A novel asymptotic-analysis-based homogenisation approach towards fast design of infill graded microstructures

Abstract

Graded microstructures have demonstrated their values in various engineering fields, and their production becomes increasingly feasible with the development of modern fabrication techniques, such as additive manufacturing. With the use of asymptotic analysis, we propose in this article a homogenisation framework to underpin the fast design of devices filled with quasi-periodic microstructures. With the introduction of a mapping function which transforms an infill graded microstructure to a spatially-periodic configuration, the originally complicated cross-scale problem can be asymptotically decoupled into a macroscale problem within a homogenised media and a microscale problem within a representative unit cell. For a given graded microstructure, the stress field and overall compliance computed by the proposed method are shown, both theoretically and numerically, to be consistent with the underlying fine-scale results. Upon linearisation, the computational cost associated with the proposed formulation is found to be as low as that in existing asymptotic-analysis-based homogenisation approaches, where only spatially periodic microstructures are considered. The present framework also exhibits interesting features in several other aspects. Firstly, smooth connectivity within graded microstructures is automatically guaranteed. Secondly, the configuration obtained here is naturally characterised by a finite length scale associated with the resolution of fabrication. The proposed approach effectively reproduces the optimal microstructure for the case of uniaxial loading where explicit solutions are available, and other numerical results are further provided.