Arrangement of electrodes

condition. For a number (N_b) of specified nodal degrees of freedom over the boundary surface of RVE, Eq. (2.41) can be written in general form as,

Τ

1

( )

Nb

r r r i i

i

U X

 



 ^{X K X} 



^{P (2.42)}

Employing the principle of minimum potential energy, Eq. (2.42) can be written as,

1 Nb

r r i i

i

X



 



K X P (2.43)

Equation (2.43) represents the electro-elastic FE model of RVE under the specified nodal electric potentials and/or displacements. Using Eq. (2.43), the nodal solutions for electric potential and displacement fields within the RVE corresponding to every set of boundary conditions (Eqs. (2.23)-(2.31)) can be obtained. Subsequently, the volume-average (RVE/phase) strain and electric fields can be computed according to Eq. (2.18) using these nodal solutions.

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

Fig. 2.5(b) Top/bottom surface of the SPFC in cylindrical coordinates with electrodes.

Figure 2.5(b) shows the top/bottom surface of the SPFC lamina (Fig. 2.1) over which the surface-electrodes are provided at the radial gap of any two consecutive short fibers. The uniform polarity of the external voltage is considered for the top and bottom surface-electrodes lying on the same radial location, while any two consecutive electrodes along the radial direction are of opposite polarity. Any two consecutive electrodes of opposite polarity are denoted as a pair of electrodes.

Fig. 2.6 RVE of SPFC in cylindrical coordinates with surface-electrodes.

Across all the pairs of electrodes, uniform applied voltage is considered, and it yields uniform magnitude of the electric field within all such pairs. But, the radial component of the electric field within a pair of electrodes is in the opposite direction to that of the same within the consecutive pairs of electrodes. For obtaining the electrically induced actuation force in the same direction from all pairs of

l

c

l

^f

(l

^c

- l

^f

) (l

c

- ^l

^f

^)/2

(+) Electrode

(-) Electrode

(+) Electrode

(-) Electrode

h

f

h

c

(h

c

-h

f

)/2

+TSE

TSE

+BSE -BSE

(l

c

-l

f

)/2 (l

c

-l

f

)/2 l

f

l

c

TSE: Top surface electrode, BSE: Bottom surface electrode

electrodes, the corresponding piezoelectric fibers are poled along the radial direction in an alternate manner. According to this arrangement of surface- electrodes, the macroscopic behaviour of the SPFC actuator can be estimated by defining an elemental volume as illustrated in Fig. 2.6.

The volume in Fig. 2.6 is basically a volume of RVE (Fig. 2.2) with surface- electrodes. Similar to Figs. 2.5(b) and 2.6, the arrangement of electrodes for the SPFC in rectangular coordinates is illustrated in Figs. 2.7(a) and 2.7(b). In this arrangement of electrodes, it seems too small in-plane gap between any two consecutive surface-electrodes. But it facilitates to achieve a significant magnitude of the externally induced electric field in effect of the small applied voltage across the pairs of surface-electrodes. The difficulty may arise in the fabrication of the SPFC actuator because of the small in-plane gap between any two consecutive surface-electrodes. But this gap can be increased by increasing the length of fibers or fiber aspect ratio. The fiber aspect ratio does not have much effect on the magnitude of effective magnitude of induced electric field or driving electric field for an applied voltage across the pairs of electrodes in the use of the SPFC as an actuator. In this evaluation of driving electric field, the difficulty arises due to the material heterogeneity.

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

Fig. 2.7(a) Arrangement of electrodes over the top and bottom surfaces of SPFC in rectangular coordinates, (b) the corresponding RVE with electrodes.

The effective properties or constitutive behaviour of the smart composite is valid only for specially statistically homogeneous fields which can be produced by means of homogeneous kinematic/kinetic boundary conditions over the surface of large composite (Hasin, 1970). The present applied electric potentials over the surface- electrodes do not satisfy the conditions for homogeneous kinematic boundary conditions. So, in a strict sense, the corresponding statistically homogeneous electric field would not be adequate to the effective constitutive relation. As a consequence, for a situation where the aforesaid specially statistically homogeneous fields do not arise or cannot be produced, an assumption of local averages of fields over the volume of RVE may be made to salvage the analysis (Hasin, 1970). Following this assumption in the present analysis, the local electric field within the elemental volume (RVE) with surface-electrodes (Fig. 2.6 or Fig.

2.7(b)) is considered as the local volume-average electric field over the same volume (RVE). Without this assumption, it is difficult to model the induced electric field within a local heterogeneous volume of the large smart composite actuator.

However, as per this consideration, the induced electric field due to an external voltage can be taken as its volume-average quantity over the volume of RVE according to Eq. (2.12). Now, for structural applications of the smart actuator under the assumption of small strain, the induced electric field due to the strain of overall structure is of very small magnitude as compared to the large magnitude of the applied electric field by means of external voltage through electrodes. So, the magnitude of volume-average electric field (E) over the volume of RVE may be

(b) b^c

b^f a^c

lf

lc

y z

x

(TE)

(BE) (TE)

(BE)

(lc

-

lf

)/

2

(b^c

-

^bf

)/

2 (lc

-

lf

)/

2

+



+



-



-



considered as a function of applied voltage (V ) only. Since one value of applied voltage yields one magnitude of E within the actuator, E is a single-valued function of applied voltage (V). This functional relation for the SPFC/CPFC in cylindrical coordinates may be represented by the following expressions (Eq.

(2.44a)), where G_r( )V , G_( )V and G_z( )V are the functions of applied voltage (V ) corresponding to the components of volume-average electric field (E_r, E_, E_z).

r Gr( )

E V V_, E_ G_( )V V_, E_z G_z( )V V (2.44a) Similar expressions may also be written for the SPFC/CPFC in rectangular coordinates as given in Eq. (2.44b).

xGx( )

E V V_, E_y G_y( )V V_, E_zG_z( )V V _(2.44b) It may be noted here that the same arrangement of surface-electrodes may be considered for the consideration of the continuous fibers instead of the short fibers, and also the same procedure as considered for SPFC actuator may be followed for CPFC actuator for its mathematical modelling. The foregoing demonstration is for a lamina of present SPFC/CPFC actuator. For achieving greater actuation force in an application, several SPFC/CPFC laminas can be used in the form of a laminate with the proper alignment of surface-electrodes as illustrated in Fig. 2.8 or Fig. 2.9 for the SPFC/CPFC in cylindrical or rectangular coordinates, respectively.

Fig. 2.8 Schematic diagram for the alignment of electrodes in forming a laminate of several layers of SPFC in cylindrical coordinates.

 

 

layer 1

layer 1