses of FGM plates.

In Part I submitted by Chi and Chung (2006), FG plates are analyzed under mechanical

loading and series solutions for different kinds of plates based on various micromechanical

models are proposed. Extensive numerical problems are performed based on the solutions

presented in Part I by Chi and Chung (2006) as Part II. The graded properties of the FGM are

described by power-law, sigmoid and exponential functions. The CPT and Fourier series are

employed in the study to define the displacement field and closed form solutions, respectively.

They concluded that the location of the neutral surface of the FGM plates depends on the ratio

E1/E2 for particular material distribution or variation of material properties in the thickness

29  

direction, while it was independent of the aspect ratio or the external loads. The stresses in the

FGM plates are not linearly proportional to z and function of the product z. E(z). The tensile

stresses are observed to be maximum at the bottom of plate, while the location of maximum

compressive stress moves towards the inner side, rather than at the top surface of the plate.

A 3D based solution was proposed by Uymaz and Aydogdu (2007) for vibration

analysis of functionally graded plates based on the small strain linear elasticity theory. The

Chebysheve displacement functions combined with Ritz method was employed to solve the

vibration problem of FGM plates. Prominence has been given to sketch the influence of various

boundary conditions on frequency response of FG plates. For all the boundary conditions, rise

in a/h ratio increases the frequency parameter and converges at a/h=50. Also, constant density

with variable Young’s modulus produce lower frequency values compared to the case, where

both Young’s modulus and density are treated as variables.

The free vibration study of thin FGM cylindrical shells having ring support and made of

stainless steel/nickel alloy was studied by Najafizadeh and Isvandzibaei (2007). To perform the

analysis, ring supports are arbitrarily placed along the shell, which impose zero deflection in

the lateral direction. The TSDT and Love’s shell theory are used to represent the kinematics

and kinetic field and the final governing equations are derived based on Rayleigh-Ritz method.

Type I and type II FG cylindrical shells as considered by Loy et al. (1999) are considered to

execute the numerical part. The frequency behavior of FG shells indicates a lower fundamental

frequency mode for linear value of volume fraction index. Further, a quite opposite frequency

response was visualized between type I and type II cylindrical shells with respect to

circumferential wave number. A remarkable influence of ring support location on frequency

parameter was noticed. When the ring support was placed at the center, the frequency is

maximum for simply supported FG cylindrical shell, while the frequency tends to show

decreasing trend as the ring support move towards either of its ends.

The free vibration study of FG cylindrical shell with ring supports by Najafizadeh and

Isvandzibaei (2007) was extended to study the effect of various shear deformation theories on

free vibration response of FG cylindrical shells by Najafizadeh and Isvandzibaei (2009).

Numerical studies are performed for cylindrical shells having different types of boundary

conditions (simply supported-simply supported, clamped-clamped, free-free, clamped-simply

supported, clamped-free and free-simply supported boundary conditions). The TSDT proposed

by Reddy was accomplished to establish the kinematic field and further modified in to FSDT

30  

by means of appropriate substitution of variables in the displacement field. Symmetric pattern

of the natural frequency curve is demonstrated, provided the symmetric conditions are chosen

at both the ends along with the location of ring support at center of the cylindrical shell.

The coupled thermo-elasticity problem of functionally graded cylindrical shells was

solved by Bahuti and Eslami (2007) based on second-order shear deformation shell theory that

considers the influence of transverse shear strain part. The thermal problem was solved by

Laplace technique in time domain, while Galerkin finite element method was employed for

space domain. The heat flux was considered at the inner portion of the shell to cause maximum

temperature. The distribution of axial force for the shell having pure ceramic material was

minimum, while it becomes maximum for pure metal plates. For FG shells, the axial stresses

are higher than pure ceramic and metal plates. Further the stress distribution was found be

linear during the shock occurrence, but becomes periodic in nature after the shock ends and this

period follow the pattern of radial displacement.

The buckling of functionally graded circular plates (FGCP) based on HSDT (Reddy and

Khdeir 1989) was carried out by Najafizadeh and Heydari (2008) under uniform radial

compression. They compared their numerical results with the FSDT and CPT results for

different cases and concluded that the HSDT results accurately predict the buckling behavior of

plates, while CPT and FSDT overestimates the buckling loads. They also observed that

mechanical instability of FGM plates are lower than pure ceramic plates at volume fraction

value equal to zero. They concluded that the effect of transverse shear deformation should be

considered as far as thick plates are concerned. Further, the critical buckling load of FGCP

tends to reduce at higher value of volume fraction index parameter.

A C0 higher order formulation has been employed to study the free vibration analysis of

FG curved panels by Pradtumna and Bandyopadhyay (2008). Third order term of thickness was

assumed in the in-plane fields, while constant variation of transverse displacement (Tarun and

Kare 1997) was considered. An element with nine nodal unknowns was adopted and Sander’s

approximation for doubly curved shell was incorporated in the formulation. Results are

presented for cylindrical, spherical and hypar shells by considering various values for

curvature, thickness and volume fraction index. The frequency declines with respect to lower

values of volume fraction index and R/a ratio. Further dominance of stiffness was observed

over mass which results in to rise in stiffness parameter. In some cases, contribution of both

stiffness and mass are observed to dictate the frequency response of FG panels. For simply

31  

supported hypar shells, with the increase of c/a ratio, abrupt increase of frequency value was

observed (c/a=0.5), beyond this slow increasing trend of frequency was noticed. Further, the

superiority of the hypar shells was established compared to spherical and cylindrical shell while

keeping the other common parameters as constant.

The effects of rotary inertia, normal and transverse shear deformation was considered

by Matsunaga (2008) to analyze the natural frequencies and buckling stresses of FGM plate.

The 2D higher order theory and Hamilton’s principle was used to derive the governing

equilibrium equations. Modal displacements and stresses in the thickness direction are obtained

by satisfying the surface boundary conditions. Integration of three-dimensional equations of

motion has been done to obtain the modal transverse stresses. Modal displacements and stresses

are plotted for FG plates considering different thickness ratios. In addition, magnitudes of

internal and external work done for first fundamental vibration mode are established. Negative

sign for internal work was observed due to the effect of thickness changes in FG plates. Under

in-plane stress, lowest displacement mode gives the critical buckling stress for thin FG plates,

while higher displacement modes are responsible for critical buckling stress in thick FG plates.

The wave propagation technique was employed by Iqbal et al. (2009) to study the

vibration response of circular FGM cylindrical shells. Expressions for strain and curvature

deformations are adapted from Love’s (Love 1952) theory. The magnitude of frequency was

found to be lower for volume fraction index equal to 2 and 3, and for any other choices of

volume fraction index, only a minute variation of frequency was noticed. Depending on the

concentration of stainless steel, nickel and zirconia materials on inner and outer of shell

surface, six categories of FG cylindrical shells are incorporated in the numerical investigation.

It was illustrated that the increment and decrement trend of shell frequency was dependent on

the ratios of Young’s modulus and Poisson’s ratio of the two constituent materials selected to

form an FG shell. But the ratio of density of two materials does not seem to affect the

frequency with regard to value of volume fraction index.

The FSDT in the frame work of element free kp-Ritz method was employed to study the

buckling of functionally graded plates under mechanical and thermal loading conditions by

Zhao et al. (2009a). The exponential variation of effective properties was considered along the

thickness direction. To avoid any shear locking problem encountered in thin plates, shear and

membrane terms are computed using a direct nodal integration technique, while bending part

32  

was evaluated using nodal integration techniques. Different features such as plate with arbitrary

geometry and contain square and circular holes at the center are investigated. For simply

supported and clamped boundary cases, initial rising trend of critical buckling temperature was

observed for volume fraction index (n) equal to 0 to 2, when n rises further and attain the value

equal to 5, negligible temperature change was noticed. Also, for a certain hole dimension and

volume fraction index, critical buckling load for the first mode declines initially, in proportion

to hole size and buckling load exhibit unstable trend as the hole sizes increases.

As an extension of the work by Zhao et al. (2009a), the FSDT combined with element

free kp-Ritz method was performed to study the thermo-mechanical buckling response of FG

shells by Zhao and Liew (2009a). The non-linear through the thickness distribution of

temperature profile was incorporated in the study. Buckling mode shapes are shown for

composite panels also having different stacking sequences. The temperature rise drops for

higher value of volume fraction index, and the slope of declination curve becomes gentle as

volume fraction index becomes greater than 2. For the value of volume fraction index equal to

zero, the panel endures a linear temperature field, and buckling temperature tends to elevate for

all the modes as the volume fraction index approaches higher values.

Zhao et al. (2009c) studied the thermo elastic and vibration analysis of functionally

graded cylindrical shells based on Sander’s FSDT. Variation of axial-stress distribution was

exposed for different boundary conditions of aluminium/zirconia and Ti-6Al-4V/aluminium

oxide plates for various value of volume fraction index. Influence of constant and modified

shear co-efficient on frequency parameter was demonstrated. It was observed that for R/h=50,

the variation between the shear co-efficient was negligible and for higher values of R/h=100

and 200, the discrepancy is even smaller. The top surface of the shells ensures tension and

bottom surfaces shows compression nature of stresses. Further, the maximum tensile stress

occurs at volume fraction index equals 5, while minimum value was noticed at volume fraction

index equals zero. In addition, mode shape plots were given for different FGM shells

considering various boundary conditions, thickness ratio and curvature values.

Thermo elastic analysis of FGM plates was performed by Lee et al. (2009) based on

FSDT and element kp-Ritz method. To show the versatility of the method skew and

quadrilaterial plates are also considered. At top maximum compressive stresses are confirmed

for volume fraction index equals 2.0 and at bottom surfaces pure ceramic plate (n=0)

experiences maximum tensile stresses. In addition, the difference among various stress patterns

33  

corresponding to different volume fraction index was not significant. The authors concluded

that the effect of length-to-thickness ratio on displacement was independent of the volume

fraction index chosen for the problem. Under thermal loading, negative deflection was recorded

due to the higher thermal expansion of the constituent at the top surface. When skew angle is

large, the higher magnitude of axial stress was induced in the plate.

Tornabene (2009) presented the FSDT based GDQ method for the free vibration

analysis of conical, cylindrical shell and annular plate structures made of FGMs. Two different

kinds of FGM profiles (FGM1 and FGM2) are proposed based on four-parameters exist in the

power law distribution. With various combinations of these parameters classical, symmetric

and asymmetric volume fraction profile through the thickness can be achieved. Also such

profiles lead to the combination of ceramic and metal at different location of geometry other

than the conventional one. Frequencies are tabulated for first ten frequencies by varying

boundary condition; shell geometry and the parameters exist in the power law formula. Mode

shapes were plotted for the different cases of plate/shell geometry. It has been seen that for

specific values of volume fraction index, FGM1 frequencies are greater than FGM2 model.

Because of the curvature effect, increasing the shell thickness leads to more frequency

deviation between FGM1 and FGM2 models. But, due to lack of principle curvature in the

annular plate structures, this effect was not pronounced. Finally, it has been concluded that the

appropriate choice of parameters in the power law was essential to define the appropriate

constituent present at the top and bottom of the shell/plate structures.

Buckling analysis of thick functionally graded rectangular plates was performed by

Bodaghi and Saidi (2010) using higher-order shear deformation theory. The governing stability

equations are divided into two uncoupled partial differential equations in terms of boundary

layer function and transverse displacement and Levy type solution are employed to solve the

governing equations. When the aspect ratio elevates, the number of half waves in x direction of

critical load gets changed. Since the theory incorporates the shear deformation effect,

considerable influence of thickness parameter was noticed for different thickness values (h/b).

For fixed thickness-side ratio, the buckling load of FG plates lies between buckling load of

ceramic and metal plates, as observed in earlier buckling studies.

The Carrera’s unified formulation for single layer and layer-wise description was

implemented to study the thermo-mechanical response of simply supported FGM shells by

Cinefra et al. (2010). One dimensional Fourier heat conduction equation was solved to obtain

34  

the non-linear distribution profile of the temperature distribution through the thickness. The

principle of virtual displacements was applied to obtain the governing thermo dynamic

equations. The Mori-Tanaka model was employed in the study. Through-the-thickness

variation of transverse displacement component was not constant for thermal loading, and

constant for pure mechanical loading case. Hence for thermal case constant variation was not

valid and this is true for even thin shell cases. In addition the need of higher order shell theories

was exploited to capture the all the possible effects of displacement and stress distributions. In

particular, the assumption of higher order thickness was established under thermal loading

conditions.

A three dimensional static solutions are obtained for thick FG plates by Vaghefi et al.

(2010) by assuming the exponential variation of Young’s modulus along the thickness

direction. The 3D equilibrium equations are utilized to arrive for local weak symmetric

formulation and the field variables are approximated using the least square (LS) approximation.

In addition, more nodes are incorporated in the thickness direction to increase the accuracy of

the 3D solutions. A wide range of numerical results are presented by considering different

combination of boundary constraints under uniformly and sinusoidal loading patterns. It was

observed that the maximum compressive stress of FG plates occurs close to the top surface of

the plate having low Young’s modulus ratio and maximum tensile stress of FG plates occurs at

bottom of the plate having high Young’s modulus ratio.

The buckling study performed by Zhao and Liew (2009b) was further extended to the

buckling of conical shell panels by Zhao and Liew (2011) based on FSDT and mesh-free kernel

particle functions. The effective mechanical properties of the FGM conical panels are assumed

to obey power law distribution. Temperature dependent properties of aluminium/zirconia and

stainless steel/silicon nitride panels are considered for the thermal analysis. When the volume

fraction exponent deviates from 0 to 0.5, a fall-off trend was observed for critical temperature

and declines further as the volume fraction exponent represent the metal segment. When the

thickness ratio and semi vertex angle increases, critical buckling temperature of the panel

records declining tendency. This statement was observed to be common for all the types of

boundary conditions. It was stated that the volume fraction index, boundary conditions,

thickness ratio and semi-vertex angle are the vital parameters that affect the stability of conical

panel under thermal loading.

35  

The free vibration analysis of thick functionally graded plates is carried out by Zhao and

Liew (2011). A local Kringing meshless method based on Petrov-Galerkin weak formulation

and combined with Kronecker delta functions was incorporated in the study. Square, skew and

quadrilateral plates are considered in the numerical analysis. The first six mode shapes are

presented for different combinations of volume fraction index and boundary condition. A

pronounced drop in frequency parameter was observed when the skew angle transform from

30° to 60°, and for other higher values of skew angle, the frequency drop become insignificant.

