Effective electro-elastic properties of the smart composite

In the theory of linear piezoelectricity at room temperature, the coupled interaction between the electric and elastic fields is described by four different piezoelectric

l

c

l

f

a

^c

b

^f

b

^c

a

^f

z

y

x Piezoelectric fiber

Epoxy matrix

(b)

Chapter 2: Design of SPFC/CPFC in cylindrical/rectangular coordinates

constitutive formulations (Eq. (1.1)). Among these four different constitutive formulations, the piezoelectric stress formulation is commonly employed when the strain and electric fields are considered as the natural variables. According to this constitutive formulation, the natural variables (strain and electric fields) are related to the stress and electric displacement fields by,

ij Cijkl kl e Esij s







 , D_i e_{ikl kl}



_is E_s (2.1)

where, i j k l s, , , , 1,2,3; C_ijkl, e_{sij and} _is denote the elements of stiffness, piezoelectric and permittivity tensors, respectively;



_ij,



_kl, D_{i and} E_s are the elements of stress, strain, electric displacement and electric field tensors, respectively. In Eq. (2.1), the mathematical objects are symmetric in i and j, and also in k and l. So, using Voigt notation, ij kl/ _for i j k l, , , 1,2,3 can be represented as, 111, 222, 333, 23 / 324, 13 / 315 and 12 / 216. Using this notation, Eq. (2.1) can be rewritten as,

C e E

   







 , D e_{ }



_ E_{ (2.2)} where,

 

, 1,2,3,4,5,6 and ,



1,2,3. In Eq. (2.2), the elements of stiffness, piezoelectric and permittivity matrices are defined by (Cady, 1946; Furukawa, 1989),

C_ ^







 

  

E

, e

E

 



 

  



or D

e_



 

  

E

, D

 E



 

   



(2.3)

The superscript E (Eq. (2.3)) indicates zero or constant electric field and the superscript  (Eq. (2.3)) indicates zero or constant strain field. The constitutive relations and material constants in Eqs. (2.2) and (2.3), respectively are given for a homogeneous piezoelectric solid. Analogous to this homogeneous piezoelectric solid, the present piezoelectric composite is considered to be a macroscopically homogeneous piezoelectric solid, and its macroscopic behavior can be defined by the effective constitutive relations according to standard micromechanical theories for composites. These effective constitutive relations are valid only for specially statistically homogeneous fields within the composite, which (fields) can be produced within a heterogeneous body through the imposition of homogeneous boundary conditions over the boundary surfaces of the body (Hasin, 1965; Hasin, 1970). Making use of this analogy, the effective material properties of

asymptotically homogeneous composites can be estimated through the application of volume-average strain field and/or electric field by means of the homogeneous kinematic boundary conditions (displacement and/or electric potential) (Aboudi et al., 2013). Following the same, presently the effective material constants of the piezoelectric composite are estimated by applying volume-average strain field and/or electric field over the volume of RVE. The volume-average field quantities over a volume (V_d) are defined by,

‘

1

d d

d V

V dV

 

 



 ^{, 1} _Vd ^d

d

V dV

 

 



 ^, 1

d d

d V

E E dV

V



^, D ¹ VD dV

 V





(2.4) where, the over-bar signifies volume-average quantity. The constitutive relations for fiber ( f ) and matrix (m) phases within the RVE can be written in terms of the volume-average field quantities as,

f f f f f

C _ e E

  







 (2.5)

m m m

 C 







(2.6)

f f f f f

D e_



_ _ E_ (2.7)

m m m

D _ E_ (2.8)

The volume-average field quantities of the RVE can be expressed in terms of the similar quantities of constituents according to the rule of mixtures as,

(v_f ^f v_m ^m)

  











(2.9)

( _f ^f _m ^m)

D  v D v D (2.10)

(v_f ^f v_m ^m)

  











(2.11)

( _f ^f _m ^m)

E_  v E_ v E_ (2.12)

Substituting Eqs. (2.5)-(2.6) in Eq. (2.9), and then using Eq. (2.11), the following expression can be obtained,

