Summation of Virtual Work Contributions

In Sections 3.4.2 and 3.4.5, the virtual work contributions of a mechanical system being either static and deformable or dynamic and rigid were derived. They can be combined to picture a deformable dynamic system with the terms of Eqs. (3.45) and (3.59) as follows  

Alternatively, this formulation may be deduced starting from the interior conservation of momentum, Eq. (3.61), which upgrades the interior mechanical equilibrium, Eq. (3.14), with the inertia contributions of d Alembert s principle, Eq. (3.56b). Thereby the derivation steps can be transferred from Section 3.4.2. 

The electric contributions from the principle of virtual potential, as derived in Section 3.4.4 and given by Eq. (3.53), still have to be incorporated. This can be achieved equivalently by the addition or subtraction of Eqs. (3.60b) and (3.53). In conformance with Allik and Hughes [4] and in view of the symmetry properties of the not yet introduced constitutive relation, the electrostatic expressions will be subtracted from the mechanical ones. The virtual work of external contributions takes the following form  

Since the forces fg and charges q A on the boundary are zero apart from their respective working surface, the surface integrals may be summarized. Then the integrands can be collated in vector form, as shown in the last line. Similarly, the virtual work of internal contributions can be formulated, where the vectors of virtual strains 6e and virtual electric field strength SE, as well as the vectors of actual stresses r and actual electric flux density D, be merged  

Here the negative sign has been incorporated into the representation of the electric field strength by setting E = —E. 

---