PIEZOELECTRIC MATERIALS
4.5 DYNAMIC FORCE AND MOTION SENSING
4.6.2 Bending 31 Piezoelectric Devices
–f Y1E wpd13v
2d13Lpv/tp x
Ψe >> 1
Ψe = 1
Ψe << 1
Figure 4.22 Force–deflection trade-off for an extensional actuator as a function of the relative stiffness parameter.
Solution The expression for the free strain is shown in equation (4.124) normalized with respect to the free strain in the unconstrained piezoelectric layers. The free strain in the unconstrained piezoelectric layers is
d13E3 =(320×10−12m/V)(0.5×106V/m)
=160µstrain.
Brass shim is assumed to have a modulus of 117 GPa. Recognizing thattpin equation (4.125) is thetotalthickness of the piezoelectric layers, we can compute the relative stiffness parameter:
e= (117×109N/m2)(0.25×10−3m) (50×109N/m2)(0.5×10−3m)
=1.17.
The free strain in the composite extensional actuator is computed from equation (4.124):
S1= 160µstrain
1+1.17 =73.7 µstrain.
v
v
tp/2
tp/2
ts x
z
Figure 4.23 Electrical connections for a piezoelectric bimorph.
.
that the electric field is in the same direction as the poling direction in one of the layers, whereas in the second layer the electric field is in the direction opposite the poling direction. This is illustrated in Figure 4.23.
Application of an electric field to both layers produces extension in one of the layers and contraction in the other. The net result is a bending of the material. Assuming a perfect bond between the inactive layer and the piezoelectric layers, and assuming that the piezoelectric layers are symmetric about the neutral axis of the composite, the bending will result in the deformed shape shown in Figure 4.24.
Under the assumption that the field is in the poling direction in the top layer and opposite to the poling direction in the bottom layer, we can write the constitutive equations of the composite as
S1(z)=
1
Y1ET1(z)+d13E3 ts
2 ≤z≤ 1
2(ts+tp) 1
Ys
T1(z) −ts
2 ≤z≤ ts
2 1
Y1ET1(z)−d13E3 −1
2(ts+tp)≤z≤ −ts
2.
(4.128)
du3/dx
u3
v
v Figure 4.24 Bending induced in a symmetric piezoelectric bimorph.
.
Assuming that the Euler–Bernoulli beam assumptions are valid, the relationship be- tween the strain and the curvatureκis
S1(z)=κz. (4.129)
Substituting equation (4.129) into the constitutive relations and rewriting, we obtain Y1E(κz)=T1(z)+Y1Ed13E3
Ys(κz)=T1(z) (4.130)
Y1E(κz)=T1(z)−Y1Ed13E3.
The equilibrium expressions for the moment are obtained by multiplying equation (4.130) byzand integrating over the domain inyandz. The result is
Y1Ewpκ t3p
24+t2pts
8 +tpts2 8
=
y,z
zT1dy dz+Y1Ewpd13 t2p
8 +tpts
4
E3 Ysκwpts3
12 =
y,z
zT1dy dz (4.131)
Y1Ewpκ t3p
24+t2pts
8 +tpts2 8
=
y,z
zT1dy dz+Y1Ewpd13
t2p 8 +tpts
4
E3. Adding the results from the three domains together yields
Y1Ewpκ tp3
12+t2pts
4 +tpts2 4
+Ysκwpts3 12
=
y,z
zT1(z)dy dz+Y1Ewpd13
t2p 4 +tpts
2
E3 (4.132)
The integration of the stress component on the right-hand side of the expression is the moment resultant from externally applied loads. If this moment resultant is zero, we can solve for the curvature as a function of
κ = Y1E
tp2/4+tpts/2 Y1E
t3p/12+t2pts/4+tpts2/4 +Ys
ts3/12d13E3. (4.133)
A nondimensional expression for the curvature of the composite beam due to piezo- electric actuation is obtained by dividing the numerator and denominator by the inertia per unit width,Y1Et3p/12, and making the substitutionτ =ts/tp. The result is
κ ts
2d13E3
= 3τ/2+3τ2 1+3τ +3τ2+
Ys/Y1E
τ3. (4.134)
0 5 10 15 20 0
0.2 0.4 0.6 0.8 1
ts /tp κ t s/2d 13E 3
E = 1
E = 1/2 Ys/Y1
Ys/Y1 Ys/Y1
E = 2
Figure 4.25 Variation in nondimensional curvature for a composite bimorph.
