Virtual Power Term Names

It is possible to give a physical name to each of the terms in the virtual power equation. This will be useful in the development of finite element equations. The nodal forces in the finite element equations will be identified according to the same physical names.

The first term can be successively written:(1)

\frac{\partial (δ v_{i})}{\partial x_{j}} σ_{j i} = (δ L_{i j}) σ_{j i} = (δ D_{i j} + δ W_{i j}) σ_{j i} = δ D_{i j} σ_{j i}

You have used the decomposition of the velocity gradient $L$ into its symmetric and skew symmetric parts and that $δ W_{i j} σ_{j i} = 0$ since $δ W_{i j}$ is skew symmetric and $σ_{j i}$ is symmetric.

The latter relation suggests that

δ D_{i j} σ_{j i}

can be interpreted as the rate of internal virtual work or virtual internal power per unit volume. The total internal power

δ P^{int}

is defined by the integral of

δ D_{i j} σ_{j i}

:(2)

δ P^{int} = \int_{Ω} δ D_{i j} σ_{j i} d Ω = \int_{Ω} \frac{\partial (δ v_{i})}{δ x_{j}} σ_{j i} d Ω \equiv \int_{Ω} δ L_{i j} σ_{j i} d Ω

The second and third terms in Virtual Power Principle, Equation 5 are the virtual external power:(3)

δ P^{e x t} = \int_{Ω} δ v_{i} ρ b_{i} d Ω + \int_{Γ_{σ}} (δ v_{i}) τ_{i} d Γ

The last term is the virtual inertial power:(4)

δ P^{i n e r t} = \int δ v_{i} ρ {\dot{v}}_{i} d Ω

Inserting Equation 2, Equation 3 and Equation 4 into Equation 5, the principle of virtual power can be written as:(5)

δ P = δ P^{int} - δ P^{e x t} + δ P^{i n e r t}

for all $δ v_{i}$ admissible.

We can show that virtual power principle implies strong equations of equilibrium. So it is possible to use the virtual power principle with a suitable test function as a statement of equilibrium.

The virtual power principle has a simple physical interpretation. The rate of work done by the external forces subjected to any virtual velocity field is equal to the rate of work done by the equilibrating stresses on the rate of deformation of the same virtual velocity field. The principle is the weak form of the equilibrium equations and is used as the basic equilibrium statement for the finite element formulation. Its advantage in this regard is that it can be stated in the form of an integral over the volume of the body. It is possible to introduce approximations by choosing test functions for the virtual velocity field whose variation is restricted to a few nodal values.