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Higher-order theory for functionally graded materials

J. Aboudia, M.-J. Pinderab,*, S.M. Arnoldc

aTel-Aviv University, Ramat-Aviv 69978, IsraelbUniversity of Virginia, Charlottesville, VA 22903, USA

cNASA Glenn Research Center, Cleveland, OH 44135, USA

Received 18 June 1999; accepted 12 August 1999

Abstract

This paper presents the full generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials

developed by the authors during the past several years. This theory circumvents the problematic use of the standard micromechanicalapproach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites

by explicitly coupling the local (microstructural) and global (macrostructural) responses. The theoretical framework is based on volumetric

averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the

subvolumes used to characterize the composites functionally graded microstructure. The generalization outlined herein involves extension

of the theoretical framework to enable the analysis of materials characterized by spatially variable microstructures in three directions.

Specialization of the generalized theoretical framework to previously published versions of the higher-order theory for materials functionally

graded in one and two directions is demonstrated. In the applications part of the paper we summarize the major findings obtained with the

one-directional and two-directional versions of the higher-order theory. The results illustrate both the fundamental issues related to the

influence of microstructure on microscopic and macroscopic quantities governing the response of composites and the technologically

important applications. A major issue addressed herein is the applicability of the classical homogenization schemes in the analysis of

functionally graded materials. The technologically important applications illustrate the utility of functionally graded microstructures in

tailoring the response of structural components in a variety of applications involving uniform and gradient thermomechanical loading.

1999 Elsevier Science Ltd. All rights reserved.

Keywords: Higher-order theory; Functionally graded materials; Microstructural tailoring/optimization

1. Introduction

Functionally graded materials (FGMs) are a new genera-

tion of engineered materials wherein the microstructural

details are spatially varied through non-uniform distribution

of the reinforcement phase(s), by using reinforcement with

different properties, sizes and shapes, as well as by inter-

changing the roles of reinforcement and matrix phases in a

continuous manner [1]. The result is a microstructure thatproduces continuously or discretely changing thermal and

mechanical properties at the macroscopic or continuum

scale. Examples illustrating different types of functionally

graded microstructures are presented in Fig. 1 (where the

terminology to describe the gradation type is used to reflect

the spatial variation of macroscopic properties). This new

concept of engineering the materials microstructure marks

the beginning of a revolution both in the materials science

and mechanics of materials areas as it allows one, for the

first time, to fully integrate the material and structural

considerations into the final design of structural compo-

nents. The rapidly growing interest in FGMs, originally

conceived in Japan, is evidenced by the large number of

recent conferences and special issues of technical journals

devoted to the analysis, design and fabrication of these

materials [27].

Functionally graded materials are ideal candidates forapplications involving severe thermal gradients, ranging

from thermal structures in advanced aircraft and aerospace

engines to computer circuit boards. In one such application,

a ceramic-rich region of a functionally graded composite is

exposed to hot temperature while a metallic-rich region is

exposed to cold temperature, with a gradual microstructural

transition in the direction of the temperature gradient,

Fig. 1(a). By adjusting the microstructural transition appro-

priately, optimum temperature, deformation and stress

distributions can be realized. This concept has been success-

fully employed to enhance the thermal fatigue resistance

Composites: Part B 30 (1999) 777832

1359-8368/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.

PII: S1359-8368(99) 00053-0

www.elsevier.com/locate/compositesb

* Corresponding author. Tel.: 1-804-924-1040; fax: 1-804-982-2951.

E-mail address: [email protected] (M.-J. Pindera)


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and life of ceramic thermal barrier coatings [8,9]. Micro-

structural grading through non-uniform reinforcementspacing, Fig. 1(b), or through the use of different types of

reinforcement, Fig. 1(c), can also be effectively used to

reduce the mismatch in the thermomechanical properties

between differently oriented, adjacent plies in a laminated

plate. Thus, the reduction of thermally induced interlaminar

stresses at the free edge of a laminate (which result from the

large property mismatch between the adjacent plies) can be

realized by using the functional grading concept to smooth

out the transition between dissimilar plies. Along similar

lines, joining of dissimilar materials can be made more effi-

cient through the use of functionally graded joints [10 12].

Other benefits to be realized from the use of functionally

graded architectures include fracture toughness enhance-ment in ceramic matrix composites through tailored inter-

faces [13], and/or introduction of a second phase that creates

compressive stress fields in critical, crack-prone regions.

Many more applications of FGMs can be found in the afore-

mentioned conference proceedings and special issues of

technical journals, including the most recent focus on the

solar energy conversion devices [14], dental implants [15],

and naturally occurring biological FGMs [16,17].

Owing to the many variables that control the design of

functionally graded microstructures, full exploitation of the

FGMs potential requires the development of appropriate

J. Aboudi et al. / Composites: Part B 30 (1999) 777832778

Nomenclature

dpa ; h

qb ; l

rg dimensions of the subcell (ab g) in the (p,q,r)th generic cell

eabgijl;m;n coefficients of the total strain series expansion in the subcell (ab g)

kabgi coefficients of heat conductivity of the material in the subcell ( abg)

p; q; r indices used to identify the cell (p; q; r)q

abgi components of the heat flux vector in the subcell ( ab g)

vpqra;b;g volume of the subcell (abg) in the (p,q,r)th generic cell

uabgi displacement components in the subcell (abg)

xa1 ; x

b2 ; x

g3 local subcell coordinates

Cabgijkl elements of the stiffness tensor of the material in the subcell (abg)

1abgij local strain components in the subcell (abg)

1inabgij local inelastic strain components in the subcell (a bg)

Np, Nq, Nr number of cells in the x1, x2 and x3 directions, respectively

Qabgil;m;n

average values of the subcell heat flux component qabgi

when l

m

n

0; higher-order heat flux terms for

other values of l,m,n

Pn Legendre polynomial of order n

Rabgijl;m;n inelastic strain distribution components in the subcell (abg)

Sabgijl;m;n average values of the subcell (a bg) stress components s

abgij when l m n 0; higher-order stress compo-

nents for other values of l,m,n

T(abg) temperature field in the subcell (abg)

Tabgl;m;n temperature at the center of the subcell (abg) when l m n 0; coefficients associated with higher-order

terms in the temperature field expansion within the subcell ( abg) for other values of l,m,n

Wabgil;m;n displacement components at the center of the subcell (abg) when l m n 0; coefficients associated with

higher-order terms in the displacement field expansion within the subcell (abg) for other values of l,m,n

a ,b,g indices used to identify the subcell (abg)

m(abg) shear modulus of an isotropic material in the subcell (abg)

sabgij local stress components in the subcell (abg)

sTabgij local thermal stress components in the subcell (abg)

tabgijl;m;n coefficients of the stress series expansion in the subcell (ab g)

tTabgijl;m;n coefficients of the thermal stress series expansion in the subcell (abg)

Gabgij elements of the thermal tensor of the material in the subcell (abg)


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modeling strategies for their response to combined thermo-

mechanical loads. Presently, most computational strategies

for the response of FGMs do not explicitly couple the mate-

rials heterogeneous microstructure with the structural

global analysis. Rather, local effective or macroscopic prop-

erties at a given point within the FGM are first obtained

through homogenization based on a chosen micromechanics

scheme, and subsequently used in a global thermomechani-

cal analysis, Fig. 2. In this approach, the local micromecha-nical analysis is carried out in an independent fashion from

the global macromechanical analysis, essentially decou-

pling the influence of the surrounding spatially inhomoge-

neous microstructure on the local response of the equivalent

(homogenized) continuum point. This is what is meant by

the absence of micro-macrostructural coupling in this type

of an analysis, as will be elaborated in the next section. The

exclusion of the possibility of coupling between local and

global effects often leads to potentially erroneous results in

the presence of macroscopically non-uniform material prop-

erties and large field variable gradients. This is particularly

true when the temperature gradient is large with respect to

the dimension of the inclusion phase, or the characteristic

dimension of the inclusion phase is large relative to the

global dimensions of the composite and the number of

uniformly or non-uniformly distributed inclusions is rela-

tively small. Perhaps the most important objection to

using the standard micromechanics approach based on the

concept of a representative volume element (RVE) in

the analysis of FGMs is the lack of a theoretical basis for

the definition of an RVE, which clearly cannot be unique in

the presence of continuously changing properties due to

non-uniform inclusion spacing [1820].

As a result of the limitation of the standard micromecha-

nics approaches in FGM applications, a new higher-order

micromechanical theory (HOTFGM), which explicitly

couples the local (microstructural) and global (macrostruc-

tural) effects, has been developed and applied to function-

ally graded composites. The theoretical framework is based

on volumetric averaging of the various field quantities,

satisfaction of the field equations in a volumetric sense,and imposition in an average sense of boundary and inter-

facial conditions between the subvolumes used to character-

ize the composites functionally graded microstructure. The

need for development of such a theory has been demon-

strated by comparison with results obtained using a standard

micromechanics approach that neglects the micro-macro-

structural coupling effects.

