# Virtual Power Principle

The preceding expression is the weak form for the equilibrium equations, traction boundary conditions and interior continuity conditions. It is known as the principle of virtual power.

The basis for the development of a displacement finite element model is the introduction of some
locally based spatial approximation to parts of the solution. The first step to
develop such an approximation is to replace the equilibrium equations by an
equivalent weak form. This is obtained by multiplying the local differential
equation by an arbitrary vector valued test function defined with suitable
continuity over the entire volume and integrating over the current
configuration.(1)

$$\underset{\Omega}{\int}\left(\delta {v}_{i}\left(\frac{\partial {\sigma}_{ji}}{\partial {x}_{j}}+\rho {b}_{i}-\rho {\dot{v}}_{i}\right)\right)}d\Omega =0$$

The first term in Equation 1 is then
expanded.(2)

$$\underset{\Omega}{\int}\left(\delta {v}_{i}\frac{\partial {\sigma}_{ji}}{\partial {x}_{j}}\right)}d\Omega ={\displaystyle \underset{\Omega}{\int}\left[\frac{\partial}{\partial {x}_{j}}\left(\left(\delta {v}_{i}\right){\sigma}_{ji}\right)-\frac{\partial \left(\delta {v}_{i}\right)}{\partial {x}_{j}}{\sigma}_{ji}\right]}d\Omega $$

Taking into account that stresses vanish on the complement of the traction boundaries, use the
Gauss's theorem.(3)

$$\underset{\Omega}{\int}\left(\frac{\partial}{\partial {x}_{j}}\left(\left(\delta {v}_{i}\right){\sigma}_{ji}\right)\right)d\Omega ={\displaystyle \underset{{\Gamma}_{\sigma}}{\int}\left[\left(\delta {v}_{i}\right){n}_{j}{\sigma}_{ji}\right]}d\Gamma$$

Replacing Equation 3 in Equation 2
gives:(4)

$$\underset{\Omega}{\int}\left(\delta {v}_{i}\frac{\partial {\sigma}_{ji}}{\partial {x}_{j}}\right)d\Omega ={\displaystyle \underset{\Gamma}{\int}\left(\delta {v}_{i}\right){\tau}_{i}d\Gamma -{\displaystyle \underset{\Omega}{\int}\frac{\partial \left(\delta {v}_{i}\right)}{\partial {x}_{j}}{\sigma}_{ji}d\Omega}}$$

If this last equation is then substituted in Equation 1, you
obtain:(5)

$$\underset{\Omega}{\int}\left(\frac{\partial \left(\delta {v}_{i}\right)}{\partial {x}_{j}}\right){\sigma}_{ji}d\Omega -{\displaystyle \underset{\Omega}{\int}\delta {v}_{i}\rho {b}_{i}d\Omega -{\displaystyle \underset{\Gamma}{\int}\left(\delta {v}_{i}\right){\tau}_{i}d\Gamma +{\displaystyle \underset{\Omega}{\int}\delta {v}_{i}\rho {\dot{v}}_{i}d\Omega =0}}}$$

The preceding expression is the weak form for the equilibrium equations, traction boundary conditions and interior continuity conditions. It is known as the principle of virtual power.