On the Elastic Deformation of Symmetric Periodic Structures

Abstract

SummaryWe discuss locally anisotropic periodic elastic materials in R n which are symmetric with respect to a hyperplane. The cell problem for finding the effective elastic parameters is considered. It is shown that the components of the displacement satisfy either Neumann or Dirichlet conditions on the sides of the cell of periodicity parallel with the symmetry plane. We also prove that the corresponding homogenized material is monocyclic with respect to this plane. The proof for the general case is based on variational and weak formulations of the elasticity problem. For the special case of plates with a square array of holes or perfectly-bonded inclusions we present an independent proof based on Kolosov-Muskhelishvili analytic potentials. In computational practice the derived boundary conditions permit us to easily incorporate the periodic boundaryvalue problem into standard numerical schemes.

Nucleation of cracks in two-dimensional periodic cellular materials

“…As noted in [15], determining a periodic elastic stress field in a periodic composite material with cells possessing two perpendicular axes of symmetry can always be formulated as a relatively simple problem for a unit cell. The four topologies considered in this work possess this attribute and the periodic stress states were thus determined analytically.…”

On domain symmetry and its use in homogenization

“…which resembles, but is not identical to Equation 31. 10 Componentwise, 9 Since the periodicity cell was chosen to be a hexagon, now we have three pairs of "opposite faces" wich relate to each other through either of τ 2 , τ 3 or τ 4 . Note that, for this same periodicity group, it is possible to choose a different periodicity cell, e.g.…”

“…Surprisingly, group theory has not played a significant role in the homogenization computations over unit cells, despite the fact that many unit cells exhibit numerous symmetries. The symmetric unit cell topic is addressed in [9,10], however group theory is not applied and only simple orthotropic and reflective symmetries are considered.…”

Estimating the Vigdergauz constants for regular hexagonal honeycombs

“…[1], [2], [3], [4], [10] and [11]. We also want to refer to [9] concerning simplification in cases of structural symmetries. Concerning the homogenization method in general, see e.g.…”