Smart sandwich beam

A balanced laminate of piezoelectric fiber composite for improved shear piezoelectric actuation of beams

Chapter 4: A balanced laminate of PFC

4.3 Smart sandwich beam

The shear actuation capability of the balanced laminate of PFC is investigated by utilizing it for actuation of bending deformation of a smart sandwich beam as shown in Fig. 4.4 where the shear actuator patches made of the balanced laminate of PFC are located at the core of foam. A number (n_p) of uniform segments with the length of L_s is considered over the length (L, L(n_pL_s)) of the smart sandwich beam where every segment contains an actuator patch with the length of L_p at its (segment) centre (Fig. 4.4). The patches are considered to be closely spaced along the length of the smart sandwich beam ((L_sL_p)L_s), and the corresponding coverage of a patch within a segment is denoted by the ratio as, L_p/L_s. The other geometric dimensions of the smart sandwich beam

Chapter 4: A balanced laminate of PFC

are shown in Fig. 4.4. The face layers and foam are considered to be made of the isotropic materials while the material properties of the actuator patches are considered as the homogenized material properties of the BL-PFC.

According to these constructional features of the smart sandwich beam (Fig. 4.4), the material properties do not vary along the y-direction at any point on the xz-plane. So, the rectangular cross-section of the overall beam has a vertical plane (xz-plane) of symmetry. The boundary conditions at the ends of the smart sandwich beam are considered symmetric with respect to this plane of symmetry. The patches of balanced laminate of PFC are activated by the external transverse electric field across their top and bottom fully electrode surfaces, and the corresponding shear actuation force in the xz-plane of any actuator patch can be assumed as uniformly distributed over its (patch) volume because of the homogenized material properties of the patches. So, the shear actuation force in the actuator patches is also symmetric with respect to the aforesaid plane of symmetry. Under these conditions, the smart sandwich beam has no tendency for twist and undergoes bending deformation in the vertical plane (xz-plane) of symmetry due to the shear actuation force. Now, the smart sandwich beam is taken with a narrow rectangular cross-section (bh, Fig.

4.4). Also, there is no applied force on the boundary surfaces of the smart sandwich beam across its thickness (b, in y-direction). Under these geometrical properties, material properties, boundary conditions and applied load, the state of stress at any point in the smart sandwich beam can be assumed as the state of plane stress in the xz-plane. So, the analysis of the smart sandwich beam can be carried out by taking a typical xz-plane. Accordingly, the state of strain and the state of stress at any point in the xz-plane are,



_x _{z xz}



^T

 

 ,



_x _z ^_xz



^T

  (4.15)

The linear strain-displacement relations at any point in the xz-plane can be written as,

Ld

 , ^d



^{u w}



^T,

0 T

0

x z

z x

 

 

 

 

   

 

 

L

(4.16)

Chapter 4: A balanced laminate of PFC

where, u and w are the displacements at any point on the xz-plane of the smart sandwich beam along the x ^and z directions, respectively; d is the displacement vector and L is the operator matrix. The constitutive relations for the isotropic face layers and foam at the core can be written as,

kCk

 ,

2

1 0

1 0

1 ( )

0 0 (1 ) / 2

k k

k k

k

k

E



 



 

 

  

  

  

 

C , k1, 2 ^(4.17a)

where, E^k and ^k are Young’s modulus and Poisson’s ratio, respectively for the kth material. As the shear actuator patches are activated by applying the external voltage (V) across their top and bottom fully electrode surfaces, they are subjected to a dominant electric field (E_z) along their thickness direction.

