Effective coefficients of SPFC/CPFC

2.6 Results and discussions

2.6.1 Effective coefficients of SPFC/CPFC

The materials for fiber and matrix phases of the SPFC are considered as PZT-5H and Epoxy, respectively. Table 2.2 presents the material properties of these constituent materials. For the specified constituent materials, the electro-elastic properties of SPFC are mainly dependent on the area-ratio (A_r) and length-ratio (L_r ) through the relation, v_f (A_rL_r). Further, since the SPFC acts as a material of distributed actuator by the application of electric field along the poling direction of fibers, the important effective piezoelectric coefficients are, e₁₁, e¹² and e₁₃. So, the present analysis is carried out mainly for investigating the effects of these parameters (A_r,L_r) on the magnitudes of effective coefficients, e₁₁,e₁₂ and e₁₃.

Figure 2.11 represents the variations of the effective piezoelectric coefficients with the area-ratio (A_r) for continuous piezoelectric fibers (L_r 1). For the variation of the area-ratio (A_r) without altering the thickness (h_c) and circumferential span (

c) of RVE, the fiber dimensions (h_f ,



_f ) are varied following the relations,

f c r

h h A and



_f 



_c A_r . It may be observed from Fig. 2.11 that the magnitude of the effective coefficient (e₁₁) significantly increases with the increasing area-ratio.

But, the magnitude of other coefficient (e₁₂/e₁₃) varies insignificantly. The magnitude of e₁₂/e₁₃ is also indicatively lesser than that of e₁₁. This result infers that the use of the smart composite as a material of distributed actuator would be based on the coefficient (e₁₁), and its (e₁₁) magnitude can be increased by increasing the area-ratio.

Table 2.2 Material properties of constituent materials (Ray, 2006a).

Fiber/Matrix C¹¹ (GPa)

C12

(GPa) C23

(GPa) C22

(GPa) C44

(GPa) Epoxy 3.86 2.57 2.57 3.86 0.64

PZT-5H 124 96 98 151 26.5

Fiber/Matrix e¹¹ (C/m²)

e12/e₁₃ (C/m²)

e26/e₃₅ (C/m²)

11

(C/Vm) (×10^-09)

22 / 33

 

(C/Vm) (×10^-09)

Epoxy 0 0 0 0.079 0.079

PZT-5H 27 -5.1 17 13.27 13.4

Fig. 2.11 Variations of effective piezoelectric coefficients (e₁₁,e₁₂,e₁₃) with the area ratio (A_r).

For the use of short piezoelectric fibers (L_r 1), the variations of the same coefficients (e₁₁,e₁₂,e₁₃) with the length-ratio (L_r) are illustrated in Fig. 2.12 for different values of area-ratio (A_r). In this computation, the fiber-length is varied keeping the constant values of other geometrical parameters in such a manner that the central location of short fiber within the RVE does not alter.

For any value of the area-ratio (A_r), it may be observed from Fig. 2.12 that the magnitude of coefficient e₁₁ increases significantly with the increasing length- ratio (L_r) and reaches to its (e₁₁) maximum value for, L_r 1 (continuous fiber). The other coefficient (e₁₂/e₁₃) also increases with the increasing length ratio (L_r). But, after a certain value of, L_r, its (e₁₂/e₁₃) magnitude decreases to the minimum value for continuous fiber (L_r 1). Similar to the case of continuous fibers (L_r 1, Fig.

2.11), the magnitude of e₁₂/e₁₃ is much lesser than that of e₁₁ (Fig. 2.12). Thus, for both the forms (short and continuous) of piezoelectric fibers within the present smart composite, the coefficient e₁₁ would be chosen as the major piezoelectric coefficient in its (SPFC/CPFC) use as a distributed actuator. In case of SPFC, Fig.

2.12 shows a higher value of L_r (close to 1) for the improved magnitude of e₁₁. The same figure (Fig. 2.12) also indicates significant effect of area-ratio on the magni-

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

Fig. 2.12 Variations of the effective piezoelectric coefficients (e₁₁,e₁₂,e₁₃) with the length-ratio (L_r) for different values of the area-ratio (A_r).

-tude of e₁₁ for a higher value of length-ratio. So, for SPFC, higher values of both the area-ratio and length-ratio are to be considered for the improved magnitude of e11. On the basis of these results, a higher value of area-ratio is chosen at present as 0.6 for both the SPFC and CPFC, while a higher value of length-ratio for SPFC is chosen as 0.95. Basically, the CPFC is modified into an SPFC by reducing the value of length-ratio from 1 to 0.95. This modification causes a little decrease in the magnitude of e₁₁ (Fig. 2.12(a)) from its maximum possible magnitude (at L_r 1).

But, an advantage of greater flexibility and conformability of the smart composite can be achieved, and that is useful in its structural applications.