A plot of equation (4.134) is shown in Figure 4.25 for three different values of Ys/Y1E. For a constant value ofts, we see that the curvature will reach a maximum at a specific value ofts/tp. Increasing the substrate thickness relative to the piezoelectric layer thickness will produce a decrease in the induced curvature.
The nondimensional expression in equation (4.134) has physical significance if we examine the strain induced through the thickness of the bimorph. The strain induced at the interface between the substrate and the piezoelectric layers is equal toκts/2;
therefore, we can write the strain at the interface as a normalized expression:
S1|z=ts/2
d13E3 = 3τ/2+3τ2 1+3τ+3τ2+
Ys/Y1E
τ3. (4.135)
The plot in Figure 4.25 can now be examined as the ratio of the induced bending strain to the extensional strain induced in the piezoelectric by the application of the electric field. As expected, this value is always less than 1. At large values ofτ, we note that the induced strain is small due to the fact that the substrate layer is much thicker than the piezoelectric layers. As small values ofτ, the induced strain at the interface becomes very small because the thickness of the substrate layer is small and the interface is becoming very close to the neutral axis of the composite bimorph.
The strain at the outer surface of the composite bimorph can also be obtained by evaluating S1atz=12(ts+tp). The result in nondimensional form is
S1|z=ts/2+tp/2
d13E3
= (3τ/2+3τ2) (τ +1) τ
1+3τ +3τ2+ Ys/Y1E
τ3. (4.136) A plot of equation (4.136) for a value ofYs/Y1E=1 is shown in Figure 4.26.
The solid curve illustrates the variation in strain at the outer fibers of the composite
0 5 10 15 20 0
0.25 0.5 0.75 1 1.25 1.5
ts /tp S 1 /d 13E 3
Figure 4.26 Variation in strain at the outer fibers of the composite bimorph (solid) and at the substrate–piezoelectric interface (dashed) forYs/Y1E= 1.
bimorph normalized with respect to the free strain produced in extension, d13E3. Also plotted is the normalized strain at the interface between the substrate and the piezoelectric layer (the dashed curve). The figure illustrates that these two values converge for large values ofτ, due to the fact that the thickness of the piezoelectric layer becomes small. There is a large difference in the induced strain for small values ofτbecause the thickness of the piezoelectric layer is large compared to the substrate thickness.
Example 4.10 A symmetric piezoelectric bimorph is constructed from 2-mm-thick brass shim with piezoelectric thicknesses of 0.25 mm for each layer. Plot the variation in the strain as a function of thickness through the bimorph and label the strain at the substrate–piezoelectric interface and at the outer fibers of the bimorph. The piezoelectric material is APC 856 and the applied field is assumed to be 0.5 MV/m.
Solution The variation in the strain through the thickness is given by equation (4.129), and the expression for the nondimensional curvature is obtained from equa- tion (4.134). The free strain in extension is computed from
d13E3=(260×10−12m/V)(0.5×106V/m)=130µstrain.
The thickness ratio in the bimorph is computed to be τ = 2 mm
0.5 mm =4.
The nondimensional curvature is computed from equation (4.134):
κ ts
2d13E3 = (3)(4/2)+(3)(42)
1+(3)(4)+(3)(42)+(117 GPa/66.7 GPa)(43)
=0.3117. The curvature is computed from
κ =(0.3117)
2
2×10−3m 130×10−6m/m
=0.0405 m−1.
The strain through the thickness is computed from equation (4.129). The two strain values of interest are
S1|z=ts/2=(0.0405 m−1)2×10−3m
=40.5µstrain. 2 S1|z=ts/2+tp/2=(0.0405 m−1)
2×10−3m+0.35×10−3m 2
=45.6µstrain.
The results are illustrated in Figure 4.27. The diagonal line represents the strain through the thickness of the bimorph with the values labeled at the outer fibers and at the substrate–piezoelectric interface. Also shown to scale is the free extensional strain produced by the piezoelectric layers at the electric field value specified. Note that the free strain is approximately three times that of the maximum strain in the bimorph.
40.5 µstrain 45.6 µstrain
130 µstrain
Figure 4.27 Variation in strain through the thickness of a symmetric bimorph for example 4.10.
This is due to the fact that the piezoelectric layers must overcome the bending stiffness of the inactive substrate to produce curvature in the bimorph.