Rami Haj-Ali

Georgia Institute of Technology, Atlanta, Georgia 

Jacob Aboudi

Tel-Aviv University, Ramat-Aviv, Israel 

A New and General Formulation of the 

Parametric HFGMC Micromechanical Method 

for Three-Dimensional Multi-Phase Composites

NASA/CR—2012-217715

October 2012
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Rami Haj-Ali

Georgia Institute of Technology, Atlanta, Georgia 

Jacob Aboudi

Tel-Aviv University, Ramat-Aviv, Israel 

A New and General Formulation of the 

Parametric HFGMC Micromechanical Method 

for Three-Dimensional Multi-Phase Composites

NASA/CR—2012-217715

October 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared under Contract NNC07BA13B
Acknowledgments

The support from the Kahanoff Foundation and the European-Union Marie-Curie IRG grant are greatly acknowledged.

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Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by expert reviewer(s).
NASA/CR—2012-217715 1

A New and General Formulation of the Parametric HFGMC

Micromechanical Method for Three-Dimensional 

Multi-Phase Composites 

Rami Haj-Ali

Georgia Institute of Technology

Atlanta, Georgia 30332

Jacob Aboudi

Tel-Aviv University 

Ramat-Aviv 69978, Israel 

Abstract

The recent two-dimensional (2–D) parametric formulation of the high fidelity generalized method of 

cells (HFGMC) reported by the authors is generalized for the micromechanical analysis of 

three-dimensional (3–D) multiphase composites with periodic microstructure. Arbitrary hexahedral subcell 

geometry is developed to discretize a triply periodic repeating unit-cell (RUC). Linear 

parametric-geometric mapping is employed to transform the arbitrary hexahedral subcell shapes from the 

physical space to an auxiliary orthogonal shape, where a complete quadratic displacement expansion is 

performed. Previously in the 2–D case, additional three equations are needed in the form of average 

moments of equilibrium as a result of the inclusion of the bilinear terms. However, the present 3–D

parametric HFGMC formulation eliminates the need for such additional equations. This is achieved by 

expressing the coefficients of the full quadratic polynomial expansion of the subcell in terms of the side or 

face average-displacement vectors. The 2–D parametric and orthogonal HFGMC are special cases of the 

present 3–D formulation. The continuity of displacements and tractions, as well as the equilibrium 

equations, are imposed in the average (integral) sense as in the original HFGMC formulation. Each of the 

six sides (faces) of a subcell has an independent average displacement micro-variable vector which forms 

an energy-conjugate pair with the transformed average-traction vector. This allows generating symmetric 

stiffness matrices along with internal resisting vectors for the subcells which enhances the computational 

efficiency. The established new parametric 3–D HFGMC equations are formulated and solution 

implementations are addressed. Several applications for triply periodic 3–D composites are presented to 

demonstrate the general capability and varsity of the present parametric HFGMC method for refined 

micromechanical analysis by generating the spatial distributions of local stress fields. These applications 

include triply periodic composites with inclusions in the form of a cavity, spherical inclusion, ellipsoidal 

inclusion, discontinuous aligned short fiber. A 3–D repeating unit-cell for foam material composite is 

simulated.

1.0 Introduction

Early analytical three-dimensional micromechanical models have been concerned with 

homogenization methods able to generate the effective elastic properties of composites. A classical 

approach employs the Eshelby (1957) single inclusion elasticity solution in a homogenization scheme, e.g., 

the self-consistent scheme and its generalizations, Christensen (1979), and the Mori-Tanaka (Mori and 

Tanaka (1973)) methods. The latter approach has been widely used in the literature and extensions from 

single to double or nested inclusions have been reported, e.g., Hori and Nemat-Nasser (1993). Highlights of 

further extensions to compute the effective viscoelastic and inelastic composites has been given by Wang 

and Weng (1992) and Gavazzi and Lagoudas (1990), for example. Additional extensions to composites 

with inelastic matrix and progressive failure including debonding have been proposed by Ju and Tseng 
NASA/CR—2012-217715 2

(1996). The latter models have been generalized by Ju and Sun (2001) to develop a 3–D statistical 

micromechanical damage modeling framework for multiphase brittle composites with combinations of 

interacting microcracks and inclusions. Finally, combined damage 3–D micromechanics have been 

formulated and applied to functionally graded materials by Paulino et al. (2006).

A different 3–D micromechanics approach is to use discrete based subvolumes where expansion of the 

displacement is carried out in order to capture the local fields. To this end, the classical displacement-based 

nonlinear FE method has been widely utilized, e.g., Levy and Papazian (1990). Jain and Ghosh (2008) and 

Jain and Ghosh (2009) used an extension from their 2–D to 3–D microstructural RVE model with Voronoi 

cell elements including damage. Multi-level variational formulation has been achieved to construct 3–D

micromechanics with and without damage, e.g., Fish et al. (1999). Finally the previous micromodels and 

other simplified 3–D micromechanics have been used for local-global structural analysis. Towards that end, 

the lower-order GMC has been developed for 3–D composites. Thus, Haj-Ali et al. (2001) developed a 

nested 3–D micromodels for nonlinear multi-scale pultruded composite structures. Muliana and Kim 

(2007) employed a simplified 3–D micromodel using relatively small number of subcells for the 

viscoelastic multi-scale analysis of composite structures. The GMC has been utilized by Pahr and Arnold 

(2002) for the analysis of metal matrix composites with discontinuous fibers, by Bednarcyk and Arnold 

(2003) for the modeling of woven composites, and by Liu et al. (2011) for a multiscale analysis of triaxially 

braided composites. The higher-order regular orthogonal 3–D-HFGMC has been employed by Bednarcyk 

et al. (2008) in their investigation of the foam insulation of the space shuttle.

Refined micromechanics is a class of nonlinear micromechanical models capable not only to predict the 

instantaneous effective stiffness of a periodic heterogeneous medium, but also to accurately predict the 

local spatial distributions of the deformations within the repeating unit-cell (RUC). The latter requirement 

is needed in order to use such modeling for nonlinear material response along with potential localized 

damage, e.g., microcracking, fiber-matrix interface debonding, material defects and imperfections, among 

other modes.

The HFGMC method is a higher-order refined micromechanical theory and has evolved from its 

lower-order predecessors, the method of cells (MOC), Aboudi (1982) and Aboudi (1991), and the 

generalized method of cells (GMC), Paley and Aboudi (1992), respectively. The HFGMC micromechanical 

model is well suited for detailed nonlinear analysis of multiphase composites, Aboudi (2004). Unlike 

general classical numerical methods such as finite element (FE) and finite-volume (FV), the HFGMC is 

tailored and specialized to micromechanics of periodic composites. It is explicitly based on 

micromechanical variables needed to establish the elastic and inelastic concentration tensors of the phases 

in addition to the detailed local fields. The formulation of the HFGMC is performed using average 

equilibrium, traction and displacement continuity equations. This is because it aims to strike a balance 

between simplicity and accuracy without the need for excessive number of discrete sub-volumes (subcells). 

The computational affordability is especially important when HFGMC is integrated within a nonlinear 

analysis of composite structures including localized effects, often called multi-scale or local-global 

modeling.

