Graded effective elastic properties

FiberMatrix

5.3 Graded effective elastic properties

Since every layer is comprised of identical fiber-matrix packs (Fig. 5.1(b)), a representative volume (RV) as shown in Fig. 5.2 may be chosen as an elemental volume for predicting the effective graded elastic properties of the composite lamina. A boundary face of RV is defined by its normal direction. So, its six boundary faces are denoted by, x, x, y, y, z and z planes. The effective graded elastic properties of the present composite lamina are estimated following a concept of homogeneous model of FG materials proposed by Reiter and Dvorak (1997, 1998). The original work of Reiter and Dvorak (1997, 1998) is for the particulate micro-structure of a two-phase graded material with single composition gradient. In this proposition (Reiter and Dvorak 1997, 1998), the medium of FG solid is modeled by stacking several parallel homogeneous layers along the direction of variation of composition gradient. The constituent volume fractions of layers are assigned following the single composition gradient of the FG solid. Then, the effective properties of every layer are determined using either Mori-Tanaka method or Self-consistent scheme depending on the location of that layer within the FG solid. However, following this proposition, every fiber- matrix pack of RV (Fig. 5.2) of present composite is assumed as a homogeneous layer while the layers within the RV are of different FVFs. In the present prediction of effective properties of every homogeneous layer, first volume-average elastic properties of RV are estimated. Subsequently, the effective properties of every homogeneous layer are extracted from the volume-average properties of RV. Since the layers are coupled within the RV, the prediction of effective properties of homogeneous layers from the volume-average properties of RV includes the effect of coupled interaction among the layers on their effective properties. It should be noted that the effective properties of homogeneous layers in the work of Reiter and Dvorak (1997, 1998) are estimated considering every layer as a discrete representative volume element (RVE). However, for the present graded fiber-reinforced composite, negligibly small effect of this coupled

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114

interaction among the layers on their effective properties is observed as it is quantified in later section.

The volume-average stress ({ }σ ) and strain ({ }ε ) field quantities over the volume of RV can be expressed in terms of the similar field quantities ({^},{^k}) of layer-volumes as follows,

(2 1)

1

{ }

  

{ }









^{ }

 Nc

,

(2 1)

1

{ }

  

{ }









^{ }

 Nc

,

^ V^ /V

(5.1) where, V^ and V are the volumes of ^th layer and RV, respectively. Similar relations can also be written for ^th layer as follows,

 

{



^}

 

^_f{ ^_f}

 

_m^{ _m^} ,

 

{ }



^ 

 

^_f{ }_f^ 

 

_m^{ }_m^ ,

^_f V_f^ /V^,



_m^ V_m^/V^

(5.2)

where, V_f^ and V_m^ are the volumes of fiber and matrix phases in the ^th layer;

{^_f}/{_f^} and {_m^}/{_m^} are the volume-average stress/strain vectors for fiber and matrix phases of ^th layer. Substituting Eq. (5.2) in Eq. (5.1), the following expression for the volume-average stress vector ({ }σ ) of RV can be obtained,

^{(2 1)}

 

1

{ }

   

{ }

 

{ }









^{  }  ^{ }

 Nc

f f m m (5.3) The constitutive relations for the fiber and matrix phases within a layer () can be written as,

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115

{



^_f} [ C_f]{ }



_f^ ,

{



_m^} [ C_m]{ }



_m^ (5.4) where, [C_f] and [C_m] are stiffness matrices for fiber and matrix phases, respectively.

Substituting Eq. (5.4) in Eq. (5.3), the following expression of { }σ can be obtained,

^{(2 1)}

 

1

{ }

  

[ ]{



}



[ ]{



}









^{  }  ^{ }

 Nc

f Cf f m Cm m (5.5) The volume-average strain vector ({ε^}) for ^th layer is connected with that ({ }ε ) of RV by,

{^} [ A^]{ } ,

(2 1)

1

[ ] [ ]









^{ } 

 Nc

A I

(5.6) where, [A^] is the volume-average strain concentration matrix for ^th layer and [ ]I is the unity matrix. Similarly, the volume-average strain vectors ({ }



_f^ ,{_m^}) for fiber and matrix phases within a layer () can be expressed in terms of the volume-average strain vector ({ }ε ) of RV as,

{ } [



_f^  A^_f]{ }



, {_m^} [ A_m^]{ } ,

(2 1)

1

( [ ] [ ]) [ ]

  





 



^{    }

 Nc

f Af m Am I

(5.7) where, [A^_f] and [A_m^] are the volume-average strain concentration matrices of fiber and matrix phases in ^th layer. Substituting Eq. (5.2) in Eq. (5.6) and then using Eq. (5.7), the following expression of the volume-average strain concentration matrix ([A^]) for

^thlayer can be obtained,

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116

^[A^]

 ^

^f^[A^f^]

^

m^[Am^]



(5.8) Introducing Eq. (5.7) in Eq. (5.5) and then using Eq. (5.8), the following relation between the volume-average stress and strain vectors of RV can be obtained,

{ } [ ]{ }



 C



,

 

(2 1)

1

[ ]

 

[ ] [ ] [ ] [ ][ ]





 

  

 



^{   }

 Nc

f f m f m

C C C A C A

(5.9) Equation (5.9) describes the volume-average elastic properties of RV. For the assumption of homogenized layers (), these average properties (Eq. (5.9)) would be the average of effective properties of all layers within the RV. For a homogeneous layer () within the RV, the effective constitutive relation can be written as,

{^} [ C^]{^} (5.10) where, [C^] is the effective stiffness matrix for the ^th homogeneous layer. Now, since all homogeneous layers are coupled within the RV, Eq. (5.10) can be inserted into Eq.