The original formulation of HOTFGM was developed for

the thermoelastic analysis of metal matrix composite plates

functionally graded in the thickness direction, constrained

from deforming in the plane of the plate, and subjected to a

through-thickness temperature gradient [2123]. Thisversion of the higher-order theory is known as HOTFGM-

1D as it reflects the one-directional grading character of the

materials microstructure. Extensive comparison between

the predictions of the one-directional version of the

higher-order theory and the results generated using the

finite-element analysis [24] and the boundary-element

analysis [25] established the theory as a viable tool for the

analysis of functionally graded composites. Subsequent

incorporation of inelastic constitutive models with tempera-

ture-dependent parameters for the response of the constitu-

ent phases into HOTFGM-1Ds theoretical framework

made possible the analysis of through-thickness function-

ally graded metal matrix composites over a wide tempera-ture range [26]. Relaxation of the constraint on the inplane

deformation through a partial homogenization procedure in

the non-functionally graded (periodic) directions extended

the theorys applicability to practical structural problems by

allowing simulation of generalized plane strain inplane

boundary conditions [27].

The analytical framework of HOTFGM, which results in

a closed-form system of equations for the unknown micro-

variables that govern the distribution of internal tempera-

ture, displacement and stress fields at each point within the

heterogeneous material, greatly facilitates the incorporation

J. Aboudi et al. / Composites: Part B 30 (1999) 777832 779

Fig. 1. Examples of different types of functionally graded microstructures.


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of HOTFGM into an optimization algorithm as demon-

strated by Aboudi et al. [28] using HOTFGM-1D coupled

to the commercially available optimizer DOT [29]. In that

investigation, optimum through-thickness fiber distributions

were determined that minimized the inplane force and

moment resultants in unidirectionally and bidirectionally

reinforced MMC plates subjected to a through-thickness

thermal gradient.The recently developed two-directional version of the

higher-order theory, HOTFGM-2D, allows coupled micro-

macrostructural analysis of orthogonal composite laminates,

with finite dimensions in the plane containing arbitrarily

spaced inclusions or continuous fibers, subjected to

combined thermomechanical loading [30,31]. In particular,

the technologically important interlaminar stress fields in

laminated composites in the vicinity of the free edge were

analyzed and the advantages of functionally graded micro-

structures in reducing the free-edge stress concentrations

were demonstrated. In the most recent application of the

two-directional theory to functionally graded thermal

barrier coatings subjected to thermal gradient cyclic load-

ing, the creep induced stress redistribution was shown to

depend on the level of microstructural refinement, further

illustrating the shortcomings of the homogenization-based

analysis [32,33]. Summaries of selected developments of

the higher-order theory during the periods 1992 through

1995 and 1995 through 1997 presented in the above refer-ences have been provided by Pindera et al. [34,35].

As the development of the higher-order theory was driven

not only by our desire to address fundamental issues related

to microstructural effects in composites but also by various

technologically important applications, the notation

employed in constructing the theory was not always

uniform. In particular, the notation employed in construct-

ing the one-directional version was influenced by the nota-

tion used in developing the first-order homogenization

theory known in the literature as the generalized method

of cells [36,37]. In contrast, the extension of the theory


Fig. 2. Homogenization-based micromechanical analysis of functionally graded materials.


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that enables the analysis of functionally graded microstruc-

tures in two directions necessitated the adaptation of a more

general notation. This most recent notation makes further

generalization of the higher-order theory to three dimen-

sions relatively straightforward.

One of the objectives of this review article, therefore, is to

present the full generalization of the higher-order theory to

three dimensions from which the two-directional and one-

directional versions are obtained as special cases using a

unified notation. In addition, the major results that illustrate

both the fundamental issues related to the influence ofmicrostructure on macroscopic and microscopic quantities

governing the response of composites and the technologi-

cally important applications are summarized. A major

theme that the authors wish to highlight is the issue of the

applicability of the classical homogenization schemes in the

analysis of functionally graded materials. The technologi-

cally important applications summarized herein, on the

other hand, illustrate the utility of functionally graded

microstructures in tailoring the response of structural

components under spatially uniform and non-uniform ther-

momechanical load histories.

2. Applicability of classical homogenization schemes

An alternative solution methodology for the thermal and

mechanical fields in functionally graded materials involves

a combination of the standard micromechanics and conti-

nuum approaches in which local (micromechanical) and

global (macromechanical) analyses are carried out inde-

pendently of each other, i.e. in a decoupled manner. In

this approach it is assumed that the heterogeneous micro-

structure of an FGM can be replaced by an equivalent

continuum with a set of macroscopic properties that vary

with spatial coordinates in a manner commensurate with

the materials heterogeneity, Fig. 2. The macroscopic prop-

erties at each point of the equivalent continuum are gener-

ated by assuming that it is possible to define an RVE in the

presence of spatially variable microstructure and then

carrying out an RVE-based micromechanical analysis

using a chosen micromechanics model. That is, the macro-

scopic properties are generated on the premise that no

coupling exists between local and global responses asthese properties are calculated without explicitly taking

into account the influence of the adjacent spatially variable

microstructural details. Subsequently, the continuum or

macrolevel results are obtained by solving the given

boundary-value problem of a homogenized medium with

spatially variable equivalent material properties calculated

in the manner described above. As stated previously, this

decoupled two-step procedure is the standard micro-macro-

mechanical approach currently employed by most

researchers working in the area of functionally graded

materials. With the knowledge of the macroscopic thermo-

mechanical fields, the stresses in the individual phases at acontinuum point may then be estimated by applying

average thermomechanical field quantities at the selected

point (or region), treating it as an RVE within the frame-

work of the chosen micromechanical model. A problem

arises, however, when the microstructural scale is large

relative to the thermomechanical field gradients so as to

invalidate the basic assumptions on which the RVE

concept is based. The situation becomes more complicated

when the microstructure changes continuously as in this

case, the micromechanical concept of an RVE, and thus

material property, is not well defined.

The various micromechanical approaches used to calcu-

late effective properties of composites include use of simpleReuss and Voigt hypotheses, self-consistent schemes and

their generalizations, differential schemes, the Mori

Tanaka method, concentric cylinder models, bounding tech-

niques and approximate (e.g. the generalized method of

cells) or numerical (e.g. finite-element or finite-difference)

analyses of periodic arrays of inclusions or fibers in the

surrounding matrix phase [38]. The central assumption in

applying these well-established techniques is the existence

of an RVE and the ability to apply homogeneous boundary

conditions to such an element, Fig. 3. These homogeneous

boundary conditions can be specified either in terms of


Fig. 3. Calculation of macroscopic moduli based on the concept of a repre-

sentative volume element.


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surface displacements

uiS 10ijxj 1

or in terms of prescribed surface tractions

Ti

S

s0ijnj

2

where ni is the unit outward normal vector on the boundary

surface S of the composite, xi are the Cartesian coordinates

of the surface, 10ij and s0ij are constants, and repeated index

implies summation. For a heterogeneous medium the

constants 10ij and s0ij are the volume averaged strains and

stresses under the prescribed boundary conditions given

by Eqs. (1) and (2), respectively. This result is a conse-

quence of the following relations:

1ij 1

V

V

1ijxk dV 1

V

S

1

2uinj ujni dS 3

sij 1

V

V

sijxk dV 1

V

S

1

2Tixj Tjxi dS 4

where Vis the volume enclosed by the surface S. The above

relations hold provided that: the displacements ui are contin-

uous; the tractions Ti are continuous at all interfaces of the

heterogeneous medium; and body forces vanish.

In practice, the average strains and stresses that result

from the application of homogeneous boundary conditions

are calculated for an RVE whose macroscopic behavior is

indistinguishable from the behavior of the composite-at-

large. By applying the homogeneous boundary conditions

to the bounding surface of the RVE, which are the same as

the boundary conditions applied to the entire composite, its

average behavior can be calculated. This average behavior,

in turn, defines the composites macroscopic properties. To

qualify as an RVE, the volume of the element used to calcu-

late average composite behavior must meet two criteria.

First, it must be sufficiently small with respect to the dimen-

sions of the composite-at-large in order to be considered a

material point in the equivalent homogeneous continuum

(i.e. hp H; see Fig. 4). Second, it must be sufficiently

large with respect to the inclusion phase (i.e. dp h) so

that to the first order the elastic strain energy induced by

both sets of homogeneous boundary conditions is the same,rendering the effective elastic properties independent of the

manner in which boundary conditions are applied [18]. In

this case, the microstructure of the RVE can be replaced by

a fictitious homogeneous material with effective or homo-

genized moduli or compliances, Cijkl or Sijkl, respectively,

that connect the average stress and strain quantities as

follows:

sij Cijkl 1kl 5

1ij Sijkl skl 6

where Sijkl C1ijkl ; Fig. 3. In the case of periodic fiber arrays,the repeating unit cell is interpreted as an RVE provided that

the homogeneous boundary conditions are replaced by

either symmetry conditions on the deformation of the unit

cell or periodic boundary conditions, depending on the type

of loading. It should be mentioned that the use of periodic

boundary conditions in conjunction with a multi-scale

asymptotic expansion of the displacement field in the

repeating unit cell forms the basis of the so-called homo-

genization methods for estimating the macroscopic proper-

ties of periodic composites [39,40].

