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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2017016848
pages 35-78


Yosra Rahali
Institut Préparatoire aux Etudes d'Ingénieur de Bizerte, 7000, Tunisie
F. Dos Reis
LEMTA, Université de Lorraine, 2, Avenue de la Foret de Haye, BP 90161, 54505 Vandoeuvre Cedex, France
Jean-François Ganghoffer
LEMTA, Université de Lorraine, 2, Avenue de la Foret de Haye, BP 90161, 54505 Vandoeuvre Cedex, France


We presently construct effective second-order grade anisotropic continuum models equivalent to initially discrete periodic beam lattices. This entails the construction of a second-order grade continuum with effective mechanical properties at the first and second order, accounting for the impact of the underlying microstructure on the overall effective mechanical response. % of the effective continuum. Starting from the weak form of the equilibrium equations of the lattice and writing the expansion of the discrete displacement up to the second-order gradient of the continuum displacement field, the expressions of the Cauchy stress and the hyperstress tensors of the effective second-order grade continuum are identified versus the first and second-order gradients of the displacement field. Three models of increasing complexity of the beam kinematics are presented: a first model relying on the displacement as the sole kinematic variable, a second model incorporating a local microrotation in addition to the displacement as kinematic descriptors, and a third model accounting for the network curvature considering a general parameterization of the material points with curvilinear coordinates, and two hierarchical scales of the microstructure. The consideration of the local microrotation is shown to strongly improve the quality of the homogenized second-order gradient continuum, when comparing the effective first- and second-order moduli with the corresponding FE computed moduli. The relevance of the third more complex model is illustrated by two examples showing a strong effect of the microstructured beams on the evaluated second-order effective elastic properties.