The warping, the torsion and the Neumann problems in a quasi-periodically perforated domain

Fading memory effects in elastic-viscoelastic composites

“…The case where the size and shape of the holes varies from cell to cell, is called quasi-periodic and has been treated in [11]. We briefly summarize here the homogenization results obtained in [11], our present setting being slightly different, but more adapted to the elassieal methods of eontrol in domains and including the N dimensional case.…”

“…We briefly summarize here the homogenization results obtained in [11], our present setting being slightly different, but more adapted to the elassieal methods of eontrol in domains and including the N dimensional case. The same proofs hold, with minor modifications (cf.…”

“…More precisely, we will define a microstructure as an element Once defined the characteristic funetion %, we consider problem (2.3)-(2.4), corresponding to the viscoelastic matrix occupying Q\ with elastic inclusions occupying Q\Q e , and where the matrices A £ and B 8 are symmetrie and satisfy the ellipticity conditions (2.5). Also for the séquence (B e ) of degenerate matrices, there exists a homogenized limit matrix B = (b f ), in the sense first introduced in [5], for the periodic case, in [11], for the quasi-periodic case, and recently generalized in [3], under the name of H°-convergenee: where the matrices A, B and C(X) have their coefficients given by (4.5), (4.13) and (4.8), respectively Consequently, K is the homogenized kernel that appears in the limit équation (3.9).…”

On the optimization of non periodic homogenized microstructures

“…We deal with « quasi-periodic » structures, as defined and studied by Mascarenhas and Polisevski [13]. The basic idea is to use periodic cells on which non-periodic holes are included.…”

“…The basic idea is to use periodic cells on which non-periodic holes are included. For a given microstructure (which is a law giving the holes in each cell, for each size of the cell), Mascarenhas and Polisevski [13] give with a mathematical proof the homogenized équations for the problem of the torsion of a rod (real void-material structures are allowed, no approximation needs to be done). The homogenized équations can be solved theoretically and numerically for any microstructure belonging to a broad class.…”