Clearly, the range of applicability of the aforementioned

micromechanical approaches is limited to composites rein-

forced by numerous fibers with very small diameters such

as graphite or carbon fibers. In such composites, a typical

RVE contains a sufficiently large number of fibers while

occupying a very small volume of the entire composite,

allowing one to disregard boundary-layer effects near the

bounding surfaces of the RVE upon application of either

type of homogeneous boundary conditions. As a result,even in the presence of highly inhomogeneous deformation

gradients within the composite-at-large, the field quantities

within the RVE will not vary significantly, thereby permit-

ting the definition of a material property at a point in the

equivalent homogeneous continuum. In contrast, in compo-

sites with few and relatively large-diameter fibers (with

respect to the thickness of a single ply), the variation of

the quantities of interest within the RVE (assuming that it

can be defined) invalidates the basic assumptions on which

the concept of effective properties is based. These local

variations of the field quantities within the RVE may

give rise to unexpected phenomena rooted in the local-global coupling which is neglected in the traditional micro-

mechanical homogenization schemes. For instance, differ-

ent thermal conductivities of the individual phases together

with their directional arrangement may produce thermal

gradients in the individual phases which are quite different

from the thermal gradients in the homogeneous composite

with equivalent effective properties subjected to identical

boundary conditions, Fig. 4. This, in turn, may alter the

local conductivity characteristics and produce unexpected

effects such as localized hot spots for instance. The size

of the RVE in relation to the thickness of the composite

and the temperature gradient obviously will play an impor-

tant role in the above scenario.The preceding discussion raises questions about the

applicability of the traditional micromechanical approach

based on the concept of an RVE in the presence of large

thermal gradients and coarse or spatially variable micro-

structure. In light of this discussion, the current practice of

decoupling the local response from the global response by

calculating pointwise effective thermoelastic properties of

functionally graded materials without regard to whether the

actual microstructure admits the presence of an RVE, and

subsequently using these properties in the global analysis of

the heterogeneous material, remains to be justified. These



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issues were discussed qualitatively as early as 1974 by

Pagano [41] with regard to mechanical loading of macro-

scopically homogeneous composites. Most recently, discus-

sions regarding the admissibility of the classical RVEconcept in the homogenization analysis of random heteroge-

neous media can be found in the works of Huet [42], Ostoja-

Starzewski and Schulte [43], Ostoja-Starzewski et al. [44]

and Ostoja-Starzewski [45]. To account for the materials

random microstructure, these authors used the concept of a

material window containing different numbers of inhomo-

geneously distributed inclusions in order to study the effect

of the window size on the local macroscopic properties.

In order to resolve the issue of the applicability of the

classical (in the sense of Hill) RVE concept in the analy-

sis of FGMs, a model is required that explicitly couples

the microstructural and macrostructural analyses. The

higher-order micromechanical theory, developed by the

authors during the past several years and presented in

the following section, is a step in this direction for appli-cations involving, but not limited to: composites with

uniformly or non-uniformly spaced; large-diameter fibers

subjected to through-thickness thermal gradients; finite

cross-section metal matrix composite laminates subjected

to combined thermomechanical loading; and thermal

barrier coatings. The generalization of this higher-order

theory for materials functionally graded in three orthogo-

nal directions is outlined next. Finally, we mention that

the present approach is based on deterministic considera-

tions, in contrast with the statistical approaches mentioned

above.


Fig. 4. Applicability of the representative volume element for heterogeneous materials in the presence of a thermal gradient.


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3. Generalized three-directional higher-order theory

3.1. Model overview

The generalized higher-order theory is based on the

geometric model of a heterogeneous parallelepiped occupy-

ing the region 0 x1 D; 0 x2 H; 0 x3 L

(Fig. 5). The loading applied on the bounding surface of

the parallelepiped may involve an arbitrary temperature or

heat flux distribution and mechanical effects represented by

a combination of surface displacements and/or tractions

consistent with global equilibrium requirements. Thecomposite may be reinforced by arbitrarily spaced planar

arrays of continuous fibers (with arbitrary in-plane spacing)

oriented along any of the three axes x1, x2, x3, or finite-length

inclusions that are arranged arbitrarily in the three planes.

The microstructure of the heterogeneous composite is

discretized into Np, Nq and Nrcells in the intervals 0 x1

D; 0 x2 H; 0 x3 L; respectively. The generic cell

(p,q,r) used to construct the composite, highlighted in Fig. 5,

consists of eight subcells designated by the triplet (abg),

Fig. 6, where each index a ,b,g takes on the values 1 or 2

which indicate the relative position of the given subcell

along the x1, x2 and x3 axis, respectively. The indices p, q

and r, whose ranges are p 1; 2; ;Np; q 1; 2; ;Nq andr 1; 2; ;Nr; identify the generic cell in the three planes.The dimensions of the generic cell along the x1, x2 and x3axes, d

p1 ; d

p2 ; h

q1 ; h

q2 ; and l

r1 ; l

r2 ; can vary in an arbitrary

fashion such that

D Np

p1dp1 dp2 H

Nqq1

hq1 hq2

L Nrr1

lr1 lr2

Given the applied thermomechanical loading, an approx-

imate solution for the temperature and displacement fields is

constructed based on volumetric averaging of the field equa-

tions together with the imposition of boundary and continu-

ity conditions in an average sense between the subvolumes

used to characterize the materials microstructure. This is

accomplished by approximating the temperature and displa-

cement fields in each subcell of a generic cell using a quad-

ratic expansion in the local coordinates xa; xb; xg

centered at the subcells mid-point. A higher-order repre-

sentation of the temperature and displacement fields is

necessary in order to capture the local effects created bythe thermomechanical field gradients, the microstructure

of the composite and the finite dimensions in the function-

ally graded directions. This is in sharp contrast with

previous treatments involving fully periodic composite

media which employed linear expansions [38]. We note

that it is possible to specialize this temperature and displa-

cement field representation to model composites that are

functionally graded in one direction with the remaining

directions periodic, or composites that are functionally

graded in a plane with periodicity in the out-of-plane direc-

tion. In either case, the temperature and displacement fields

in the periodic directions are approximated using a linear

expansion in local coordinates. In the former case, the one-directional version of the theory is obtained, while the latter

case yields the two-directional theory. Both specialized

cases will be elaborated upon in Section 4.

The unknown coefficients associated with each term in

the temperature and displacement field expansions are

obtained by constructing systems of equations that satisfy

the requirements of a standard boundary-value problem for

the given field variable approximations. That is, the zeroth,

first, and second moments of the heat and equilibrium equa-

tions are satisfied in a volumetric sense. Similarly, the ther-

mal and heat flux, as well as the displacement and traction,


Fig. 5. Schematic of a composite functionally graded in three directions.


9/56

continuity conditions (the mechanical analogues) within agiven generic cell, and between a given generic cell and its

neighbors, are imposed in an average sense across the inter-

facial planes. The solution for these coefficients in the

generalized higher-order theory follows the general frame-

work for the solution of the corresponding two-directional

thermoelastic and thermoinelastic problems discussed

previously [30,31]. Therefore, only a summary of the

governing equations for the temperature and displacement

fields in the individual subcells within the rows and columns

of cells considered in solving the outlined boundary-value

problem is given in the following sections. Detailed deriva-

tion of these equations can be found in the above references.

3.2. Thermal analysis

Let the functionally graded parallelepiped be subjected to

steady-state temperature or heat flux distributions on its

bounding surfaces. Under steady-state heat conduction,

the heat flux field in the material occupying the subcell

(abg) of the (p,q,r)th generic cell, in the region xa1 12

dpa ; x

b2

12

hqb ; x

g3

12

lrg ; must satisfy:

21qabg1 22q

abg2 23q

abg3 0 7

where 21 2=2 xa1 ; 22 2=2 xb2 ; 23 2=2 xg3 : The

components qabg

i of the heat flux vector in this subcellare obtained from the Fouriers heat conduction law foranisotropic materials,

qabgi kabgij 2jTabg; i;j 1; 2; 3 8

where kabgij are the coefficients of heat conductivity of the

material in the subcell (abg), with kabgij kabgi dij (no sum

on i) for orthotropic materials, and no summation is implied

by repeated Greek letters in the above and henceforth.

The temperature distribution in the subcell (a bg) of the

(p,q,r)th generic cell, measured with respect to a reference

temperature Tref, is denoted by T(abg). We approximate this

temperature field by a second-order expansion in the local

coordinates xa1 ; xb2 ; x

g3 as follows:

Tabg Tabg000 xa1 Tabg100 xb2 Tabg010 xg3 Tabg001

1

23 xa21

dp2a

4

2 3T

abg200

1

23 x

b22

hq2b

4

2 3T

abg020

1

23 x

g23

lr2g

4

2 3T

abg002 9


Fig. 6. Generic cell (p,q,r) of a composite functionally graded in three directions showing the dimensions and the designation convention of the internal subcells

(a b g).


