HENCKY–PRANDTL NETS AND CONSTANT PRINCIPAL STRAIN MAPPINGS WITH ISOLATED SINGULARITIES

Abstract. The work presented in this paper is motivated in large measure by the appearance of Hencky–Prandtl nets (HP-nets) in the context of planar quasi-isometries with constant principal stretching factors (cps-mappings) and by compelling analogies between such mappings and those given by analytic functions of one complex variable. We study the behavior of HP-nets in the vicinity of isolated singularities and use the results of this analysis to show that if an HP-net is regular in the entire plane except for isolated singularities, then it can have at most two of them, and that all possibe nets of this kind fall into five classes each of which depends on a small number of parameters. In light of the relationship between HP-nets and cps-mappings it follows that an analogous statement holds for the latter as well, and this connection is further exploited to prove that HP-nets regular except for isolated singularities in smoothly bounded Jordan domains have nontangential limits in the appropriate sense at almost all boundary points. The treatment includes, in addition, an interpretation of cps-mappings with isolated singularities as deformations produced by the cryptocrystalline solidification with microscopic flaws of a planar film and a discussion of the problem of just how the singularities of such mappings can actually be distributed in a given domain