( )

m f m f f f

f f

C v C C v e E

      







 



 (2.13)

Chapter 2: Design of SPFC/CPFC in cylindrical/rectangular coordinates

where, 1,2,3,4,5,6. Similarly, substituting Eqs. (2.7)-(2.8) in Eq. (2.10), and then using Eq. (2.12), the following expression can be obtained,

( )

m f m f f f

f f

D _ E_ v  _{ } E_ v e_



_ (2.14)

where,



1,2,3. According to the definitions of piezoelectric constants (Eq. (2.3)), the expressions for the effective electro-elastic coefficients of the piezoelectric composite can be obtained as,

( )

f f

m f m f

f f

C_ C_ v C_ C_ ^ v e_ E^

 



 

 

   

        

for E_ 0 _(2.15)

( )

f f

f m f

f f

e_ v _{ } E^ v e_ ^

 



 

   

   

    

     

   

for E_ 0 _(2.16)

( )

f f

m f m f

f f

v E v e

E E

 

    

 

   

 

        

 

   

 

for



_0 _(2.17) According to Eqs. (2.15)-(2.17), the effective electro-elastic constants of the composite can be determined by computation of the volume-average field quantities for RVE and fiber phase, and it is presently performed through the imposition of volume-average strain and electric fields over the volume of the RVE by means of applying homogeneous displacement and electric potential boundary conditions.

For this computation, an electro-elastic analysis of the RVE is to be carried out, and that is presently done using FE procedure. It may be noted here that the effective properties of the smart composite can be estimated using the standard micromechanics theories like Mori-Tanaka method, Self-consistent method, etc.

The same can also be estimated numerically using FE procedure. The FE procedure is very time consuming and expensive procedure. But, it may provide more realistic results in the prediction of electro-elastic constants of piezoelectric composites (Odegard, 2004). So, the FE procedure is utilized at present through the derivation of a three-dimensional FE model of the RVE as presented in the next section. The solutions from FE model of RVE for the applied homogeneous kinematic boundary conditions yield the volume-average field quantities according to the following expressions,

1

1 ^Vd

i d

N

i i

V d d i

V dV

 

 



 

  



 

 ^,

1

1 ^Vd

i d

N

i i

V d d i

E E dV

 V 



 

  



 

 ^(2.18)

where,

Vd

N is the number of elements within a volume V_d (RVE/fiber phase/matrix phase);



_ⁱ /E_ⁱ is a component of strain/electric field vector within i^th _element having the elemental volume of V_dⁱ. In order to determine the effective coefficients of the composite/RVE, nine sets of homogeneous kinematic boundary conditions are presently applied over the boundary surface of the RVE. The RVE is initially considered as a stress/strain/electric field free solid, and then every set of boundary conditions over its boundary surfaces are applied separately. Every set of boundary conditions yields only one nonzero element of strain (



_ at E_ 0) or electric (E_ at



_ 0) field vector. Corresponding to this nonzero element (say,

 1

  for



1;



_ 0 for



2,3,…6; E_ 0 or E_ E₁ for



1; E_ 0 for



 2,3;  _ 0 ), the derivatives in Eqs. (2.15)-(2.17) can be written as,

1 1

f f

 

 

 

 

 ^for E_ 0 ^(2.19)

1 1

f f

E_ E_

 

 

 ^for E_ 0 ^(2.20)

1 1

f f

E E

 

 

 

 ^for



_0 ^(2.21)

1 1

f f

E E

E E

 

 ^for



_0 ^(2.22)

Introducing these terms in Eqs. (2.15)-(2.17), the effective coefficients, C_1 , e₁ and ₁ can be obtained in terms of the volume-average field quantities over the RVE and the fiber phase. A similar computation can be performed for all the sets of boundary conditions in order to obtain all the effective coefficients of the composite/RVE. The nine sets of kinematic boundary conditions are illustrated in the following points, where the displacements at any point along r, _and z directions in the cylindrical coordinate system (or x,y and z directions in the rectangular coordinate system) are denoted by u, v and w, respectively. The electric potential at any point within the volume of the RVE is denoted by



. (a) Effective constants ( C_1, e₁):

Boundary conditions for RVE in cylindrical coordinates:

Chapter 2: Design of SPFC/CPFC in cylindrical/rectangular coordinates

r 0

u_  , u_r (



₁⁰l_c) , v_ 0, w_z 0, _{, ,} 0

r  z



_{  }  .