Further, it was observed that for quadrilateral plates the in-plane and out-of-plane modes are

coupled.

The free vibration behavior of shear deformable functionally graded plates was studied

by Talha and Singh (2011) based on higher order theory and power law variation of material

properties in the thickness direction was assumed. The governing equations are derived based

on variational approach. A C0 element with thirteen degrees of freedom was employed to

accomplish the results. To generate the new results, the combination of various boundary

conditions, thickness ratio, aspect ratio, material constituents and volume fraction index are

incorporated. Highest frequency parameter was observed for CCCC plates, while lowest

frequency was discerned for SSSS FGM plates. Also, the frequency parameter reduces at

higher temperature due to weaker Young’s modulus of the material. The effect of a/h ratio on

frequency was more pronounced up to a/h=20, beyond that the changes are negligible. For a/b

greater than 1, the influence of volume fraction index on frequency becomes insensitive.

Janghorban and Zare (2011) studied the influence of thermal load on free vibration

analysis of Aluminium/alumina FGM plates having different cutouts in their geometry.

Different geometry of the plate (square, skew and trapezoidal) and different cutout shapes

(circular and rectangular) with different sizes are studied using SOLID 45 and SOLID 70

elements. Under conventional loading conditions, frequency tends to rise for

square/skew/trapezoidal plates, when the temperature on the upper surface was raised. Further,

it was noticed that increasing the film coefficients of fluid decrease the natural frequencies of

skew plate.

The Navier type analytical solution was proposed for static analysis of functionally

graded plates by Mantari et al. (2012b) based on HSDT. Since the theory incorporate the effect

of shear deformation, use of shear correction factor has been eliminated. The principle of

virtual work was employed to derive the governing differential equations. Uniform and

36  

sinusoidal loading conditions are considered to analyze the FGM plates. The in-plane stress

variation for thick and thin plates was observed to be almost same, and negligible deviation was

observed for thick plate case having a/h value equal to 5.

An efficient and simple refined theory that accounts for quadratic variation of the

transverse shear strains across the thickness was proposed by Thai and Choi (2012) to perform

buckling analysis of functionally graded plates. The principle of minimum potential energy was

applied to derive the final governing equations. The variation of buckling load was observed to

be sensitive for lower values of volume fraction index, due to the higher deposition of ceramic

component. Also critical buckling load decreases for increase in volume fraction index, and

increases as metal-ceramic modulus ratio increases. While plotting the results for influence of

aspect ratio (a/b) on buckling load, the variation is not smooth due to change of critical

buckling mode under uni-axial compression, and becomes smooth under bi-axial compression.

The bending response of functionally graded plates and doubly curved shells was

performed using higher order shear deformation theory and Fourier series based solution

methodology by Oktem et al (2012). The variation of mechanical properties of the plate and

shell model was incorporated by means of power law function of volume fraction of the

constituents. The displacement model and strain equations similar to the one proposed by

Reddy and Lie (1985) was assumed. When the plot of transverse displacement vs. spherical

shell curvature was plotted, the decreasing tendency of deflection was observed as the shell

geometry approaches to plate geometry. This effect was due to the predominance of membrane

effects observed in the shell panel. The magnitude of in-plane normal stress was markedly

higher in thin shell panel compared to that of thick shells. Once again, the membrane effect to

reduce the magnitude of stress components was discerned. The magnitude of transverse shear

stress was higher for plates compared to its spherical counterparts. If the thickness ratio was

increases, transverse shear stress deceases for shells and this is not true for its plate counterpart.

Except the value of volume fraction index equals 0.5, the in-plane stress variation through the

thickness was smooth and sharp trend near the bottom surface was observed in case of both

plate and shells. Further, the effect of curvature shifts the axial stress from the compressive

zone to tensile zone through-the-thickness. It was quite interesting to observe the variation of

in-plane shear stress for moderately deep shell (R/a=10), where for values of volume fraction

index equals 0.5, 1 and 2, the magnitude was maximum at the top of the panel and decreases at

the bottom to approach zero. In addition, in homogeneity also increases the in-plane shear

37  

stress close to the top surface. As a final observation, the predominance effect of curvature was

observed in shells which plays vital role in predicting the static response of the shell panel.

Tornabene and Viola (2013) obtained the static response of functionally graded shell

and laminated composite shells based on GDQ procedure. Unlike the earlier study performed

by the author (Tornabene 2011), the displacement model has been improved to consider the

geometry of the shell by means of curvature effect in the kinematic as introduced by Toorani

and Lakis (2000). The grading of material properties exist in the shell layer are defined by a

generalized four parameter power law distribution. Two kinds of power law distributions are

shown for the modeling which contains the four variables that define the material properties of

the shell layer at a particular point. GDQ rule was implemented in the generalized displacement

components to estimate the strain and stress resultants. Further, the 3D elasticity equilibrium

equations are solved to get the thickness profile of the transverse shear and normal stress

components. The shell panel consists of aluminium and zirconia combination of material

constituents and six stress components are found for these isotropic materials. Also, two types

of power law equations assumed in the study that generate quite different results in comparison

with each other. At the end, the authors concluded that the higher order terms are necessary in

the kinematic model to catch the realistic static behavior of shell and plate structures.

A higher order theory that accounts for through-the-thickness deformation has been

considered based on radial basis collocation technique by Neves et al. (2013b) for free

vibration response of FGM plates. The principle of virtual work and Carrera’s unified

formulation are combined to arrive for the equations of motion and the boundary conditions.

Results are shown for cylindrical and spherical shells containing simply supported and clamped

boundary conditions. Cubic and quadratic variation of thickness is considered in in-plane and

transverse component of displacement, respectively. Results are tabulated by considering and

without considering the thickness stretching effect. As the shell geometry transform in to plate

geometry, the fundamental frequency reduces for all the values of volume fraction indexes. The

model without incorporating thickness stretching part, records lower values of frequency

compared to the model that include thickness stretching effect. But for thick plates, the effect of

thickness stretching has to be considered.

The finite element formulation for bending and vibration study of functionally graded

plates was presented by Thai and Choi (2013a), by employing various shear deformation

theories. These theories display strong similarity with the CPT and leads to four unknows in the

38  

displacement field. The primary variables present in the in-plane and transverse displacement,

respectively, are described by Lagrange and Hermitian interpolation functions. The

contribution of bending and shear component was incorporated in the transverse displacement

field. The in-plane field has been represented by shape function associated with the derivatives

of transverse displacement. Three different kinds of shape functions described by Shimpi

(2002) to include polynomial functions, Touratier (1991) to include sinusoidal functions, and

Soldatos (1992) to include hyperbolic sine functions are considered in the work. Regardless of

boundary conditions and thickness ratios, the frequency decreases and deflection increases as

the volume fraction index was chosen to represent the bottom of the plate (pure metal).

The buckling problem of thin rectangular FG plates subjected to biaxial compressive

loading with arbitrary edge supports was analyzed by Latifi et al. (2013). The displacement

equations are based on CPT that assumes the form of Fourier double series (Chung 1981) and

physical neutral plane (Zhang and Zhou 2008) was considered to derive the stability equations.

The derivatives of Fourier double series was performed by Stroke’s formulation. It was

illustrated that any possible combination of boundary conditions could be incorporated in the

study without imposing any conditions on Fourier series. The FG plate considered was

elastically restrained by means of translational and rotational springs at the four sides. As

expected, they observed that increasing additional constraints on the boundary increases the

buckling load. The results shows that shortening of the FG plate in the direction parallel to

loading direction gives rise to higher value of buckling load.

A higher order displacement model to include shear and normal deformation effect was

considered to obtain the stress and free vibration response of FG rectangular plates by Jha et al.

(2013a). They considered the material properties in the graded manner along the thickness

direction. Minimization of total potential energy was applied to derive the governing

differential equilibrium equations. They observed the fall-off tend in frequency parameter for

increase in values of aspect ratio (b/a) and pure ceramic plates ensures maximum frequency

parameter. Further, the efficiency of the theory with regard to FSDT and CPT theory was

demonstrated in the study in detail.

The static analysis of functionally graded plates is performed by Castellazzi (2013),

based on nodal integration plate element and FSDT. The power law distribution was assumed

in the study to estimate the mechanical properties of the plate at a specified height. The

interesting outcome from the study was that when the difference between the material

39  

properties of the two material constituent increases, the less clustered stress profiles are

observed about the line of pure ceramic and metal plates. Also, the bottom and top surfaces of

the plate shows more clustered profile of the stress distribution for FG plates having

compressive and tensile nature of stresses at top and bottom of the plate, respectively.

Tran et al. (2013) proposed isogeometric (IG) formulation for thermal buckling of

functionally graded rectangular and circular plates based on TSDT. To achieve geometric

representation and higher order approximations, non uniform rational B-spline (NURBS)

functions of arbitrary continuous order are employed as basis functions, which also fulfill the

C1 requirement of the HSDT. Due to the stiffness degradation offer by enrichment of metal

part, critical buckling temperature reduces for higher values of volume fraction index. Under

uniform temperature rise, this change was rapid for volume fraction index nearly equal to 2,

and for further values it becomes independent. Also under non uniform temperature

distribution, the FGM plates sustain higher buckling load compared to uniform temperature

distribution. Bifurcation type of buckling has been observed in clamped plates since it

neutralizes the bending-stretching coupling. Also, for homogeneous rectangular plates, non

uniform thermal distribution results in linear response of buckling parameter.

A three dimensional solution was proposed for arbitrarily thick functionally graded

rectangular plates incorporating general boundary conditions by Jin et al. (2014). They

incorporated closed-form auxiliary functions to eliminate all the discontinuities related to the

displacements and its derivatives at the edges of FG plate. Rayleigh-Ritz procedure was

implemented to obtain the exact solution by the energy functions of the FG plate. The plot of

variation of volume fraction through the thickness demonstrates that the volume fraction varies

abruptly near the bottom and top surfaces of the plate for volume fraction index nearly less than

or greater than equal to one. The response of FG plate with several combinations of realistic

boundary conditions was considered in the numerical segment.

A Navier closed form solution based on higher order SSDT developed by Levy and

widely adopted by Touratier (1991) was proposed by Mantari and Soares (2014) to predict the

bending response of functionally graded plates and shells. Stretching effect is incorporated in

the kinematic model and their shear strain shape functions are described by the two arbitrary

parameters m and n which are to be chosen by appropriate displacements and stress functions.

The detailed procedure was explained to extend the theory to non-polynomial HSDT in FEM.

The accuracy of the SSDT is proved in terms of constant unknowns in the displacement field

40  

with reduced error compared to conventional SSDT. The appropriate values suggested for m

and n is 4h (h is the total thickness of the panel) for some shell cases, while m=n=h/π for some

other cases.

A local Kringing meshless method based on Petrov-Galerkin weak formulation

combined with Kronecker delta functions was applied to study the mechanical and thermal

buckling behavior of FG plates by Zhang et al. (2014). To simplify the weak form of governing

equations at the internal boundaries, the cubic spline functions are employed. Uni axial

compression, bi-axial compression, a combination of bi-axial tension and compression, in-

plane shear and thermal loadings are considered as in-plane forces. Under thermal loading, two

types of thermal loads, thermal gradient and heat flux are considered. It was observed that the

buckling temperature attains maximum value for the FGPs correspond to volume fraction index

equal to zero. In addition, higher possibility of existence of buckling is expected for the case of

steeper distribution of temperature in a FGP and buckling temperature difference between

different FGPs increases as the volume fraction parameter decreases. The discrepancy with

respect to buckling parameter between FGP having temperature-dependent and temperature

independent properties was noticeable for higher values of volume fraction index.

The free vibration problem of a two dimensional (2D) functionally graded circular

cylinders was solved by Ebrahimi and Najafizadeh (2014) based on Love’s first approximation

CST. The generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ)

are used to discretize the equations of motion and boundary conditions, respectively. Two

micromechanical models based on Voigt method and Mori-Tanaka approach are used and the

frequency results are tabulated by considering both the models. It has been observed that the

results obtained by both the methods are virtually same, and shows only a minor deviation with

respect to high value of circumferential wave number. Also, the frequency assessments are

prepared for 1D and 2D cylindrical shells, and it was observed that the 2D FGM shells exhibit

improved performance compared to conventional 1D FGM by means of ensuring high

frequency values in all the cases, and this observation is more obvious for higher value of

circumferential wave number.

A brief discussion performed on analysis of FGM reveals the fact that the responses

(static, dynamic and stability) of the FGM plate/shell under thermal and mechanical loading are

greatly influenced by the parameter that dictate the material profile variation termed as volume

41  

fraction index in addition to other parameters like boundary constraints, geometry, loading

condition, thickness ratio, aspect ratio and curvature ratio.

A sinusoidal higher order shear deformation theory is proposed for the bending analysis

of functionally graded shells by Mantari and Soares (2014). The stretching effect was

incorporated in the theory and their strain functions are described by two arbitrary parameters.

A Navier form solution and principle of virtual work are assumed in the analysis. Extensive

numerical results are generated based on sinusoidal HSDT and compared with the FSDT and

other quasy 3D hybrid type HSDT results. Deflection and stress plots are presented based on

the various values of volume fraction index and geometrical properties of shells.

2.4 Static, dynamic and buckling responses of skew plates/shells

In general, research studies on skew plates are limited in number owing to the

complexity involved in the analysis. Various solution methods such as trigonometric series

(Echasz 1946; Mirsky 1951), Power series (Raju and Shah 1966; Coull 1967; Iyengar and

Srinivasan 1971), polynomial series (Reissner and Stein 1951; Reissner 1952; Stavsky 1963),

complex seires (Dorman 1953), biharmonic eigen functions (Morley 1961; Morley 1962),

Fourier series (Kennedy and Huggins 1964; Kennedy 1965) have been employed for analysis

of plates having skew geometry, in the past. The technique of finite difference (Morley 1963;

Jenson 1941; Naruoka and Ohmura 1959) has also been successfully used for the analysis of

skew plates, but they have limited accuracy for small skew angles. In addition, the finite

element method found its extensive application in skew plate analysis. Different studies based

on Kirchhoff plate bending elements (Rames et al. 1973; Rossow 1978; Vora and Matlock

1979; Wang et al. 1984; Felippa and Bergan 1987; Ming and Song 1987) and based on Mindlin

elements (Monforton and Michail 1972; Kolar and Nemec 1973; Ahmed and Mathers 1977;

Hughes et al. 1978; Pulmano and Lim 1979; Hughes and Tezduvar 1981; Belytschko and Tsay

1983; Owen and Figueiras 1983; Zienkiewicz and Lefebvre 1988; Prathan and Somashekar

1988) have proved better performance for the analysis of skew plates. In addition to above

mentioned techniques, some other techniques are also available in the literature for the analysis

of skew plates. Among the different methods, variational solution (Morley 1963; Morley 1964;

Kennedy 1968; Hadid et al. 1979), electrical analogy (Ruston 1964; Harden and Ruston 1967),

point matching (Warren 1964; Sattinger and Conway 1965), conformal mapping (Aggarwal

1966; Aggarwal 1967), equivalent grid method (Yettram 1972), finite strip method (Brown and

42  

Ghali 1974; Brown and Ghali 1975; Mukhopadhyay 1976; Cheung and Z. Dashan 1987) are

few that are reported in the literature.

