where [D.sub.2n], [D.sub.1n] are the normal components of the electric displacement
vector: [D.sub.2n] = [[epsilon].sub.0][[epsilon].sub.2] [E.sub.2n], [D.sub.1n] = [[epsilon].sub.0][[epsilon].sub.1] [E.sub.1n] [epsilon].sub.0] = 8.85 [10.sup.-12] F/m is the electric constant; [[epsilon].sub.1], [[epsilon].sub.2] are the dielectric permeabilities of dielectrics, a is the surface density of electric charges; [E.sub.1n], [E.sub.2n] are the normal, [E.sub.1t], [E.sub.2t] are the tangential components of the electric field strength vector of the first and the second dielectrics, respectively.
where p after a comma, for p = i, j, k, l, denotes partial differentiation with respect to [x.sub.p] * [u.sub.i] and [[sigma].sub.ij] are the components of the phonon displacement vector and stress tensor, [w.sub.i] and [H.sub.ij] are the components of the phason displacement vector and stress tensor, and [D.sub.i] and [E.sub.i] are the components of the electric displacement
Because the electric displacement
[D.sub.3] does not change with the thickness, the above equation is subjected to homogenization in the thickness direction:
The time-dependent stretch, nominal electric field-stretch, and nominal electric field-nominal electric displacement
curves of representative applied voltage frequency are presented in Figure 3.
[9,10], we can derive the next formulas for stress tensor and electric displacement
When there is no contact at a point on the surface (i.e., [u.sub.v] < h), the gap is assumed to be an insulator, there are no free electrical charges on the surface and the normal component of the electric displacement
In what follows, we use the equations of mechanical equilibrium and small deformation in the strain constitutive relation (2), to account for the induced stresses and strains and then "eliminate" the stress dependence from the constitutive relations for electric displacement
and magnetic induction fields.
According to the constitutive relations in (12), and considering the displacement-strain relations, the electric displacement
can be given in the following form:
The maximum electric displacement
obtained from the dipolar D-E loops of composite films under the breakdown field was summarized in Fig.
As mentioned in , it must be emphasized that in the x, y transverse plane, the magnetic field is a purely gradient field [nabla] x H = 0 (with open field lines, beginning and ending on different wires), and that the electric displacement
vector is a purely solenoidal field [nabla] x D = 0 (with closed field lines, embracing one or several wires).
where [D.sub.i] is electric displacement
, [E.sub.i] is electric field and P is polarization vector.
The electric displacement
may be obtained as a linear combination of the strain and electric field as follows [17, 18]