Boundary conditions for RVE in rectangular coordinates:

X 0

u_  , u_X  (



₁⁰l_c), v_Y 0, v_Y 0, w_Z 0, w_Z 0, _{, ,} 0

X Y Z



_{  }  .

Elements (



_, E_ ):



₁



₁⁰;



_ 0 for  2,3,4,5,6; E_ 0. Effective material constants:

1 1 0 0

1 1

( )

f f

m f m f

f f

C_ C_ v C_ C_ ^ v e_ E^

 

 

 

 

      

(2.23)

1 0 0

1 1

( )

f f

f m f

f f

e v _{ } E^ v e_ ^

 

   

   

    

   

   

(b) Effective constants ( C_2, e₂):

Boundary conditions for RVE in cylindrical coordinates:

r 0

u_  , v_ 0, v_ (

 

₂⁰ _c), w_z 0, _{, ,} 0

r  z



_{  }  . Boundary conditions for RVE in rectangular coordinates:

X 0

u_  , u_X 0 , v_Y 0, v_Y (



₂⁰a_c), w_Z 0, w_Z 0,



_{X, Y, Z} 0. Elements ( _, E_ ):



₂



₂⁰;



_ 0 for  1,3,4,5,6 ; E_ 0.

Effective material constants:

2 2 0 0

2 2

( )

f f

m f m f

f f

C_ C_ v C_ C_ ^ v e_ E^

 

 

 

 

      

(2.24)

2 0 0

2 2

( )

f f

f m f

f f

e v _{ } E^ v e_ ^

 

   

   

    

   

   

(c) Effective constants ( C_3, e₃):

Boundary conditions for RVE in cylindrical coordinates:

r 0

u_  , v_ 0, w_z 0, w_z (



₃⁰h_c), _{, ,} 0

r  z



_{  }  . Boundary conditions for RVE in rectangular coordinates:

X 0

u_  , u_X 0, v_Y 0, v_Y 0, w_Z 0, w_Z (



₃⁰b_c), _{, ,} 0

X Y Z



_{  } 

Elements ( _, E_ ):



₃



₃⁰;



_ 0 for  1,2,4,5,6; E_ 0. Effective material constants:

3 3 0 0

3 3

( )

f f

m f m f

f f

C_ C_ v C_ C_ ^ v e_ E^

 

 

 

 

      

(2.25)

3 0 0

3 3

( )

f f

f m f

f f

e v _{ } E^ v e_ ^

 

   

   

    

   

   

(d) Effective constants ( C_4, e₄):

Boundary conditions for RVE in cylindrical coordinates:

z 0

v_  , v_z  (12₄⁰h^c), w|_0, w_ (12₄⁰^c),_{  r, , z} 0. Boundary conditions for RVE in rectangular coordinates:

Z 0

v_  , (1 ₄⁰ )

2 ^c

v_Z 



b , w_Y 0, (1 ₄⁰ )

2 ^c

w_Y 



a ,

, , 0

X Y Z

_{  }  .

