Exact relations for effective moduli of polycrystals.

                                 Part I

                 Rank one plus a null-Lagrangian is an
                 inherited property of two-dimensional
                compliance tensors under homogenization.

                                   by

                             Yury Grabovsky

                    JTB 120 3:20pm, Monday, January 27, 1997

                                Abstract

  Assume that the local compliance tensor of an elastic composite in two
  space dimensions is equal to a rank-one tensor plus a null-Lagrangian
  (there is only one symmetric one in 2-D). The main result is that that
  the effective compliance tensor has the same representation: rank-one
  plus the null-Lagrangian. This statement generalizes the well-known
  result of Hill that a composite of isotropic phases with a common
  shear modulus is necessarily elastically isotropic and shares the same
  shear modulus. It also generalizes the recent discovery that under a
  certain condition on the pure crystal moduli the shear modulus of an
  isotropic polycrystal is uniquely determined. The present paper sheds
  light on this effect by placing it in a more general framework and
  using some elliptic PDE theory rather than the translation method. Our
  results allow us to calculate the polycrystalline G-closures of the
  special class of crystals under consideration. This talk discusses the
  first part of a project on discovering exact relations on effective
  moduli of polycrystals.

Order reprints via email to yuri@math.utah.edu.