On electromechanical instability in semicrystalline polymer under
unequal biaxial prestrain
Huadong Yong
Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of
Education of China, and Department of Mechanics, Lanzhou University, Lanzhou 730000,
People’s Republic of China
Email: yonghd@lzu.edu.cn
ABSTRACT
Semicrystalline polymers are promising materials for actuators and capacitors. In response to the
electric field, the polymer can undergo large deformation. Based on a simple model, the critical
electric field in the polymer is investigated in the present paper. The polymer is assumed to be
incompressible and specified by the power law relation. Using the stability condition of the
determinant of the Hessian, the critical electric field can be obtained. Comparing the results from
prestress with prestrain, it is shown that the critical electric field is related to the hardening
exponent N and may be restricted by the necking instability. To enhance the critical electric
field, the anisotropic polymer is discussed based on the simple free energy function. The results
show that the anisotropic parameter can increase the critical electric field.
Keywords: semicrystalline polymer; anisotropic; electromechanical instability; Hessian
1
1. Introduction
High energy density dielectrics are widely used as hybrid electric vehicles and
microelectronics [1-3]. Due to low cost and high breakdown strength, semicrystalline polymers
have become the material of choice for an increasing number of high energy density dielectrics.
Polymers exhibit promising and impressive properties. Although the permittivity in the polymer is
relatively low, the structural simplicity makes it convenient to control the device behavior.
However, the major problem of such polymer is that electromechanical instability occurs during
the application of the electric field.
Semicrystalline polymers deform under the application of an electric field and the material
expands in plane and reduces in thickness. The reduction in thickness unavoidably leads to larger
electric field. Due to the positive feedback, the electric field can locally reach the critical electric
field of the material so that pull-in instability takes place [4]. Another failure mode of the polymer
is electric breakdown [5-7]. If the polymer is prestretched and fixed, the electric breakdown occurs
in the polymer as the electric field reaches the electric breakdown strength [8]. In addition, unlike
dielectric elastomer, the semicrystalline polymer may exceed the elastic limit and undergo plastic
deformation. Thus, the occurrence of the necking instability in tension is expected for the
semicrystalline polymer [9-11].
There have been many efforts to understand the complicated nonlinear behavior of dielectrics
[12-17]. The electromechanical instability has been studied and realized as a failure mode of
polymer insulators over the past decades. Stark and Garton studied the pull-in instability and
necking instability earlier [18]. Based on the nonlinear field theory of deformable dielectrics, Zhao
and Suo formulated a method to analyze the electromechanical stability [4]. The critical electric
2
field of a polar fluoropolymer was investigated by taking into consideration of the plastic
deformation [19]. Recently, the critical conditions for the electromechanical instability of a
semicrystalline polymer were calculated [20]. The electromechanical instability of semicrystalline
polymer was analyzed based on the exponential model with two different material constants [21].
In this work, we study the critical electric field of a semicrystalline polymer which is subjected
to an electric field and a fixed prestrain. For a stiff dielectric which is prestrained, the
electromechanical instability of the dielectric is limited by critical electric field and necking
occurring during plastic deformation. Based on the formulation of nonlinear electroelasticity and a
relatively simple model, the critical electric field for prestrain is calculated. We then discuss the
effect of the hardening exponent N , stretch ratio 2 1 and anisotropic parameter on the
critical electric field. Finally, we present the conclusions of this article.
2. Basic Equations
Figure 1 shows the sample geometry and loading conditions. Consider a semicrystalline
polymer with the dimensions of L1 L2 L3 in the reference configuration. The electrodes are coated
on both faces of the semicrystalline polymer. The polymer deforms in the presence of an electric
field and mechanical loading. In the deformed configuration, the polymer has the dimension
l1l2 l3 . In analyses of the deformation of polymer a common simplification is to ignore the
piezoelectric effects [22].
L3
l3
L2
l2
P1
L1
undeformed
l1
deformed
3
P2
V
Fig. 1. (a) A semicrystalline polymer with two compliant electrodes in the undeformed state. (b)
The semicrystalline polymer deforms under a mechanical loading and a voltage.
The principal stretches are defined by
1  l1 L1 , 2  l2 L2 , 3  l3 L3
(1)
The analysis is based on use of the energy function, and the energy formulations for
semicrystalline polymers can be motivated by the deformations occurring in response to the
applied electric field and mechanical loading. For the polymer which is subjected to a mechanical
loading and a voltage, the free energy is [23]
W  Wc  We
(2)
where Wc is the strain energy density function of semicrystalline polymer, and We is the
dielectric energy. It is to be noted that the effect of crosslinks on polarization is neglected in the
polymer [24].
When the semicrystalline polymers are subjected to the prestress and electrical field, the
materials may undergo pull-in instability and necking instability [18, 20]. For the finite strain and
plastic deformation analysis, the most appropriate measure of strain in the formulations is the
logarithmic strain which has principal values [25]
 i  ln  i  ,  i  1, 2,3
With
m 
1
 1   2   3 
3
as
the
mean
strain
of
(3)
the
logarithmic
strain
and
 i   i   m  i  1, 2,3 as the strain deviator, the effective strain is taken to be [25]
2
 i i
3
e 
(4)
To obtain the analytical results we assume that the polymer is characterized by a general power
4
law relation [25]
Wc 
K
 eN 1
N 1
(5)
where K is the strength index and N is the strain hardening index. The parameters K and
N can be determined by fitting a curve to a uniaxial tension test. It is to be noted that N is
normally in the region 0.1  0.6 for polymers [20]. The failure mechanism of the dielectric
materials depends on the electro-elastic interactions, so we adopt the model of the ideal dielectric
elastomer which was proposed by Zhao and Suo [4]
D 2 3
We 
2 12
(6)
where D is the nominal electric displacement.
For the formulation based on the energy function W , the true stress
 and nominal
electric field E are given as [23]
σF
W
W
, E
F
D
(7)
in which F is deformation gradient tensor, I is the second order unit tensor. Inserting eq. (2)
into eq. (7), we can obtain the components of the true stress and nominal electric field.
The counterpart of the nominal stress tensor denoted here by s is defined by
s  F 1σ
(8)
In general, it is appropriate to adopt the incompressibility constraint for the polymer 12 3  1 .
Due to the condition of incompressibility, we may regard the polymer as a system of two degrees
of freedom: 1 and 2 . First, the polymer is subjected to the prestresses in the direction L1
and L2 . Applying the general power law model and boundary conditions, one has [26]
 E 2 K   2 ln 1   1 s1 K  14
N


