Equilibrium configurations of defective crystals
Davini's model [\textit{C. Davini}, Arch. Ration. Mech. Anal. 96, 295-317 (1986; Zbl 0623.73002)] for a slightly defective crystal uses a tensor field to represent locally a triad of `average' lattice vectors, a scalar field to represent density variations due to vacancies or extra atoms and a set of functionals which, in some sense, measure the defects and remain unchanged when the crystal is deformed elastically. Defect-preserving (neutral) deformations may involve also atomic rearrangements representing the slip mechanism of plasticity theory. Pursuing earlier work on this model by \textit{C. Davini} and the second author [Proc. R. Soc. Lond., Ser. A 439, No. 1906, 247-256 (1992; Zbl 0766.73057)], the authors, in this very interesting paper, study equilibria of defective crystals within a variational framework; they take the view that equilibria correspond to minimizers of the elastic energy functional \(E\) when the class \(A\) of admissible variations includes all neutral changes of state. The issue is not the existence of reasonably regular minimizers (actually they may be measure-valued), rather it is the study of minimizing sequences leading to values of energy and stresses. The conclusion is startling: the Cauchy stress must be pressure; this apparent paradox reminds one of the d'Alembert paradox for perfect fluids. If one will be able to eschew the paradox through the artifice of adding somehow friction (as the authors suggest) or if one need invent rather some missing deeper concept (as was the boundary layer concept in the case of fluids), only further study will tell.
scientific article; zbMATH DE number 78054
scientific article