PIEZOELECTRIC MATERIALS
4.5 DYNAMIC FORCE AND MOTION SENSING
4.6.1 Extensional 31 Piezoelectric Devices
Piezoelectric materials are often used in a multilayer composite as extensional or bending actuators. The composite consists of one or two layers of piezoelectric ma- terial and an inactive substrate made from a material such as brass, aluminum, or steel. The poling direction of the piezoelectric material is parallel with the thickness direction of the piezoelectric layer and the desired extension is perpendicular to the poling direction. Therefore, the 31 mode of the piezoelectric material is utilized in these applications. In a composite extensional actuator, the amount of strain and stress produced is a function not only of the piezoelectric material properties, but also the properties of the inactive layer.
In this section, expressions for the strain and stress produced by these composite actuators are derived as a function of the piezoelectric material properties and the ma- terial properties of the inactive layer. The derivation will focus on a typical composite
z
tp/2 ts tp/2
modulus Y1E modulus Ys
wp upper piezoelectric
layer elastic substrate
lower piezoelectric layer
x 3
1
Figure 4.19 Composite actuator consisting of an elastic substrate and two piezoelectric layers.
layup that consists of piezoelectric layers attached to the surfaces of an inactive sub- strate as shown in Figure 4.19. The elastic substrate has a thicknesstsand a modulus ofYs. Each piezoelectric layer has a thickness oftp/2 and a short-circuit modulus of Y1E. For simplicity we assume that the layers are symmetric about the neutral axis of the composite and that the active and inactive layers are equal in width. The width of the piezoelectric materials and the substrate is denotedwp.
Consider the case in which the voltage applied to the piezoelectric layers is aligned with the poling direction of both piezoelectric layers (Figure 4.20). Without a substrate and with no restraining force, the strain in the piezoelectric layers would be equal d13E3. To determine the strain produced in the piezoelectric composite, first write the constitutive relationships for each of the three layers within the composite. These are
S1=
1
Y1ET1+d13E3
ts
2 ≤z≤ 1
2(ts+tp) 1
Ys
T1 −ts
2 ≤z≤ ts
2 1
Y1ET1+d13E3 −1
2(ts+tp)≤z≤ −ts
2
(4.118)
v
v
tp/2
tp/2 ts z
x
Figure 4.20 Electrical connections for a piezoelectric extender actuator.
Multiplying by the modulus values and integrating over the yandzdirections for their respective domains produces the expressions
wptp
2 Y1ES1=
y,z
T1dy dz+wptp
2 Y1Ed13E3 (4.119) wptsYsS1=
y,z
T1dy dz (4.120)
wptp
2 Y1ES1=
y,z
T1dy dz+wptp
2 Y1Ed13E3. (4.121) Assuming that the strain in all three regions of the beam is the same, which is equivalent to assuming that there is a perfect bond and no slipping at the boundaries, these three expressions can be added to obtain the equation
Yswpts+Y1Ewptp
S1=
y,z
T1dy dz+Y1Ewptpd13E3. (4.122) The externally applied force f is in equilibrium with the stress resultant. If the force applied externally is equal to zero, the expression for the strain can be solved from equation (4.122):
S1= Y1Etp
Ysts+Y1Etp
d13E3. (4.123)
The termd13E3is recognized as the free strain associated with the piezoelectric ma- terial if there were no substrate layer. The coefficient modifying the free strain can be rewritten as a nondimensional expression by dividing the numerator and denominator byY1Etp:
S1
d13E3 = 1 1+e
, (4.124)
where
e= Ysts
Y1Etp
. (4.125)
Equation (4.125) illustrates that the variation in the free strain of the piezoelectric extender is a function of the relative stiffness between the piezoelectric layer and the substrate layer,e. A plot of equation (4.124) is shown in Figure 4.21. As the relative stiffness approaches zero, indicating that the stiffness of the piezoelectric is large compared to the stiffness of the substrate layer, the free strain of the composite approaches the free strain of the piezoelectric layers. The stiffness match point at whiche=1 produces a free strain in the composite that is one-half that of the free strain in the piezoelectric layers. Increasing the stiffness of the substrate layer such
10–2 10–1 100 101 102 0
0.2 0.4 0.6 0.8 1
S1/d13 E3
Ψe
Figure 4.21 Variation in free strain for a piezoelectric extender as a function of the relative stiffness parameter.
thate1 produces an extensional actuator whose free strain is much less than the free strain of the piezoelectric layers.
The deflectionu1of a piezoelectric extender of total lengthLcan be expressed as a function of the voltage by noting that S1=u1/Lpand that the electric field is equal to the applied voltagevdivided by the piezoelectric layer thicknesstp/2. Substituting these relationships into equation (4.125) produces the expression for the deflection:
u1 = 2 1+e
d13Lp
tp
v . (4.126)
Replacing the stress resultant in equation (4.122) with an applied force f divided by the area of the composite and solving for the blocked force yields
fbl=2Y1Ewpd13v. (4.127) The trade-off between force and deflection for an extensional actuator is shown in Figure 4.22, which illustrates that the relative stiffness parameter will change the deflection of the extensional actuator but not the blocked force. The blocked force is equivalent to that of a piezoelectric layers combined, but the deflection is reduced as the relative stiffness parameter increases from much smaller than 1 to much larger than 1.
Example 4.9 A piezoelectric extensional actuator is fabricated from two 0.25-mm layers of PZT-5H and a single layer of 0.25-mm brass shim. Compute the free strain in the device when the applied electric field is 0.5 MV/m.
–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.