Second-order homogenization of 3-D lattice materials towards strain gradient media: numerical modelling and experimental verification

Danial Molavitabrizi, Sergei Khakalo, Rhodel Bengtsson, S. Mahmoud Mousavi (Corresponding Author)

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)


The literature in the field of higher-order homogenization is mainly focused on 2-D models aimed at composite materials, while it lacks a comprehensive model targeting 3-D lattice materials (with void being the inclusion) with complex cell topologies. For that, a computational homogenization scheme based on Mindlin (type II) strain gradient elasticity theory is developed here. The model is based on variational formulation with periodic boundary conditions, implemented in the open-source software FreeFEM to fully characterize the effective classical elastic, coupling, and gradient elastic matrices in lattice materials. Rigorous mathematical derivations based on equilibrium equations and Hill–Mandel lemma are provided, resulting in the introduction of macroscopic body forces and modifications in gradient elasticity tensors which eliminate the spurious gradient effects in the homogeneous material. The obtained homogenized classical and strain gradient elasticity matrices are positive definite, leading to a positive macroscopic strain energy density value—an important criterion that sometimes is overlooked. The model is employed to study the size effects in 2-D square and 3-D cubic lattice materials. For the case of 3-D cubic material, the model is verified using full-field simulations, isogeometric analysis, and experimental three-point bending tests. The results of computational homogenization scheme implemented through isogeometric simulations show a good agreement with full-field simulations and mechanical tests. The developed model is generic and can be used to derive the effective second-grade continuum for any 3-D architectured material with arbitrary geometry. However, the identification of the proper type of generalized continua for the mechanical analysis of different cell architectures is yet an open question.

Original languageEnglish
Pages (from-to)2255-2274
Number of pages20
JournalContinuum Mechanics and Thermodynamics
Publication statusPublished - Nov 2023
MoE publication typeA1 Journal article-refereed


The authors D.M. and S.M.M. acknowledge the financial support by the Starting Grant (2018-03636) from the Swedish Research Council (Vetenskapsrådet). The access to U-PRINT 3D-printing facilities at Uppsala University is also acknowledged.


  • Experimental validation
  • Gradient elasticity
  • Lattice materials
  • Second-order homogenization
  • Size effect


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