10/56

where Tabg000 which is the temperature at the center of the

subcell, and Tabglmn l; m; n 0; 1; or 2 with l m n 2

are unknown coefficients which are determined from condi-

tions that will be outlined subsequently. It should be noted

that no terms of the form xi x

j i j appear in the tempera-ture field representation, as well as in the displacement field

representation given in the sequel, due to the averaging

procedure employed in our higher-order theory.

Given the seven unknown quantities associated with each

subcell (i.e. Tabg000 ; ; T

abg002 ) and eight subcells within each

generic cell, 56NpNqNr unknown quantities must be deter-

mined for a composite with Np, Nq, and Nrrows and columns

of cells containing arbitrarily specified materials. These

quantities are determined by first satisfying the heat conduc-

tion equation, as well as the first and second moment of this

equation, in each subcell in a volumetric sense in view of the

temperature field approximation given by Eq. (9). Subse-

quently, continuity of heat flux and temperature is imposed

in an average sense at the interfaces separating adjacent

subcells, as well as neighboring cells. Fulfillment of thesefield equations and continuity conditions, together with the

imposed thermal boundary conditions on the bounding

surfaces of the composite, provides the necessary 56NpNqNrequations for the 56NpNqNr unknown coefficients in the

temperature field expansion.

We begin the outline of steps to generate the required

number of equations by first considering an arbitrary

(p,q,r)th generic cell in the interior of the composite (i.e.

p 2; ;Np1; q 2; ;Nq1; and r 2; ;Nr1: Thisproduces 56Np 2Nq 2Nr 2 equations. The addi-tional equations are obtained by considering the boundary

cells (i.e. p 1;Np

;q 1

;Nq and r 1

;Nr). For thesecells, most of the relations obtained from considering the

interior cells will also hold, with the exception of some of

the interfacial continuity conditions between adjacent

interior cells which will be replaced by the specified

boundary conditions.

In the course of satisfying the steady-state heat equation

in a volumetric sense, it is convenient to define the follow-

ing flux quantities:

Qabgil;m;n

1

npqrabg

dpa=2d

pa=2

hqb=2h

qb=2

lrg=2l

rg=2

xa1 l xb2 m

xg3 nqabgi d xa1 d xb2 d xg3 10

where l; m; n 0; 1; or 2 with l m n 2; and vpqrabg d

pa h

qb l

rg is the volume of the subcell (a bg) in the (p,q,r)th

generic cell. For l; m; n 0; Qabgi0;0;0 is the volume averagevalue of the heat flux component q

abgi in the subcell,

whereas for other values of (l,m,n) Eq. (10) defines

higher-order heat fluxes. These flux quantities can be eval-

uated explicitly in terms of the coefficients Tabglmn by

performing the required volume integration using Eqs. (8)

and (9) in Eq. (10). This yields the following non-vanishing

zeroth-order and first-order heat fluxes in terms of the

unknown coefficients in the temperature field expansion:

Qabg10;0;0 kabg1 Tabg100 11

Qabg11;0;0 kabg1

dp2a

4T

abg200 12

Qabg

20;0;0 kabg

2 Tabg

010 13

Qabg20;1;0 kabg2

hq2b

4T

abg020 14

Qabg30;0;0 kabg3 Tabg001 15

Qabg30;0;1 kabg3

lr2g

4T

abg002 16

Subsequently, satisfaction of the zeroth, first and second

moment of the steady-state heat equation results in the

following eight relationships among the first-order heat

fluxes Qabgil;m;n

in the different subcells (abg) of the

(p,q,r)th generic cell, after some involved algebraic manip-

ulations (see [21,22]):

Qabg11;0;0=d2a Qabg20;1;0=h2bQabg30;0;1=l2gp;q;r 0 17where the triplet (abg) assumes all permutations of the

integers 1 and 2.

The continuity of heat fluxes at the subcell interfaces and

between individual generic cells, imposed in an average

sense, is ensured by:

12Q1bg11;0;0=d1 Q2bg10;0;0 6Q2bg11;0;0=d2p;q;r

Q2bg

10;0;0 6Q2bg

11;0;0=d2p1;q;r

0 18

Q1bg10;0;0 12 Q2bg10;0;0 3Q

2bg11;0;0=d2p;q;r

12Q2bg10;0;0 6Q2bg11;0;0=d2p1;q;r 0 19

12Qa1g20;1;0=h1 Qa2g20;0;0 6Qa2g20;1;0=h2p;q;r

Qa2g20;0;0 6Qa2g20;1;0=h2p;q1;r 0 20

Qa1g20;0;0 12 Qa2g20;0;0 3Q

a2g20;1;0=h2p;q;r

12 Qa2g20;0;0 6Qa2g20;1;0=h2p;q1;r 0 21

12Qab130;0;1=l1 Qab230;0;0 6Qab230;0;1=l2p;q;r

Qab230;0;0 6Qab230;0;1=l2p;q;r1 0 22

Qab130;0;0 12 Qab230;0;0 3Q

ab230;0;1=l2p;q;r

12Qab230;0;0 6Qab230;0;1=l2p;q;r1 0 23

Eqs. (18)(23) provide us with 24 additional relations

among the zeroth-order and first-order heat fluxes. These



11/56

relations, together with Eq. (17), can be expressed in terms

of the unknown coefficients Tabglmn by making use of

Eqs. (11)(16), providing a total of 32 of the required 56

equations necessary for the determination of these

coefficients in the interior (p,q,r)th generic cell.

An additional set of 24 equations necessary to deter-

mine the unknown coefficients in the temperature field

expansion is subsequently generated by the thermal

continuity conditions imposed on an average basis at

each subcell and cell interface. Imposing the interfacial

thermal continuity conditions we obtain the following

relations:

T1bg000 12 d1T1bg100

14

d21 T

1bg200 p;q;r

T2bg000 12 d2T2bg100

14

d22T

2bg200 p;q;r 24

T2bg000 12 d2T2bg100

14

d22 T

2bg200 p;q;r

T1bg

000 1

2 d1T1bg

100 1

4 d

2

1T1bg

200 p1;q;r

25Ta1g000 12 h1T

a1g010

14

h21T

a1g020 p;q;r

Ta2g000 12 h2Ta2g010

14

h22T

a2g020 p;q;r 26

Ta2g000 12 h2Ta2g010

14

h22T

a2g020 p;q;r

Ta1g000 12 h1Ta1g010

14

h21T

a1g020 p;q1;r 27

Tab1000 12 l1Tab1001

14

l21T

ab1002 p;q;r

Tab2

000 12 l2T

ab2

001 14 l

2

2Tab2

002 p;q;r

28

Tab2000 12 l2Tab2001

14

l22T

ab2002 p;q;r

Tab1000 12 l1Tab1001

14

l21T

ab1002 p;q;r1 29

which comprise the required additional 24 relations.

The steady-state heat equations, Eq. (17), together with

the heat flux and thermal continuity equations, Eqs. (18)

(23) and Eqs. (24)(29), respectively, form 56 linear

algebraic equations which govern the 56 field variables

Tabglmn in the eight subcells (abg) of an interior generic

cell

p; q; r

; p

2; ;Np1; q

2; ;Nq1; r

2; ;Nr1: For the boundary cells p 1;Np; q 1;Nqand r 1;Nr a different treatment must be applied. For

p 1; for instance, the above relations are operative, butrelations (18) and (19), on the other hand, which follow

from the continuity of heat flux between a given generic

cell and the preceding one are not applicable. They are

replaced by the condition that the heat flux at the interface

between subcell (1bg) and (2bg) of the cell (1,q,r) is

continuous, as well as the applied temperature relation at

the bottom surface x1 0 (see Fig. 5). For the cell p Np;the previous equations are applicable except Eq. (25) which

are obviously not operative. These equations are replaced by

the specific temperature applied at the top surface x1 D:The boundary conditions at these surfaces are

T1bg1;q;r Tbottomx2;x3; x11 12 d

11 30

T2bgNp ;q;r Ttopx2;x3; x21 12 d

Np2 31

where q 1; ;Nq; r 1; ;Nr:Similar reasoning holds for the cells q 1 and q Nq:

That is, the temperature in the cells (p,1,r) and (p,Nq,r) at the

x2 0 and x2 H(i.e. left and right) surfaces, respectively,must equal the applied temperature,

Ta1gp;1;r Tleftx1;x3; x12 12 h

11 32

Ta2gp;Nq ;r Trightx1;x3; x22 12 h

Nq2 33

where p 1; ;Np; r 1; ;Nr:Finally, the temperature in the cells (p,q,1) and (p,q,Nr) at

the x3 0 and x3 L (i.e. front and back) surfaces, respec-tively, must equal the applied temperature,

Tab1p;q;1 Tfrontx1;x2; x13 12 l

11 34

Tab2p;q;Nr Tbackx1;x2; x23 12 l

Nr2 35

where p 1; ;Np; q 1; ;Nq: Alternatively, it ispossible to impose mixed-boundary conditions involving

temperature and heat flux at different portions of the

boundary.

The governing equations for the interior and boundary

cells form a system of 56NpNqNr algebraic equations in the

unknown coefficients Tabglmn : Their solution determines the

temperature distribution within the functionally gradedcomposite subjected to the boundary conditions (30)(35).