Several applications and implementations of the HFGMC have been performed to model different 

physical effects in periodic composites, such as coupled electro-magneto-thermo-elastic (Aboudi (2001)), 

viscoelastic-viscoplastic micromechanics Aboudi et al. (2002), Aboudi et al. (2003) and Aboudi (2005), 

bond damage of fiber, Bednarcyk et al. (2004) and Ryvkin and Aboudi (2007). A review of the method and 

application for HFGMC applied to smart materials can be found in Aboudi (2007) and Aboudi and Freed 

(2010). Damage evolution effects have been introduced within HFGMC by Haj-Ali and Aboudi (2009), 

Bednarcyk et al. (2010) and Aboudi (2011). These include cohesive growth between the subcells, cell 

extinction damage (CED), isotropic and anisotropic damage evolutions in unidirectional composites.

The original HFGMC formulations have been performed in conjunction with orthogonal array of 

subcells used to depict the geometry of the phases (regular array). This limitation requires a relatively large 

number of subcells if the objective is to accurately capture refined geometrical features and the spatial 

variations of the local solution fields. Despite this limitation, the previous applications have demonstrated 

the effectiveness of the method to generate the local solution fields by using sufficiently large number of 
NASA/CR—2012-217715 3

rectangular subcells. It should be emphasized that few number of rectangular subcells is sufficient to 

generate the effective linear and nonlinear response for the overall composite with high accuracy. In fact, 

this is one of the advantages of using the HFGMC in a local-global analysis of composite structures with 

selective geometrical refinement for the microstructure, (Haj-Ali and Aboudi (2009)).

A natural extension that overcomes the limitations of the regular orthogonal array is to employ a 

parametric mapping. To this end, Haj-Ali and Aboudi (2010) used linear geometrical mapping of the 

subcells to map the geometry of the phases of the composite. This linear and parametric geometric mapping 

can be applied for a unit-cell with general phase geometry using arbitrary quadrilateral cell shapes that are 

transformed to an auxiliary uniform square shape (natural coordinates). It is important to note that while the 

geometric mapping is linear, the subcell displacement expansion is of a full quadratic form (subparametric). 

In addition, the complete form of the Jacobian transformation has been employed without approximation 

which necessitates a numerical integration of the HFGMC equations. The utilization of a full quadratic 

expansion vector with its bilinear terms in a subcell required three additional equations which were chosen 

as the average moment of the equilibrium equations. It was emphasized by authors that this choice is not 

optimal and further research is needed to reach a proper form of the needed relations.

In the present investigation, a generalization of the parametric HFGMC to 3–D multiphase composites 

with periodic microstructure is offered. To this end, general hexahedral subcells are introduced to model 

triply-periodic composites. Here too, the full quadratic displacement expansion is still used within the 

parametric HFGMC. However, we now show that the average displacement vectors at the edges (or faces) 

of the subcell can be expressed in terms of the coefficients of the expansion to form an independent subset 

of side-based variables. Furthermore, it is shown that corresponding transformed average-traction at the 

edges are conjugate pairs to the average displacements. In this way, two outcomes can be achieved. In the 

first, relations for the bilinear terms can be established with the quadratic coefficients of the displacements 

and thus no need for additional moment of equilibrium relations. In the second outcome, the energy 

conjugate pair result in symmetric linearized form of the overall HFGMC equations with obvious 

computational advantages. The present new 3–D parametric HFGMC generalization allows the analysis of 

composites with general shape of inclusions. The 2–D parametric HFGMC can be obtained as a special case 

of the present formulation.

This paper is organized as follows. In Section 2.0, the general theory to 3–D parametric HFGMC is 

presented including the 2–D-HFGMC as a special case. In Section 3.0, computational implementation 

aspects are described and discussed. Wide range of applications are given in Section 4.0, followed by 

concluding remarks in Section 5.0.

2.0 General Formulation

The HFGMC micromechanical method is presently extended for general parametric formulation 

suitable for three-dimensional (3–D) analysis of multiphase composite materials. Figure 1 schematically 

illustrates a triply-periodic multiphase material system having a global coordinates (x1, x2, x3). The 

repeating unit cell (RUC) of this periodic composite can be identified and described by using the coordinate 

system (y1, y2, y3). The goals of the HFGMC micromechanical method, based on the homogenization 

technique for periodic composites, are to predict the overall effective properties as well as the spatial local 

deformation fields. In the HFGMC framework, the RUC is divided into array of cells, often denoted as 

subcells. In the present section, this method is extended to analyze an RUC of a triply-periodic composite 

with irregular array of generally shaped hexahedral subcells. Figure 1 shows a schematic RUC domain 

which is discretized into a general assembly of hexahedral subcells to represent the different phase 

geometries. Figure 2 shows a general hexahedral subcell isolated and shown in its physical coordinates 

(y1, y2, y3). This subcell is mapped to a uniform parametric coordinate system (r, s, t) using the classical 

linear transformation 
NASA/CR—2012-217715 4

( , , ) ( , , ) , 1,2,3

8

1

=∑ =

=

y r s t H r s t y i k ki

k

i (1)

where yi, i = 1, 2, 3, are the coordinates of a general point within the subcell (β) mapped from the parent 

parametric coordinates to the physical RUC coordinates, and 

(1 )(1 )(1 ) 8

1 (1 )(1 )(1 ), 81

(1 )(1 )(1 ) 8

1 (1 )(1 )(1 ), 81

(1 )(1 )(1 ) 8

1 (1 )(1 )(1 ), 81

(1 )(1 )(1 ) 8

1 (1 )(1 )(1 ), 81

4 8

3 7

2 6

1 5

H r s t H r s t

H r s t H r s t

H r s t H r s t

H r s t H r s t

= − + − = − + +

= + + − = + + +

= + − − = + − +

= − − − = − − +

(2)

and yki are the corner coordinates of the subcell.

Figure 1.—Schematic illustration of a triply periodic array in 

the global (x1, x2, x3) space of multiphase composite media 

with its repeating unit-cell (RUC) having hexahedral 

subcells, defined with respect to its (y1, y2, y3) local 

coordinate system.
NASA/CR—2012-217715 5

Figure 2.—A hexahedral shaped subcell in its physical and 

natural coordinates. Numbering for the faces and corners are 

shown.

In the present 3–D HFGMC formulation, the complete quadratic form of the displacement expansion in 

the subcell, is given by 

( ) ( ) (3 1) 2

1 3 1

2

1 3 1

2

1 2

002

2

020

2

200

000 100 010 001 110 101 011

0

+ − + − + −

= ⋅ + + + + + + +

r s t

r s t rs rt st

W W W

u ε x W W W W W W W

(3)

where the applied remote global-scale displacement field is given by u0 ≡ ε

0 ⋅ x. This expansion is a direct 

extension of the full quadratic expansion which has been used by Haj-Ali and Aboudi (2010) in the case of 

the parametric HFGMC for doubly-periodic composites. However, in the present parametric formulation, 

the mixed bilinear terms of this expansion are shown to be dependent variables. This resolves the open 

question that has been raised by Haj-Ali and Aboudi (2010) on the optimal method that should be used to 

determine these terms. Previously, the bilinear terms have treated as independent variables and 

consequently additional moment of equilibrium equations have been introduce to determine them. To this 

end, let us define the average displacement vector on the six faces of the hexahedral subcell: 

( , ) , 1,2,..,6

4

1 ( ) 1 ( ) 1

1

1

1

( ) ( ) = = ξ η ξ η = β

− −

β β

∫ ∫ ∫ dA d d k

A k k k k k A k k

k u u y u (4)
NASA/CR—2012-217715 6

where (ξk, ηk) are the surface parametric integration variables of the k-th edge or face of the subcell denoted 

by (βk). The above integration provides the six vectors of the surface-average displacements expressed in 

terms of the expansion coefficients W(lmn) as follows: 

























































































