Vol. 6, No. 3, 2018
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ISSN: 2325-3444 (e-only)
ISSN: 2326-7186 (print)
Strain gradient and generalized continua obtained by homogenizing frame lattices

Houssam Abdoul-Anziz and Pierre Seppecher

Vol. 6 (2018), No. 3, 213–250
DOI: 10.2140/memocs.2018.6.213
Abstract

We determine the effective behavior of periodic structures made of welded elastic bars. Taking into account the fact that flexural and torsional stiffnesses are much smaller than the extensional one, we bypass classical homogenization formulas and obtain totally different types of effective energies. We work in the framework of linear elasticity. We give different examples of 2D or 3D microstructures which lead to generalized 1D, 2D, or 3D continua like the Timoshenko beam, Mindlin–Reissner plate, strain gradient, or Cosserat or micromorphic continua.

Keywords
strain gradient, generalized continua, homogenization, lattices, $\Gamma$-convergence
Mathematical Subject Classification 2010
Primary: 35B27, 74Q05, 78M40
Milestones
Received: 22 December 2017
Revised: 27 March 2018
Accepted: 28 April 2018
Published: 26 July 2018

Communicated by Pierre Suquet
Authors
Houssam Abdoul-Anziz
Institut de Mathématiques de Toulon
Université de Toulon
Toulon
France
Pierre Seppecher
Institut de Mathématiques de Toulon
Université de Toulon
Toulon
France