The final form of this system of equations is symbolically

represented below

kT t 36where the structural thermal conductivity matrix k contains

information on the geometry and thermal conductivities of

the individual subcells (abg) in the NpNqNr cells spanning

the x1, x2 and x3 directions, the thermal coefficient vector T

contains the unknown coefficients that describe the tempera-

ture field in each subcell, i.e.

T T111

111 ; ; T222

NpNqNrwhere

Tabgpqr T000; T100; T010; T001; T200; T020; T002abgpqrand the thermal force vector t contains information on the

boundary conditions.

3.3. Mechanical analysis

Given the temperature field generated by the applied

surface temperatures and/or heat fluxes obtained in the

preceding section, we proceed to determine the resulting



12/56

displacement and stress fields. This is carried out for arbi-

trary mechanical loading, consistent with global equili-

brium requirements, applied to the surfaces of the

composite.

The stress field in the subcell (a bg) of the (p,q,r)th

generic cell generated by the given temperature field must

satisfy the equilibrium equations

21sabg1j 22s

abg2j 23s

abg3j 0; j 1; 2; 3 37

where the operator 2ii 1; 2; 3 has been definedpreviously. The components of the stress tensor, assuming

that the material occupying the subcell (abg) of the

(p,q,r)th generic cell is orthotropic, are related to the

strain components through the familiar generalized

Hookes law:

sabgij Cijkl1abgkl 1inabgkl Gabgij Tabg 38

where Cijkl are the elements of the stiffness tensor, 1in

abg

klare the inelastic strain components, and the elements

Gabgij of the so-called thermal tensor are the products

of the stiffness tensor and the thermal expansion coeffi-

cients. In this paper, we consider either elastic ortho-

tropic materials or inelastic materials which are

isotropic in both elastic and inelastic domains. Hence,

Eq. (38) reduces to

sabgij Cijkl 1abgkl 2mabg1inabgij sTabgij 39

where m (abg) is the elastic shear modulus of the mate-

rial filling the given subcell (a bg), and the term

sTabgij ; henceforth referred to as thermal stress, standsfor the thermal contribution G

abgij T

abg: The compo-

nents of the strain tensor in the individual subcells

are, in turn, obtained from the strain-displacement rela-

tions

1abgij 12 2iu

abgj 2ju

abgi ; i;j 1; 2; 3 40

The displacement field in the subcell (abg) of the

(p,q,r)th generic cell is approximated by a second-order

expansion in the local coordinates xa1 ; xb2 ; and x

g3 as

follows:

uabg1 Wabg1000 xa1 Wabg1100 xb2 Wabg1010 xg3 Wabg1001

1

23 xa21

dp2a

4

2 3W

abg1200

1

23 x

b22

hq2b

4

2 3W

abg1020

1

23 x

g23

lr2g

4

2 3W

abg1002 41


1

23 xa21

dp2a

4

2 3W

abg2200

12

3 xb22 h

q2b4

2 3Wabg2020

1

23 x

g23

lr2g

4

2 3W

abg2002 42


1

23 xa21

dp2a4

2 3W

abg3200

1

23 x

b22

hq2b4

2 3W

abg3020

1

23 x

g23

lr2g

4

2 3W

abg3002 43

where Wabgi000 ; which are the displacements at the center of

the subcell, and the higher-order terms Wabgilmn i 1; 2; 3

must be determined from conditions similar to those

employed in the thermal problem. In this case, there are

168 unknown quantities in a generic cell (p,q,r). The deter-

mination of these quantities parallels that of the thermalproblem. Here, the heat conduction equation is replaced

by the three equilibrium equations, and the continuity of

tractions and displacements at the various interfaces

replaces the continuity of heat fluxes and temperature.

Finally, the boundary conditions involve the appropriate

mechanical quantities. As in the thermal problem, we start

with the internal cells and subsequently modify the govern-

ing equations to accommodate the boundary cells p 1;Np; q 1;Nq; and r 1;Nr:

In the perfectly elastic case, the quadratic displacement

expansion, Eqs. (41) (43), produces linear variations in

strains and stresses at each point within a given subcell. In

the presence of inelastic effects, however, a linear strainfield generated by Eqs. (41)(43) does not imply the linear-

ity of the stress field due to the path-dependent deformation.

Thus the displacement field microvariables must depend

implicitly on the inelastic strain distributions, giving rise

to a higher-order stress field than the linear strain field

generated from the assumed displacement field representa-

tion. In the presence of inelastic effects, this higher-order

stress field is represented by a higher-order Legendre poly-

nomial expansion in the local coordinates. Therefore, the

strain field generated from the assumed displacement

field, and the resulting mechanical and thermal stress fields,



13/56

must also be expressed in terms of Legendre polynomials:

1abgij

l0

m0

n0

1 2l1 2m1 2n

p

eabgijl;m;nPlza1 Pmzb2 Pnzg3 44

sabgij

l0

m0

n0

1 2l1 2m1 2n

p

tabgijl;m;nPlza1 Pmzb2 Pnzg3 45

sTabgij

l0

m0

n0

1 2l1 2m1 2n

p

tTabgijl;m;nPlza1 Pmzb2 Pnzg3 46

where the non-dimensionalized variables z

i s, defined in theinterval 1 zi 1; are given in terms of the local

subcell coordinates as za1 xa1 =dpa =2; zb2 xb2 =hqb =2; and zg3 xg3 =lrg =2: For the given displace-

ment field representation, Eqs. (41)(43), the upper limits

on the summations in Eq. (44) become 1, while for the given

temperature distribution, Eq. (9), the upper limits on the

summations in Eq. (46) become 2. Alternatively, the

upper limits on the summations in Eq. (45) are chosen so

that an accurate representation of the stress field (which

depends on the amount of inelastic flow) is obtained within

each subcell. This will be discussed in Section 3.5 and

Appendix A where the solution of the resulting equations

for the unknown coefficients appearing in the displacementexpansions is outlined. The coefficients e

abgijl;m;n; t

abgijl;m;n;

tTabgijl;m;n in the above expansions are determined as described

below.

The strain coefficients eabgijl;m;n are explicitly determined in

terms of the displacement field microvariables of Eqs. (41)

(43), using orthogonal properties of Legendre polynomials.

Similarly, the thermal stress coefficients eTabgijl;m;n can be

expressed in terms of the temperature field microvariables

Tabglmn of Eq. (9). For example,

eabg110;0;0 Wabg1100

and

tTabg110;0;0 Gabg11 Tabg000

The complete sets of non-zero strain and thermal stress

coefficients, eabgijl;m;n and t

Tabgijl;m;n; respectively, are given in

Appendix A.

The stress coefficients tabgijl;m;n are expressed in terms of

the strain coefficients, the thermal stress coefficients and the

unknown inelastic strain distributions, by first substituting

the Legendre polynomial representations for 1(abg), s(abg),

and sT(abg) into the constitutive equations, Eq. (39), and

then utilizing the orthogonality of Legendre polynomials:

tabgijl;m;n Cijkleabgkll;m;n tTabgijl;m;n Rabgijl;m;n 47

The Rabgijl;m;n terms represent inelastic strain distributions

calculated in the following manner,

Rabgijl;m;n mabgLlmn 1 1

1 1

1 1

1inabgij Plza1

Pmzb2 Pnzg3 dza1 dzb2 dzg3 48

where

Llmn 141 2l1 2m1 2n

pIt should be noted that the choice of the Legendre poly-

nomials in the above expansions is motivated by the simpli-

city of their orthogonal properties (i.e. the weight function is

equal to one). It is possible, of course, to use any other

orthogonal set of polynomials.In the course of satisfying the equilibrium equations in a

volumetric sense, it is convenient to define the following

stress quantities:

Sabgijl;m;n

1

npqrabg

dpa =2d

pa =2

hqb

=2

hqb

=2

lrg =2 l

rg =2

xa1 l xb2 m

xg3 nsabgij d xa1 d xb2 d xg3 49

For l m n 0; Eq. (49) provides volume average stres-ses in the subcell, whereas for other values of ( l,m,n) higher-

order stresses are obtained that are needed to describe the

governing field equations of the higher-order continuum.