−

−

−

=













































≡













































−













































β

β

β

β

β

β

β

β

β

β

β

β

000

002

020

200

001

010

100

0

6

5

4

3

2

1

0,( )

0,( )

0,( )

0,( )

0,( )

0,( )

000

( )

( )

( )

( )

( )

( )

0 0 0 0 0 0 1

0 0 1 0 0 1 1

0 0 1 0 0 1 1

1 0 0 1 0 0 1

0 1 0 0 1 0 1

1 0 0 1 0 0 1

0 1 0 0 1 0 1

6

5

4

3

2

1

6

5

4

3

2

1

W

W

W

W

W

W

W

W

W

W

W

W

W

W

0

u

u

u

u

u

u

W

u

u

u

u

u

u

(5)

The inverse relation of Equation (5) is given by: 

























































































−

−

−

−

−

−

=













































0

6

5

4

3

2

1

000

002

020

200

001

010

100

0 0 0 0 0 0 2

0 0 0 0 1 1 2

1 0 1 0 0 0 2

0 1 0 1 0 0 2

0 0 0 0 1 1 0

1 0 1 0 0 0 0

0 1 0 1 0 0 0

2

1

W

W

W

W

W

W

W

W

W

W

W

W

W

W

(6)

It can be readily observed that the bilinear coefficients W110, W101, W011, do not contribute to the average 

displacements. Thus it is possible to assume that these are dependent coefficients if our aim is to use the 

average displacements as the primary independent variables. Hence, two possible solutions can be 

identified. The first trivial case is to assume that the bilinear terms to be zero: 

W110 = W011 = W101 = 0 (7)

This solution generates a truncated quadratic expansion that may affect the spatial distribution of the elastic 

fields. However, the average strain in the subcell, directly responsible for the overall elastic effective 

stiffness of the RUC, is not affected due to the vanishing integration of the bilinear terms in the hexahedral 

domain.

Alternatively, a non-trivial solution can be obtained while retaining the complete quadratic expansion 

(3), maintaining terms symmetry and frame indifference of this polynomial form. Here 

( ) ( )

( ) ( )

( ) ( ) 101 200 002 1 2 4 6 0

011 020 002 1 2 3 5 0

110 200 020 3 4 5 6 0

4

2

1

2

1

4

2

1

2

1

4

2

1

2

1

W W W W W W W W

W W W W W W W W

W W W W W W W W

= + = + + + −

= + = + + + −

= + = + + + −

(8)

It can be verified that the average displacement relations given by Equation (5) is maintained with the trivial 

and the non-trivial forms given for the bilinear terms, Equations (7) and (8), respectively. It should be noted 

that the trivial case is a direct result of the original HFGMC formulation, e.g., Aboudi (2004), for 
NASA/CR—2012-217715 7

orthogonal and regular array of subcells where both physical and natural coordinates coincide and the 

Jacobian of the mapping is constant.

As a result of the above derivations it is possible to rewrite the displacement expansion (3) in the 

following compact form: 

( ) ( ) ( )

( )( )

( )( )

( 2 )(3 1) 2

1

2 3 1

2

1

2 3 1

2

1

2

1

2

1

2

1

2

1 2 0

2

5 3 0

2

4 6 0

0 4 6 5 3 2 1

0

+ + − + + −

+ + − + + −

+ + − + + −

= ⋅ + + − + − + −

t rt st

s rs st

r rs rt

r s t

W W W

W W W

W W W

u ε x W W W W W W W

(9)

It is worth mentioning that in the special case of 2–D parametric HFGMC for doubly-periodic composites, 

the above expansion takes the form: 

( ) ( )

( )( )

( 2 )(3 1) 2

1

2 3 1

2

1

2

1

2

1

2

1 3 0

2

2 4 0

0 2 4 3 1

0

+ + − + −

+ + − + −

= ⋅ + + − + −

s rs

r rs

r s

W W W

W W W

u ε x W W W W W

(10)

where Wk follows the labeling of the quadrilateral subcell of Haj-Ali and Aboudi (2010). In this case, the 

bilinear term coefficients W11 takes the following trivial and non-trivial forms, respectively: 

( ) 11 11 20 02 2

1 W = 0, W = W + W (11)

As mentioned above, this answers the open question raised by the authors on the optimal way of handling 

the bilinear terms in the 2–D case.

In order to establish the expressions for the strain components, the displacement gradients with respect 

to the physical coordinates y are needed. The standard Jacobian of the linear transformation (1) is given by: 

( )

( ) 























= ∂

∂ ≡

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

t

y

t

y

t

y

s

y

s

y

s

y

r

y

r

y

r

y

r s t

y y y J

1 2 3

1 2 3

1 2 3

, ,

, , 1 2 3 (12)

Its inverse is denoted by ( ) ( ) 1 2 3

1 , , / , , ˆ

J ≡ J = ∂ r s t ∂ y y y − and used to relate the displacement derivatives 

as 

[ ]

( )

,

,

, ( )

( )

,3

, 2

,1

ˆ

β

β

β





















=





















t

s

r

u

u

u

J

u

u

u

(13)

The right-hand-side can be easily established by using Equation (3) yielding 
NASA/CR—2012-217715 8

r s t

r t s

s t r

t

s

r

001 101 011 002

( )

,

010 110 011 020

( )

,

100 110 101 200

( )

,

3

3

3

u W W W W

u W W W W

u W W W W

= + + +

= + + +

= + + +

β

β

β

(14)

However, in order to express these gradients in terms of the average displacement micro-variables of the six 

hexahedral faces, we first define the following subcell vector with its components lumped in the following 

order 

{ , , , , , , } 1,2,3 ( )

1 2 3 4 5 6 0

( ), = = β β W W W W W W W i i i i i i i i W T (15)

Next, Equations (6) and (8) are used for the non-trivial case to obtain, after some manipulations, the 

displacement gradients with respect to the parametric variables as follows 

[ ]

( )

0

6

5

4

3

2

1

( )

( )

,

,

,

β

β

β













































=





















W

W

W

W

W

W

W

M

u

u

u

t

s

r

(16)

where the matrix [M](β) is given by 





















+ + − + + + + − − +

− + + + + + + − − +

+ + + − + + + − − +

1 3 1 3 6 4( )

1 3 1 3 6 4( )

1 3 1 3 6 4( )

2

1

r s r s r s r t s t t r s

s t t r s r t r r t s r t

t r t s t r s s t s r s t

(17)

Similarly, Equations (6) and (7) provide the gradients for the trivial case as 

( )

0

6

5

4

3

2

1

( )

,

,

,

0 0 0 0 1 3 1 3 6

1 3 0 1 3 0 0 0 6 )

0 1 3 0 1 3 0 0 6

2

1

β

β

































































− + + −

− + + −

+ − + −

=





















W

W

W

W

W

W

W

u

u

u

t t t

s s s

r r r

t

s

r

(18)

The spatial form of the strains in a given cell (β) are defined by 

















∂

∂

+

∂

∂

ε = ε + β

i

j

j

i

ij ij y

u

y

u

r s

2

1 ( ( , )) ( ) 0 y (19)

It is possible to bring the strains into a general matrix form 
NASA/CR—2012-217715 9

(β) 0 (β) (β) ε = ε + A W (20)

where the strain vector notation ε is defined by 

= {ε11,ε22 ,ε33,2ε23,2ε13,2ε12} Tε (21)

and A(β)is the matrix that relates the strain to the displacement microvariables. The size of the vector W(β)is 

21 which includes all the microvariables of subcell (β) as shown in Equation (15). Using Equations (17) and 

(18), respectively, along with Equations (13) and (18), it is possible to derive the strain-displacement 

matrix, A(β)

. The non-zero components of this matrix are listed in the Appendix for the trivial and 

non-trivial bilinear terms.

