General Principle of Virtual Work

This fundamental principle of physics is given by the axiom of Remark 3.1 in its most general formulation, where SW is the total virtual work of the system. For mechanical fields in deformable structures as well as for electrostatic fields in dielectric domains, it can be restated by the equality of internal 51A and external 6V contributions. 

Remark 3.1. k physical system will be in equilibrium only if the total virtual work in accordance with the admissibility criteria vanishes. 

Mechanical work at every particle of a continuum results from acting force and respective displacement or local stress and strain, correspondingly. So the above axiom of Remark 3.1 actually comprises two principles involving either virtual loads or virtual displacements. A brief derivation of both will be given in the following subsections. Similarly, the electric work can be treated, but we will present only one of the variants. The different formulations of the principle of virtual work are independent of a constitutive law and may be denoted as the weak forms of equilibrium, as only the equilibrium conditions have to be fulfilled in the integral mean. Weaker requirements with regard to differentiability of the involved functions have to be fulfilled, since the order of derivatives is reduced in comparison to the equilibrium formulation of Eqs. (3.f4) and (3.34). 

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Following from the general principle of virtual work of Eq. (3.41), the equality of internal and external virtual work is also demanded for the beam and shall serve as the basis for the derivation of the equations of motion ... 

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