Addendum to ?Some examples of nonuniqueness of solutions to the equation of linear elastodynamics? (original) (raw)

1988, Journal of Elasticity

The present paper is meant as nothing more than a short note, devoted to complete the picture of nonuniqueness in linear elastodynamics given in the work quoted in the title [1]. It was originally planned as a "note added in proof" to [1], but when I received the galley proofs of [1], it was not yet ready. According to its nature of an "appendix" to [1], throughout this note we shall always refer to the notations and the formulae of the full paper. The reader should be then aware that, when quoting formula (,) or Section X or Remark Y, we mean formula (,) or Section X or Remark Y of[1]. The need of this addition to [1] may be easily understood by considering the following objection to the counterexamples given there. The one-dimensional model of the dynamical problem of linear elasticity is not quite suitable to sample the considered case in which a point o of the body B is allowed to be a singularity point for the material data. Indeed, the statement of the boundary-initial value problem, in its general form, requires that B is an "open connected set" (cf. Section 3): now, when dealing with a one-dimensional example, the presence of the singularity point o "breaks" the connection of B, which in this case turns out to be the join of two different bodies B l = (-x0, 0) and B 2 = (0, + o0) (cf. Section 5). Thus, at a first glance, it could seem that if we would construct an analogous example in the general case (i.e. in two or three dimensions), we should think, rather than of a single body B containing a singularity point o, of two different bodies B1 and B2 with a common boundary 7, where the material data are allowed to be singular. In such a problem, the dependence of the uniqueness of solutions on some conditions on 7 (e.g., the continuity of the displacement and/or the stress field across 7, or the validity of the Third Law of Dynamics for the contact interactions between B~ and B2) is to be