Next, we define average traction vector ( ) T βk on the six sides of the hexahedral subcell defined by 

( , , ) , 1,2,..,6

4

1 ( ) 1 ( ) ( ) 1

1

1

1

( ) ( ) ( ) = ⋅ = ⋅ ξ η = β β

− −

β β β

∫ ∫ ∫ dS r s t d d k

A k k k S k

k k

k

k T σ y n σ n (22)

where ( ) βk n is the unit normal vector to the k-th side of subcell (β). The stress field, σ(β), is given by 

( ) ( ) ( ) : β β β σ = C ε (23)

assuming a linear material behavior of the subcell, with C(β)being the stiffness of the material in the subcell. 

Since the geometry is interpolated using linear mapping, the normal vector ( ) βk n to each of the six faces is 

constant. Employing the established spatial strains in the cell to obtain the corresponding stresses. The latter 

are used in the expression for the average tractions, Equation (22). This provides 

[ ] [ ] ( ) ( ) 0 ( ) ( ) ( ) ( ) 0 ( ) ( ) 1

1

1

1

( )

4

1 β β β β β β β β

− −

β = + ξ η = + T k

∫ ∫ N k C ε A W d kd k N k C ε A k W (24)

with, 

( )

3 2 1

2 3 1

1 3 2

( )

0 0 0

0 0 0

0 0 0 k

k

n n n

n n n

n n n β

β





















N = (25)

and 

k k d d k = ξ η β

− −

β

∫ ∫ ( ) 1

1

1

1

( )

4

1 A A (26)

As mentioned above, due to the applied linear mapping, the matrix is constant.

Following the original HFGMC formulation, we impose the displacements and tractions continuity 

between the subcells. These conditions are enforced in an average integral sense. In addition, periodicity 

conditions are imposed between the boundary subcells of the RUC by requiring that the displacements and 

tractions be continuous at these mirrored interfaces. The third major requirement of the HFGMC method is 

the intra subcell equilibrium applied in a volumetric average form. In the following, the above transformed 

expressions for the average displacements and tractions, (4) and (24), respectively can readily be used to 

impose the HFGMC equations. The displacements and tractions continuity can be written as 
NASA/CR—2012-217715 10

( ) ( ) ( ) ( ) ( ) ( ) , k m k m

k m

β β γ γ β γ u = W = u = W T = T (27)

where (βk) denotes the k-th interface (side) of subcell (β) and (γm) is neighboring m-th interface side of 

subcell (γ). The displacement and traction periodicity conditions are imposed as in Equation (25), but with 

cell (β) and (γ) located on opposite sides of the RUC.

The equilibrium equations for each subcell are imposed in an average sense in conjunction with 

Divergence theorem in order to utilize the derived expressions for the average tractions as follows. 

0 ( )

6

1

( ) ( )

6

1

∇ ⋅ = ⋅ = = = β

=

β β

=

∫ ∫ ∑∫ ∑ k k

k

k

k

k S

k

V S

σdV σ ndS N σ dS A T (28)

where V and S are the volume and surface of the (β)-subcell, respectively, and Ak is the area of the k-th side.

The total number of unknown microvariables in the RUC is 21Nc where Nc is the total number of 

subcells every one of which has 21 microvariables, see Equation (5). The number of continuity and 

periodicity equations for displacements is 3×3Nc (only three interfaces for each subcell provide 

independent relations), see the first equality in Equation (27). Similarly, the number of traction continuity 

and periodicity equations is 3×3Nc as shown by the second equality in Equation (27). The average 

equilibrium relations, Equation (28), provide additional 3Nc equations.

The above formulation completes the full theoretical derivation of the general three-dimensional 

parametric HFGMC micromechanical method. However, it is interesting to introduce an internal force 

vector I

(β)

that form energy conjugate pairs with the average displacements of each side of the subcell, W(β). 

The internal force vectors are directly related to the average traction defined above. These force resisting 

vectors are defined using external and internal work balance: 

( ) dV dV T T

V

T T

V

(β),T (β) (β), 0, (β) (β), (β), (β)

∫ ∫ W I ≡ ε − ε σ = W A σ (29)

and since W is arbitrary, the expression for the internal resisting vector is 

dV T

V

(β) (β), (β)

∫ I = A σ (30)

The above work balance is known in the mechanics of heterogeneous media as the principal of average 

virtual work, e.g., Christensen (1979). For the linear material case, 

[ ] [ ] (β) (β), 0,(β) (β), (β) (β)

∫ ∫ I = A Cdv + A TCA dv W

V

T

V

ε (31)

We further recognize that the first six vectors within I

(β)

are the average forces, ( ) βk I , on each face or side 

of the hexahedral subcell. The use of the above form of the average internal resisting vector, ( ) βk I , instead 

of the average traction, ( ) T βk , in the traction continuity relation and average equilibrium, Equations (27), 

(28), respectively, provide enhanced computational advantage mainly expressed by using the symmetric

matrices in Equation (31).

3.0 Implementation and Computational Aspects

The assembly of the derived 21×Nc HFGMC governing equations of the RUC are solved to obtain the 

unknown 21 microvariables for each cell. In general, equations can symbolically be grouped into three parts 

in the form 
NASA/CR—2012-217715 11

{ } 



















=





















∑

+ −





















∑

+

−

+ − + −

+ −

0

0

0

D

D D

0

W

I

I I

1 1

{ } ( ) ( ) ( ) ( ) 0

( ) ( )

ε

T

T T

T

T T

u u

(32)

where the first part (row) represents the average continuity and periodicity of the displacements, the second 

part represents the continuity of the tractions in a similar fashion. The third part represents the equilibrium 

equations for all the cells. The above system of equations is solved for a given externally applied strain ε

0 to 

obtain the cell microvariables. Furthermore, the local spatial strains can be readily obtained using 

Equation (20), which leads to the spatial distribution of the stresses in the RUC. These can be used, for 

example, in a progressive analysis to determine possible damage in the constituents as well as interfacial 

debonding, see Haj-Ali and Aboudi (2010) for the parametric 2–D-HFGMC case.

Since the first part of these equations are homogeneous and each row is composed of only two terms, 

expressing interface average displacement continuity becomes a straight forward manner. A pre-analysis 

overall RUC equivalence of matching subcells' faces can be performed similar to nodal equivalence carried 

out in the FE method. This can be done at the global level of the code or even using simple preprocessing 

subroutines. Furthermore, the equilibrium equation for each subcell, Equation (28), expressed in either 

( ) T βk or ( ) βk I , can be used in conjunction with a static condensation of the internal dependent 

displacement microvariables, W0

(β)

. This leaves the six-side average displacements as the only set of 

independent variables for each subcell, which results in computational saving.

