Electric flux density unidirectional

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

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 ... 

The implications of both discussed simplifications of the constitutive relations may be illustrated by sununarizing all the shear cases. With the assumption of unidirectional electric field strength for the stresses of Eqs. (4.22), respectively, unidirectional electric flux density for the stresses of Eqs. (4.23), whereby the additional terms of the latter are furnished with the variables t 23 and U31, and Eq. (4.21), this reads as follows ... 

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 ... 

Remark 4-4- By virtue of either unidirectional electric field strength or unidirectional electric flux density aligned with the polarization direction, the shear stresses may be completely decoupled from the electric field strengths. 

Fig. 4.9. Transition between unidirectional electric field strength (vsi = 0) and unidirectional electric flux density (usi = 1) as a function of electroding geometry depicted by finite element calculations ( ) according to Eq. (4.25) and approximated...

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. 

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