Editing Virtual work (section)

===Principle of virtual displacements===
Depending on the purpose, we may specialize the virtual work equation. For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify:

* Virtual displacements and strains as variations of the real displacements and strains using variational notation such as <math> \delta\ \mathbf {u} \equiv \mathbf{u}^* </math> and <math> \delta\ \boldsymbol {\epsilon} \equiv \boldsymbol {\epsilon}^* </math>
* Virtual displacements be zero on the part of the surface that has prescribed displacements, and thus the work done by the reactions is zero. There remains only external surface forces on the part <math> S_t </math> that do work.

The virtual work equation then becomes the principle of virtual displacements:
{{NumBlk||<math display="block"> \int_{S_t} \delta\ \mathbf{u}^T \mathbf{T} dS + \int_{V} \delta\ \mathbf{u}^T \mathbf{f} dV = \int_{V}\delta\boldsymbol{\epsilon}^T \boldsymbol{\sigma} dV </math>|{{EquationRef|f}}}}

This relation is equivalent to the set of equilibrium equations written for a differential element in the deformable body as well as of the stress boundary conditions on the part <math> S_t </math> of the surface. Conversely, ({{EquationNote|f}}) can be reached, albeit in a non-trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on <math> S_t </math>, and proceeding in the manner similar to ({{EquationNote|a}}) and ({{EquationNote|b}}).

Since virtual displacements are automatically compatible when they are expressed in terms of [[Continuous function|continuous]], [[single-valued function]]s, we often mention only the need for consistency between strains and displacements. The virtual work principle is also valid for large real displacements; however, Eq.({{EquationNote|f}}) would then be written using more complex measures of stresses and strains.