Asymptotic homogenisation for strain gradient elasticity: convergence, regularisation and lattice applications

Abstract For solids and structures composed of architected metamaterials, detailed micro-level numerical modeling becomes a critical bottleneck due to both memory and processor requirements. For periodic metamaterials, computational homogenisation provides an attractive alternative, whereas multi-scale continuum theories provide an appropriate framework for capturing size effects stemming from the metamaterial architecture. This article focuses on asymptotic computational homogenisation for three-dimensional strain-gradient elasticity (SGE) by assessing its numerical performance for periodic unit cells. After revisiting the variational formulation of SGE, the derivation of homogenised fourth- and sixth-order elasticity tensors is accomplished, leading to solving micro-level corrector problems with periodic boundary conditions. Regarding verification of the corresponding numerical implementation, two stabilisation strategies are assessed: a global-constraint formulation and a Tikhonov regularisation—the latter avoids additional Lagrange multipliers and turns out to be both efficient and stable. The workflow is implemented by combining the finite element software COMSOL with MATLAB LiveLink for obtaining the homogenised (meta)material tensors. As a virtual validation phase, the homogenised constitutive tensors are involved in three-dimensional SGE simulations to solve macroscopic boundary-value problems of lattice structures via isogeometric analysis. This phase is accomplished within the open-source GeoPDEs-software via user-defined subroutines developed for SGE to obtain conforming Galerkin approximations. Verification and validation for the homogenisation approach cover p - and h -convergence studies, mesh-type comparisons between hexahedral and tetrahedral elements, sensitivity to unit-cell volume fraction, and a comparison between the global-constraint formulation and the Tikhonov regularisation. Results show that higher-order basis functions essentially accelerate convergence, mesh type differences become negligible for sufficiently rich approximation spaces, and gradient moduli peak at an intermediate range of volume fractions. Simulations for cantilever lattice beams confirm that SGE predicts bending deflections more accurately than classical elasticity due to the size effect phenomenon present in bending and shear deformations.