On optimal shapes of elastic solids in the problems of their stability by two measures (Q2771576)

For the linearized equation of motion of isotropic solids two integral measures (functionals) of stability are suggested and a definition of stability is introduced. One of the functionals is analogous to the Lyapunov function. The theory is developed for rods of variable cross-section. The problem of maximization of the critical axial load at the expense of optimization of shape of the rod cross-section under prescribed rod volume is formulated. This problem is solved using an analogue of the direct Lyapunov method for distributed systems and variational technique. The obtained results are illustrated by examples of pivots with variable cross-section under the action of compressive axial forces.