Unidirectional Electrostatic Fields

For the intended normal mode actuation or sensing with electrodes responsible for the polarization direction, the latter case is practically not possible, while the prior needs to be further examined. The complications considered in 

For unidirectional electric field strength confined to the component E3, the transverse electric flux density components D, D2 as well as shear stress 

The internal energy of an electrostatic system is represented by the product of the correlated field strength and flux density components, as exemplarily derived for the virtual work of internal charges, Eq. (3.53). The shear strain induced flux density components Di and D2, as given in Eq. (4.22), do not contribute by virtue of the above assumption. Thus, the assumption of unidirectional electric field strength is equivalent to the neglect of shear associated electrostatic energy contributions. 

For the remaining case of unidirectional electric flux density confined to the component D3, the transverse electric field strength components E, E2 may be expressed in terms of the shear strains 731, 723. Therewith Ei and E2 can be eliminated from the constitutive equations by static condensation. Thus, this modification of Eqs. (4.20) represents a purely mechanical interaction with strengthened shear stiffnesses as the result of the piezoelectric effect  

Thereby, from the energetic point of view again, the shear associated electrostatic energy contributions, so to speak, are transferred to the elastic energy. 

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For the typical applications of piezoelectric materials, simplifying assumptions with respect to the mechanical and electrostatic fields are reasonable. These may be introduced in consequence of the spatial extent and electroding of the considered structure and, namely, are assumptions of planar mechanical and unidirectional electrostatic fields. Thereby, the variants of the latter, although... 

The case of unidirectional electric field strength is expressed by U23 = U31 = 0, while the case of unidirectional electric flux density is indicated by U23 = vzi = 1. The above discussion on the influences of mismatched electroding for the shear cases identified the two unidirectional field assumptions as extremes with the actual effective properties in between. Thus, V2z and vzi may be determined as functions of the electrostatic field distributions affected by the geometry of structure and electrodes as well as the material properties in the ranges 0 < vzz < 1 and 0 < U31 < 1. This might be used to represent the macroscopic mechanical behavior of piezoelectric structures subjected to shear induced transverse electrostatic fields within the simplified framework of assumingly unidirectional electrostatic fields. Thereby the essential and beneficial consequences would be inherited as conclusively formulated ... 

By virtue of the assumptions of unidirectional electrostatic fields and planar mechanical stress, the electromechanically coupled constitutive relations have been modified significantly. In example, the formulation on the left-hand side of Eq. (4.10a) reduces to Eq. (4.28) or (4.30). A transformation of coordinates on the considered plane may be performed as a rotation around the axis normal to this plane. In the case of Eq. (4.30), the base vector C2 represents the axis of rotation and thus this planar rotation may be formulated as follows ... 

The characterization and control of electrostatic forces are of particular interest. Electrostatic forces depend on the electric charge and potential at the particle surfaces. When subjected to a uniform, unidirectional electric field E. charged colloidal particles accelerate until the electric body force balances the hydrodynamic drag force, so that the particles move at a constant average velocity v. This motion is known as electrophoresis, and v is the electrophoretic velocity. 

Ji is the electron flux vector, C is the electrochemical potential acting on the electron flux, and Js represents the total entropy density flux vector. We choose this expression, rather than a version based on Eq. (6.1.29), because we wish to treat separately the effects of temperature and of electrochemical potential. The latter involves aU the contributions associated with temperature gradients, electron density gradients, and the externally imposed electrostatic field. It is expedient to introduce a current density vector as = e Ti- Then, along one dimension, we adopt i) = T 3s T + -V( /e) as our dissipation function. For this unidirectional flow pattern, the... 

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