Principle of Virtual Electric Potential

Electric systems may be treated like mechanical ones the quantities appearing in electricity have a lower tensorial order the mechanical displacement is a vector field, the electric potential is a scalar field. The electrostatic equilibrium of an infinitesimal volume element of a dielectric domain, given by Eq. (3.34), may be multiplied by the scalar field of a virtual electric potential 6(p and integrated over the Volume A, yielding 

Separation of the terms leads to Eq. (3.51a), where the first term can be split into two parts with the aid of the divergence product rule of Eq. (3.6a), as shown in Eq. (3.51b). With the commutativity of the terms in the first integrand according to Eq. (3.1) and application of Gauss s integral theorem of Eq. (3.7) to the second integrand, the formulation of Eq. (3.51c) is reached  

The transformation of the volume integral over A into a surface integral over the volume s closed surface dA allows for the application of the surface flux boundary conditions of Eq. (3.35). The virtual electric potential gradient vSif may be represented by the virtual variant SE of the electric field strength vector of Eq. (3.39)  

With rearrangement of terms and conversion to matrix representation, the principle of virtual electric potential takes its final form as 

Here is the virtual work of external charges, and the virtual work of internal charges. As the contained virtual electric field strength vector 8E is assembled from derivatives of the virtual electric potential Sip, the latter has to be continuously differentiable. Further on, the virtual electric potential has to comply with the actual conductive boundary conditions of Eq. (3.36). The initial axiom of Remark 3.1 may now be reformulated for the virtual electric potential. 

---

When a current source is connected to the electrodes of a piezoelectric actuator, the electric potential p is unknown. Therefore, the contributions of the principle of virtual electric potential of Eq. (3.53) need to be retained. Since the current describes the derivative trend of charge with respect to time, it prescribes the area charge qsA on the electrodes. An adequate formulation of the principle of virtual work from Eqs. (3.62) and (3.63), in consideration of Eq. (4.36), reads... 

---