Least failure energy density: A comprehensive strength index to evaluate and optimize heterogeneous periodic structures

Assessing the comprehensive strength of structures under multiple loading conditions is crucial for designing microstructures. This paper proposes the use of the least failure energy density (LFED) to measure the comprehensive strength of heterogeneous periodic structures, which corresponds to the minimum energy density required to destroy a structure. To enhance the comprehensive strength of a periodic structure, the LFED can be maximized. We constructed a two-layer optimization algorithm and found that the high time consumption renders topology optimization unfeasible. We subsequently developed an approach for solving inner-layer optimization analytically and quickly so that the problem becomes a single-layer optimization. We compared the LFED of several classical structures, including plate structures, lattice structures, and TPMSs. The calculations reveal that plate structures exhibit the best performance in terms of LFED, followed by TPMSs whereas truss structures have the poorest performance. Among the three types of classical structures, the octet plate, Schwartz-D minimal surface, and octet truss structures are the best-performing types, respectively. Additionally, the LFED is combined with the BESO topology optimization method to obtain the best 2D periodical structure, a 2D curved-edge kagome structure. For optimal 3D periodical structures, rarely discussed space kagome structures (plate or lattice) are obtained with an LFED superior to that of other counterpart classical structures.