A Parallelogram-shaped (skew) plates are studied under bending by Butalia et al. (1990)

using a Mindlin nine-node quadrilateral Heterosis element. Uniformly distributed load, point

load and different support conditions are incorporated in the study. In their earlier research

works, the authors proved that the heterosis elements are better than serendipity and Lagrangian

elements with respect to accuracy in case of thin plate situations (Hughes and Cohen 1978;

Hinton and Owen 1984). The moment plots reveal the fact that, the moment in x and y

directions strongly exhibit singularity nature in the vicinity of obtuse edges having opposite

signs. The authors concluded that using H9 (Heteroris element with 9 degrees of freedom)

elements show better convergence for deflection and principal bending moments at the centre

but obtuse corner modeling rather deteriorates. Hence after comparison, the L4 (Lagraningan

element with 4 degrees of freedom) shows better performance for obtuse corner modeling, even

though, only 1/3 the total degrees of freedom as that of H9 elements are considered for the

analysis.

Reddy and Palaninathan (1999) employed triangular plate element to perform the free

vibration of laminated skew plates. The consistent mass matrix has been derived in explicit

form and the boundary conditions of the skew corners are implemented through the

transformed element matrices. The fundamental frequency was found to be small for the layer

number of laminate equals 2. Ultimately, if the laminate contains more number of layers, the

response of skew plate under free vibration tends to be that of a homogeneous orthotropic plate.

In skew laminates with simply supported boundary, when number of layers is greater than 4,

the frequency initially rises with ply angle and reaches a maximum value and decreases again.

This maximum value for frequency occurs at ply angle values 45°, 50° and 65° for skew angle

values 15°, 30° and 45°, respectively. For all the skew angle values, the frequency co-efficient

elevates with the rise of skew angle due to reduction of non skew edge distance. The symmetric

variation of frequency parameter found in rectangular and square plates gets distorted for

laminates having skew boundary. Further, the extent of the distortion has direct proportion

relation with the skew angle of the plate.

The FSDT and HSDT (Kant 1982; Kant et al. 1982) based finite element models are

proposed by Babu and Kant (1999) for stability analysis of skew laminated composite and

sandwich panels. For global degrees of freedom of nodes lying on the skew edges of the plate,

43  

the transformation operation has been performed to achieve degrees of freedom at local

domain. The angle ply and cross ply laminates with various lamination parameters, width-

thickness ratios and boundary conditions are assumed to perform the numerical section. For 90°

lamination scheme, the plate with skew angle 45° shows higher buckling strength compared to

0° lamination scheme. The buckling load factor increases with the increase in skew angle

irrespective of the thickness (for both thin and thick laminates). But the increase is negligible

due to the large transverse shear effect in thick plates. In case of thin skew laminates, the

influence of fiber orientation angle is observed to increase with the increase in skew angle. But

for the case of thick laminates, this observation is more or less remain same for skew angle

0°,15°, and 30° and reduces for plates with skew angle value 45°. As far as shear models are

concerned, HSDT results are slightly higher than FSDT results and this is true of skew angle

0°. Also, for SSSS and CCCC laminates, the effect of shear deformation increases with rise of

skew angle and decreases with the increase of laminate thickness. Regarding sandwich plates,

for hf/h 0.05 (hf-thickness of core and h-total thickness of the plate), HSDT results are the one

at lower side to that of FSDT results and this discrepancy increases with increasing hf/h ratio

and skew angle.

Hu and Tzeng (2000) performed the stability study of skew composite laminate plates

subjected to uniaxial inplane compressive loads. The finite element based software ABAQUS

has been employed to perform the bifurcation buckling analysis of skew plates. With the rise of

skew angle, critical buckling load of the plate tends to increase. Under the simply supported

boundary, for plates with fibers parallel to edges records lowest buckling load and quasi-

isotropic plates usually show highest buckling loads. When the clamped boundary was chosen,

the buckling results are quite different. Also, for plates with skew angles, the buckling modes

shows more waves in in-plane loading direction. This buckling mode waves are more for

clamped edges than simply supported edges.

A new version of the DQ method was proposed by Wang et al. (2003) for buckling

analysis of isotropic and anisotropic rectangular skew plates. The proposed DQ method differs

from the conventional DQ by means of less degree of freedom used for the corner points and

the direct estimation of weight coefficients. Different features including various skew angle,

aspect ratio (a/b) and boundary conditions are assumed in the numerical examples. The

presented DQ method can also be represented as differential quadrature element method

44  

(DQEM), and useful to solve the problems with discontinuous loads, geometry and/or mixed

boundary conditions.

Ganapathi et al. (2006) studied the influence of functionally graded materials on

buckling of skew plates in conjunction with FSDT and finite element approach. The effective

properties of the functionally graded materials was assumed to be graded in the thickness

direction and estimated by means of Mori-Tanaka approach and Voigt rule of mixture. The

skew angles ranges from 0° to 45° are considered under in-plane bi-axial loads for thin and

thick plate cases. It was manifested that the Voigt rule of mixture produce higher buckling

loads and decrease in the buckling load is significant up to volume fraction index 2. The

volume fraction index beyond 2 yields no significant reduction in critical buckling load and this

tendency is independent of aspect ratio, skew angle and thickness of the plate. Also, the rate of

increase of critical load is high for plate with a/b=1 compared to a/b value equals 2.

A technical note based on FSDT and finite element approach was presented by

Ganapathi and Prakash (2006) for thermal buckling of functionally graded skew plates. The

temperature in linear and non-linear form was considered across the thickness. The solution for

the thermal problem was obtained by means of one dimensional heat conduction equation. For

thin FGM skew plate, increase in skew angle considerably increases the buckling strength for

a/b=1compared to other higher values of aspect ratio. When comparison was made between

buckling loads under linear and non-linear temperature variation, non-linear temperature

variation yields higher value compared to linear case. For thick plate case, buckling

temperature increases with skew angle and does not show any appreciable increase with respect

to volume fraction index. Further, the degradation of buckling temperature occurs slowly for

the value of volume fraction index greater than 2.

A simple, accurate and reliable algorithm based on discrete singular convolution (DSC)

has been proposed by Civalek (2007) to get the natural frequencies and buckling loads of

composite plates. Four noded element is used to map the straight-sided quadrilateral domain

into a square domain by means of second order transformation and the equations are finally

solved by chain rule. It was observed that the frequency and buckling load increases as the

skew angle of the plate increases.

The vibration study of skew plates using moving least square Ritz method was proposed

by Zhou and Zheng (2008). Due to stress singularities at the obtuse corner of the plate for large

skew angles, the results show slow convergence results. A trial function is assumed for the

45  

transverse displacement field and Ritz method has been employed to solve for the eigen value

equation. The boundary conditions are applied by means of MLS-Ritz trial function that

satisfies the essential boundary conditions along the plate edges. To overcome the stress

singularity problem observed in skew plates more grid points are placed around the obtuse

corners of a skew plate. The authors considered the large skew angles to perform the vibration

study and modal frequencies for different skew angles are tabulated in the numerical part.

Kumar et al. (2013) studied the free vibration of skew hypar shells using C0 finite

element formulation based on HSDT. The proposed element has seven nodal unknowns per

node and the effect of cross curvature has been included in the formulation. The frequency

results are generated for different laminations schemes and skew angles. High frequency values

are recorded for skew angle equals 90° for CFCF shells and for other category of boundary

conditions, skew angle equals 45° produces maximum frequency. For hypar shells with

combination of free and clamped edges, the skew angle should be chosen between 45° and 90°,

to ensure the high frequency.

Jaberzadeh et al. (2013) investigated the buckling of functionally graded skew and

trapezoidal plates under thermal load. The element free Galerkin method was employed and

shape functions are constructed using moving least square approximation. The technique of

orthogonal transformation was utilized to enforce the essential boundary conditions in the

formulation. Different temperature variations such as linear, uniform and non-linear cases are

considered to study the thermal buckling response of FG plates. The buckling temperature of

pure ceramic plates is higher than FGPs, and this effect was more pronounced in thick plate

cases. An interesting observation was discerned regarding skew plates. As the skew angle rises,

the critical buckling temperature increases and the deviation among skew angle 60° and other

cases of skew angle was considerable under linear, uniform and non-linear cases. Only a small

change of critical buckling temperature was observed for volume fraction index equal to 5 and

beyond that no significant improvement was noticed. This fact is obvious for all the value of

skew angles considered in the problem. Regarding the response of skew angle to buckling

temperature, a similar trend observed in skew plates was concluded for trapezoidal plates also.

46  

2.5 Static, dynamic and buckling responses of sandwich

plates/shells

In general, sandwich plates are constructed by moving the load carrying face sheets

away from the neutral plane or the torsion axis by means of low strength core layer to increase

the moments of inertia of the cross section. Owing to this reason, sandwich plates registered

their application in a variety of engineering field including aircraft, construction and

transportation, where the stiff, strong and light structures are the primary requirements (Zenkert

1997). Due to the mismatch of material/thermal properties exists at the core-face sheet layer

interface, sandwich plates are susceptible to delamination/debonding type of failure modes,

especially under impact loading (Abrate 1998). In some applications, the upper layer of the face

sheet or core has to be stiffer than the bottom face sheet which necessitates the implementation

of FGM concept for the face sheet and core layers. An advanced construction of sandwich

panel consists of two FG face sheets, not necessarily be identical, are bonded to a core layer

either isotropic/FGM thereby increasing the bending rigidity of the plate at an expense of small

weight. In some cases, the piezo electric effect has been incorporated in the sandwich

construction to serve the purpose of smart materials. In such situation, piezoelectric ceramics

will act as sensors and actuators and usually placed at the mid layer of the sandwich

construction (Shen 2005). In addition, under thermal environments, the metal-rich face sheets

can alleviate the large tensile stresses on the surface at the early stages of cooling (Noda 1999).

In this connection, many research studies on static, dynamic and stability analysis of FGM

sandwich plates are available in the literature by incorporating graded distribution of material

properties either in the core or face sheet layer.

As an extension of the earlier works submitted on sandwich panels incorporating

functionally graded material under transverse loading (Anderson 2002a; Anderson 2002b),

Anderson (2003) presented an analytical 3D elasticity solution for a sandwich plate with a

functionally graded core. The transverse loading has been applied by means of a rigid spherical

indentor and the contact area and pressure distribution due to indentation was obtained by using

an iterative solution method. The conditions of continuity of traction and displacement

components between the layers are utilized to solve the equation based on Reissner’s theory.

The sandwich plate was modeled with orthotropic face sheets and isotropic core having

functional properties that have the exponential variation in the thickness direction. The plot of

in-plane normal compressive stress vs. contact force establish the fact that no reduction of

47  

stress with respect to given contact force was observed by incorporating stiff material in the

core. Also, interfacial transverse shear stress will not reduce with increase of stiffness ratio

Emax/E0 (Emax-maximum stiffness, E0-minimum stiffness). This trend is common for the other

transverse shear stress values. It was predicted that the increase in the interfacial shear stresses

is the effect of localization caused by the indentor loading to increase the stiffness of the core in

the region of core-face interface.

In part I for deflection and stresses of functionally graded sandwich plates, Zenkour

(2005a) presented the two dimensional solution for simply supported condition. The face sheets

are assumed to have a power law variation of modulus of elasticity and Poisson’s ratio through

the thickness. The core layer is made of homogeneous ceramic material and by considering the

symmetry of the layers, different sandwich plates are proposed. Various displacement models

based on CPT, FSDT, sinusoidal, and TSDT are accomplished in the study. The pure ceramic

plate records small magnitude of displacement compared to pure metallic plates. The FGM

plate undergoes deflection in between that of pure ceramic and metallic plates by virtue of its

intermediate stiffness strength. Under the application of sinusoidal pressure, the ceramic plate

ensures maximum compressive and tensile at the bottom and top of the plate, respectively.

When the plot of transverse shear stress was considered, the maximum value occurs at a point

on the mid-plane of the plot and isotropic plate shows lower value of stresses compared to

FGM plate. For FG plates, the FSDT theory provides results close to the TSDT and SSDT

particularly at the faces of the core layer.

As an extension of Part I of Zenkour (2005a), Zenkour (2005b) studied the vibration

and buckling response of functionally graded sandwich plate considering rotator inertia in the

formulation. Analytical solution based on sinusoidal shear deformation theory was incorporated

in the study. The sandwich plate modeled herein consists of pure ceramic material at the core

layer and homogeneous face sheets at the top and bottom face sheets. In the bottom segment,

the composition has been varied from a metal-rich part to a ceramic-rich part, while in the top

segment; the composition has been varied from ceramic-rich part to a metal rich part. The

generated results based on SSDT are compared with CPT, FSDT and HSDT based studies.

Although the SDPT based frequencies are marginally lower than elasticity solution; the

buckling loads and vibration frequencies obtained by the SDPT are at considerably higher side

than other theories. As the core thickness with respect to total thickness decreases and volume

fraction index increases, the values of buckling loads and fundamental frequencies are shown to

48  

have a fall-off trend. For the above statement, an exception has been observed for the value of

volume fraction index equals 5. Among the different types of sandwich plates with respect to

the symmetry, the 1-2-1 case exhibit highest sensitivity for the various parameters (aspect ratio,

thickness ratio and volume fraction index) considered in the study. As one may expect, the

uniaxial buckling load may be twice the biaxial one and this observation was independent of

kind of sandwich plate and value of volume fraction index. In general, the authors concluded

that the results corresponding to the ceramic and metal layers are respectively, the upper and

lower bound solutions of those of the sandwich functionally graded plates.