(9)
2
in which s1  s2 . By rewriting eq. (9) through 3  1 , it can be proved that the last equation
5
is equivalent to the eq. (2) in the Ref. [20].
As pointed out in [26, 27] , after obtaining the relation between voltage and stretch, it is
necessary to compare it with the critical electric field and check whether it is stable. For a fixed
prestrain, we can obtain the critical electric field. The critical stretch can be obtained at the point
where the two curves intersect.
Following the procedure given by Norris [4, 28], the critical electric field can be obtained when
the determinant of the Hessian H equal to zero. The Hessian is given as
 W

 11
 W
H
 12
 W

 1D
W
12
W
2 2
W
2 D
W 

1D 
W 

2 D 
W 

DD 
(10)
Solving the equation det  H   0 , the nonlinear algebraic equation determines the critical
electric field for given values of the prestrain. For the equal biaxial prestrain 1  2 , the critical
electric field is expressed as follows
 N 1

 4 
if   exp 
 Ec2 K   2 ln  
N 1
 N  1  2 ln   N  1  4 ln  
3 4
(11)
N 1
 N  1  2 ln   N  1  4 ln  
3 4
(12)
 N 1

 4 
if   exp 
 Ec2 K   2 ln  
The eqs. (11)-(12) are exact expressions from which the critical electric field can be calculated
once the prestrain  is known. One can see immediately that as the prestretch   1 , the
critical electric field  Ec K   . Also note that the critical electric field  Ec K is a
2
2
piecewise function.
6
3. Results for isotropic polymer
In this section, the nominal electric field and critical electric field are calculated for different
cases of electric and mechanical loading. First, we consider the equal biaxial case. The results are
shown in fig. 2, where the nominal electric field with three different prestresses is plotted which is
equivalent to the results in Ref. [20].
0.5
~2
Ec /K
N=0.3
0.4
~2
E /K
s/K=0
~2
E /K
0.3
s/K=0.3
0.2
s/K=0.5
0.1
0.0
1.0
1.1
1.2
1.3
1.4