This electric field (E_z) can be assumed as, E_z V h/ _p. Under this dominant electric field and the plane stress assumption, the constitutive relations for the actuator patches can be written as,

k k k

Ez

C e

  ,

T

( ) 33

k k k

z z

D  e  E ,

11 13 15

13 33 35

15 35 55

k k k

k k k k

k k k

C C C

C C C

C C C

 

 

  

 

 

 

C ,

31 33 35 k

k k

k

e

= e e

 

 

 

 

 

 

  e , k3

2 11^k 11^k ( 12^k ) / 22^k

C C  C C , C₁₃^k C₁₃^k (C C₁₂^k ₂₃^k ) /C₂₂^k ,

15^k 15^k ( 12^k 25^k ) / 22^k

C C  C C C , C₃₃^k C₃₃^k (C₂₃^k ) /² C₂₂^k ,

35^k 35^k ( 23^k 25^k ) / 22^k

C C  C C C , C₅₅^k C₅₅^k (C₂₅^k ) /² C₂₂^k ,

31^k 31^k ( 32 12^{k k} ) / 22^k

e e  e C C , e₃₃^k e₃₃^k (e C₃₂^k ₂₃^k ) /C₂₂^k ,

35^k 35^k ( 32^k 25^k ) / 22^k

e e  e C C

(4.17b) In Eqs. (4.17a) and (4.17b), the superscript k denotes the materials for the face layers, foam and actuator patches as per its value as 1, 2 and 3, respectively.

The symbols C_ij^k, e_ij^k and _ij^k in Eq. (4.17b) are the stiffness, piezoelectric and dielectric coefficients, respectively for the actuator patches (k3).

Chapter 4: A balanced laminate of PFC

For FE analysis of the smart sandwich beam by taking a typical xz-plane under the plane stress assumption, this plane (xz-plane) is discretized by nine- node quadrilateral element. The edges of any element are in parallel to the edges of the xz-section of the smart sandwich beam, and a typical element is made of either of the constituent materials within the xz-section. The displacement vector (d, Eq. (4.16)) at any point in an element can be expressed by the elemental nodal displacement vector (d^e) and the shape function matrix (N),

 e

d Nd ^(4.18)

For a deformation of the smart sandwich beam due to an applied electric field ( Ez) across the thickness of the actuator patches, the first variation of its total potential energy

 

TP



can be written as,

3 Τ

1

) )

k

k k

P z z k

k A

T D dA

  



 

 

   

 

 

 

^{ } _(4.19)

where,  is an operator for the first variation; A_k is the area of the k^th material on the xz-plane of the smart sandwich beam. Using Eqs. (4.17a), (4.17b), (4.16) and (4.18) in Eq. (4.19), the following expression of T_P for a typical element can be obtained,

 

( ^e)T ^{e e e}

P E z

T E

  d K d P _,



^{T Τ}



e k

e k

e k

A





dA

K N L C LN ,



^{T Τ}



e k

e k

e k

E A





dA

P N L e (4.20)

In Eq. (4.20), A_k^e is the area over an element within the k^th material on the xz- plane of the smart sandwich beam. According to the principle of minimum potential energy (T_P0), the governing equations of a typical element can be obtained from Eq. (4.20) as,

e e e

EEz

 d P

K (4.21)

Assembling the elemental equations (Eq. (4.21)), the governing equations of the smart sandwich beam under the plane stress assumption can be obtained as,

1 np

i i E z i

E







^P

KX (4.22)

Chapter 4: A balanced laminate of PFC

where, Kis the global stiffness matrix; X is the global nodal displacement vector; P_Eⁱ is the electro-elastic coefficient vector that is obtained by assembling the elemental vectors (P_E^e) for the elements in i^th actuator patch; Eⁱ_z is the applied electric field to the i^th actuator patch; n_p is the number of actuator patches within the smart sandwich beam.

The ends of the smart sandwich beam are considered as fixed ends ( 0

u w at x0,L). The applied electric field (E_z) to the actuator patches in the left half (0 x ( / 2)L ) of the smart sandwich beam is taken in the opposite directionto that for the actuator patches in the right half (( / 2)L  x L) of the same beam. Accordingly, the positive (E_z) and negative (E_z) electric fields are applied to the actuator patches in (0 x ( / 2)L and (( / 2)L  x L), respectively, and these applied electric fields are considered to be of equal magnitude. This kind of applied electric field to the actuator patches is considered to cause the bending deflection of the smart sandwich beam either in upward (z) or in downward (z) direction.