A higher order based triangular element was employed by Das et al. (2006) to study the

deflection and stress pattern in sandwich plate having homogeneous and FGM layer as core

part. To reduce the computational cost offered by the layer wise theories, a single layer theory

has been proposed in which the field variables with weighted average accurately capture the

deformation modes in the thickness direction. To satisfy the inter element continuity

requirement, a hybrid energy functional has been employed and non-uniform variation of

temperature was assumed on the top surface. In homogeneous core sandwich plates, due to

mismatch of properties at the core and face sheets, high magnitude of stresses was observed at

the layer interfaces. Further, steep stress gradients are developed at the interfaces and to

minimize the shear and peeling stresses at the layer interfaces, graded properties are considered

depending upon the temperature distribution dictated by the extreme environment. In case of

sandwich plates with FGM core, due to the occurrence of high temperature at the top face

sheets the plate tend to bulge outwards in all the cases. The plate corresponding volume

fraction index equal to 0.2 and 0.5 experience a global bending, where as for the case n equals

1.0 and 2.0, downward expansion of face sheets happens reflecting the swelling characteristics

of the panel. Due to the phenomenon of high values of thermal strain, the in-plane displacement

attains large values at the center of the core. But top face sheets undergo maximum transverse

stresses in case of n=2.0 and this magnitude was considerably lesser than the one observed in

homogenous core model. It was put forward that the discontinuity of strain observed at the

interfaces can be effectively minimized by the reducing the difference in properties of the face

sheet and core materials.

A sandwich functionally graded rectangular plates with simply supported and clamped

boundaries are analyzed by Li et al. (2008) based on 3D elasticity solution. Two types of

models viz. the first model with homogeneous core and FGM face sheet and the later model

49  

with homogeneous face sheet and FGM core are considered for the analysis. The displacement

functions are expanded by a series of Chebyshev polynomicals (Cheung and Zhou 2002; Zhou

et al. 2002) multiplied by appropriate functions that satisfy the essential boundary conditions

are assumed. Due to the increase of volume fraction index in type A or the decrease of volume

fraction index in type B simply supported and clamped plates, the natural frequency decreases

with respect to the decrease of material rigidity. The material rigidity factor plays vital role in

thin plates compared to thick plate and further this effect was little larger for simply supported

boundary compared to clamped plates. In the absence of homogeneous core layer (1-0-1 type

plate), the effect of volume fraction index was more significant than the sandwich plate (1-8-1)

having homogeneous hard core. Also, the role played by volume fraction index is significant in

case where the core was modeled with hard core than that with soft core. When the

displacement plot along the thickness direction was plotted, flexural and extensional modes are

observed and for flexural modes the displacement is non uniform in nature which implies the

existence of normal stress in the thickness direction. For the extensional modes, the deformed

plate retains the same thickness but the in-plane displacement components are symmetrical

about the mid-plane.

The stability study of truncated conical shells has been carried out by Sofiyev et al.

(2008) under uniform pressure. The material properties of the three-layered functionally graded

conical shell having FGM core vary in graded fashion through the thickness. This gradation

variation may be arbitrary in nature and combines the volume fraction of ceramic and metal

constituents. The closed form solutions based on Galerkin method are obtained for the stability

analysis of conical shells. The volume fraction of ceramic is often chosen as a function of

linear, quadratic and inverse quadratic term of thickness co-ordinate. As the ratio of the total

thickness to FG layer (h/2a) increases, the dimensionless external pressure increases for the

case of linear and quadratic variation of compositional profile, however decreases for inverse

quadratic compositional profile. But external pressure becomes insensitive for h/2a greater than

3. On the other side, number of circumferential waves does not vary with respect to h/2a ratio.

When the case of three layer conical shell are compared with corresponding homogenous

conical shell, highest effect was encountered for the quadratic compositional profile (24.46%),

while the lowest effect for the inverse quadratic case (19.15%). The ratio h/2a have constant

effect on critical buckling load for homogeneous case, but have considerable influence on three

layer conical shell. For example, when h/2a ratio equals 1.1, the effect was 24.46% and 19.15%

50  

for quadratic and inverse quadratic change of compositional profile, respectively; further for

higher values of h/2a i.e., h/2a = 6, the effect was 21.44% and 21.18% for kopquadratic and

inverse quadratic change of compositional profile, respectively.

Part et al. (2008) presented the dynamic response of skew sandwich plate with

laminated composite faces based on HSDT. They have made an attempt to sketch the influence

of skew angle, layup sequence on dynamic response. The authors emphasize the fact that the

including higher order terms is necessary to analyze skew laminates, due to the contributions

made by the non-linear shear deformation effects through the laminate thickness. For skew

angle 0° and 15°, the displacement curves are close to each other, where for the skew angle 30°

extremely lower values are noticed. The flexural rigidity of the plate reduces with the rise in

skew angle of the plate. Finally, the authors conclude that (90°/0°/core), layout may be the best

choice while designing cross-ply skew sandwich laminates.

Brischetto (2009) proposed equivalent single layer and layer wise theories for sandwich

plates with functionally graded core under mechanical loading. The theories are based on

principle of virtual displacement and Reissner’s mixed variational theorem. In the case of layer

wise theories transverse shear/normal stresses are used as primary variables. For thick plates

layer wise models seem to be essential, while equivalent single layer models are suitable for

thin plates with the assumption of higher order expansions. Also, the use of layer wise models

was proven to be good enough to predict the normal stress in the z direction. If the value of

volume fraction index chosen was 10, the use of mixed models seems to be better choice. The

addition of FGM core exhibit the continuous distribution of stress components in the z

direction, which is otherwise not possible in case of conventional sandwich plates. The

discontinuity offer by FGM in stress case is due to the application model based on principle of

virtual displacement theory, which can be alleviated efficiently by means of mixed model.

A three-dimensional elasticity solution was presented by Kashtalyan and Menshykova

(2009) for sandwich panels under transverse loading. The core layer has been modeled with

two options; in the first option the core and face sheets are modeled with homogeneous

material having different shear modulus values and the latter option employs functionally

graded material in the core part, while the face sheets are assumed to be homogeneous in

nature. The layer with graded properties has exponential variation of material properties in the

thickness direction. From the plot of through-the-thickness variation of transverse shear stress,

the reduction in stress magnitude was observed in the face sheet/core interface, provided the

51  

core portion was modeled by functionally graded material. But this reduction was observed at

the expense of increased transverse shear stresses in the core part of the sandwich plate. The

influence of type of material in the core is more pronounced in thin panels by means of

considerable reduction of in-plane normal and shear stresses at layer interfaces and face sheets.

Also, due to high stiffness offered by the functionally graded panels the model with such

material will considerably reduces the deflection. The authors emphasize the statement that, the

use of functionally graded material in the sandwich panels, in general, eliminates the deflection

and stresses in face sheets and at layer interfaces.

A three-dimensional elasticity solution presented by Kashtalyan and Menshykova

(2009) for functionally graded sandwich panels was further extended to study the panel under

different loading configurations (Woodward and Kashtalyan 2011). The uniformly distributed,

patch, line, point and hydrostatic loadings are considered in the analysis. The models similar to

the one considered in Kashtalyan and Menshykova (2009) are incorporated in the analysis.

Both the models (core with homogeneous material and core with functionally graded material),

exhibits similar trend as far as variation of out-of-plane normal stress was considered. This

observation is due to the mechanical properties of face sheets which have major contribution in

dictating the stress variation. Further, this variation has common effect on different types of

loading conditions considered in the study. However, for the panels under point and line loads

sharp changing pattern of stresses are manifested in the core part near to upper face sheets.

Further, the magnitude of transverse shear stresses is reduced if the core with FGM was chosen,

and this is true under all the forms of loading conditions. Under distributed form of loading

patterns (hydrostatic, udl and sinusoidal), the homogenous core shows maximum transverse

shear stresses (σ13) at the center of the core than the one with FGM core. But for point and line

loads, the stresses are maximum at the upper face sheets, and this condition is regardless of the

type of material exist in the core portion. Further, for the case of point and line loadings the

maximum transverse displacement in the homogeneous core increases sharply to attain the

maximum value at the upper portion of the panel, while the assumption of FGM core in the

model diminishes this peak point.

The simple refined theory developed by Shimpi (2002) for isotropic plates and further

extended by Shimpi and Patel (2006a; 2006b) for orthotropic plates was implemented by

Abdelaziz et al. (2011) for functionally graded sandwich plates under mechanical loading. To

derive the governing differential equations, PVD is used and to obtain the closed form solution

52  

of the functionally graded plate with simply supported condition Navier’s method was

employed. Two models the one with FGM core and the later model with homogenous core and

having the different thickness ranges of layer such as 1-0-1, 2-1-2, 1-1-1, 2-2-1, and 1-2-1are

considered. For the case of pure ceramic plates, different kinds of plates show identical bending

behavior. The variation of axial stress is observed to be very sensitive to the change of volume

fraction index. As a general observation, the pure ceramic plats give smallest shear stresses and

deflections and largest axial stresses. As the value of core thickness with respect to the total

thickness of the plate increases, deflection, axial and shear stresses decreases. Among the

different kinds of sandwich plates, 2-2-1 type FGM ensures smallest magnitude of axial

stresses. When the plot of through-the-thickness distribution of axial stress in x direction was

plotted for plate with FGM face sheets, the stresses are tensile at the top and compressive at the

bottom surface of the plate. This stress variation has linear profile for isotropic plates and non-

linear profile for FGM plates. The plot of shear stress for homogeneous soft core reveals the

maximum value at the mid-plane of the plate and its magnitude is small for FGM plates than

homogeneous metal plate.

Meiche et al. (2011) assumed a sandwich model based on hyperbolic shear deformation

theory for buckling and free vibration study of FGM plates. The final displacement form

proposed by the theory leads to four nodal unknowns in the kinematics model and the grading

technique of the FGM layer follows a simple power law distribution. Hamilton’s principle and

Navier solution are used to derive the governing equations. The fundamental frequency

increases as the core thickness to the total thickness of the plate decreases and the frequency

results are maximum for ceramic plates and minimum for metal plates. The influence of core

thickness has similar impact on buckling response as that of frequency response. Further, the

buckling load increases smoothly in the plate as the ceramic content in the plate increases.

Merdaci et al. (2011) proposed two refined shear deformation models (RSDT1 and

RSDT2) for the bending response of functionally graded sandwich plates. The theory

developed was variationally consistent and leads to four nodal unknowns in the formulation.

The parabolic variation of transverse shear stresses was assumed and the shear stress conditions

are satisfied at the top and bottom of the plate. The core layer was modeled with pure isotropic

(ceramic) material, while the skin layers are assumed to have graded material properties in the

thickness direction. The obtained results for bending response of functionally graded sandwich

plates are compared with the parabolic shear deformation plate theory, exponential shear

53  

deformation plate theory, sinusoidal shear deformation plate theory and FSDT. Even though all

these theories lead to the different form of displacement field, they provide identical results

(either deflection or stresses) for pure ceramic plates. As the aspect ratio of the plate rises, it

tends to elevate the deflection parameter and this irrespective of the sandwich plate type. The

plate having pure ceramic material ensures highest value for axial stresses in the x-direction

and this magnitude increases with the increase in volume fraction index. On contrary, the

ceramic plates have smallest shear stress values compared to other isotropic and FGM

sandwich plates. The maximum value of shear stresses occurs at the mid-plane and its

magnitude for homogeneous plate (ceramic and metal) is smaller than FG plates. The

transverse shear stress variation of ceramic and metal plates are identical in nature and this is

due to the reason that these plates are fully homogeneous and the stresses do not depend on the

modulus of elasticity of these materials.

After the Part I (Zenkour 2005a) and Part II (Zenkour 2005b) submission with respect

to static, vibration and buckling analyses of functionally graded materials, recently, Zenkour

(2011) presented the solution for thermal buckling of functionally graded plates. The sandwich

plate configuration similar to the one modeled in Zenkour (2005a) and Zenkour (2005b) was

assumed. In addition to the effective mechanical properties, the thermal expansion was

assumed to be graded as per material power law. The non-linear distribution of temperature

profile was considered in the thermal analysis. Buckling results are presented only for

symmetric configuration of sandwich plates. For FGM plates, a sharp decrement trend of

critical buckling temperature was observed with increase in thickness ratio (a/h) and aspect

ratio (b/a) values. But as far as homogenous plate case was concerned, the decrement was

observed in gradual fashion. When the non-linear temperature variation was opted, the critical

buckling temperature produce higher results compared to uniform variation of temperature.

Also, the results corresponding to the linear temperature change produces the intermediate

results between non-linear and uniform variation cases. When the thickness of the core

becomes half the plate thickness, rapid decreasing trend was observed in buckling temperature

to reach minimum values and then increases gradually as per the variation of in homogeneity

parameter and this tendency seem to be exemption for 1-1-1 and 1-2-1 plate cases.

Hadji et al. (2011) presented a four variable refined plate theory (RPT) to get the

frequency response of functionally graded material rectangular sandwich plates. As the name

suggests, the theory contains only four nodal unknowns which was quite less number compared

54  

to other existing shear deformation theories. The sandwich plate having homogeneous/FGM

core with the combination of FGM/homogeneous face sheets are considered for the free

vibration study. Navier’s method and Hamilton’s principle are utilized to derive the governing

equations. It was manifested that the fundamental frequency of the sandwich plate get

decreased with the decrease of material rigidity and the cause for the observation is the increase

and decrease of volume fraction index in type A and type B plates, respectively. Also, the

volume fraction index parameter has vital role in controlling the frequency parameter in case of

thin plates compared to thick plates. When homogeneous layer of core was considered in the

example, the maximum and minimum results, respectively, are corresponds to ceramic and

metal plates. Regarding the other observations pertains to frequency analysis of functionally

graded plates, the results similar to those of Li et al. (2008) are observed.

Alipour and Shariyat (2012) presented bending and stress analysis of the circular

functionally graded sandwich plates having specific material properties and edge conditions.

The governing equations are derived based on the elasticity-equilibrium equations in the

framework of zig zag theory and each layer of the sandwich plate is assumed to be made of

functionally graded material. The continuity conditions of the transverse stresses at the layer

interfaces are taken to predict the global and local response of the sandwich plates and also the

local variations of the displacements are considered. The so formed governing equations are

then solved by a Maclaurin-type power series solution. The plot of radial displacement

distribution indicates that the plate undergoes global clockwise bending in addition to

counterclockwise local rotation occurs in the core. Since the ratio of shear to bending

deflections was considerable in thick plates, the global rotation angle will be remarkably

affected by the shear forces. Further, as the thickness of the core increases, the resulting

stresses reduce and this reduction was observed to be more in plate with stiffer core. Similarly,

rise in modulus of elasticity of the core leads to increased stresses in the core and leads to

parabolic distribution of the stresses.