(5.1). Subsequently, using the relation given in Eq. (5.6), the following expression can be obtained,

(2 1)

1

{ }



( [



][ ]){ }











^{  }

Nc

k

C A (5.11) Comparing Eq. (5.9) with Eq. (5.11), the following expression for [C^] can be obtained,

^[C^]

^

^f



^[Cf^{] [} Cm^{] [}



A^f^][A^]1[Cm^] (5.12) Equation (5.12) represents the expression of effective elastic matrix ([C^]) of ^th homogeneous layer within the RV. The strain concentration matrices ([A^_f],[A^]) are the key factors for evaluation of [C^]. Since these strain concentration matrices are defined with respect to the volume-average strain of RV, the effect of coupled interaction among

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117

the layers within the RV on their (layers) effective properties is included. However, the strain concentration matrices ([A^_f]_,[A^]) for different layers in RV may be evaluated using available micromechanics theories like Mori-Tanaka method, Self-consistent method etc. Although these micromechanics theories provide quick predictions of effective properties of the composite, but the same could also be evaluated numerically using finite element (FE) procedure that may provide more realistic predictions of effective properties of the composite (Odegard 2004). So, the FE procedure is followed in this work by deriving a three-dimensional FE model of RV as presented in the next section. According to the theorem of average strain for composite materials, the homogeneous displacement boundary conditions over the boundary surfaces of RV are applied in order to compute its (RV) volume-average strain ({ } ) using the following expression,

   

1



1





 

  



 

  

 NRv

V dV

V

(5.13a) where, N_Rv is the total number of elements within the FE model of RV; { }



_ is the strain vector at any point within the ^th element; V is the volume of ^th element. Similar to Eq. 5.13(a), the volume-average strains of a layer () and corresponding fiber counterpart can be computed from the FE model of RV using the following expressions,

  ^{ }

1

 1 



 

  



 









 



 N

V dV

V ,

  ^{ }

1



1





 

  

 



 









 



 Nf

f V

f

V dV

(5.13b) where, N^ is the number of elements within the ^th layer; N^_f is the number of elements within the fiber phase of ^th layer. Six types of homogeneous displacement boundary conditions are considered as those are demonstrated in Table 5.1. Every type of these

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118

boundary conditions provides one non-zero strain component of { } that yields the corresponding columns of [A^] and [A^_f] according to Eqs. (5.6) and (5.7), respectively.

Thus, all columns of the concentration matrices ([A^]_, [A^k_f]) can be computed by applying the six types of boundary conditions (Table 5.1) separately on the FE model of RV. However, in order to estimate the effective elastic properties of homogeneous layers using the FE model of RV, there are basically four aforesaid expressions (Eqs. (5.6), (5.7), (5.12) and (5.13)) are to be utilized in association with the appropriate boundary conditions (Table 5.1).

Table 5.1 Boundary conditions for determining elements of

  ^

Boundary conditions Elements of

  ^

 0

u x , u_x(ε⁰_x_c) , _ 0

v y , 0 vy ,

 0

w z , _ 0 w z

0, 0



_x

 

_{x y} 



_z 



_yz 



_xz



_xy 

 0

u x , _ 0

u x , _ 0

v y , v_y(ε⁰_ya_c),

 0

w z , 0 wz

0, 0



_y

 

_{y x} 



_z 



_yz 



_xz 



_xy 

 0

u x , _ 0

u x , _ 0

v y _{, } 0 v y _,

 0

w z , w_z(ε⁰_zh_c)

0, 0



_z

 

_{z x}



_y 



_yz 



_xz 



_xy 

 0

v z , (1 ⁰ )

 2 _yz _c

vz γ h , _ 0 w y ,

0

4 (1 )

 2 _yz _c

w γ a

0, 0



_yz



_yz



_x 



_y 



_z 



_xz 



_xy 

 0

u z , ^ (12 ⁰_xz _c)

u z γ h , _ 0 w x , 1 0

( 2 )

  _xz_c

w x γ

0, 0



_xz



_xz



_x 



_y 



_z 



_yz 



_xy 

 0

u y _{, (}^{1 0} ₎

 2 _xy _c

uy γ a , _ 0 v x , 1 0

( )

_x 2 _xy_c

v γ

0, 0



_xy



_xy



_x 



_y 



_z 



_xz 



_yz 

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