The Cosserat brothers’ ingenuous and powerful idea (presented in several papers in the Comptes Rendus at the turn of the 20th century) of solving elasticity problems by considering the homogeneous Navier equations as an eigenvalue problem is presented. The theory was taken up by Mikhlin in the 1970’s who rigorously studied it in the context of spectral analysis of pde’s, and who also presented a representation theorem for the solution of the boundary-value problems of displacement and traction in elasticity as a convergent series of the ( orthogonal and complete in the Sobolev space H1) Cosserat eigenfunctions. The feature of this representation is that the dependence of the solution on geometry, material constants and loading is provided explicitly. Recent work by the author and co-workers obtains the set of eigenfunctions for the spherical shell and compares them with the Cosserat expressions, and further explores applications and a new variational principle. In cases that the loading is orthogonal to some of the eigenfunctions, the form of the solution can be greatly simplified. Applications, in addition to elasticity theory, include thermoelasticity, poroelesticity, thermo-viscoelasticity, and incompressible Stokes flow; several examples are presented, with comparisons to known solutions, or new solutions.