Neves et al. (2012c) studied the bending and free vibration response of an isotropic and

sandwich functionally graded plate considering the through-the-thickness variation of

deformations. The in-plane field represents the hyperbolic sine terms of transverse component

and transverse displacement contain the quadratic term of some unknown functions. Carrera’s

unified formulation (Carrera 1996; Carrera 2001) has been utilized and the interpolation

operation was performed based on radial basis collocation technique. An extensive plots and

55  

tables are presented for isotropic and sandwich plates (FGM as core) where the material

properties have polynomial material law variation (Zenkour 2006). The results incorporated in

the study highlight the importance of thickness stretching effect to be considered in the

kinematic model to accurately predict the displacement as well as normal stress component.

The bending and free flexural vibration of sandwich functionally graded material

incorporating FGM as core/face sheet has been investigated by Natarajan and Manickam

(2012) by employing QUAD-8 shear flexible element under mechanical and thermal

environment. The assumed kinematic field incorporates the cubic and quadratic terms in the in-

plane and transverse displacement components, respectively, in addition to the zig-zag function

in the in-plane fields (Ali et al. 1999; Ganapathi and Makhecha 2001; Makhecha et al. 2001).

The zig-zag function is piecewise linear at the interfaces and address the slope discontinuities

of u and v at the sandwich interface. The effect of rotary inertia and in-plane terms are

considered for the vibration response of sandwich plates. Results are presented based on four

displacement models (three HSDT models with 13, 11 and 9 nodal unknowns and one FSDT

model). The stresses and displacements reduce with increase of thickness of core layer and

increases with rise of volume fraction index. The increase in ceramic and metal component is

the attributed reason for this change in flexural stiffness of the plate. The first two HSDT

models produce identical results for displacements and stresses, while the latter models (HSDT

9 and FSDT model) cannot predict the displacements and stress, accurately. For the response of

plate under mechanical loading higher order and lower order models yield identical results for

stresses and displacements. Due to the variation of thermal expansion co-efficient, the models

show different stress variation under mechanical loading. Also, the fundamental frequency

parameter decrease with decreasing gradient index for type B plates, while the fundamental

frequency parameter decrease with increasing gradient index for type A plates due to the

material rigidity difference of the layers. In type A sandwich plates, the material rigidity

decreases with volume fraction index and for type B sandwich plates due to the larger volume

fraction of ceramic material rigidity tends to boost up.

The study performed by Neves et al. (2012a) has been expanded to study the static, and

free vibration response of functionally graded sandwich plates. The Carrera’s unified

formulation, PVD and higher-order shear deformation theory that accounts for the extensibility

in the thickness direction was incorporated in the work. Different kinds of sandwich plates

similar to other earlier works have been considered with FGM as core and face sheet layers. As

56  

the thickness ratio increases, the in-plane axial stress increases and this change was abrupt for

the a/h value beyond 10. Also, displacement decreases as the thickness ratio increases for the

different types of sandwich plates and the influence of volume fraction index is to elevate the

deflection parameter. The first ten modes of natural frequencies are shown for sandwich plates

considering thickness stretching effect in the model. Regarding the buckling analysis

observation, fully ceramic plate has higher buckling strength and this strength decreases as the

volume fraction index increases. In addition, if the core to total thickness of the plate increases

the buckling strength of the sandwich plate increases. Finally, the inclusion of thickness

stretching effect in predicting the static, free vibration and buckling response was established in

the study through various numerical examples and stress variation plots.

Neves et al. (2012a) included the Murakami’s Zig-Zag term (Murakami 1986) to

address the slope discontinuities exist in the functionally graded sandwich plates under bending

by incorporating hyperbolic sine term for the in-plane fields and quadratic variation term in the

transverse displacement field. The Carrera’s unified formulation combined with the radial basis

functions was adopted. The graded properties are considered either in the core layer or in the

face sheets. As the volume fraction index increases the displacement component increases for

simply supported sandwich plate. But the displacement results again depends on considering or

neglecting the warping effect in the thickness direction. The significant contribution of the

warping effect was visualized in the thick plates, as expected. The transverse displacement has

significant rise as the core to total thickness ratio increases.

An improved higher order theory was implemented by Khalili and Mohammadi (2012)

for the free vibration analysis of sandwich pales consists of functionally graded face sheets

under thermal environment. The temperature dependent material properties are assumed for the

face sheet and core materials by a third-order non-linear function of temperature (Reddy 1998)

and further the distribution of volume fraction was estimated by power law equation.

Unsymmetric and symmetric sandwich plates are analyzed by Hamilton’s principle. The core

in-plane stresses are considered in the vibration response of the sandwich structures. It was

manifested that the fundamental frequency elevates with the increase in the thickness of the

face sheets. This observation is due to large amount of ceramic material and thus increases in

the structural stiffness of face sheets. The magnitude of fundamental frequency exhibit rising

trend for lower value of temperatures and for higher value of volume fraction index. If the non-

linear strains of the face sheets are not considered in the analysis, the influence of temperature

57  

on the frequency parameter tends to fall-off with higher face sheet thickness. This phenomenon

is due to the fact that the thick FG face sheets have more amount of silicon nitride (ceramic)

than the thin face sheets. Also, the temperature has plays more dominant role over pure metal

component (stainless steel), than over ceramic component (silicon nitride). When the non-linear

terms of the face sheets are considered in the formulation, the frequency has a tendency to

increase with increase of face sheet thickness. The final conclusion from the study was that the

fundamental frequency increases for higher value of volume fraction index of soft core and

declines for higher value of volume fraction index of hard core sandwich plates.

The bending study of functionally graded sandwich plates is performed by Houari

(2013) based on higher order shear and normal deformation plate theory. The theory considers

the sinusoidal variation of displacements through the thickness and satisfies the stress free

boundary conditions at the top and bottom surface of the plate and thus the elimination of any

shear correction factor. The core is made of isotropic ceramic material, while the face sheets are

made of two-constituent phase of functionally graded material that obeys simple power law

equation. Each displacement (u, v and w) contains the term for bending, shear and stretching, in

which bending and shear terms are functions of x and y; while stretching part was a function of

x, y and z directions. By considering the symmetry of the plate, there kinds of sandwich plates

with notation 1-0-1, 1-1-1 and 1-2-1 are incorporated in the numerical segment. The influence

of shear deformation theories becomes least significant for fully ceramic plates under the

condition of neglecting stretching effect. When the stretching effect was incorporated in the

formulation, the plate become stiff and hence reduces the deflection of sandwich plates. The

axial stress values are found to be lower in plates that consider the thickness effect than the

plates neglecting the effect, and axial stress increases with the raise in the value of volume

fraction index.

A refined trigonometric shear deformation theory that involves four nodal unknowns

was proposed by Tounsi et al. (2013) for bending analysis of functionally graded sandwich

plates under mechanical and thermal loading conditions. The parabolic variation of transverse

shear stress was assumed in the study and satisfies the stress boundary conditions at the top and

bottom of the plate. The sandwich plate having FGM core and homogeneous skin layers was

incorporated in the study. By considering the symmetry of the layers with respect to mid-plane

different thickness schemes are proposed in the numerical part. For all kinds of sandwich

plates, deflection declines as the aspect ratio rises. The difference between the results based on

58  

different shear deformation theories show stable tendency for isotropic case and this fact is

irrespective of the types of sandwich plate. A large variation of response was concluded among

different kinds of sandwich plates when the case of thermal loading was considered. The theory

based on trigonometric functions produce identical results with sinusoidal shear deformation

theory while almost identical to those of parabolic shear deformation theory. For all the kinds

of sandwich plates, the pure isotropic case produce smallest values of stresses and deflection,

when graded properties are encountered in the plate layer, all the quantities shows rising trend

and this trend depends on the value of volume fraction index. Under various types of sandwich

plates, the plate with symmetric thickness ratio (2-1-2) ensures smallest transverse shear

stresses and the plate with non-symmetric thickness ratio (2-2-1) endures smallest value of

axial stresses.

A meshless based collocation technique has been employed by Xiang et al. (2013) for

the free vibration of sandwich plate made of functionally graded face sheet and homogeneous

core. The proposed meshless method enables to approximate the governing equations in the

plate domain using all the nodes. The modified form of Reddy third order theory known as nth-

order theory has been developed for the purpose. The sandwich plates of 1-1-1, 2-1-2 and 1-8-1

thickness schemes are considered to generate the numerical results. Different combinations of

volume fraction index and boundary conditions are considered to tabulate the frequency values.

A 2-D Ritz models are proposed by Dozio (2013) for the free vibration response of

functionally graded sandwich plates having functionally graded core. The formulation becomes

general due to the admissible functions of Ritz variables and the assumption of invariant

properties with respect to the kinematic theory. To generate the results the boundary conditions

other than simply supported and clamped are considered. It was noticed that irrespective of the

different parameters considered in the study, the frequency parameter generally deceases with

the increase in the value of volume fraction index. This is due to the smaller volume fraction of

ceramic component to reduce the stiffness of the plate. When thin sandwich plates are

considered, this effect becomes smooth. When the value of volume fraction index becomes

greater than 5, the first modes for SCSC and CFFF sandwich plates are unaffected. The

frequency was found to be higher for the SCSC plates than the corresponding cantilever plates.

This fact is owing to the reason of high number of constraints imposed in the SCSC sandwich

59  

plates. The authors concluded that the eigen frequencies tend to crowd together with increasing

mode number for CFFF plates.

Yasin and Kapuria (2013) employed four-node quadrilaterial element for static and free

vibration response of multi layered composite and sandwich shells based on efficient layer wise

zig zag theory. The requirement of C1 continuity has been circumvented by means of improved

discrete Kirchhoff technique. Comprehensive examples are performed by considering various

parameters such as boundary conditions, curvature ratio, aspect ratio and ply angle values. The

theory proposed by authors was proved to be more accurate than three-layer higher order layer

wise theories. For moderately thick sandwich shells, there observed a high level of error of the

order (> 60%) for the fundamental natural frequencies, when equivalent single layer theories

are incorporated.

2.6 Geometrically non-linear responses of laminated and functionally graded plates/shells In general, the non-linear problem that involves deformations of the order of the

thickness of the plate/shell is called large deformation problem. These problems are addressed

by the assumption of non-linear strain-displacement relations, since the deformation in the

elastic body can have a magnitude that does not overstrain the material. Because this process

was deformation dependent, it was classified as geometric non-linear problems. In most of the

literature works, strain-displacement field was considered by means of Green-Lagrange strain

relation. Further, von Kármán assumptions are imposed on the strain field by retaining the

quadratic terms in the slopes of the deflection and neglecting other non-linear terms (Reddy

1997), thus leading to final non-linear strain equations. From the past literatures, even for the

case of laminated (Baskar et al. 1993; Vu-Quoc and Tan 2003; Balah and Al-Ghamedy 2002),

homogeneous and isotropic shells unpredictable response was observed under large

deformation situation. Hence, it becomes vital to study the non-linear response of

inhomogeneous materials like FGM plates/shells.

Srinivasan and Bobby (1976) performed the non-linear analysis of skew plates using

finite element method. For the analysis, a high precision confirming triangular plate bending

element was used. The assumed triangular element was initially reported by Cowper et al.

(1970) and extended for non-linear plate analysis by Hwang et al. (1972). The skew angle of

the plate ranges from 0°, 30° and 45° are analyzed under large deformation and it was estimated

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that the computational effort increases as the skew angle increases and also less number of

elements are found to be sufficient for the clamped plates than simply supported plate. Except

for the case of skew angle 45°, four elements are required for the analysis of quarter plate. It

was manifested that as the skew angle increases the central deflection decreases for both simply

supported and clamped boundary conditions. This is the observation similar to the one derived

in the case of linear analysis. Under the application of higher loads, the bending stress increases

as the skew angle increases and this trend more marked for the simply supported case. In

addition, the minor principal stress found to be decreased as the skew angle increased and this

behavior is common for both the boundary conditions. But the membrane stresses at the centre

are less for simply supported boundary compared to the clamped one. Since the simply

supported skew plate transfer the load by bending action and hence the reduction of membrane

stresses was observed.

Pica et al. (1980) performed the geometric non-linear analysis of plates using Mindlin’s

theory and finite element formulation. The solution algorithm for the obtained non-linear

equations was based on Newton-Rpahson method that combines the series of linear solutions.

Various numerical examples are performed considering linear, Serendipity, Lagrangian and

Heterosis element for square, skew, circular and elliptical geometry of plates under distributed

and point loading. An irregularity sense of performance was observed in case of quadratic

Heterosis (QH) element by means of providing better stresses at the edge than at the centre. For

rectangular type of mesh, the QH elements are emerged as best among all the other element

cases. But the curved boundaries present in the mesh lead to the inconsistent behavior of all the

elements, particularly in case of predicting the stress values.

Kant and Kommineni (1992) employed the higher order shear deformation theory for

the linear and non-linear finite element analysis of fibre reinforced composite and sandwich

laminates. The transverse shear stresses are assumed to have parabolic distribution and Green’s

strains are considered in the von Kármán sense to account for large deformations, small strains

and moderate rotations. For the finite element analysis, a simple nine noded Lagrangian

quadrilateral element with nine degrees of freedom was considered. The displacement field

considered in the study of Kant and Pandya (1988) and Reddy (1982) was assumed in the work.

The cubic variation of thickness in the in-plane fields and constant variation of transverse

displacement was considered in the kinematic mode. Results are generated by considered

various material and geometric parameters of sandwich laminates. The close range exists

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between HSDT and FSDT results for thin cases, and significant deviation has been observed

for the case of thick and moderately thick plates.

Rao et al. (1993) obtained the finite element formulation for the large deflection

analysis of stiffened plates using the eight noded isoparametric quadratic stiffened plate

bending element. To derive the fundamental equations of the plate, the Mindlin’s hypothesis

was considered. The deflection equations under large deformations are based on von Kármán

theory. The obtained non-linear equilibrium equations were based on the Newton-Raphson

iteration technique. The formulation was made for general case, so that the stiffener can

accommodate anywhere in the plate other than nodal line. Further, the stiffener properties are

taken at the Gauss points in the tangential direction of the stiffener. This leads to the different

local axes system which then converted to global axes. The effect of stiffener in non-linear part

was neglected, since it will not have much variation at the global level. The proposed non-

linear formulation based on Mindlin’s hypothesis was incorporated in the computer code

FORTRAN 77 to generate the results. Different problems such as clamped skew stiffened plate,

clamped DRES (Defense Research Establishment, Suffield) panel, clamped rectangular plate

with single stiffener and square clamped plate are considered under large deformation.