The effective elastic properties of the multiphase composite can be obtained from the cell strain 

concentration tensors G(β)

which relates the average strain in the subcell to the externally applied strain. The 

latter is established by considering the average strain of the cell having the form 

( )dV

V

dV

V V V yu u y = y = + ∇ + ∇ ∫ ∫ β

2

1 ( ) ( ) 1 0 ε ε ε (33)

By using the Divergence theorem and the linear mapping, Equation (30) takes the form 

( ) ( )

( ) ( ) 0

6

1

( ) 0 0 :

2

1

2

1

ε ε ε ε β β

=

β = + ⊗ + ⊗ = + ∑ ⊗ + ⊗ ≡ ∫ u n n u k u n n u k G

k

S

A

V

dS

V (34)

It is observed from Equation (34) that the concentration tensor G(β) depends on the average displacement 

vectors on the surfaces of the cell. Those can be computed only after solving the entire HFGMC system of 

equations for the RUC as discussed above. Once the concentration tensors G(β)

have been obtained, the 

effective elastic stiffness tensor C* is evaluated in the form 

( ) ( ) ( )

1

β β β

β

β=

∗ C =∑v C G

Nc

(35)

where vβ = V(β)/VRUC, and VRUC being the total volume of the RUC.

It is also possible to use the intact overall system of equations, Equation (34), without side-based 

condensation between the displacement microvariable. This approach allows the formulation of a new 

micromechanical damage approach, termed cell-extinction-damage (CED). As has been demonstrated by 

the 2–D-HFGMC, Haj-Ali (2009) and Haj-Ali and Aboudi (2010), this cohesive micromechanical 

approach has the ability to perform subcell degradation along with general traction-separation between the 

subcells. In addition, the structure of exclusive interface-based equations unlike nodal-based FE, makes it 
NASA/CR—2012-217715 12

easier to remove the subcells from the problem and/or generate average-based traction-free surfaces for 

crack propagation within the RUC without introducing computational instabilities.

As mentioned, linear geometric mapping and quadratic displacement interpolation (subparametric 

formulation) has been carried out in the previous section. This formulation has been done in order to 

simplify the numerical integration. Having said that, extension to full quadratic geometric mapping 

(isoparametric formulation) is straightforward. In both of these cases, numerical integration of the 

equations is needed since the Jacobian matrix is not constant. However, a computational implementation of 

the 2–D parametric HFGMC has been reported in the literature under the name “FVDAM”, e.g., Bansal and 

Pindera (2006). This latter 2–D method assumes a volume-based constant Jacobian of the linear mapping 

for the geometry of the subcells. Their assumption imposes severe limitation and restriction on the 

parametric HFGMC method as the shape of the subcells need to be close to rectangular or parallelograms in 

order to maintain accurate depiction of the geometry. The use of an a priori constant Jacobian amounts to a 

one center-point quadrature and may lead to large solution errors especially in the displacement gradients. 

Furthermore, the assumption of a constant Jacobean is not applicable in the general 3–D HFGMC that has 

full quadratic displacements. Thus, the proposed 3–D parametric HFGMC has been formulated in a direct 

and general manner, i.e., retaining the full quadratic expansion of the displacement together with the 

complete Jacobian. The level of computational accuracy is determined by the number of integration points 

specified during the numerical integration of the governing equations.

It is important to draw the distinctions between the proposed parametric HFGMC and the classical 

displacement-based finite-element (FE) method, where a common misconception has been to link the 

HFGMC to FE. In the FE formulation the displacement continuity between two adjacent and connected 

elements is satisfied in a pointwise manner by sharing the same nodal degrees of freedom at the sides of an 

element. However, the HFGMC quadratic displacement expansion is nonconforming and displacement 

continuity is satisfied in an average sense between adjacent cells. The latter is an approximation that allows 

the HFGMC to explicitly use additional stress variables in the formulation and directly apply average 

traction continuity. The advantage of this approach is that both traction and displacement interface 

continuity between different materials can be directly employed. Further, cohesive damage modeling can 

be easily employed as mentioned before for the 2–D case. Unlike FE, the HFGMC is a specialized 

micromechanical modeling framework in which the remote fields are directly tied to the micromechanical 

formulation (e.g., the remote average strain is directly used in the HFGMC formulation). On the other hand, 

the periodicity in the FE formulation is imposed only through displacement boundary conditions, while in 

the HFGMC, the periodic boundary conditions are imposed directly using both the average displacement 

and tractions. The latter differences are illustrated in the next application section where examples for RUCs 

with both traction and displacement periodic conditions are compared to FE RUCs modeled with applied 

far-field nodal displacements. Having said that, it is possible to draw an analogy between the FE and 

HFGMC method by considering the average displacement microvariables as face or side-based generalized 

displacements variables. This analogy allows the use of the average virtual work principal to define the 

internal force vector and its associate symmetric matrices which are useful for the solution of the overall 

system of equations.

4.0 Applications

Several applications are performed to show the effectiveness of the parametric HFGMC method in 

capturing the spatial stress field distributions within the phases of triply periodic composites. To this end, 

RUCs are generated with coarse and refined resolutions using hexahedral subcells. Selected stress fields are 

compared with solutions of RUCs using displacement based FE models. In order to compare the parametric 

HFGMC and the FE solutions, the full periodic conditions between the boundary surfaces of the RUC, i.e., 

both traction and displacement continuity on mirrored boundary surfaces, were not applied. It is important 

to mention that the full periodic conditions are an integral part of the HFGMC formulation, however, full 

periodicity is not a straightforward using the displacement-based FE. Thus, the boundary conditions 

imposed in all compared cases, unless stated otherwise, are: 
NASA/CR—2012-217715 13

u = ⋅ x 0 0 (S) ε (36)

where (S) here is the outside periodic surface of the RUC, ε

0 is the applied average strain and x are the 

coordinates of the surface points. In the case of the FE-RUC, constrains were written for the boundary 

nodes that link their degrees of freedom to the applied average strain. However, for the case of the RUC 

modeled with the HFGMC, the average displacement microvariables on the subcells with boundary faces 

were set to zero. Select cases for HFGMC-RUCs with full periodic boundary conditions are presented in 

order to show the different spatial distribution of the local fields that the can be achieved using the HFGMC.

The first RUC model is for a medium with repeated spherical cavities, thus forming a porous material. 

Figure 3 shows the geometry of this problem in the form of a half RUC cut at the mid-plane and another 

with a quarter portion removed. It should be mentioned that that while it is possible to apply symmetric 

conditions, the full geometry is simulated. The dimensions of the RUC were taken as D = H = L =1 and the 

radius of the spherical cavity was R = 0.4. The material Young's modulus and Poisson's ratio elastic 

properties are E = 4.8 GPa, ν = 0.34. The composite is subjected to a remote uniaxial strain loading of 

0.01 0 ε11 = . The number of hexahedral subcells used to map the geometry was Nc = 3584. This discretization 

of the RUC is not considered as a refined one compared to the number of subcells needed to capture the 

local spherical shape curvature. This was the choice in order to demonstrate the ability of the HFGMC to 

accurately solve for the local fields despite the use of average traction and displacements. Figure 4(a) and 

(b) show the σ11 stress distributions for the HFGMC and the FE models, respectively. In these two cases, 

only the displacement-based periodic conditions were enforced. It is clear that the stress distribution from 

the HFGMC solution is quite identical to the one obtained from the FE model despite the use of relatively 

smaller number of independent solution variables. As mentioned, each subcell has 21Nc total independent 

average displacement microvariables in the case of the HFGMC among of which many dependent 

displacements are condensed due to subcells inter-continuity at the global system of equations as discussed 

in the implementations section. Therefore, the element that was selected for the FE RUC cases is the 

8-nodes linear brick which has 8×3×Nc = 24Nc number of variables having both RUCs close in their 

problem size. Figure 4(c) shows the stress distribution generated by the HFGMC, however, the full periodic 

conditions were applied in this case. It is evident that the local spatial distribution is affected at the 

boundaries of the RUCs if accurate periodicity needs to be captured. In the case where the effective elastic 

properties of the entire medium are sought, applying full or partial periodic conditions may not yield 

pronounced differences. Our aim in the parametric HFGMC is to provide an analysis tool capable of 

accurately predicting the local fields.