These stress quantities can be evaluated explicitly in terms

of the unknown coefficients Wabgilmn by performing the

required volume integration upon substituting Eqs. (39),

(40) and (41)(43) in Eq. (49). This yields the following

non-vanishing zeroth-order and first-order stress compo-

nents in terms of the unknown coefficients in the displace-

ment field expansion:

Sabg110;0;0 Cabg11 Wabg1100 Cabg12 Wabg2010 Cabg13 Wabg3001

Gabg1 T

abg000 R

abg110;0;0 (50)

Sabg111;0;0

1

4d

p2a C

abg11 W

abg1200

1

12d

p2a G

abg11 T

abg100

1

2

3p dpa Rabg111;0;0 51

Sabg110;1;0

1

4h

q2b C

abg12 W

abg2020

1

12h

q2b G

abg11 T

abg010

1

23p hqb Rabg110;1;0 52



14/56

Sabg110;0;1

1

4lr2g C

abg13 W

abg3002

1

12lr2g G

abg11 T

abg001

1

2

3p lrg Rabg110;0;1 53

with similar expressions for Sabg220;0;0; S

abg221;0;0; S

abg220;1;0;

Sabg

220;0;1; and Sabg

330;0;0; Sabg

331;0;0; Sabg

330;1;0; Sabg

330;0;1 and

Sabg120;0;0 Cabg66 Wabg1010 Wabg2100Rabg120;0;0 54

Sabg121;0;0

1

4d

p2a C

abg66 W

abg2200

1

2

3p dpa Rabg121;0;0 55

Sabg120;1;0

1

4h

q2b C

abg66 W

abg1020

1

2

3p hqb Rabg120;1;0 56


Sabg131;0;0 14 dp2a C

abg55 W

abg3200 1

2

3p dpa Rabg131;0;0 58

Sabg130;0;1

1

4lr2g C

abg55 W

abg1002

1

2

3p lrg Rabg130;0;1 59


Sabg230;1;0

1

4h

q2b C

abg44 W

abg3020

1

2

3p hqb Rabg230;1;0 61

Sabg23

0;0;1

1

4

lr2g C

abg44 W

abg2002

1

23p lrg R

abg23

0;0;1

62

where contracted notation has been employed for the stiff-

ness elements Cabgijkl :

Subsequently, satisfaction of the zeroth, first, and second

moments of the equilibrium equations results in the follow-

ing 24 relations among the volume-averaged first-order

stresses Sabgijl;m;n in the different subcells (abg) of the

(p,q,r)th generic cell, after lengthy algebraic manipulations

(see Refs. [21,22]):

Sabg1j1;0;0=d2a Sabg2j0;1;0=h2b Sabg3j0;0;1=l2gp;q;r 0 63where j

1; 2; 3 and, as in the case of Eq. (17), the triplet

(abg) assumes all permutations of the integers 1 and 2.The continuity of tractions at the subcell interfaces and

between adjacent cells, imposed in an average sense, is

ensured by the following relations:

12S1bg1j1;0;0=d1 S2bg1j0;0;0 6S2bg1j1;0;0=d2p;q;r

S2bg1j0;0;0 6S2bg1j1;0;0=d2p1;q;r 0 64

S1bg1j0;0;0 12 S2bg1j0;0;0 3S

2bg1j1;0;0=d2p;q;r

12S2bg1j0;0;0 6S2bg1j1;0;0=d2p1;q;r 0 65

12Sa1g2j0;1;0=h1 Sa2g2j0;0;0 6Sa2g2j0;1;0=h2p;q;r

Sa2g2j0;0;0 6Sa2g2j0;1;0=h2p;q1;r 0 66

Sa1g2j0;0;0 12 Sa2g2j0;0;0 3S

a2g2j0;1;0=h2p;q;r

12 S

a2g2j0;0;0 6S

a2g2j0;1;0=h2

p;q1;r 0 67

12Sab13j0;0;1=l1 Sab23j0;0;0 6Sab23j0;0;1=l2p;q;r

Sab23j0;0;0 6Sab23j0;0;1=l2p;q;r1 0 68

Sab13j0;0;0 12 Sab23j0;0;0 3S

ab23j0;0;1=l2p;q;r

12Sab23j0;0;0 6Sab23j0;0;1=l2p;q;r1 0 69

where j 1; 2, and 3.Eqs. (64)(69) provide us with 72 additional relations

among the zeroth-order and first-order stresses. These rela-tions, together with Eq. (63), can be expressed in terms of

the unknown coefficients Wabgilmn by making use of Eqs.

(50)(62), providing a total of 96 of the required 168 equa-

tions necessary for the determination of these coefficients in

the (p,q,r)th generic cell.

The additional 72 relations necessary to determine the

unknown coefficients in the displacement field expansion

are subsequently obtained by imposing displacement conti-

nuity conditions on an average basis at each subcell and cell

interface. This produces,

W1bgj000 12 d1W1bg

j100 14

d21 W

1bgj200p;q;r

W2bgj000 12 d2W2bg

j100 14

d22 W

2bgj200p;q;r 70

W2bgj000 12 d2W2bg

j100 14

d22 W

2bgj200p;q;r

W1bgj000 12 d1W1bg

j100 14

d21 W

1bgj200p1;q;r 71

Wa1gj000 12 h1Wa1g

j010 14

h21W

a1gj020p;q;r

Wa2gj000 12 h2Wa2g

j010 14

h22W

a2gj020p;q;r 72

W

a2gj

000

12

h2Wa2g

j

010

14

h22W

a2gj

020

p;q;r

Wa1gj000 12 h1Wa1g

j010 14

h21W

a1gj020p;q1;r 73

Wab1j000 12 l1Wab1

j001 14

l21W

ab1j002p;q;r

Wab2j000 12 l2Wab2

j001 14

l22W

ab2j002p;q;r 74

Wab2j000 12 l2Wab2

j001 14

l22W

ab2j002p;q;r

Wab1j000 12 l1Wab1

j001 14

l21W

ab1j002p;q;r1 75

which comprise the required additional 72 relations.



15/56

The equilibrium equations, Eq. (63), together with the

traction and displacement continuity equations Eqs. (64)

(69) and Eqs. (70)(75), respectively, form 168 equations in

the 168 Wabgilmn unknowns which govern the equilibrium of a

subcell (abg) within the (p,q,r)th generic cell in the inter-

ior. As in the thermal problem, a different treatment must be

adopted for the boundary cells (1,q,r), (Np,q,r), and (p,1,r),

(p,Nq,r), and (p,q,1), (p,q,Nr). For (1,q,r), the above relations

are operative, except Eqs. (64) and (65), which follow from

the continuity of tractions between a given generic cell and

the preceding one. These 12 equations must be replaced by

the conditions of continuity of tractions at the interior inter-

faces of the cell (1,q,r) and by the applied tractions at x1 0: For the cell (Np,q,r), the previously derived governing

equations are operative except for the 12 relations given

by Eqs. (71), which are obviously not applicable. These

are replaced by the imposed traction conditions at the

surface x1 D: Similar arguments hold for boundary cells(p,1,r), (p,Nq,r), and (p,q,1), (p,q,Nr).

The traction vector in the cells (1,q,r) and (Np,q,r) at thebounding surfaces must equal the applied surface loads,

s1bg1i

1;q;r tbottomx2;x3; x11 12 d11 76

s2bg1i

Np ;q;r ttopx2;x3; x21 12 dNp2 77

where q 1; ;Nq; r 1; ;Nr; and tbottom and ttop describethe spatial variation of these loads at the bottom and top

surfaces. Similar conditions hold for the traction vector in

the cells (p,1,r) and (p,Nq,r) at the left and right surfaces,

respectively,

sa1g

2i p;1;r

tleftx1;x3; x12 12 h

11 78

sa2g2i

p;Nq ;r trightx1;x3; x22 12 hNq2 79

where p 1; ;Np; r 1; ;Nr; and tleft and tright describethe spatial variation of these loads at the left and right

surfaces. Similar traction boundary conditions can be writ-

ten for the front and back faces. Alternatively, if the front or

back surfaces are rigidly clamped (say), then

uab1i

p;q;1 0; x13 12 l11 80

uab2i

p;q;Nr

0; x2

3

1

2lNr2

81

where p 1; ;Np; q 1; ;Nq: For other types of bound-ary conditions, Eqs. (76)(81) should be modified accord-

ingly.

Consequently, the governing equations for the interior

and boundary cells form a system of 168NpNqNr algebraic

equations in the unknown coefficients Wabgilmn : The final

form of this system of equations is symbolically represented

by

KU f g 82where the structural stiffness matrix K contains information

on the geometry and thermomechanical properties of the

individual subcells (abg) within the cells comprising the

functionally graded composite, the displacement coefficient

vector U contains the unknown coefficients that describe the

displacement field in each subcell, i.e.

U

U

111111 ; ; U

222NpNqNr

where

Uabg

pqr Wi000; Wi100; Wi010; Wi001;

Wi200; Wi020; Wi002abgpqr i 1; 2; 3and the mechanical force vector f contains information on

the boundary conditions and the thermal loading effects

generated by the applied temperature. In addition, the

inelastic force vector g appearing on the right-hand side of

Eq. (82) contains the inelastic effects given in terms of the

integrals of the inelastic strain distributions that are repre-

sented by the coefficients Rabgijl;m;n

: These integrals depend

implicitly on the elements of the displacement coefficient

vector U, requiring an incremental solution of Eq. (82) at

each point along the loading path.

3.4. Higher-order theory vs finite-element analysis

Despite some similarities between the finite-element

method and the outlined coupled higher-order theory, the

higher-order theory contains many unique features not

found in the traditional displacement-based finite-element

analyses. The common features include discretization of the

microstructure into subvolumes wherein the displacement

field is approximated by certain functions (in this casesecond-order expansions in the local subcell coordinates).

