Chapter 2 Analytical model
2.5 Breakdown phenomenon
In Figure 2–8, the dielectric elastomer model with compliant electrodes is compressed with the increase of applied voltage. With a higher applied voltage, the thickness of elastomer is going to be thinner. Due to equation (2. 41), electric field is proportional to applied voltage and inversely proportional to thickness between the compliant electrodes. With the increase of applied voltage increases and decrease of thickness decreases, electric field is going to increase.
3 30
V V
E x x (2. 41)
In equation (2. 58), V is applied voltage across compliant electrodes, and E is electric field.
With a high enough applied voltage, electrons can be freed from the atoms of dielectric elastomer, resulting in a high current through that material as Figure 2–14.
The voltage is called breakdown voltage and the electric field is also called breakdown electric field. Because that the high current through the material, resulting in remnant charges in the material. The high current and remnant charges are going to make the properties of dielectric elastomer alter and destroy its structure. Thus, dielectric breakdown is an irreversible phenomenon. We can observe that the point of the high current occurs and find when the breakdown phenomenon generates by experiment.
Discussion of experiment will be mentioned in chapter 4. In this section, the physical meaning and derivation is the priority.
In Figure 2–15, the blue part of curve expresses that the sample is compressed with the increase of applied voltage. It reaches the critical point with the increasing voltage,
and the pull-in effect should be observed at the point. The red straight line represents the breakdown phenomenon and the pull-in effect occurs at the same time, then we can obtain the meaning of red line by the following derivation.
Due to equation (2.31. a) and (2. 41), the equation can be rewritten as
Where e is normalized electric field. According to Figure 2–15 and definition of slop, the slop of line in Figure 2–15 expresses as
slop 1 slop can be expressed as below.
1 1 obtain the normalized electric field which is equal to normalized electric field at pull-in point, eebd . As ep ebd , where ebd represents normalized electric field of breakdown, the normalized electric field at breakdown and pull-in point are the same. In other words, red straight line represents breakdown phenomenon and pull-in effect occurs at the same time.
Because of the meaning of red straight line, we can say that breakdown phenomenon occurs before or after pull-in effect. According to Figure 2–15, the figure is divided into two regions by red straight line. We discuss separatelyⅠ- region and Ⅱ- region by Case1 and Case2.
Case1:
In Ⅰ- region, the slop of straight line is bigger than red straight line From Figure 2–15. It can be written as Because normalize electric field at breakdown point is smaller than normalize electric field at pull-in point, breakdown phenomenon occurs before pull-in effect in Ⅰ- region.
Investigating the condition in this situation by the result, ebd ep. Normalize electric field at pull-in point can be expressed as
p p
p
e
(2. 46)
According to equation (2. 31), (2. 46) can be rewritten as
3 normalize electric field at pull-in point. Therefore, we can obtain condition by equation (2. 47) below.
2
According to equation (2. 42), (2. 48) can be rewritten as
10 01 breakdown electric field and constants of Mooney-Rivlin material. Equation (2. 48) and (2. 49) are general condition in Ⅰ- region. If the data at breakdown phenomenon by experiment conform to equation (2. 49), the breakdown point is certainly located in Ⅰ- region. In that breakdown phenomenon occurs before pull-in effect.
Case2:
In Ⅱ- region, the slop of straight line is smaller than red straight line from Figure 2–15. It can be written as Because normalized electric field at breakdown point is bigger than normalized electric field at pull-in point, breakdown phenomenon occurs after pull-in effect in Ⅱ- region.
Investigating the condition in this situation by the result, ebd ep. Derivation is similar with case1, the relationship below is obtained from equation (2. 47).
2
Similarly, equation (2. 51) can be rewritten as
10 01
2
2 1
0.3361 ( )( ) ( )
bd p p
r p
E C C V
m
(2. 52)
Equation (2. 51) and (2. 52) are the condition in Ⅱ- region. If the data at breakdown phenomenon by experiment conform to equation (2. 52), the breakdown point is certainly located in Ⅱ- region. In that breakdown phenomenon occurs after pull-in effect.
From equation (2. 49) and (2. 52), we can determine that breakdown phenomenon occurs before or after pull-in effect occurs. Equation (2. 49) and (2. 52) are the general forms for electromechanical coupling system depending on mechanical properties of hyper-elastic model (Mooney-Rivlin model) and electric properties of parallel-capacitor.
Figure 2–14. A schematic diagram of the dielectric elastomer model with a compliant electrodes, shows that the current through the elastomer occurred between the compliant electrodes with applied voltage increases.
Figure 2–15. Normalize thickness vs. Normalize voltage. Shows that red line expresses as breakdown phenomenon and pull-in effect occur at the same time. In Ⅰ- region, slop of line is bigger than the slop of red line and expresses as breakdown phenomenon occurs before pull-in effect. Oppositely, slop of line is smaller than the slop of red line in Ⅱ- region and expresses as breakdown phenomenon occurs after pull-in effect.
Current
V
Electrodes
Elastomer x3
Ⅰ
Ⅱ Pull-in
Normalize voltage
Normalize thickness
2.5.2 Discussion of VHB sample at breakdown
In chapter 2.5.1, we present the general condition used in determining breakdown phenomenon which occurs before or after pull-in effect. Form equation (2. 34), (2. 35), (2. 37) and Table 2-3, we can simplify the conclusion as below.
Case1:
Meanwhile, normalized electric field at pull-in point can also be
0.5934
In chapter 4, we aim at Mooney-Rivlin material: VHB, we can observe the point of the high current occurs and find when the breakdown phenomenon generates by experiment. The data which was obtained by experiment is consistent with equation (2.
54) or (2. 56). Breakdown phenomenon which occurs before or after pull-in effect can also get conclusion.