Topology optimization of periodic microstructures for load-independent in-plane strength

Topology optimization is proved to be an invaluable tool in designing novel materials with extreme or prescribed properties. However, most existing approaches to strength optimization are confined to prescribed loading conditions. This inherently limits the resulting designs to perform optimally only under specific load cases, reducing their applicability in scenarios involving uncertain or multidirectional loading. Achieving load-independent strength (microstructures that perform well across the full range of possible stress states) remains an open challenge. This work proposes a topology optimization framework for designing microstructures with enhanced, load-independent strength. The approach treats the yield locus of the homogenized material as the design objective, directly targeting the admissible stress domain rather than the response to any single load. By analyzing the microstructure under a set of unit test strains and exploiting linear elasticity prior to yielding, macroscopic and microscopic stresses are expressed as linear combinations of these responses. This enables an implicit mapping between homogenized stress states and local stresses, providing means to systematically control and expand the homogenized yield locus. Sensitivities are derived for general cases and the framework is embedded within a topology optimization algorithm. Numerical examples demonstrate the capability of the proposed approach to tailor microstructural strength characteristics in a load-independent manner. The results show that while optimal designs in the principal stress plane are consistent with established findings, incorporating stress states with non-zero shear leads to markedly different, and non-trivial, optimal topologies, highlighting the critical importance of accounting for the full yield surface in strength-driven material design.