Strain and electric fields

For many problems it is convenient to separate the piezoelectric (i.e., strain induced) polarization P from electric-field-induced polarizations by defining D = P + fi , where s is the permittivity tensor. For uniaxial strain and electric field along the 1 axis, when the material is described by Eq. (4.1) with the E term omitted. 

Fig. 4.1. As shown on the left, the configuration of conducting plates on the flat faces of piezoelectric disks produces one-dimensional strain and electric field conditions with a guard-ring arrangement. On the right, the typical electrostatic conditions are shown. The axis through the thickness of the disk is chosen as the x axis.

Optical and electrical DC properties of AlGaN/GaN HEMTs fabricated from heterostructures grown on Si (111) substrates by MOCVD were studied. The correlation between the optical properties and electrical performances of the structures was established. An increased concentration of nonradiative centers, strains and electric field in the GaN layer correspond to worse 2DEG properties. The main origin of the increased defect density is assumed to be insufficient compensation of the stress caused by the Si substrate. 

Electrostriction is related to the converse piezoelectric effect. At modest electric field strengths, the piezoelectric equations given previously are adequate and there is a linear relationship between strain and electric field. However, at higher electric field strengths, these equations need to be extended to include a further term quadratic with respect to the electric field. The strain is now given by... 

Based on Equations (16.9) and (16.15), the Maxwell effect and electrostrictive effect result in the same relationship between the strain and electric field and they therefore share some common features. For instance, an apparent piezoelectric effect can be observed when a DC bias is applied the strain response can be enhanced by the nonuniformity of the electric field, which can be created either by employing nonuniform materials or by the presence of the space (trapping) charge. Due to the electrostrictive effect and the appearance of the space charge, an insulation material can exhibit piezoelectricity and is known as an electret [9, 10]. The piezoelectric constant of an electret depends on the space charge and its distribution as well as the nonuniformity in the elastic properties and electrostrictive coefficient of the materials. 

Z, Z column matrix of strains and electric field strengths... 

Mean stresses and electric flux densities Y as well as mean strains and electric field strengths Z are composed of the corresponding mean fields in the inclusion and matrix phase as indicated by the superscripts i and m. Crossconnecting these fields, the constitutive relations of the homogenized composite, as well as of the individual material phases, may be given as follows ... 

The unique dependence of strains and electric field strengths as well as stresses and electric flux densities in the individual material phases upon the overall fields of the composite may be formulated with the aid of the concentration... 

The overall strains and electric field strengths Z of the composite can be either interpreted as a response to the applied stresses and electric flux densities or are the direct implication of the boundary conditions. In absence of the inclusion, they would prevail throughout the domain A. The resulting strains and electric field strengths Z inside the inclusion may be assembled... 

So the mechanical relation of Eq. (7.31) can be extended to incorporate the electric case. Thus, the strain and electric field strength measures of the wall,... 

In the lower line, use has been made of the relation between strain and electric field strength measures of wall and beam, provided by Eq. (7.33) and adapted to the virtual expressions. 

The discretization of the mechanical and electric degrees of freedom as well as of mechanical strains and electric field strengths may be... 

The analogous matrix B(xj) of interpolation functions for the mechanical strains and electric field strengths arises, when the discretization of the degrees of freedom, as given by Eq. (9.18a), is substituted into the vector of... 

X strain and electric field strength column matrix... 

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