However, the present approach is based on the volumetric

averaging of the field equations (heat conduction and equi-

librium) in the individual subcells which, in turn, involves

the satisfaction of the zeroth, first, and second moments of

these equations in a volumetric sense. In contrast, the tradi-

tional finite-element analysis involves the minimization of

either an energy expression or a weak form of the field

equations taken over a subvolume (but not their higher

moments). The satisfaction of the higher moments of the

field equations in the present approach naturally gives rise to

the higher moments of the heat flux and stress quantities

given in Eqs. (10) and (49). The higher-order quantitiesappear in the higher-order moments of the field equations

that must be satisfied. The volume-averaged zeroth and

first-order quantities are similar to the force and moment

resultants introduced in the standard plate theories except

here we average throughout each subcells volume rather

than through the entire plates thickness. Such higher-order

quantities are not employed in the traditional finite-

element analysis based on the second-order displacement

expansion.

Additional differences between the present approach and

the traditional finite-element analysis involve the manner in



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which the continuity of displacements and tractions are

satisfied between the various subvolumes. In the present

formulation, both displacement and traction continuity

between the individual subcells is satisfied in an integral

(average) sense, in contrast with the traditional finite-

element analysis based on a second-order displacement

expansion where only the displacement continuity is speci-

fied between adjacent elements. Moreover, in the present

approach, exact displacement field continuity between

adjacent domains is not required, whereas finite-element

analysis requires satisfaction of exact displacement con-

tinuity at the nodes (even in the case of non-conformable

elements).

Finally, in the present theory, the governing field

equations and the displacement and traction continuity

conditions are always satisfied in a volumetric sense

irrespective of the subvolume discretization. Naturally,

more detailed (accurate) results are obtained upon further

discretization, but further refinement is not necessary to

ensure that the traction continuity is satisfied in an integralsense. This is in sharp contrast with the traditional dis-

placement-based finite-element analysis where refinement

typically leads to better satisfaction of the traction continu-

ity in a point-wise fashion, and thus to more accurate point-

wise results. The present theory makes possible consistent

analysis of different microstructures at different levels of

volume discretization, and the fully analytical nature of

the approach allows us to do this efficiently in a volumetric

sense, although not as accurately as the point-wise finite-

element analysis.

3.5. Solution of the governing equations

The choice of an appropriate technique for the solution of

Eq. (82) depends on the inelastic constitutive model

employed to calculate the inelastic strain distributions in

each subcell from which the coefficients Rabgijl;m;n can be

generated. For instance, if the classical incremental plasti-

city (PrandtlReuss) equations are employed to model the

inelastic response of the matrix phase, then Mendelsons

iterative method of successive elastic solutions is an appro-

priate technique for the determination of the plastic strains

needed in the solution of Eq. (82) at each increment of the

applied load [46]. This method has been employed by

Pindera et al. [47,48], in investigating the thermoplasticresponse of unidirectional metal matrix composites

subjected to axisymmetric loading for those situations

where rate effects can be neglected. An advantage of this

solution technique is its efficiency and relative quick

convergence even for relatively large load increments

[49]. If, on the contrary, a unified viscoplastic constitutive

theory is employed to model the inelastic response of the

matrix phase, then either an implicit or an explicit technique

can be employed to integrate the viscoplastic rate equations

at each increment of the applied load. The integration of

viscoplastic constitutive equations, however, may require

a substantial computational effort due to the potentially

stiff behavior of this class of equations.

In the examples presented in Section 7, three constitutive

models are employed to describe the inelastic response of

the matrix phase, namely the classical incremental plasticity

theory, the BodnerPartom unified viscoplasticity theory,

and the power-law creep model, briefly outlined in

Appendix A. The plasticity theory is employed to efficiently

model the inelastic constitutive response of the matrix phase

when rate effects can be neglected, whereas the more

computationally intensive BodnerPartom theory and the

power-law creep model are employed for those situations

where rate-dependent deformation must be taken into

account. It should be noted that the present formulation is

sufficiently general to accommodate other types of unified

viscoplastic theories.

4. Specialization of the higher-order theory

The generalized (i.e. three-directional) higher-order

theory can easily be specialized for materials functionally

graded in a given plane or along a specific direction. In the

former case we obtain the two-directional version of the

higher-order theory referred to in previous communications

as HOTFGM-2D, and in the latter case we obtain the one-

directional version referred to as HOTFGM-1D. Herein, we

briefly show how the generalized higher-order theory

presented in the preceding section can be specialized to

the above two cases.

4.1. Two-directional higher-order theory

To maintain consistency with our previous communica-

tions, we consider a composite functionally graded in the

x2x3 plane (see Fig. 7). Such a composite has a finite thick-

ness H and finite length L in the functionally graded direc-

tions and extends to infinity in the periodic x1 direction. In

the x2x3 plane, the composite is reinforced by an arbitrary

distribution of infinitely long fibers oriented along the x1axis or finite-length inclusions that have periodic spacing

in the direction of the x1 axis. Both aligned and arbitrary

fiber or inclusion architectures in the x2x3 plane, Fig. 7(a)

and (b), respectively, are admissible. The microstructure of

the heterogeneous composite is discretized into Nq and Nr

cells in the intervals 0 x2 H; 0 x3 L in the x2x3plane. The indices q and r of the generic cell p; q; r thatidentify its location in the x2x3 plane have the same ranges

as in the generalized higher-order theory, while the range of

the index p is infinite due to the microstructures periodicity

in the x1 direction. This index can therefore be either

suppressed or retained with the understanding that its

value is indeterminate. The dimensions of the generic cell

along the x1 axes, d1, d2, are fixed for a given configuration

whereas the dimensions along the x2 and x3 axes, hq1 ; h

q2 ;

and lr1 ; lr2 ; can vary in an arbitrary fashion as previously

discussed.



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The temperature distribution T(abg)

in the subcell (ab g)of the (p,q,r)th cell is approximated as follows:

Tabg Tabg000 xb2 Tabg010 xg3 Tabg001

1

23 xa21

d2a

4

2 3T

abg200

1

23 x

b22

hq2b

4

2 3T

abg020

1

23 x

g23

lr2g

4

2 3T

abg002 83

In contrast with the generalized theory, the above represen-tation does not contain a linear term in the local coordinates

xa1 ; i.e.

Tabg100 0 84

(see Eq. (9)). This follows directly from the assumed peri-

odicity in the x1 direction and symmetry with respect to the

lines xa1 0 for a 1 and 2. In the two-directional theory,therefore, the temperature field within the subcell (abg) is

described by six unknown coefficients instead of seven.

Thus for a composite with Nq rows and Nr columns,

48NqNr unknown thermal coefficients must be determined


Fig. 7. Composite with non-periodic fiber distributions in the x2x3 plane: (a) aligned inclusion architecture, (b) arbitrary inclusion architecture.


18/56

in order to be able to describe the temperature field in every

subcell (abg) of every generic cell p; q; r:The determination of these coefficients follows the same

methodology as outlined previously. The system of

equations for the unknown coefficients within the interior

subcells can be obtained from the generalized solution by

using Eq. (84) and suppressing those equations which

involve the variations associated with the index p appearing

in the superscripts, i.e. p 1; q; r; p; q; r and p; q; r;p 1; q; r: Thus the satisfaction of the zeroth, first, andsecond moment of the steady-state heat equation, Eq. (7),

results in eight relations among the first-order heat fluxes

Qabgil;m;n in the different subcells (abg) of the (p,q,r)th

generic cell which are the same as Eq. (17). Twenty addi-

tional relations are obtained among the zeroth and first-order

heat fluxes upon satisfaction of the continuity of heat fluxes

at the subcell interfaces and individual cells, whereas the

same number of additional equations is obtained upon

imposing thermal continuity at the subcell interfaces and

individual cells. The boundary subcells are dealt with inthe same manner as discussed previously. The above

relations provide the required 48NqNr equations for the

48NqNr unknown coefficients Tabg000 ; T

abg010 ; T

abg001 ; T

abg200 ;

Tabg020 ; T

abg002 ; and can be expressed in the same form as

Eq. (36).

Next, the displacement field in the subcell (abg) of the



g3 as

follows, taking into account symmetry considerations:

uabg1 Wabg1000 xa1 Wabg1100 85

uabg2 Wabg2000 xb2 Wabg2010 xg3 Wabg2001

1

23 xa21

d2a

4

2 3W

abg2200

1

23 x

b22

hq2b

4

2 3W

abg2020

1

23 x

g23

lr2g

4

2 3W

abg2002 86

uabg3 Wabg3000 xb2 Wabg3010 xg3 Wabg3001

1

23 xa21

d2a

4

2 3W

abg3200

1

23 x

b22

hq2b

4

2 3W

abg3020

1

23 x

g23

lr2g

4

2 3W

abg3002 87

The first equation does not contain the linear terms in the

local coordinates xb2 and x

g3 ; i.e.

Wabg1010 Wabg1001 0 88

This follows from the assumed periodicity in the x1 direction

and symmetry with respect to xa1 0 a 1; 2: Further,the presence of the constant term W

abg

1000 in the firstequation, that represents subcell center x1 displacements,

produces uniform composite strain 111 upon application of

a partial homogenization scheme described in the next

section. This partial homogenization, which couples the

present higher-order theory and an RVE-based theory,

leads to an overall behavior of a composite, functionally

graded in the x2 and x3 directions, that can be described as

a generalized plane strain in the periodic x1 direction. Plane

strain behavior in the periodic direction is obtained by

setting the constant term to zero [27]. The absence of the

higher-order terms in the first equation, i.e.

