Principle of Minimum Potential Energy

Let a body be in equilibrium under the action of specified body forces and surface tractions. The surface tractions are prescribed over a part $\Gamma_1$ of the boundary, and on the remainder part $\Gamma_2 = \Gamma - \Gamma_1$ the displacements are assigned. Let u denote the displacement of the body in the equilibrium state.

Consider a class of arbitrary displacements $u_i + \delta u_i$ where $\delta u_i$ vanishes on $\Gamma_2$, and $u_i + \delta u_i$ result in infinitesimal strains in the body. Displacements $\delta \mathbf{u}$ are called virtual. The virtual work $\delta U$ done by external forces f_i and g_i in a virtual displacement $\delta u_i$ is given by

$$\delta U = \int_\Gamma f_i\delta u_i d\Gamma + \int_\Omega\rho g_i\delta u_id\Omega. \tag{9.1}$$ (9.1)

During the virtual displacement of the body, the forces f_iand g_i are assumed to remain unchanged. Thus

$$\delta U = \delta \left(\int_\Gamma f_iu_id\Gamma + \int_\Omega\rho g_iu_id\Omega\right). \tag{9.2}$$ (9.2)

Combining this with (8.9) we obtain

$$\delta\int_\Omega Wd\Omega = \delta \left(\int_\Gamma f_iu_id\Gamma + \int_\Omega \rho g_iu_id\Omega \right)\tag{9.3}$$ (9.3)

or

$$\delta\left(\int_\Omega Wd\Omega - \int_\Gamma f_iu_id\Gamma - \int_\Omega\rho g_iu_id\Omega\right) = 0.\tag{9.4}$$ (9.4)

That is, during an admissible virtual displacement of the body the expression in parentheses has a stationary value. The quantity

$$V = \int_\Omega Wd\Omega - \int_\Gamma f_iu_id\Gamma - \int_\Omega \rho g_iu_id\Omega \tag{9.5}$$ (9.5)

is called the potential energy of the body, and (9.4) implies that

$$\delta V = 0.\tag{9.6}$$ (9.6)

Since the displacements u_i were assumed to correspond to an equilibrium state, therefore V takes on a stationary value when the displacements u are for a state of equilibrium. Actually, the following stronger result holds.

Principle of Minimum Potential Energy: Of all displacements satisfying the given boundary conditions those which satisfy the equilibrium equations make the potential energy an absolute minimum.

We will use this principle to find an equilibrium configuration of the body by the finite element method.