Like area-ratio and length-ratio, the dimensions like l_c, h_c and _c are also to be mentioned for complete geometrical properties of RVE. These dimensions (l_c,h_c,_c) of RVE are directly related to the similar dimensions (l_f ,h_f ,



_f ) of corresponding fiber by the relations, l_f L l_{r c}, h_f h_c A_r and h_f h_c A_r . So, the dimensions of fiber can also be specified instead of those of RVE. For the aforesaid selected values of area-ratio and length-ratio (A_r 0.6, L_r 0.95), the cross-sectional dimensions (

hf ,



_f ) are considered to have small values as compared to the value of

longitudinal dimension (l_f ). This consideration results in negligibly small effects of variations of these parameters (h_f ,



_f ) on the magnitudes of effective coefficients (

e11,e₁₂,e₁₃) when other geometrical parameters remain with their constant values.

So, for simplicity, these dimensions (h_f ,



_f ) can be represented by a single parameter, A_f (A_f h_f



_f). Now, for specified values of A_r and L_r, the values of A_f and l_f may be assigned arbitrarily. But, this arbitrary assignment may have an effect on the magnitudes of the effective coefficients (e₁₁,e₁₂,e₁₃), and that implies a relation among these parameters (A_f ,l_f ) for the improved magnitude of, e₁₁. This relation is estimated at present by defining a parameter as fiber aspect ratio (

f f / f

S l A ). Presently, two values of A_r are considered by varying A_f with a constant value of A_c (A_ch_{c c} , A_r A_f /A_c). For every value of A_r or A_f, S_f is varied through the variation of l_f. It should be noted here that the value of l_c would vary with l_f for the constant value of, L_r.

Fig. 2.13 Variations of effective piezoelectric coefficients (e₁₁,e₁₂,e₁₃) with the fiber aspect ratio (S_f ) (L_r0.95).

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

Figure 2.13 illustrates the variations of coefficients (e₁₁,e₁₂,e₁₃) with the fiber aspect ratio (S_f ) for different values of area-ratio (A_r). It may be observed from Fig.

2.13 that there is a little variation of every coefficient (e₁₁/e₁₂/e₁₃) with the piezoelectric fiber aspect ratio (S_f ) for any value of A_r or A_f . Also, the nature of variation of any of the coefficients (e₁₁,e₁₂,e₁₃) does not alter for different values of area-ratio. Thus, for specified values of area-ratio and length-ratio, the value of S_f can be chosen from this result (Fig. 2.13), and that is taken at present as 1.2 10 ⁴ for an improved magnitude of, e₁₁. According to the present geometrical properties (SPFC, L_r 0.95, A_r0.6, S_f 1.2 10 ⁴) of RVE, the effective electro-elastic properties are illustrated in Table 2.3. The same table also illustrates similar properties when the continuous piezoelectric fiber of RVE is considered (CPFC,

r 1

L  , A_r 0.6). It may be observed from Table 2.3 that the magnitude of the major effective piezoelectric coefficient (e₁₁) decreases for the use of short piezoelectric fibers (L_r 0.95) instead of continuous fibers (L_r 1). This decrease in the magnitude of e₁₁ is an important concern when one intends to use SPFC actuator instead of CPFC actuator. The corresponding change in the actuation capability is studied in the next chapter considering the aforesaid arrangement of surface- electrodes.

Table 2.3 Effective electro-elastic properties of SPFC/CPFC in cylindrical coordinates.

Smart Composite

e11

(C/m²)

12/ 13

e e (C/m²)

26/ 35

e e (C/m²)

11

(C/Vm) (×10^-09)

22/ 33

 

(C/Vm) (×10^-09) SPFC 13.818 -1.158 1.220 0.95 0.302 CPFC 18.379 -0.231 1.429 8.75 0.28 Smart

Composite C11

(GPa) C12

(GPa) C23

(GPa) C22

(GPa) C44

(GPa) C55

(GPa) SPFC 31.635 4.734 4.201 12.623 1.628 2.229 CPFC 36.019 6.794 4.34 13.056 1.649 2.503

The foregoing results are evaluated for the SPFC/CPFC in cylindrical coordinates. Similar results are also evaluated for the SPFC/CPFC in rectangular coordinates, and the same characteristics in the electromechanical behaviour are

obtained. This may be due to the fact that the geometrical and material properties of the SPFC/CPFC remain the same for its construction in both the coordinate systems.

The values of the length-ratio (L_r), area-ratio (A_r) and fiber aspect ratio (S_f ) for the SPFC/CPFC in cylindrical coordinates are not altered for the SPFC/CPFC in rectangular coordinates. The corresponding properties of the SPFC/CPFC in rectangular coordinates are furnished in Table 2.4. In comparison to Table 2.3, Table 2.4 shows negligibly small change in the magnitudes of the effective coefficients due to the alteration of the coordinate system.

Table 2.4 Effective coefficients of SPFC/CPFC in rectangular coordinates.

Smart

Composite ¹¹ e (C/m²)

e21/e₃₁ (C/m²)

e62/e₅₃ (C/m²)

11

(C/Vm) (×10^-09)

22/

33

(C/Vm) (×10^-09) SPFC 13.486 -1.032 1.222 0.534 0.295 CPFC 18.380 -0.230 1.394 8.113 0.200 Smart

composite ¹¹ C

(GPa) ¹² C

(GPa) ²³ C

(GPa) ²² C

(GPa) ⁴⁴ C

(GPa) ⁵⁵ C (GPa) SPFC 29.979 5.008 4.150 12.533 1.607 2.224 CPFC 36.014 6.793 4.325 13.070 1.637 2.458