This paper presents a framework for systematically optimizing the fracture toughness of periodic beam lattice materials using topology optimization. We introduce a new normalization factor for fracture toughness based on the unit cell size arguing that it offers a more consistent and unambiguous basis for comparing different lattice microstructures than the conventional beam-length-based normalization. Our analysis demonstrates that the relative performance ranking of lattice topologies is significantly affected by the choice of normalization. Notably, when evaluated using this proposed unit-cell-based normalization, classical triangular and Kagome structures consistently demonstrate remarkably high fracture toughness, outperforming a demi-regular structure that appear superior under conventional beam-length normalization. This reinforces their established efficacy as high-performance lattice designs. The proposed optimization framework is applied to design lattice structures at low ( ρ ̄ = 1 % ) and moderate ( ρ ̄ = 15 % ) relative densities. Interestingly, the framework did not yield structures that surpassed the performance of the Kagome or triangular lattices when assessed with the proposed normalization factor. However, it is remarkable that when evaluated using the conventional beam-length-based normalization from the literature, the framework is able to generate a design that significantly outperforms the triangular and Kagome lattices, as well as a tension-dominated demi-regular structure, at moderate relative densities ( 5 % < ρ ̄ < 20 % ). This work highlights the critical influence of normalization choices on performance assessment and underscores the inherent efficiency of classical lattice topologies.

Soft materials can undergo large deformations while exhibiting highly nonlinear behavior. In soft particulate composites, this nonlinearity arises predominantly from geometry-mediated particle interactions and the intrinsic stiffening of the soft phases, which together govern the elastic instabilities and subsequent buckling patterns. In this study, we investigate how particle interaction–induced stiffening modulates elastic instabilities in soft particulate composites subjected to finite strains. We use the Gent material model, a non-Gaussian framework that captures the stiffening behavior through a single parameter associated with the limiting extensibility of polymer chains. Our results reveal that material stiffening modulates both the onset of instability and the transition between buckling modes, with outcomes strongly dependent on the initial geometry of the unit composite. These effects arise from variations in particle interactions along the loading and transverse directions. Thus, by strategically designing soft particulate composites via particle interaction-induced stiffening with tailored material properties and geometrical parameters, elastic instabilities can be effectively controlled and manipulated.