The next application deals with a composite with spherical inclusions, thus forming a particulate 

composite. Figure 5 shows the discretized RUC with D = H = L =1 and the radius of the spherical inclusion 

is R = 0.4, which corresponds to inclusion volume fraction of 26.8 percent. An effort has been made to use 

a refined mesh at the interface of the spherical inclusion. The elastic properties of the inclusion and the 

matrix are: E = 380 GPa, ν = 0.1 and E = 4.8 GPa, ν = 0.34, respectively. The far field applied strain is 

0.01 0 ε33 = . The number of hexahedral subcells used for both HFGMC and FE is Nc = 74088. Figure 6(a) 

and (b) show the Mises stress distributions as predicted by the HFGMC and the FE, respectively, in the 

direction of the applied strain. Here, excellent agreement between the two solutions is exhibited. As a 

continuation of this problem, we consider an ellipsoidal inclusion, the mesh of which is shown in Figure 7. 

It illustrates a half RUC with removed inclusion and the ellipsoidal inclusion. A coarser mesh is used for the 

latter in order to minimize the size of the problem: Nc = 54872. The dimensions of the RUC are D = H = L

=1 and the major and two minor axes of the ellipsoid are 0.8, 0.4 and 0.4 (spheroidal shape), respectively. 

The same elastic properties for the inclusion and matrix as well as the magnitude of the applied far field are 

chosen as in the previous problem. Figure 8 displays the resulting stress σ33 distribution in the major-axis 

direction of the ellipsoid for both the HFGMC and FE solutions and for the full RUC with quarter portion 

removed. Again, a very good agreement exists.
NASA/CR—2012-217715 14

Figure 3.—Parametric HFGMC meshes for an RUC with spherical cavity.

Figure 4.—Stress distributions in the RUC with a spherical cavity. (a) HFGMC results (periodic displacements), 

(b) FE results (periodic displacements), and (c) HFGMC results (periodic displacements and tractions).
NASA/CR—2012-217715 15

Figure 5.—Parametric HFGMC 3–D meshes for an RUC with spherical 

inclusion using hexahedral subcells.

Figure 6.—Stress distributions in the RUC with a spherical inclusion. (a) HFGMC 

results, (b) FE results.
NASA/CR—2012-217715 16

Figure 7.—Parametric HFGMC 3–D meshes for an RUC with 

spheroidal inclusion using hexahedral subcells.

Figure 8.—Stress distributions in the RUC with a spheroidal inclusion. (a) HFGMC results, 

(b) FE results.

The next studied problem analyzes a composite with a system of reinforcements in the form of 

discontinuous aligned circular fibers (short-fiber composite). Figure 9 displays the refined mesh of the 

RUC together with the short fiber itself and its cross-section. The ratios of the dimensions of the RUC are 

D = 5 and H = L =1. The axial length of the fiber is 4.53 and its diameter is equal to 0.4, resulting in a fiber 

volume fraction of 11.4 percent. The RUC is subjected to remote axial strain 0.01 0 ε11 = . The values of the 
NASA/CR—2012-217715 17

fiber and matrix elastic properties are the same as those of the inclusion and matrix properties in the 

previous problem. The size of the problem in this case is Nc =17984. The resulting Mises stress distribution 

within the RUC as predicted by the HFGMC and the FE are shown in Figure 10(a) and (b), respectively. It 

can be observed that the HFGMC is capable in capturing the local mechanism of stress-transfer at the 

discontinuous edge of the fiber to the matrix. Both solutions are very close as shown.

The final example considers a periodic foam-like material with a relatively complex geometry of the 

microstructure. Figure 11 shows the selected geometry of the RUC. It is evident that this is a relatively 

complex geometry due to the spatial curvature and shape of the cuts. Accurate modeling of the material in this 

case requires a refined 3–D mesh able to represent the local geometry. The choice of the RUC geometry has 

been inspired from scaffolds used in tissue engineering and orthopedic micro-implants. Its dimensions are D = 

H = L =1 and was constructed by removing material enclosed by three orthogonal cylinders, the diameter of 

each is 0.8. It is worth observing the local geometry where the removed cylinders intersect. The material 

elastic properties are E = 4.8 GPa, ν = 0.34. The total number of subcells is Nc =34560 and the remote axial 

strain is 0.01 0 ε11 = . Figure 12(a) and (b) exhibit the Mises stress distribution in the 3–D HFGMC compared 

with the FE solution, respectively. Excellent agreement exists especially at the inner corners. It is important to 

mention for this case, that an explicit average traction-free conditions are imposed by the HFGMC on the 

inner surfaces. Finally, the Mises stress variations are shown in Figure 13 for the same RUC but when full 

periodicity conditions are applied between the RUCs by the HFGMC model. It is clearly observed that the 

form of the stress distribution is impacted by imposing the full micromechanical boundary conditions. This 

demonstrates the micromechanical capability embedded in the formulation of the HFGMC. By introducing 

inelastic effects, it would be possible to extend the 3–D parametric HFGMC to analyze metallic foams.

Figure 9.—Parametric HFGMC 3–D meshes for an RUC, with discontinuous 

aligned fiber having a circular cross-section, using hexahedral subcells.
NASA/CR—2012-217715 18

Figure 10.—Stress distributions in the RUC with the discontinuous fiber. (a) HFGMC 

results, (b) FE results.
NASA/CR—2012-217715 19

Figure 11.—Parametric HFGMC mesh for an RUC of a foam material.

Figure 12.—Stress distributions in the RUC of the foam material. (a) HFGMC results (periodic displacements), 

(b) FE results (periodic displacements).
NASA/CR—2012-217715 20

Figure 13.—Stress distributions in the RUC of the foam 

material showing HFGMC results with full periodic 

conditions (displacements and tractions).

5.0 Conclusions

A new 3–D parametric micromechanical formulation of the HFGMC method is presented using linear 

geometric mapping with hexahedral subcells. A complete quadratic expansion of the displacement is used 

for the subcells where its unknown coefficients are in the form of average displacements of the six sides of 

the subcells in addition to an internal coefficient that can be statically condensed using the integral-based 

equilibrium equations. The independent average displacement micro-variable vector for each side of the 

subcell forms an energy-conjugate pair with the transformed average-traction vector. Trivial and non-trivial 

relations for the coefficients of the bilinear terms are found and their corresponding strain-displacement 

matrices are derived without the need for additional equations in the form of moments of the equilibrium. 

