Betti's reciprocal theorem

Remarks

By the divergence theorem and the equilibrium equations, the left side equals $\int_\Omega \sigma^{(1)} : \varepsilon(u^{(2)}) \, dV = \int_\Omega \varepsilon(u^{(1)}) : C : \varepsilon(u^{(2)}) \, dV$. The major symmetry of $C$ makes this expression symmetric in the two states, which gives the right side. In words: the work done by the first system of loads through the displacements of the second equals the work done by the second through the displacements of the first. For point loads on a structure it gives Maxwell's reciprocity. The displacement at a point $A$ due to a unit load at $B$ equals the displacement at $B$ due to a unit load at $A$, in the corresponding directions. Without the major symmetry, which holds when the material has a stored energy, the theorem fails in general.

Going deeper

For a student

The theorem is the basis of the boundary integral equations of elastostatics. Taking the second state to be the Kelvin point-force solution gives the Somigliana identity. This identity expresses the displacement inside the body by integrals over the boundary. It also underlies influence lines in structural analysis and the symmetry of stiffness and flexibility matrices.