The non-linear transient thermo elastic response of functionally graded plates was

studied by Praveen and Reddy (1998) accounting for transverse shear strains, moderate

rotations and von Kármán strain assumptions. As a general observation it was manifested that

the response of FGM plates are not intermediate to the response of pure ceramic and metal

plates. In the investigation, the shear deformable element developed by Reddy (1984 b) was

implemented for the von Kármán strain equations. By imposing the constant surface

temperatures at the ceramic and metal rich surfaces, thermal analysis was performed and the

temperature variation was assumed to vary in the thickness direction. Two combinations of

ceramic-metal constituents were considered in the study so that they have wide variation of

thermal conductivity ratio between the two materials. Thus even the same values of

temperatures are prescribed on the top and bottom surfaces, the temperature variation for the

two chosen cases differ in appreciable manner. At top, the temperature of 300°C and 20°C at

the bottom was applied in addition to mechanical loading at the top. The temperature at any

location of the plate for alumina-zirconia plates was lesser than the aluminium-alumina plates.

The thermal distribution was linear for pure isotropic plates and non-linear trend was observed

for FGM plates. This non-linear trend will reach its maximum in terms of the average behavior

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and again turn back to linear behavior under some value of volume fraction index. When the

intensity of mechanical load increases, the non dimensional deflection of the plates tends

towards negative side of the plate, but for thermal case the deflection was positive. Due to

higher thermal expansion at the top surface results in the upward deflection of the plate. The

various observations concluded from the numerical example insist the fact that the deflection

parameters depends on the product of the thermal expansion and the imposed temperature. This

is the reason why the FGM plate does not have intermediate response between pure isotropic

plates. Since various FGM plates have close temperature profile among them, the deflection

response was also close to each other. Under the application of mechanical loading, the axial

stresses are compressive at top and tensile at the bottom surface. For different values of volume

fraction index, the FGM plate having n equal 2.0 (represents high content of ceramic) ensures

the maximum compressive stresses at the top.

Sheikh and Mukhopadhyay (2000) obtained the geometric non-linear analysis of

stiffened plates using spline finite strip method. The finite element formulation was based on

Lagrangian coordinate system and the nonlinear equations are formed on the basis of von

Kármán’s plate theory. The final non-linear equations are solved by the Newton Raphson

method and the whole plate was mapped into square domain. Then the mapped domain was

discretised into a finite number of strips where the spline functions are used in the longitudinal

direction and finite element shape functions are utilized in the other direction. The orientation

of stiffener in the plate and the eccentricity are incorporated in the formulation, so that it can be

accommodated anywhere in the plate geometry. The results obtained are lower than the finite

element results, due to the fact of neglecting the effect of shear deformation in the finite strip

method. A wide variety of problems such as a square plate, circular plate, an annular sector

plate, rectangular orthotropic plate, a two bay rectangular stiffened plate, a five bay Defense

Research Establishment, Suffield, Canada DRES stiffened panel (Houlston and Slater 1986),

and stiffened skew plate are treated under this topic.

An analytical solution for FGM plates and shallow shells was provided by Woo and

Meguid (2001) under thermo-mechanical environment. The solution for the equations was

obtained by means of Fourier series von Kármán assumptions are implemented in the strain

field. A combination of aluminium and alumina was adopted and aluminium plates are the one

that undergoes large deflection. As discussed in the earlier paragraphs, the aluminium plates

undergo larger deflection due to the lower modulus of elasticity. But for n=2.0, even though the

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plate represents lower proportion of alumina having high stiffness, it shows better performance

in terms of stiffness compared to aluminium plates. Also, the stress distribution was linear for

the case of pure aluminium and alumina plates, while exhibit non-linear response for other

FGM cases. The snap-thorough responses of rectangular shallow shallow shells are visualized

under uniform transverse loading. The shells with initial less curvature exhibit high stiffness

and this stiffness diminishes further for higher value of curvature, and when the slope of load-

deflection curve approaches zero value, the shell ultimately undergoes buckling. Due to higher

thermal expansion, the plate deflects in the negative direction when temperature field alone was

considered. When the coupling effects are considered in the non-linear analysis, the

compressive stresses are increased about 7.49% at the top of the panel.

Wu et al. (2006) obtained the explicit solution for the non-linear static and dynamic

responses of the functionally graded rectangular plates based on FSDT and von Kármán non-

linear assumptions. For the case of tempeoral discretization Houbolt time marching scheme and

finite double Chebyshey series for spatial discretization are employed. The highest

displacement was observed for aluminium plate and the lowest for alumina plate, because the

alumina plate has higher modulus of elasticity than aluminium plate. As the value of volume

fraction index increases, the value of displacement increases and hence indicating the stiffness

degradation. When pure ceramic plates or FGM plates are considered with volume fraction

index equals 5, the plate with all the edges clamped records lower displacement parameter.

When CCSS and CSCS boundary conditions are considered, the deflection of the CSCS plate

was higher under the value of n=0.0 (pure ceramic case). But when n approaches the value 5,

the displacement for both the boundary conditions was almost identical. The center

displacement rises with the volume fraction index regardless of the boundary constrains

imposed on the plate boundaries. Also, the maximum amplitude of vibration was higher in case

of linear response compared to non-linear responses for all the FGM plates and the difference

in response increases as the n value increases. Similarly, the deviation in the maximum

amplitude of motion between linear and non-linear responses elevates with rise in n value and

this difference becomes highest for metal plates and lowest for ceramic plates. As a final

observation, the volume fraction index equals 2.0, has more significant effect on the

displacement parameter and this observation can be used as a guiding factor in the design of

FGM plates.

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Arciniega and Reddy (2007a) presented the non-linear geometric problem of

functionally graded shells that consists of two constituents ceramic and metal graded through

the thickness direction. A finite element method with tensor based formulation having

curvilinear coordinates and FSDT are used to model the FGM shell panel. To avoid the

problems of shear, membrane, and thickness locking higher order interpolation functions are

implemented in the formulation. In the Lagrangian formulation, the second Piola-Kirchhoff

stress tensor was used and it conjugates to the rate of Green strain tensor (Reddy 2004).

Different types of problems viz., rollup of functionally graded plate strip, annular FGM plate

under end shear force, pull-out of a functionally graded cylindrical shell and FGM cylinder

under internal pressure are solved in the analysis. By taking the symmetry of the shell only an

octant of the shell has been analyzed in the computational domain under non-linear response.

As a general observation, the shell corresponding to lower values of volume fraction index

(more ceramic) shows higher response than those of lower values of volume fraction index

(more metal). Also the Newton-Raphson scheme converges below some load level and beyond

that it diverges. In general, FGM shells exhibit identical behavior to that of isotropic and

homogeneous counterparts. The bending response of FGM shells was found to have

intermediate response between pure ceramic and metal panels.

Kordkheili and Naghdabadi (2007) employed updated Lagrangian approach to obtain

the non-linear thermo elastic solution for functionally graded material plates and shells. The 2nd

Piola-Kirchhoff stress was formulated as second-order functions in terms of a through-the

thickness parameter. The heat transfer equation was non-linear through the thickness by

Rayleigh-Ritz method. Under the application of central point load, the pure aluminium shell

ensures greatest amount of deflection, while for FG shell containing n=0.5, the shell undergoes

much less deflection under the same load.

Yang et al. (2008) investigated the non-linear local bending of sandwich plates modeled

as two composite laminated face sheets and graded code under the application of patch load.

The graded core layer has power law variation of material properties along the thickness

direction. The von Kármán non-linearity was incorporated and the kinematic model was based

on the assumption of FSDT. The interaction between the loaded face sheet and graded core was

modeled as an elastic plate resting on a Vlasov-type elastic foundation. The non-linear and

bending response was adopted by perturbation technique and Galerkin method. The final

observation from the study was that the use of FGM as core part will considerably reduce the

65  

deformation and local interfacial shear stresses. This was due to the higher equivalent

supporting stiffnesses of the graded core compared to the corresponding homogeneous core.

The interfacial shear stresses (σyz) and the deflection are maximum at the plate center and tend

to reduce towards the plate edge. The sandwich plate with graded core having n=20.0 has the

lowest deflection at the center. Always, the linear solutions over predict the interfacial normal

stress and the deflection parameter but considerably underestimate the interfacial shear stress.

This trend of estimation was found to be more for the case of higher value of volume fraction

index. The non-linear local response of the SSSS and SCSC sandwich plates are almost similar,

while lowest deflection was discerned for CCCC boundary. Both the deflection and interfacial

shear stresses are get affected by the load location and boundary conditions, particularly at their

peak values. As the load moves towards either of the supports, deformed zone and peak of

interfacial stress distributions shift toward the support.

The mesh free kp-Ritz method has been employed to study the non-linear response of

functionally graded ceramic-metal plates under mechanical and thermal loads by Zhao and

Liew (2009b). The von Kármán strains are incorporated in strain part to account for the small

strains and moderate rotations. To model the displacement components, the FSDT proposed by

Reddy (2004) was assumed. To estimate the graded properties of FG material in the thickness

direction, the power law form of distribution was considered. A stabilized confirming nodal

integration method (Chen et al. 2001) was employed to evaluate the plate bending stiffness

instead of Gauss integration. Such an integration will considerably increase the computational

efficiency and also eliminates the problem of shear locking occurs in case of thin plates. The

modified Newton-Raphson method combined with the arc-length method (Crisfield 2000) was

used to track the complete load-deflection equilibrium path. As the magnitude of load ranges

from 0 to 7 N/m2, the central deflection of the plates increases with different values of volume

fraction exponents. The domination of stiffening effect may be cited as the reason for this

response. But under this load magnitude, the response was not linear. Further, as the load range

increased from 0 to 20 N/m2, a pronounced non-linear response was observed, whilst

maintaining the similar response of volume fraction index. A close observation to the axial

stress distribution reveals the fact that the top surface of the plate experiences compressive

nature of stress and at the bottom surface a tensile nature of stresses. As a second observation

when the value of volume fraction index equals 2, the plate experiences a maximum

compressive stress. On the bottom portion of the plate, the maximum tensile stress occurs in

66  

isotropic plates, while minimum value observed for FGM plate with n=0.2. When the

maximum temperature of 400°C was imposed on the top of the plate considering different

values of volume fraction index, the FGM plate with n=0, experiences the maximum deflection

and n=1.0 corresponds to minimum deflection. As observed from the earlier research works,

the results from the non-linear analysis are always higher than those of linear analysis results.

When the bottom temperature of 20°C and top temperature of 300°C are prescribed on the

plate, an upward deflection of the plate was visualized due to the higher thermal expansion of

the plate at the top. But as the load increases, the upward tendency of deflection gradually

diminishes and results in down ward deflection similar to the one observed in mechanical

loading.

The non-linear study performed by Zhao and Liew (2009b) was further extended to

study the geometric non-linear of plates using local Petrov-Galerkin approach based on the

moving Kringing interpolation technique by Zhu et al. (2014). The shape functions constructed

by Kringing interpolation method possesses the property of Kronecker delta function and hence

avoids the use of any other special techniques to enforce the essential boundary conditions. Due

to the dependency of thermal conductivity of the material on temperature was assumed, a non-

linear partial differential heat conduction equation has been solved. The tangent stiffness matrix

was explicitly developed by meshless technique and the incremental form of non-linear

equations was obtained by the Taylor series expansion. When the load-deflection graph was

drawn, a pronounced non-linear behavior was discerned as the load reaches a certain level of

magnitude. The load-deformation plots obtained for simply supported and clamped boundary

conditions are identical in nature. But as far magnitude was concerned, to get the same order of

magnitude of deflection, the applied loading of the order of eight to ten times than that of

simply supported boundary was imposed. Since the modulus of elasticity of metal was less than

ceramic, the magnitude of deflection increases with the rise in volume fraction index. When the

volume fraction attain very low values (not zero), abrupt change of axial stresses are observed

at the bottom surface occurs, while it happens for top surface for higher values of n. Further,

when the length-to-thickness ratio varies from 5 to 10, the rapid fall-off tendency of deflection

parameter was observed beyond which only little changes are observed. At the higher values of

length-to-thickness ratio, an asymptotic response can be confirmed. Since the analysis was

performed under large deformation case, the portion of deformation resulting from applied

temperature is minor relative to the mechanical response. However, the stress profiles have

67  

considerable effects under thermal loading. Considering the thermal dependent properties gives

rise to more accurate thermal field and hence the possible reconstruction of stresses distribution

within the FGP.

As an elaboration of the earlier study by Zhao and Liew (2009 b), the non-linear study

was performed for FGM shells by Zhao and Liew (2009 c) under thermal and mechanical

loading conditions. The Sander’s non-linear shell theory in combination with von Kármán

strain was incorporated in the formulation. The solution methods that are already discussed in

the work of Zhao and Liew (2009a) are incorporated in the work. Because of the low stiffness

of the panel at higher values of volume fraction index the deflection shows increasing trend.

Except for the case of pure ceramic and metal plates, there was not much difference for the

lower limit loads for FGM panels of various volume fraction indexes. The deflection response

of the FGM panels lies in between the responses of ceramic and metal panels. The maximum

compressive and tensile stresses occurs for the values n=2.0 and n= ∞, respectively. Similarly,

the minimum compressive and tensile stresses are observed for the values n=0 and n=0.2. Also,

the drastic stress changes are observed at the bottom compared to top of the panel. When a

temperature of 200°C was imposed on the top of the panel, the panel experiences the

compressive stresses through the thickness with the exemption of portion near the bottom

where the stresses are almost zero. Again, the response under thermal environment was

intermediate to that of isotropic and homogeneous plates.

Wankhade (2011) presented the geometric non-linear analysis of skew plates which

require more computational effort due to the existence of singularities involved at the obtuse

corner with varying skew angle. The finite element formulation used in the study considered

the transverse shear effect by considering Reissener/Mindlin thick plate theory. The deflection

of the skew plate increases as the skew angle elevates and the load deflection curve for 60°

angle approximates to straight line. Hence increasing the skew angle increases the rigidity of

the plate and hence increases the overall strength of the structure. The variation of membrane

stress was almost straight lines for all the skew angles and considerable deviation in the

membrane stresses are observed for skew angle 0° and 60° under the same magnitude of

loading. But the curve of bending stresses exhibit the curved tendency for all the skew angles.

