OUTLINE

In this chapter the classical variational principles of elasticity are introduced and applied. The solution of the elastic problem is shown to furnish an extremum value of a scalar potential, either the minimum of the strain energy potential or, respectively and simultaneously, the maximum of the complementary energy potential. These principles provide a complete characterisation of solutions of the regular (well-posed) elasticity problem defined in the preceeding chapters. Moreover, they allow the introduction of classes of approximate solutions through the optimisation of potentials over a subspace of the space of admissible fields. Equations of elastostatics are considered first, followed by a discussion of free vibrations.

7.1 STRAIN ENERGY AND COMPLEMENTARY ENERGY

Strain energy

Suppose that the elastic body ? is characterised by a positive definite symmetric tensor of elastic moduli C.

Define the space of kinematically admissible displacement fields as displacement fields that obey displacement boundary conditions prescribed on a part ?? ^d of the boundary ??:

Here ?? ^d ?? denotes that part of the boundary ?? where displacements u ^D are prescribed. We recall briefly that in a regular elasticity problem the boundary ?? is subject to complementary partition into with prescribed displacements and ?? ^t with prescribed tractions in each direction x _i , as discussed previously in Chapter 4.

Strain energy is equal to the actual mechanical work done by internal and external forces between the initial and the actual state.