On the traction problem of dead loading in linear elasticity with initial stress
By a linear elasticity with initial stresses the authors mean a general theory that treats small elastic deformations from a prestressed configuration. For a traction problem mentioned in the title, the authors show that if an incremental elasticity tensor \(L\) is suitably restricted and the initial stress \(T_0\) is sufficiently small, the problem has a solution \(u\) unique up to the addition of an arbitrary displacement in a finite-dimensional space \(N\). In the classical theory where \(T_0 =0\), \(N\) is a set Ird of infinitesimal rigid displacements. An example with \(T_0\neq 0\) is given in which \(N\supset \text{Ird}\). Also, it is shown that \(N\subset \text{Ird}\) if \(T_0\) is sufficiently small, and \(N= \text{Ird}\) if \(T_0\) is ``residual'' and is sufficiently small. The paper deals with a static initial stress traction problem in terms of the displacements only, so an analysis of the problem in a natural pure stress language as well as an extension of the analysis to include a pure stress treatment of linear elastodynamics with initial stresses would be useful [cf. \textit{J. Ignaczak}, New problems in mechanics of continua, SM Stud. Ser. 17, 115-127 (1983; Zbl 0564.73021)].
scientific article; zbMATH DE number 754036
scientific article