When the membrane stresses are plotted for different aspect ratios (b/a), the membrane stresses

are not affected by large value of skew angles.

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The non-linear response of FGM plates was reported by Singha et al. (2011) using a

four node high precision plate bending finite element. The FSDT theory with the exact location

of neutral plane was incorporated and shear correction factors are estimated by using energy

equivalence principle. The value of in-plane stresses evaluated from the constitutive relation

and three-dimensional equilibrium equations are used to estimate the transverse shear and

transverse normal stress components. To predict the load-displacement response curve, the

Newton-Raphson iteration method was employed. To avoid shear locking, shear strains are

considered as nodal unknowns. When the in-plane normal stress variation was plotted, the

isotropic plates exhibit linear variation, while FGM plates show non-linear variation due to the

variation of Young’s modulus through the thickness. The cubic fashion of isotropic plates was

observed for transverse normal stress through-the-thickness which qualitatively similar to FGM

plate. A hardening type of non-linearity was observed for simply supported FGM plates, and

this non-linearity was less with the increase in the value of n due to the low stiffness of the

plate. Depending on the value of volume fraction index, the maximum transverse shear stress

decreases with the increase in load parameter. The in-plane stresses are compressive in nature

at the top and tensile in nature at the bottom of the plate. With the increase of transverse

displacement, the in-plane stress variation becomes highly non-linear depending upon the value

of volume fraction index. For immovable in-plane boundary, the degree of hardening non-

linearity was more compared to movable in-plane boundary, as anticipated.

A four variable refined plate theory proposed by ABDELAZIZ (2011) has been

accomplished by Fahsi et al. (2012) to study the non-linear cylindrical bending of functionally

graded plates under thermal and mechanical loadings. The material properties are assumed to

vary in the thickness direction according to the simple power law distribution in terms of

volume fraction of material constituents. To account for the effect of geometric non-linearity

von Karman assumptions are incorporated in the work. The minimization of total potential

energy was implemented to get the final governing equation for non-linear analysis. Different

examples are performed by considering various material and geometric parameters.

Malekzadeh and Heydarpour (2012) obtained the thermoelastic transient response of

functionally graded cylindrical shells under moving boundary pressure and heat flux. The

temperature dependent material properties are considered in the radial direction. To

incorporate the non-Fourier effect, hyperbolic heat conduction equation was utilized that

include the influence of finite heat wave speed. A combination of GDQ and FEM was

69  

employed to solve the governing system of equations. The resulting non-linear equations are

solved using Newmark’s time marching scheme in temporal domain. It was observed that the

volume fraction index has significant effect over radial displacement, tangential and axial

normal stress components. But the influence of radial stress component was small and can be

ignored, especially in the regions that are far from the inner portion of the shell. Except on the

radial stress component, increase in the length of the FG cylinder has considerable influence

over other parameters. The effect of the length of the cylinder on transient response was much

more significant than steady state response. As far as the influence of thickness parameter has

concerned, its response increases with the increased elapsed time. When the effect of thermo

mechanical load front velocity was studied for clamped FG cylindrical shells, increasing the

velocity increases the results (displacement and stresses). Also, as the time level increases, the-

thorough-thickness variation of clamped FG cylinder results approaches to steady state values.

A finite element solution incorporating shear and normal deformation effect in the soft

core sandwich plate model was presented by Madhukar and Singha (2013) for geometrically

non-linear and vibration response. The von Kármán assumptions are introduced in the strain

equations and in-plane and rotary inertia was considered in the equations of motion. Newton-

Raphson iteration technique and harmonic balance method are employed, respectively, for

static and vibration analysis. The displacement model proposed by Kant and Swaminathan

(2001) and Kant and Owen (1982) was assumed in the model. The components of shear strain

vectors are taken as separate nodal unknowns and thus avoiding the problem of shear locking.

If the thickness of the soft core elevates, the displacement parameter also tends to rise for the

problem under consideration. Due to the influence of more shear and normal deformation

effect, the central displacement increases with the increase of core-to-face thickness and span-

to-thickness ratio, for a fixed load parameter. Regarding non-linear vibration analysis, the

frequency ratio is more for thick plates compared to the case of thin plates. If the total thickness

of the plate has kept constant, the non-linear frequency ratio gets decreased with the increase of

core thickness. Also, the excitation frequency if increases from zero or decreases from higher

value, the flexural vibration amplitude increases. Since structural damping was not considered

in the study, the non-linear flexural vibration amplitude increases in rapid manner as the

excitation frequency approaches the linear flexural vibration of the plate from either side.

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Upadhyay and Shukla (2013a) investigated the nonlinear static and dynamic response of

functionally graded skew plates using HSDT and von Kármán non-linear kinematics. The chain

rule of differentiation and linear mapping was performed to transform from physical domain

into computational domain. The displacement model proposed by Kant and Pandya (1988) was

implemented in the study. It was noticed that the aluminum plate exhibit less stiffness behavior

with respect to the FGM skew plates. The difference in displacement parameters of the 90° and

60° plates (23.35%) was considerably lower than the displacement difference between the 90°

and 30° (83.07%) skew plates. When compared to square plate the deflection decreases by

29.1% and 88.45%, respectively for 60° and 3°0 skew plates for the linear variation of volume

fraction index. In addition, for 30° skew plates irrespective of the value of volume fraction

index the displacement decreases. The deflection pattern of the skew plate considering various

types of boundary conditions was similar in the pattern. Under transient response, amplitude

and the time period of motion decreases with the fall-off value of skew angle. But this behavior

was no longer exists for higher value of volume fraction index i.e., with increase in volume

fraction index, time period of motion decreases and amplitude tends to be increased. Due to the

coupling effect of stiffness during increase and decrease of volume fraction index, the

stiffening effect of the plate decreases with the skew angle. When different patterns of loading

are considered, rectangular pulse loading ensures highest motion of amplitude, while the lowest

ensured for exponential pulse, depending upon the loading curve area. Upon removal of the

load, the amplitude corresponding to sine pulse increases, while it remains same for other

loading forms.

Kaci et al. (2013) solved the non-linear bending problem for sigmoid functionally

graded plates in which the variation of material properties is considered in the thickness

direction. The governing equations are reduced to a linear differential equation with nonlinear

boundary conditions. Under the application of pressure loading, the stresses are found to be

compressive at the bottom and tensile at the top surface. Further, high magnitude of tensile

stresses are confirmed on the top surface for linear analysis at n=2.0; while under non-linear

analysis, same location for maximum was observed but for n=0.2. The stress profiles for pure

metal and ceramic plates are always linear irrespective of the type of analyses. When FGM

plates are modeled by sigmoid distribution of material properties, response of such plates are

identical to those of homogeneous plates.

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Duc et al. (2014) employed the classical shell theory taking into account the geometric

non-linearity, geometric imperfection and Pasternak type elastic foundation for the analysis of

shallow spherical shells under mechanical and thermal conditions. The influence of elastic

foundation, external pressure, temperature, geometric and material properties on the non-linear

buckling and postbuckling of the shells was presented in detail. When immovable boundary

conditions are considered, the snap-through behavior the FGM spherical shell becomes more

unstable.  When the effect of elastic foundation was incorporated,  the snap through curve

become more stable. When temperature field was prescribed on the surface, outward deflection

was confirmed, which is the identical statement confirmed in other related studies. As soon as

the interaction of mechanical part was encountered, out ward deflection tends to reduce and

external pressure exceeds bifurcation point of load thus results in an inward deflection. It was

concluded that the ability of the system has been reduced in the presence of temperature field.

A cell based smoothed Mindlin plate element was recently proposed by Van et al.

(2014) for geometric non-linear analysis of functionally graded plates. The C0 based non-linear

formulation was developed and con Karman strains are implemented in the analysis. A simple

two step procedure was incorporated to analyze the plates under mechanical and thermal loads.

As observed in earlier research works, under the application of thermal load, an upward

deflection of the plate was observed due to the higher thermal expansion of the plate at the top

surface. When the volume fraction index approaches high concentration of metal segment, an

elevated deflection parameter was calculated. Also, when the temperature at the ceramic

surface increases, the deflection of plate becomes small. Different plots for axial stress

variation through the thickness were established for different material combination and

temperature range.

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2.7 Post buckling response of laminated and functionally graded plates/shells The geometric instability of the structure under in-plane thermal/mechanical load was

identified as buckling. Thin walled structural members are able to carry additional load after

buckling, known as post-buckling strength before failure of the structure. In order to utilize the

full strength of the plate and shell structures under mechanical and thermal environments, it is

necessary for a designer to know the actual critical buckling load (primary bifurcation) and the

corresponding post buckling strength (secondary bifurcation). When FGM plate/shells are

employed as heat-shielding components with restrains against in-plane expansion and

contraction, considerable amount of strains are induced and stresses are developed at elevated

temperatures. This situation establish a certain membrane pre-stress that may commence the

buckling and post buckling phenomena in the structure. In addition, the asymmetric material

properties with temperature functions make the post buckling response of the structure more

complicated. Hence thorough understanding of the response of the FGM plates and shells in the

post buckling region seems to be vital for optimum and effective design of FGM structures.

A finite element based Mindlin shallow shell formulation was presented by Pica and

Wood (1980) for circular and square plates under in-plane loading. In addition to in-plane axial

loads, the square plates are assumed under in-plane shear loading. The problems of cylindrical,

spherical shells and shallow shells are also treated, which records the snap through  behavior.

The variation of central moment, membrane stress, and deflection with respect to applied load

are plotted for different shell types and boundary conditions.

Yang and Shen (2003) obtained a semi-analytical approach for the large deflection and

post buckling response of functionally graded plates under in-plane and transverse loading

conditions. For the analysis, the temperature dependent material properties are considered that

obey the simple power law distribution in terms of the volume fraction of the constituents. The

CPT based model with the effect of plate foundation interaction was considered where the

Winkler elastic foundation can be treated as limiting case. The plate with two edges clamped

and the remaining two edges simply supported clamped or it may have elastic rotational edge

constraints was modelled. The plate with intermediate properties (FG plate) has intermediate

response (deflection) and the ceramic plate exhibit lowest deflection because of the highest

stiffness. When Bending moment plots were considered, this tendency of plates no longer

exists. As the in-plane compressive load ratio rises, both deflection and bending moments

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shows elevating trend. A deep insight into the load deflection and load-bending moment curves

insist that the more rigid boundary constraints ensures deflection and bending moments of

small magnitude. The post buckling response of the FGM plates are identical to that of

corresponding non-linear bending curves. In the post buckling region, the load capacity of the

FGM plate increases with the increase of edge rotational rigidity or with the increase of

foundation stiffness. When the clamped FGM plates are subjected to uni-axial compression, the

deflection deviates abruptly under the effect of initial transverse pressure. However, beyond the

post buckled range the response of initially pressurized plate was almost asymptotic to that of

pressure free plate.

An analytical solution was presented by Woo et al. (2005), for the post buckling

behavior of moderately thick plates and shallow shells under temperature field and edge

compression. The HSDT and von Kármán type non-linearity was assumed in the analysis. The

mixed Fourier series solution was obtained and the results are presented for Reissner-Mindlin

theory and CPT. The higher order theory developed by Reddy (1984b) for moderately thick

plates was assumed in the analysis. It was ascertained that to predict the buckling load in

accurate sense higher order theory should be incorporated in the analysis, in particular, when

the thickness of the plate equals the one-twentieth span of the plate. Also, considering the

shear deformation terms in the theory leads to the under estimation of critical buckling loads. In

spite of the higher order terms present in the theory, pure isotropic plates exhibit linear

response, while FGM plates shows non-linear response of post buckling curves. The critical

load value for a cylindrical shell was higher than corresponding flat plate and when the shell

reaches the critical load, the snap-through buckling happens and shell structure collapses. When

bending moments are calculated under edge compressive loads, for FGM shells, the magnitude

was not zero before the buckling because the clamped edges prevent the transverse deflection

from occurring. Under simply supported boundary condition, the isotropic plates (Ceramic and

metal plates) exhibits conventional type of buckling and heterogeneous plates deflect

transversely due to the structural asymmetry of the middle surface. At the same time, the FGM

plates show very high resistance at the early stages of deformation.

Yang et al (2006) performed the thermo-mechanical post buckling analysis of

functionally graded cylindrical panel considering the temperature-dependent properties in the

thickness direction. During the initial stage, the panel was stressed by an axial load and further

subjected to the change of temperature in uniform sense. The non-linearity was accounted in

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von-Kármán-Donnell type and the CPT based kinematic model was incorporated. To trace post

buckling equilibrium path, a semi analytical method based on differential quadrature was

utilized with an iteration process. When both TID and TD properties are assumed for the

particular problem, TID solutions are higher (about 9-18%) than the corresponding solutions

considering TD properties. In connection with this, lowest buckling temperature are observed

for metallic panel (nickel plates) compared to FGM and pure ceramic plates (Silicon

Nitride).Due to the sharp decline tendency of stiffness at higher n values, buckling temperature

increases for different values of n. The thermal buckling capacity of the cylindrical panel was

enhanced with the presence of axial tensile pre-stress; whereas the opposite behavior was

visualized under the axial compressive loads. As the known fact, the buckling temperature was

smaller when the edges are fully restrained against any in-plane movements. When the panels

are initially stressed under axial compression, the post buckling paths of the simply supported

panels are not bifurcational, and in the presence of axial core, initial deflections are induced so

the post buckling path do not start from the coordinate origin. Further, the post buckling

temperature difference between the TID and TD solutions are higher for CCCC panel than that

of the SCSC panes, but the former type panels have highest post buckling load carrying

capacity compared to its SCSC counterpart.

Wu et al. (2007) obtained the analytical solution for the post buckling response of

functionally graded plates under mechanical and thermal loading by means of fast converging

finite double Chebyshev polynomials. The mathematical model was based on the FSDT and

von-Kármán non-linear kinematics. The critical buckling temperature and buckling load

reduces with the presence of higher content of metal i.e., higher value of volume fraction index.

For all the cases performed in the study, it was manifested that up to the value of volume

fraction index equals 2, a significant impact on buckling and post-buckling response was

visualized. Also, the buckling temperature of FGM plates was found to have lesser values

compared to the pure isotropic plates (ceramic) and this is irrespective of the boundary

conditions. When the buckling and post buckling strength of the plates vs. volume fraction

index are plotted, the deviation between the buckling and reserve strength of the plate for n=5.0

and 10.0, was very less. The pure ceramic and metal plates, respectively, ensures higher and

lower buckling and post buckling strength with respect to various ranges of volume fraction

index. As the plate aspect ratio tends to elevates, the buckling responses of the plate decreases.