Wabg

1200 Wabg

1020 Wabg

1002 0 89is a direct consequence of the periodicity in the out-of-plane

direction. The last two equations, on the other hand, do not

contain linear terms in the local coordinate xa1 ; i.e.

Wabg2100 Wabg3100 0 90

in order to ensure symmetry of the deformation field in the

out-of-plane direction. In the two-directional theory, there-

fore, the displacement field within the subcell (abg) is

described by 13 unknown coefficients instead of 21, noting

that the constant term Wabg1000 in the first equation will be

replaced by the applied strain111 through homogenization.Thus for a composite with Nq rows and Nr columns, 104NqNr

unknown coefficients must be determined in order to be able

to describe the displacement field in every subcell (abg) of

every generic cell (p,q,r).

As in the thermal case, the determination of these coef-

ficients follows the same methodology as outlined

previously. The system of equations for the unknown

coefficients within the interior subcells can be obtained

from the generalized solution by using Eqs. (88) (90),

and suppressing those equations which involve the varia-

tions associated with the index p appearing in the super-

scripts. Thus the satisfaction of the zeroth, first, and

second moment of the equilibrium equations, Eq. (37),results in 16 relations among the volume averaged first-

order stresses Sabgijl;m;n in the different subcells (abg) of

the (p,q,r)th generic cell which are the same as Eq. (63).

Forty-four additional relations are obtained among the

zeroth and first-order stresses upon satisfaction of the

continuity of tractions at the subcell interfaces and indi-

vidual cells. The remaining 44 relations necessary to

determine the unknown coefficients in the displacement

field expansion are obtained by imposing a partial homo-

genization procedure in the periodic x1 direction together

with displacement continuity conditions applied on the



19/56

average basis at each subcell and cell interface in the

functionally graded x2 and x3 directions.

The equations resulting from the partial homogenization

procedure are:

d1W1bg1100 d2W2bg1100p;q;r d1 d2 111 91

where 111 is the unknown uniform far-field strain in the

x2 x3 plane that is either specified or determined. In the

latter case, the additional equation for the unknown 111 is

obtained from the imposed s11: These equations replace

Eq. (70) in the generalized higher-order theory solution

for j 1: On the contrary, the remaining equations obtainedfrom the imposition of the continuity of displacements at

each subcell interface of the (p,q,r)th generic cell, as well as

between adjacent cells, in the functionally graded directions

are obtained by specializing Eqs. (71)(75) in the manner

described previously.

Finally, the boundary subcells are also dealt with in the

same manner as discussed previously. The above relations

provide the required 104NqNr equations for the 104NqNrunknown coefficients

Wabg1100; W

abg2000; W

abg2010; W

abg2001; W

abg2200; W

abg2020; W

abg2002

and

Wabg3000; W

abg3010; W

abg3001; W

abg3200; W

abg3020; W

abg3002

and can be expressed in the same form as Eq. (82).

4.2. One-directional higher-order theory

To maintain consistency with our original communica-

tions, we consider a functionally graded composite in the x1direction. Such a composite has a finite depth D in the

functionally graded direction and extends to infinity in

the x2x3 plane, Fig. 8. The composite is reinforced by

periodic arrays of fibers in the direction of the x2 axis or

the x3 axis, or both. In the direction of the x1 axis, the fiber

spacing between adjacent arrays may vary. The reinforcing

fibers can be either continuous or finite-length, Fig. 8(a) or

(b), respectively.The microstructure of the heterogeneous composite is

discretized into Np cells in the interval 0 x1 D: The

index p of the generic cell (p,q,r) that identifies its location

along the x1 axis has the same range as in the generalized

higher-order theory, while the range of the indices q and ris

infinite due to the microstructures periodicity in the x2 and

x3 directions. These indices can therefore be either

suppressed or retained with the understanding that their

values are indeterminate. The dimensions of the generic

cell along the x2 and x3 axes, h1, h2, and l1, l2, are fixed for

a given configuration whereas the dimensions along the x1

axis, dp1 ; d

p2 ; can vary in an arbitrary fashion as previously

discussed.

The temperature distribution T(abg) in the subcell

(abg) of the (p,q,r)th generic cell is approximated as

follows:

Tabg

Tabg

000 xa

1 Tabg

100 1

2 3 xa

2

1

dp2a

42 3Tabg200

1

23 x

b22

h2b

4

2 3T

abg020

1

23 x

g23

l2g

4

2 3T

abg002 (92)

The above representation does not contain linear terms in

the local coordinates xb2 and x

g3 ; i.e.

Tabg

010

T

abg

001

0

93

because of the assumed periodicity in these directions. In

the one-directional theory, therefore, the temperature field

within the subcell (abg) is described by five unknown

coefficients instead of seven. Thus for a composite with

Np planar slices in the functionally graded direction

containing inclusions with periodic spacing in the x2

x3 plane, 40Np unknown thermal coefficients must be

determined in order to be able to describe the tempera-

ture field in every subcell (abg) of every generic cell

(p,q,r).

The system of equations for the unknown coefficients

within the interior subcells can be obtained from thegeneralized solution by using Eq. (93) and suppressing

those equations which involve the variations associated

with the indices q and r appearing in the superscripts,

i.e. (p,q 1,r), (p,q,r) and (p,q,r), (p,q 1,r) for the q

variation, and p; q; r 1; (p,q,r) a nd (p,q,r), p; q; r1 for the r variation. Thus the satisfaction of the zeroth,first, and second moment of the steady-state heat equation,

Eq. (7), results in eight relations among the first-order

heat fluxes Qabgil;m;n in the different subcells (abg) of the

(p,q,r)th generic cell which are the same as Eq. (17).

Sixteen additional relations are obtained among the zeroth

and first-order heat fluxes upon satisfaction of the conti-

nuity of heat fluxes at the subcell interfaces and individualcells, whereas the same number of additional equations is

obtained upon imposing thermal continuity at the subcell

interfaces and individual cells. The boundary subcells are

dealt with in the same manner as discussed previously.

The above relations provide the required 40Np equations

for the 40Np unknown coefficients Tabg000 ; T

abg100 ; T

abg200 ;

Tabg020 ; T

abg002 ; and can be expressed in the same form as

Eq. (36).

Next, the displacement field in the subcell (a bg) of the



g3 as



20/56

follows, taking into account symmetry considerations:

uabg1

Wabg1000

xa1

Wabg1100

1

23 xa2

1

dp2a

42 3Wabg1200

1

23 x

b21

h2b

4

2 3W

abg1020

1

23 x

g23

l2g

4

2 3W

abg1002 (94)

uabg2 Wabg2000 xb2 Wabg2010 95

uabg3 Wabg3000 xg3 Wabg3001 96

The first equation does not contain linear terms in the local

coordinates xb2 and x

g3 ; i.e.

Wabg

1010 Wabg

1001 0 97in order to ensure symmetry of the deformation field in the

periodic directions. The second equation does not contain

linear terms in the local coordinates xa1 and xg3 ; i.e.

Wabg2100 Wabg2001 0 98

while the third equation does not contain linear terms in the

local coordinates xa1 and xb2 ; i.e.

Wabg3100 Wabg3010 0 99

This follows from the assumed periodicity in the x2 and x3


Fig. 8. Composite with non-periodic fiber distributions in the x1 direction: (a) unidirectionally reinforced material, (b) particulate inclusion reinforced material.

RCS denotes the representative cross section.


21/56

directions and symmetry with respect to xb2 0 b 1; 2

and xg3 0 g 1; 2: The constant terms Wabg2000 and

Wabg3000 in these equations, that represent subcell center

displacements in the x2 and x3 directions, respectively,

produce uniform composite strains 122 and 133 upon appli-

cation of a partial homogenization scheme. This partial

homogenization leads to an overall behavior of a composite

that can be described as a generalized plane strain in the

periodic directions. Plane strain behavior in the periodic

directions is obtained by setting the constant terms to zero

[21]. The absence of the higher-order terms in the last two

equations, i.e.

Wabg2200 Wabg2020 Wabg2002 0 100

and

Wabg3200 Wabg3020 Wabg3002 0 101

is a direct consequence of the periodicity in the x2 and x3

directions. In the one-directional theory, therefore, thedisplacement field within the subcell (abg) is described

by seven unknown coefficients instead of 21, noting that

the constant terms Wabg2000 and W

abg3000 in the last two equa-

tions will be replaced by the applied strains 122 and 122through homogenization. Thus for a composite with Npplanar slices in the functionally graded direction, 56Npunknown coefficients must be determined in order to be

able to describe the displacement field in every subcell

(abg) of every generic cell (p,q,r).

The system of equations for the unknown coefficients

within the interior subcells is obtained from the generalized

solution by using Eqs. (97)(101) and suppressing those

equations which involve the variations associated with the

indices q and rappearing in the superscripts. Thus the satis-

faction of the zeroth, first, a

higher order thry

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