Fig. 2. The nominal electric field  E
2
K at several values of the prestress and the critical
2
electric field  Ec K for the case of prestrain as 1  2 .
It is observed that the critical electric field (  Ec K ) for the prestrain is a monotonic decreasing
2
function of  . Note that the critical voltage is the point of intersection between the curve
 E 2 K and  Ec2 K . For the power law relation, there is only one intersection. In this case,
the nominal electric field  E
2
K reaches the peak value at the point where the two curves
intersect. That is to say that the stretch is stable as the nominal electric field is smaller than the
critical nominal electric field.
In addition, it is to be noted that when the critical electric field decreases to zero, the critical
stretch corresponds to the critical condition of the necking instability. The features of the critical
electric field and prestrain relation with different stretch ratios are shown in figs. 3 and 4.
7
0.9
N=0.3
0.6
~2
Ec /K
2=1
2=1
0.3
2=1
0.0
1.0
1.1
1
1.2
1.3
Fig. 3. The critical electric field as a function of the 1 for several values of the stretch ratio
2  1 as N  0.3 .
As shown in fig. 3
 N  0.3 , the critical electric field decreases with the increasing of the
stretch ration 2  1 . The critical stretch where the necking instability occurs is expected to
decrease as well. Comparing the cases of the equal biaxial and unequal biaxial prestrains, the
critical electric field is finite as 1  1.0 in the unequal biaxial prestrain case.
1.00
0.75
~2
Ec /K
N=0.6
0.50
1=2
0.25
2=1
2=1
0.00
1.05
1.20
1.35
1.50
1.65
1
Fig. 4. The critical electric field as a function of the 1 for several values of the stretch ratio
2  1 as N  0.6 .
Figure 4 shows the dependence of the critical electric field as a function of prestrain 1 for the
case of N  0.6 . Usually, the higher the stretch ratio 2  1 is, the lower the critical electric
field will be. The behavior is similar to the case of N  0.3 . The unusual feature is that as
8
1  2 , the prestrain reaches the critical value and the necking instability will occur. As 1  2 ,
before the necking instability occurs, there is no real root for the determinant of the Hessian H
equals to zero. That is to say, the system will become unstable in advance. The results indicate that
the stability is sensitive to the hardening exponent N .
4. Polymer with anisotropic property
From above discussions on the isotropic polymer, it can be found that the equal biaxial prestrain
condition is more stable than the unequal biaxial prestrain condition. The semicrystalline polymer
becomes unstable before the necking instability occurs for unequal biaxial prestrain. The reason
may be that for equal biaxial prestrain condition, the system is in a relative balance state and the
wrinkling is difficult to occur. While for unequal biaxial prestrain condition, the deformations in
1 and 2 are not the same and wrinkling is easy to occur for larger deformation [29]. Thus, it
is necessary to avoid the instability for the unequal biaxial prestrain condition. There may be many
ways to control the instability of unequal biaxial prestrain condition. One way is to use the
anisotropic semicrystalline polymer to enhance the stability. As the polymer is subjected to the
unequal biaxial prestrain, the anisotropic property can remedy the difference of deformation in 1
and 2 directions [30]. Then, we will discuss the electromechanical instability in the anisotropic
polymer. It is to be noted that the anisotropic property is classified as the in-plane anisotropy and
out-of-plane anisotropy. First, we consider the in-plane anisotropic polymer. To avoid the
complexity of mathematics, we use simple free energy function and the anisotropic free energy is
defined as [25, 30, 31]
e 
2 2
 ln  1   c1 ln 2  2   ln 2  12  
3
(13)
where c1 represent the anisotropic property in 1 and 2 directions. Following the same
9
procedure given in section 2, the critical nominal electric field can be obtained for the anisotropic
polymer.
0.5
(a)
0.4
c1=0.5
1.0
2.0
3.0
~2
Ec /K
0.3
0.2
N=0.4
2=1.21
0.1
0.0
1.00
1.04
1.08
1.12
1.16
1.20
1.24
1
0.6
(b)
0.5
c1=0.5
~2
Ec /K
0.4
N=0.6
2=1.21
1.0
2.0
3.0
0.3
0.2
0.1
0.0
1.0
1.1
1.2
1.3
1.4
1
Fig. 5. The critical electric field for different anisotropic parameters c1 as N  0.4 (a) and
N  0.6 (b).
Figure 5 shows the critical electric field as a function of prestrain 1 , in which 2  1.21 .
From fig. 5, we can see that the anisotropic parameter c1 has obvious effect on the critical
electric field and critical prestrain stretch. Both the critical electric field and critical prestrain
increase with the increasing of the parameter c1 . This is due to the reason that the anisotropic
deformation remedys the difference of prestrain in 1 and 2 directions. As the anisotropic
parameter c1 increases, the polymer is near to the relative balance. Then, the stability can be
enhanced by increasing the anisotropic parameter c1 for unequal biaxial prestrain condition.
10
Second, we turn our attention to the out-of-plane anistropic polymer. For this case, the free
energy of polymer is given by [25, 31]
e 
2
 ln 2  1   ln 2  2   c2 ln 2  12  
3
(14)
where c 2 represents the anisotropic property between in-plane and out-of-plane directions.
0.8
0.7
(a)
0.6
c2=0.5
c2=1.0
~2
Ec /K
0.5
c2=2.0
0.4
c3=3.0
0.3
N=0.4
2=1.21
0.2
0.1
0.0
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1
1.0
0.8
(b)
c2=0.5
c2=1.0
c2=2.0
0.6
~2
Ec /K
c2=3.0
N=0.6
2=1.21
0.4
0.2
0.0
1.0
1.1
1.2
1
1.3
1.4
Fig. 6. The critical electric field for different anisotropic parameters c 2 as N  0.4 (a) and
N  0.6 (b).
Figure 6 shows the effect of anisotropic parameter c 2 on the critical electric field. Compared
fig 6 with fig. 5, it can be found that the in-plane anisotropy is different with out-of-plane
anisotropy. It is interesting to find that with the increasing of c 2 , the critical electric field
increases and critical prestrain decreases. In other words, a larger anisotropic parameter c 2 only
leads to a higher critical electric field. However, the deformation is limited.
11
5. Conclusions
When the polymer is subjected to the prestress and electric field, the pull-in instability and
necking instability may result from the positive feedback during large deformation. However, as
the polymer is prestretched and fixed, the electric field will be restricted by critical electric field.
In this paper, we describe the simple model for modeling the electromechanical instability in the
semicrystalline polymers. Furthermore, it is found that the electromechanical instability depends
on the hardening exponent N . Then, the results show that the anisotropic parameters have
significant influence on the electromechanical instability.
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