Elements (_, E_ ): ₄ ₄⁰;



_ 0 for  1,2,3,5,6; E_ 0. Effective material constants:

4 4 0 0

4 4

( )

f f

f f

m m

f f

C_ C_ v C_ C_ ^ v e_ E^

 

 

 

 

      

(2.26)

4 0 0

4 4

( )

f f

f m f

f f

e v _{ } E^ v e_ ^

 

   

   

   

   

   

(e) Effective constants ( C_5, e₅):

Boundary conditions for RVE in cylindrical coordinates:

| _z 0 u _  ,

u_z (1 ₅⁰ )

2 h^c , w|_r0,

w_r  (1 ₅⁰ ) 2 l^c ,

, , 0

r  z



_{  }  . Boundary conditions for RVE in rectangular coordinates:

Z 0

u_  , (1 ₅⁰ )

2 ^c

u_Z 



b , w_X 0, (1 ₅⁰ )

2 ^c

w_X 



l ,



_{X, Y, Z} 0. Elements (_, E_ ): ₅ ₅⁰;



_ 0 for



 1,2,3,4,6; E_ 0.

Effective material constants:

Chapter 2: Design of SPFC/CPFC in cylindrical/rectangular coordinates

5 5 0 0

5 5

( )

f f

f f

m m

f f

C_ C_ v C_ C_ ^ v e_ E^

 

 

 

 

      

(2.27)

5 0 0

5 5

( )

f f

f m f

f f

e v _{ } E^ v e_ ^

 

   

   

   

   

   

(f) Effective constants (C_6, e₆):

Boundary conditions for RVE in cylindrical coordinates:

0

u_  , (1 ₆⁰ )

2 ^c

u_ 

 

 , v|_r0, (1 ₆⁰ )

2 ^c

v_r 



l ,

, , 0

r  z

_{  }  . Boundary conditions for RVE in rectangular coordinates:

Y 0

u_  , (1 ₆⁰ )

2 ^c

u_Y 



a , 0

v_X  , (1 ₆⁰ )

2 ^c

v_X 



l , _{X, Y, Z} 0. Elements (



_, E_): ₆ ₆⁰;



_ 0 for  1,2,3,4,5; E_ 0.

Effective material constants:

6 6 0 0

6 6

( )

f f

f f

m m

f f

C_ C_ v C_ C_ ^ v e_ E^

 

 

 

 

      

(2.28)

6 0 0

6 6

( )

f f

f m f

f f

e v _{ } E^ v e_ ^

 

   

   

   

   

   

(g) Effective constant (₁):

Boundary conditions for RVE in cylindrical coordinates:

z 0

  , _ 0, _r 0, _r  (E₁⁰l_c), u_r0, v_ 0, w_z 0. Boundary conditions for RVE in rectangular coordinates:

| _Z 0

 _  ,



|_Y0,



|_R0, |_R (E₁⁰l_c), u_X 0, v_Y 0, w_Z 0.

Elements (_, E_): E₁ E₁⁰; E_ 0 for



 2,3;



_ 0. Effective material constants:

1 1 0 0

1 1

( )

f f

f f

m m

f f

v E v e

E E

 

  

   

   

     

 

   

  ^(2.29)

(h) Effective constant (₂):

Boundary conditions for RVE in cylindrical coordinates:

r 0

  , _z 0, _ 0, _  (E₂⁰_c), u_r 0, v_ 0, w_z 0. Boundary conditions for RVE in rectangular coordinates:

| _X 0



_  ,



|_Z0,



|_Y0, |_Y (E₂⁰a_c), u_X 0, v_Y 0, w_Z 0.

Elements (_, E_ ): E₂ E₂⁰; E_ 0 for



 1,3;



_ 0. Effective material constants:

2 2 0 0

2 2

( )

f f

f f

m m

f f

v E v e

E E

 

  

   

   

     

 

   

 

(2.30)

(i) Effective constant (₃):

Boundary conditions for RVE in cylindrical coordinates:

r 0

  , _ 0, _z0, _z  (E₃⁰h_c), u_r 0, v_ 0, w_z 0. Boundary conditions for RVE in rectangular coordinates:

| _R 0

 _  ,



|_Y0,



|_X0, |_X (E₃⁰b_c), u_X 0, v_Y 0, w_Z 0.

Elements (



_, E_ ): E₃ E₃⁰; E_ 0 for



 1,2;



_ 0. Effective material constants:

3 3 0 0

3 3

( )

f f

f f

m m

f f

v E v e

E E

 

  

   

   

     

 

   

 

(2.31)