Thus, the open question previously posed by the authors on the optimal equations for the bilinear 

coefficients is resolved in the current study. The previous 2–D parametric and orthogonal HFGMC 

formulations are special cases of the present 3–D formulation. Solution approaches to the parametric 

HFGMC equations are also discussed and numerical integration is needed due to nonconstant Jacobian of 

the linear mapping. Wide range of applications are presented to verify the 3–D parametric HFGMC as 

compared with finite-element RUC models. The proposed 3–D parametric HFGMC is general and well 

suited for nonlinear material and finite-deformations previously developed in the framework of the 

orthogonal HFGMC. Previously developed cohesive damage modeling can be integrated within the new 

formulation. These modeling features can be integrated within the proposed 3–D parametric 

micromechanics along with local-global multi-scale analysis of composite structures.
NASA/CR—2012-217715 21

Appendix

In this Appendix, the non-zero components of the strain micro-displacements matrices are listed first 

for the quadratic polynomial with zero bilinear terms (trivial case) and second for the constrained 

symmetric bilinear term coefficients (non-trivial case). The terms for first matrix are

A J r J s J t

A J J r A J r J s J t

A J J s A J J r

A J J r A J J s

A J J s A J J r

A J J t A J J s

A J J t A J J t

A J r J s J t A J J t

A J J r A J r J s J t

A J J s A J J r

A J J r A J J s

A J J s A J J r

A J J t A J J s

A J J t A J J t

A J r J s J t A J J t

A J J r A J r J s J t

A J J s A J J r

A J J r A J J s

A J J s A J J r

A J J t A J J s

A J J t A J J t

A J r J s J t A J J t

A J J s A J J r

A J J s A J J r

A J J t A J J t

A J J r A J r J s J t

A J J r A J J s

A J J t A J J s

A J r J s J t A J J t

A J J s A J J r

A J J s A J J r

A J J t A J J t

6, 20 1,1 1, 2 1,3

6,17 1,1 1,1 6,19 2,1 2, 2 2,3

6,14 1, 2 1, 2 6,16 2,1 2,1

6,11 1,1 1,1 6,13 2, 2 2, 2

6,8 1, 2 1, 2 6,10 2,1 2,1

6,5 1,3 1,3 6,7 2, 2 2, 2

6, 2 1,3 1,3 6, 4 2,3 2,3

5, 21 1,1 1, 2 1,3 6,1 2,3 2,3

5,18 1,1 1,1 5,19 3,1 3, 2 3,3

5,15 1, 2 1, 2 5,16 3,1 3,1

5,12 1,1 1,1 5,13 3, 2 3, 2

5,9 1, 2 1, 2 5,10 3,1 3,1

5,6 1,3 1,3 5,7 3, 2 3, 2

5,3 1,3 1,3 5, 4 3,3 3,3

4, 21 2,1 2, 2 2,3 5,1 3,3 3,3

4,18 2,1 2,1 4, 20 3,1 3, 2 3,3

4,15 2, 2 2, 2 4,17 3,1 3,1

4,12 2,1 2,1 4,14 3, 2 3, 2

4,9 2, 2 2, 2 4,11 3,1 3,1

4,6 2,3 2,3 4,8 3, 2 3, 2

4,3 2,3 2,3 4,5 3,3 3,3

3, 21 3,1 3, 2 3,3 4, 2 3,3 3,3

3,15 3, 2 3, 2 3,18 3,1 3,1

3,9 3, 2 3, 2 3,12 3,1 3,1

3,3 3,3 3,3 3,6 3,3 3,3

2,17 2,1 2,1 2, 20 2,1 2, 2 2,3

2,11 2,1 2,1 2,14 2, 2 2, 2

2,5 2,3 2,3 2,8 2, 2 2, 2

1,19 1,1 1, 2 1,3 2, 2 2,3 2,3

1,13 1, 2 1, 2 1,16 1,1 1,1

1,7 1, 2 1, 2 1,10 1,1 1,1

1,1 1,3 1,3 1, 4 1,3 1,3

ˆ 3 ˆ 3 ˆ 3

ˆ 3 ˆ 3 ˆ 3 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3 / 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3 ˆ 3 ˆ 3

ˆ 3 ˆ 3 ˆ 3 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3 / 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3 / 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3 ˆ 3 ˆ 3

ˆ 3 ˆ 3 ˆ 3 ˆ 3/ 2 ˆ 1/ 2

ˆ 3 / 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3 / 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3 / 2 ˆ 1/ 2 ˆ 3 ˆ 3 ˆ 3

ˆ 3/ 2 ˆ 1/ 2 ˆ 3 / 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3 / 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3 ˆ 3 ˆ 3 ˆ 3 / 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3 / 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3 ˆ 3 ˆ 3

ˆ 3/ 2 ˆ 1/ 2 ˆ 3 / 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

ˆ 3/ 2 ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 2

= − − −

= − + = − − −

= + = − +

= + = +

= − + = +

= + = − +

= − + = +

= − − − = − +

= − + = − − −

= + = − +

= + = +

= − + = +

= + = − +

= − + = +

= − − − = − +

= − + = − − −

= + = − +

= + = +

= − + = +

= + = − +

= − + = +

= − − − = − +

= + = − +

= − + = +

= − + = +

= − + = − − −

= + = +

= + = − +

= − − − = − +

= + = − +

= − + = +

= − + = +
NASA/CR—2012-217715 22

The non-zero terms for the second matrix are

( )

( )

( )

( )

( )

( )

( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

A J r ( J J J )s J J t

A J r J J J s J J t

A J J J r J J s J t

A J J J r J J s J t

A J r J J J s J J t

A J r J J J s J J t

A J r J s J J J t J

A J r J s J J J t J

A J r J s J J J t J

A J r J s J J J t J

A J J J r J J J s J J J t

A J J J r J J s J t

A J r J J J s J J t

A J J J r J J s J t

A J r J J J s J J t

A J r J s J J J t J

A J r J s J J J t J

A J J J r J J J s J J J t

A J J J r J J s J t

A J r J J J s J J t

A J J J r J J s J t

A J r J J J s J J t

A J r J s J J J t J

A J r J s J J J t J

A J J J r J J J s J J J t

A J J J r J J s J t

A J r J J J s J J t

A J J J r J J s J t

A J r J J J s J J t

A J r J s J J J t J

A J r J s J J J t J

4,15 2, 2 2,1 2, 2 2,3 2, 2 2, 2

4,14 3, 2 3,1 3, 2 3,3 3, 2 3, 2

4,12 2,1 2, 2 2,3 2,1 2,1 2,1

4,11 3,1 3, 2 3,3 3,1 3,1 3,1

4,9 2, 2 2,1 2, 2 2,3 2, 2 2, 2

4,8 3, 2 3,1 3, 2 3,3 3, 2 3, 2

4,6 2,3 2,3 2,1 2, 2 2,3 2,3

4,5 3,3 3,3 3,1 3, 2 3,3 3,3

4,3 2,3 2,3 2,1 2, 2 2,3 2,3

4, 2 3,3 3,3 3,1 3, 2 3,3 3,3

3, 21 3,3 3, 2 3,1 3,3 3, 2 3,1 3,3 3, 2 3,1

3,18 3,1 3, 2 3,3 3,1 3,1 3,1

3,15 3, 2 3,1 3, 2 3,3 3, 2 3, 2

3,12 3,1 3, 2 3,3 3,1 3,1 3,1

3,9 3, 2 3,1 3, 2 3,3 3, 2 3, 2

3,6 3,3 3,3 3,1 3, 2 3,3 3,3

3,3 3,3 3,3 3,1 3, 2 3,3 3,3

2, 20 2,3 2, 2 2,1 2,3 2, 2 2,1 2,3 2, 2 2,1

2,17 2,1 2, 2 2,3 2,1 2,1 2,1

2,14 2, 2 2,1 2, 2 2,3 2, 2 2, 2

2,11 2,1 2, 2 2,3 2,1 2,1 2,1

2,8 2, 2 2,1 2, 2 2,3 2, 2 2, 2

2,5 2,3 2,3 2,1 2, 2 2,3 2,3

2, 2 2,3 2,3 2,1 2, 2 2,3 2,3

1,19 1,3 1, 2 1,1 1,3 1, 2 1,1 1,3 1, 2 1,1

1,16 1,1 1, 2 1,3 1,1 1,1 1,1

1,13 1, 2 1,1 1, 2 1,3 1, 2 1, 2

1,10 1,1 1, 2 1,3 1,1 1,1 1,1

1,7 1, 2 1,1 1, 2 1,3 1, 2 1, 2

1, 4 1,3 1,3 1,1 1, 2 1,3 1,3

1,1 1,3 1,3 1,1 1, 2 1,3 1,3

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

= + + + + +

= + + + + +

= + + + + +

= + + + + +

= + + + − +

= + + + − +

= + + + + +

= + + + + +

= + + + + −

= + + + + −

= − − − + − − − + − − −

= + + − + +

= + + + + +

= + + + + +

= + + + − +

= + + + + +

= + + + + −

= − − − + − − − + − − −

= + + − + +

= + + + + +

= + + + + +

= + + + − +

= + + + + +

= + + + + −

= − − − + − − − + − − −

= + + − + +

= + + + + +

= + + + + +

= + + + − +

= + + + + +

= + + + + −
NASA/CR—2012-217715 23

( )

