Virtual Power Principle

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)

\int_{Ω} (δ v_{i} (\frac{\partial σ_{j i}}{\partial x_{j}} + ρ b_{i} - ρ {\dot{v}}_{i})) d Ω = 0

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

\int_{Ω} (δ v_{i} \frac{\partial σ_{j i}}{\partial x_{j}}) d Ω = \int_{Ω} [\frac{\partial}{\partial x_{j}} ((δ v_{i}) σ_{j i}) - \frac{\partial (δ v_{i})}{\partial x_{j}} σ_{j i}] d Ω

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

\int_{Ω} (\frac{\partial}{\partial x_{j}} ((δ v_{i}) σ_{j i})) d Ω = \int_{Γ_{σ}} [(δ v_{i}) n_{j} σ_{j i}] d Γ

Replacing Equation 3 in Equation 2 gives:(4)

\int_{Ω} (δ v_{i} \frac{\partial σ_{j i}}{\partial x_{j}}) d Ω = \int_{Γ} (δ v_{i}) τ_{i} d Γ - \int_{Ω} \frac{\partial (δ v_{i})}{\partial x_{j}} σ_{j i} d Ω

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

\int_{Ω} (\frac{\partial (δ v_{i})}{\partial x_{j}}) σ_{j i} d Ω - \int_{Ω} δ v_{i} ρ b_{i} d Ω - \int_{Γ} (δ v_{i}) τ_{i} d Γ + \int_{Ω} δ v_{i} ρ {\dot{v}}_{i} d Ω = 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.