Le problème de Ventcel pour le système de l’élasticité dans un domaine de \({\mathbb{R}}^ 3\). (Ventcel problem for an elastic system in a \({\mathbb{R}}^ 3\) domain). (French) Zbl 0624.73066

The author makes a theoretical study of the so-called Ventcel problem extended to linear elasticity: an elastic body is rigidly overlaid by a thin elastic layer. The intrinsic formalism used herein [the reviewer, Mechanics of continuous media and analysis of structures (1977; Zbl 0454.73003)] allows to easily write the weak and strong formulations of the problem and makes conspicuous a boundary condition in the second derivative of the tangential displacement. The transmission problem from the body to the envelop is studied in assuming the thickness \(\epsilon\) tending to zero. To that purpose, the author utilises a change of variables and of functions, now standard, which allows to pass to a domain independent of \(\epsilon\) ; it is deduced a new expression of the external forces. Assuming these values independent of \(\epsilon\), the methods of functional analysis provide a priori estimates on the solution and on the energy.
By making the thickness to tend to zero, the author shows that the solution satisfies the Kirchhoff-Love assumptions, and that the Ventcel problem may be written at the outset with zero thickness. The results are interesting. However the initial coupled problem looks academic as for the boundary forces. Moreover, the shell is implicitly supposed from the beginning to work as a plane stress (membrane), which is equivalent to Kirchhoff-Love assumptions [W. T. Koiter, Proc. Sympos. Theory Thin Elastic Shells, Delft 1959, 12-33 (1960; Zbl 0109.430)]. One will not be surprised then to finally restate this result.
Reviewer: R.Valid


74K15 Membranes
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients