A modified Gibson-Ashby model for functionally graded lattice structures

The current paper extends the classic Gibson-Ashby model of cellular solids for repetitive graded lattices. Three well-known porous unit cells, i.e., simple cubic (SC), body-centered cubic (BCC), and their combination (BCC＋), are considered and the corresponding graded lattices are geometrically defined by functional variation in the diameters of the ligaments based on a power law. The analytical expressions for the relative elastic modulus of the cells, which vary along the graded direction, are provided by structural analysis of the representative beam framework containing tapered beams using the Euler–Bernoulli theory. After that, continuous approximation functions for variation of the relative density and the relative elastic modulus are presented by curve fitting over the analytical expressions along the graded direction. Then, the graded lattices are considered to be fully homogenized and as a homogeneous media, the exact expressions for the effective relative density and the effective relative modulus are presented. The modified version of the Gibson-Ashby model is proposed to relate the effective relative modulus to the effective relative density by introducing a correction factor to the classic model. It is shown that the classic Gibson-Ashby model with the coefficients fitted on uniform lattices, is not accurate for relating the effective properties of the graded lattices. The proposed model, presented as an analytical expression with fitting parameters and a correction factor, serves as a useful guideline for designing optimized functionally graded lattice structures, accommodating a wide range of geometric variations.