Optimising truss-based metamaterials for unimodal and pentamodal behaviour

Abstract

Our study focuses on the optimisation of the internal structure of unimodal and pentamodal metamaterials, modelled as three-dimensional linear elastic lattice structures. For optimisation, we represent the metamaterials with discrete truss models of their respective Periodic Unit Cells (PUCs), whose effective response is determined by the first-order numerical homogenisation. The optimisation is formulated as an inverse homogenisation problem with objective functions comprising a ratio of selected eigenvalues of the effective stiffness matrix, which allows us to dispense with the traditional volume constraint and solve the optimisation problem with a simple gradient method combined with the line search method. We demonstrate the efficacy of the formulation with a design of a unimodal material compliant in a chosen shear deformation mode and we also show that our formulation recovers the traditional pentamodal metafluid.