The performance of square plate and plate with aspect ratio equals 1.5 are almost identical for

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n=0 and 2. Similarly, the case of aspect ratio b/a=3 and 4, indicating the beam response of the

plate beyond the value of b/a=3. When different boundary conditions are considered, the FGM

plates record lower buckling and reserve strength compared to alumina plate.

Panda and Singh (2009) employed the Green-Lagrange based HSDT for the post

buckling analysis of laminated composite cylindrical/hyprboloid shell panel subjected to

uniform temperature field. The non-linear stiffness terms exists in the Green-Lagrange

relationship was considered in the formulation. The solution of governing equations was

obtained by minimizing the total potential energy of the system. The quadratic variation of

transverse shear strains and transverse shear stresses are considered in the displacement model

(Reddy 2004). The critical buckling temperature generally decrease with the rise in R/a for all

the laminations, and increases with the increases of layer numbers. When different values of

curvature ratio (R/a) and amplitude ratios are chosen for the problem, the temperature ratio

becomes maximum for all the laminates at R/a=100. Due to the severity of non-linear effect,

the post buckling strength do not show any definite trend over the amplitude ratios (some

places shows increasing trend and at some places decreasing trend). The primary bifurcation

strength was highest for the case of anti symmetric angle-ply lamination scheme for the chosen

thickness ratios except for thick panels (a/h=10). As far as hyperboloid panels are considered,

as the aspect ratio increases, the buckling temperature parameter decreases due to the tendency

of the panel to become flat under higher aspect ratios and curvature ratios. Also, the post

buckling strength for square panels are higher than that of rectangular panels. Except for the

case where amplitude ratio 1.2 and R/a=20, the hyperboloid panels ensures higher post

buckling strength compared to cylindrical panels. As different modular ratios are chosen for

the hyperboloid panels, the buckling temperature decreases with increase in modular ratio and

further follows mixed type of trend with different values of amplitude ratio. In general, the

temperature ratio increases with the increase of amplitude ratio and decreases with the increase

of modular ratio for both cylindrical and hyperboloid panel, with few exceptional cases.

The element free kp-Ritz method in conjunction with the FSDT was adopted to solve

the post buckling problem of functionally graded plates under edge compression with

temperature dependent properties by Lee et al. (2010). The displacement fields are assumed by

means of kernel particle functions and direct nodal integration method was employed to

evaluate membrane and shear terms thus to avoid any shear locking problem. In conjunction

with the modified Newton-Raphson method the arc-length iterative algorithm was considered

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for the solution of non-linear equations. The highest and lowest magnitude of load factors are

ensured for ceramic and metal plates, while the frequency of FGM plate with n=0.5, 1.0 and 2.0

fall between those of pure isotropic plates. When the influence of boundary conditions are

considered on post buckling responses, initial curve shows no significant different, as the load

increases, the remarkable deviation between the curves was discerned. Under thermal analysis,

a prescribed temperature of 20°Cand 100°C was applied on the bottom and top of the shell

panel, respectively. When the plate subjected to temperature field in addition to the temperature

field, the plate experiences the higher post buckling deformations due to the thermal force and

moment resultants that reduces the stiffness of the plate. When different temperatures are

imposed on the top of the plate, the plate with highest temperature undergoes large initial

deflection and ensures a greater amount of post buckling deformation. At high temperature, the

stability of the plate tends to decrease and thus leads to the larger deformation at post buckling

stages.

Liew et al. (2012) presented the post buckling analysis of functionally graded

cylindrical shells under thermal loads and axial compression based on FSDT and element free

kp-Ritz method. The strain field based on von Kármán assumptions and power law distribution

of mechanical properties in the thickness direction was considered in the analysis. The other

solution techniques similar to that of Lee et al. (2010) are incorporated in the study. When the

plot of central deflection vs. volume fraction index was studied, the deflection increases slowly

at initial load condition for n=0 and shows fast increasing trend and finally shows

monotonically increasing trend for higher load values. The degree of displacement at higher

volume fraction index was large due to the effect of stiffness degradation. Under the plot of end

shorting vs. load curves, after certain point of loading, the curve exhibit negative slope. From

the various numerical examples presented in the study, it was manifested that the curvature

effect of panels plays a vital role in predicting the post buckling response of FGM shells.

The buckling and post buckling responses of laminated composite plates was performed

by Dash and Singh (2012) using HSDT in conjunction with Green-Lagrange strain-

displacement relationship. All the higher order terms present in the Green-Lagrange relations

are included in the analysis. The load ratio (post buckling load/critical buckling load) results for

a simply supported plate for different stacking sequences indicate that the load ratio increases

with the increase in amplitude ratios. As the two layered square plate was chosen, the buckling

resistance was found to be less up to the value of amplitude ratio 0.8. When load ratio pattern

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for different amplitude ratios are tabulated for clamped plate, initially the load ratio increases

with the amplitude ratio and beyond that no definite trend was discerned. This trend was not

same for the different kinds of laminated plates chosen for the problem. Next, the influence of

orthotropicity on the load ratio was studied for cross ply square laminates. The buckling

resistance increases with the increase of orthotropy and amplitude ratio chosen for the plate.

Further, no definite trend was observed regarding the influence of orthotropicity upto the

amplitude ratio 1.0. But, it shows increasing tend beyond the value 1.0.

Considering the randomness in the material properties, Lal et al. (2013) presented the

post buckling response of functionally graded plates under thermo-mechanical environment.

Two variables i.e., material properties of each layer and volume fraction index are assumed as

independent random input parameters. A C0 based HSDT and von Kármán non-linear

kinematics was incorporated in the study. To solve the issue of C1 continuity of the HSDT, the

authors assumed the Co continuous element as incorporated in the work of Singh et al. (2002).

Thus the artificial constraints should be enforced variationally through the approach of penalty

method, but as per the study of Shankar and Iyenger (1996), accurate results for C0 formulation

are presented without enforcing any penalty approach. Under the thermal part, the material with

temperature independent properties (TID) and temperature dependent properties (TD) are

incorporated. The co-efficient of variation (COV) with TD material properties are highly

sensitivity compared to TID material properties. The post buckling response of FGM plates are

more affected by the random change in COV of different parameters viz. Ec, Em and n.

Therefore, the strict control of these parameters was vital to achieve the reliability of the FGM

plate. The FGM plate with TID material properties shows high value of mean dimensionless

post buckling load, while lower value of mean dimensionless post buckling temperature was

confirmed for TD material properties. For different amplitude ratios, the post buckling load and

temperature increases with the rise in the value of amplitude ratio.

Upadhyay and Shukla (2013b) presented the buckling and post buckling response of

laminated composite and sandwich skew plates based on HSDT and von Kármán strains. The

linear mapping technique was employed to transform the physical domain into computational

domain. The governing equations and boundary conditions are discretisized in spatial domain

using finite double Chebyshev series solution. The displacement model proposed by Babu and

Kant (199), where the in-plane fields have cubic term and constant for transverse displacement

are considered in the formulation. Under the action of uni-axial compression, as the skew angle

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of the plate rises, both the buckling and post buckling strength of the skew plate increases and

response of symmetric cross-ply skew plates are higher than anti-symmetric plates. At low

skew angles, the lamination scheme of the plate has more pronounced effect on post buckling

strength. But no deviation in the responses (buckling and pot buckling strength) was observed

for cross-ply and angle-ply plates, when the plate was subjected to uni-axial compression.

Further when the case of positive in-plane loading was considered, except at skew angle value

45°, the cross and angle ply laminates shows no deviation in the post buckling response.

Under negative in-plane shear loading, anti-symmetric skew plates exhibit higher

buckling strength that that of symmetric skew plates. Also, the post buckling strength of the

skew plates are higher for the case of positive in-plane shear loading than that under negative

shear loading. This may be due to the development of tensile stresses at the acute corners of the

skew plate under positive in-plane loading. In addition, the influence of boundary conditions

and thickness ratio on post buckling strength was also studied for laminated skew plates. The

sandwich skew plats are treated under uni-axial compression, bi-axial compression, and

positive and negative in-plane shear loads. For the case of uni-axial compression, no significant

improvement has been observed for the skew plates having high core thickness and high skew

angle. In conclusion, the sandwich plates even though exhibit higher buckling strength under

in-plane loads, no significant improvement was observed. Hence to get the higher reserve

strength for sandwich skew plates, the ratio of material properties of core to face sheet should

not be too high.

2.8 Appraisal from previous research works

A lucid collection of literatures dealing with the responses of FGM structures by

considering different parameters has been presented in the previous sections of this chapter. A

deep insight in to the available literatures reveals the availability of different studies the static,

dynamic (free/forced vibration) and stability of plate and shell structures made of FGM

components. In addition, a finite volume of research works can also be located in the literatures

on geometric non-linearity and post buckling problems of FGMs. A very limited number of

studies are presented considering skew geometry and multilayer concept in FGMs. However,

the critical review of literature manifests the need for efficient and appropriate model to

analyze the graded structures in terms of suitable kinematic model that incorporates the realistic

structural response. Hence in the present work prominence has been given to study the

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structural response of FGM skew plate/shells and sandwich plate/shells under linear and

nonlinear analysis.

The following interpretations have been drawn from the literature study.

FGM structures are generally characterized by their spatial variation of material properties

in the predetermined fashion to achieve the desirable material properties in the chosen

direction. The primary intention behind the development of these types of materials is to

sustain large temperatures and high temperature gradients.  Due to their anti-symmetric

nature of properties with respect to middle plane, bending-stretching coupling is

anticipated. In order to capture the realistic variation of shear deformation, a suitable

higher order theory should be employed in the analysis.

In heterogeneous materials like FGM, selection of appropriate homogenization scheme

plays important role in defining the material properties in proportion to their volume

fraction values. The choice of proper homogenization approach should be based on the

gradation relative to the extent of a typical representative volume element (RVE). If the

graded material properties are relatively slow-changing functions of spatial coordinates,

standard homogenization methods that disregard the heterogeneous nature of material

property at global level can be accomplished. In this regard, the averaging techniques like

Mori-Tanaka and self consistent schemes which include the interaction among the

neighboring inclusions may be the better options. Most of the literatures incorporate the

variation of material properties by means of simple rule of mixture; while very few

literatures are concerned with the averaging methods.

FGM materials are mainly developed to serve the purpose in high temperature

environments, thus necessitate the accurate prediction of thermal distribution in the

structures. Hence most of the studies performed in the literature consider the thermal

profile in linear and nonlinear form with temperature dependent material properties. But

the studies on FGM skew plates/shells under thermal environment based on higher order

model seem to be missing in the body of literature.

General FGM structures undergo failure due to buckling, large amplitude deflections and

excessive stresses caused by the thermal and combined thermo-mechanical loading. In

such case, the strain-displacement equations should be incorporated in the non-linear

sense. To analyze the FGM structures under large deflection with small strains and

moderate rotations, the non-linear strain model was considered in Green-Lagrange sense

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by many researchers. All the studies presented so far employs the assumption of von

Kármán to obtain the final form of strain equations. Further, when the non-linearity is

severe, it becomes mandatory to consider all the non-linear terms appear in the Green-

Lagrange equations. Only few studies are reported on the geometric non-linear analysis of

FGM plates and shells and no literature has been located regarding geometric non-linear

analysis of FGM sandwich plates/shells.

FGM plate/shell structure show reserve strength after the critical load is reached which was

described as post buckling strength. Further, the sudden change of equilibrium from one

state to another involving large deformation should be investigated properly from design

point of view.  In addition, the variation of material properties makes the analysis more

complicated. Keeping these aspects in view, few studies are performed to address the post

buckling behavior of FGM plates and shells. Again the availability of literature document

to address the post buckling response of FGM sandwich plates through proper higher order

kinematic model was missing in the literature. 

In order the overcome the drawbacks that are elaborated in Section 2.7; a brief

framework of the present investigation was discussed here. From the short discussion made on

published literatures, it was manifested to propose an appropriate model to analyze the FGM

skew single layer/sandwich plate/shell under thermo-mechanical loading. To incorporate the

non-linear nature of geometric stiffness matrix associated with buckling phenomenon, it is

significant to consider all the non-linear terms in the strain model for post-buckling analysis.

Finally, a non-linear model that accurately predicts the transverse displacement variation by

means of quadratic thickness term was accomplished in the present analysis.

Various features that are accommodated in the present research are presented below.

A higher order displacement based model that accommodate the cubic and quadratic

variation of thickness term, respectively, in the in-plane and transverse displacement field

has been employed to accurately predict the bending stretching coupling exists in FGM

structures. Such a model includes the normal strain and its derivative in the kinematics thus

overall response of the plate can be efficiently accomplished.

A higher order non-linear model was assumed for geometric nonlinear and post buckling

analysis of sandwich plate/shells by employing C0 formulation.

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Two kinds of homogenization approaches viz., Mori-Tanaka and rule of mixture are

employed and comparison statement has been drawn for FGM skew plates based on these

methods.

To accurately predict the non-linear response of FGM structures, all the non-linear terms

present in the Green-Lagrange equations are incorporated in the non-linear finite element

formulation.

A suitable finite element code has been developed in FORTRAN 90 environment for static,

dynamic and stability analyses of FGM skew plates. Further, due to the problem of

numerical stability observed in FORTRAN 90, to include the transverse displacement

variation, a MATLAB (R2013b) code was developed for sandwich plates/shells.

Finally, a wide range of numerical problems are solved in the framework of developed codes

that dictate the response of FGM plate/shell structures considering linear and non-linear

strain-displacement relations.

2.9 Summary

In this chapter, an extensive research works performed on FGM plates/shells by considering

linear/nonlinear strain-displacement relations are studied in detail. In particular, works related

to static, free/forced vibration, buckling response of FGM plates/shells are discussed while

giving a brief explanation about the methodology and displacement model incorporated in

different studies. Various important conclusions that have been arrived from different studies

are also discussed. In addition, literature survey has been performed for static, free vibration

and buckling analysis of FGM sandwich plates. Finally, available literature studies on

geometric nonlinearity and post buckling response analyses of FGM plates have been discussed

in detail. From the brief literature survey carried out in Chapter 2, research gaps are identified

which formed the basis for the present work. In the end, a concise summary of the important

research works that serves as background for the present research work have been provided.