( )

( ) ( ) ( )

( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( )

( ) ( ) ( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( )

A ( J J J )r ( J J J )s ( J J J )t

A J J J r J J J s J J J t

A J J J r J J s J t

A J J J r J J s J t

A J r J J J s J J t

A J r J J J s J J t

A J J J r J J s J t

A J J J r J J s J t

A J r J J J s J J t

A J r J J J s J J t

A J r J s J J J t J

A J r J s J J J t J

A J r J s J J J t J

A J r J s J J J t J

A J J J r J J J s J J J t

A J J J r J J J s J J J t

A J J J r J J s J t

A J J J r J J s J t

A J r J J J s J J t

A J r J J J s J J t

A J J J r J J s J t

A J J J r J J s J t

A J r J J J s J J t

A J r J J J s J J t

A J r J s J J J t J

A J r J s J J J t J

A J r J s J J J t J

A J r J s J J J t J

A J J J r J J J s J J J t

A J J J r J J J s J J J t

A J J J r J J s J t

A J J J r J J s J t

6, 20 1,3 1, 2 1,1 1,3 1, 2 1,1 1,3 1, 2 1,1

6,19 2,3 2, 2 2,1 2,3 2, 2 2,1 2,3 2, 2 2,1

6,17 1,1 1, 2 1,3 1,1 1,1 1,1

6,16 2,1 2, 2 2,3 2,1 2,1 2,1

6,14 1, 2 1,1 1, 2 1,3 1, 2 1, 2

6,13 2, 2 2,1 2, 2 2,3 2, 2 2, 2

6,11 1,1 1, 2 1,3 1,1 1,1 1,1

6,10 2,1 2, 2 2,3 2,1 2,1 2,1

6,8 1, 2 1,1 1, 2 1,3 1, 2 1, 2

6,7 2, 2 2,1 2, 2 2,3 2, 2 2, 2

6,5 1,3 1,3 1,1 1, 2 1,3 1,3

6, 4 2,3 2,3 2,1 2, 2 2,3 2,3

6, 2 1,3 1,3 1,1 1, 2 1,3 1,3

6,1 2,3 2,3 2,1 2, 2 2,3 2,3

5, 21 1,3 1, 2 1,1 1,3 1, 2 1,1 1,3 1, 2 1,1

5,19 3,3 3, 2 3,1 3,3 3, 2 3,1 3,3 3, 2 3,1

5,18 1,1 1, 2 1,3 1,1 1,1 1,1

5,16 3,1 3, 2 3,3 3,1 3,1 3,1

5,15 1, 2 1,1 1, 2 1,3 1, 2 1, 2

5,13 3, 2 3,1 3, 2 3,3 3, 2 3, 2

5,12 1,1 1, 2 1,3 1,1 1,1 1,1

5,10 3,1 3, 2 3,3 3,1 3,1 3,1

5,9 1, 2 1,1 1, 2 1,3 1, 2 1, 2

5,7 3, 2 3,1 3, 2 3,3 3, 2 3, 2

5,6 1,3 1,3 1,1 1, 2 1,3 1,3

5, 4 3,3 3,3 3,1 3, 2 3,3 3,3

5,3 1,3 1,3 1,1 1, 2 1,3 1,3

5,1 3,3 3,3 3,1 3, 2 3,3 3,3

4, 21 2,3 2, 2 2,1 2,3 2, 2 2,1 2,3 2, 2 2,1

4, 20 3,3 3, 2 3,1 3,3 3, 2 3,1 3,3 3, 2 3,1

4,18 2,1 2, 2 2,3 2,1 2,1 2,1

4,17 3,1 3, 2 3,3 3,1 3,1 3,1

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ 1/ 2 ˆ 3/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 4

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ 3 ˆ ˆ

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

ˆ 1/ 4 ˆ 1/ 4 ˆ 1/ 2 ˆ 1/ 4 ˆ 1/ 4 ˆ 3/ 2

= − − − + − − − + − − −

= − − − + − − − + − − −

= + + − + +

= + + − + +

= + + + + +

= + + + + +

= + + + + +

= + + + + +

= + + + − +

= + + + − +

= + + + + +

= + + + + +

= + + + + −

= + + + + −

= − − − + − − − + − − −

= − − − + − − − + − − −

= + + − + +

= + + − + +

= + + + + +

= + + + + +

= + + + + +

= + + + + +

= + + + − +

= + + + − +

= + + + + +

= + + + + +

= + + + + −

= + + + + −

= − − − + − − − + − − −

= − − − + − − − + − − −

= + + − + +

= + + − + +
NASA/CR—2012-217715 24

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Eshelby, J.D., 1957. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related 

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A New and General Formulation of the Parametric HFGMC Micromechanical Method for 

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14. ABSTRACT

The recent two-dimensional (2-D) parametric formulation of the high fidelity generalized method of cells (HFGMC) reported by the authors is generalized for the 

micromechanical analysis of three-dimensional (3-D) multiphase composites with periodic microstructure. Arbitrary hexahedral subcell geometry is developed to discretize a 

triply periodic repeating unit-cell (RUC). Linear parametric-geometric mapping is employed to transform the arbitrary hexahedral subcell shapes from the physical space to an 

auxiliary orthogonal shape, where a complete quadratic displacement expansion is performed. Previously in the 2-D case, additional three equations are needed in the form of 

average moments of equilibrium as a result of the inclusion of the bilinear terms. However, the present 3-D parametric HFGMC formulation eliminates the need for such 

additional equations. This is achieved by expressing the coefficients of the full quadratic polynomial expansion of the subcell in terms of the side or face average-displacement 

vectors. The 2-D parametric and orthogonal HFGMC are special cases of the present 3-D formulation. The continuity of displacements and tractions, as well as the equilibrium 

equations, are imposed in the average (integral) sense as in the original HFGMC formulation. Each of the six sides (faces) of a subcell has an independent average 

displacement micro-variable vector which forms an energy-conjugate pair with the transformed average-traction vector. This allows generating symmetric stiffness matrices 

along with internal resisting vectors for the subcells which enhances the computational efficiency. The established new parametric 3-D HFGMC equations are formulated and 

solution implementations are addressed. Several applications for triply periodic 3-D composites are presented to demonstrate the general capability and varsity of the present 

parametric HFGMC method for refined micromechanical analysis by generating the spatial distributions of local stress fields. These applications include triply periodic 

composites with inclusions in the form of a cavity, spherical inclusion, ellipsoidal inclusion, discontinuous aligned short fiber. A 3-D repeating unit-cell for foam material 

composite is simulated.

15. SUBJECT TERMS

Microstructure; Micromechanics; Finite element 

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