Concurrent multi-scale variable stiffness design optimization of fiber-reinforced composite laminates with the Tsai-Wu failure criterion

Strength design is a critical aspect of ensuring the safety and reliability of composite structures in service, as stress concentration will lead to fracture, damage, and fatigue. Multi-scale variable stiffness design optimization of fiber-reinforced composites, through concurrent optimization of structural topology and fiber orientation, enhances their potential for lightweight design and stress reduction. When stress-related constraints are introduced, multi-scale variable stiffness optimization of composite materials faces additional challenges. These arise from the complex failure modes of anisotropic materials, where a single stress measure cannot adequately represent failure mechanisms; the inherent coupling between macro-scale topology and micro-scale fiber orientation, which demands efficient characterization methods for coordinated optimization; and intrinsic difficulties of variable stiffness design. Therefore, Based on the first-order shear deformation theory, the normal distribution fiber optimization interpolation scheme is employed, this study develops a multi-scale variable stiffness optimization framework for composite structures under Tsai-Wu failure constraints. This study is based on the first-order shear deformation theory, thereby enabling a more general and consistent assessment of stress and failure in laminated composite structures. To address the large-scale constraint issue, the p-norm approach is adopted to aggregate the Tsai–Wu failure criterion into a reduced set of stress-related constraints, and a stress penalization method is applied to mitigate stress singularities. Using the adjoint vector method, explicit sensitivities of macro-scale topology and micro-scale fiber angle design variables are derived for both with respect to the compliance minimization and the Tsai-Wu failure constraints. Numerical studies on a single-layer L-shaped beam, a multi-layer L-shaped laminated plate, and a single-layer corbel-shaped beam for multi-scale optimization demonstrate the proposed framework. Comparisons between designs with and without stress-related constraints are performed in terms of stress distribution, Tsai–Wu failure factor, objective function value, macro-scale topology configuration, and micro-scale fiber orientation, to validate its effectiveness. The results provide a theoretical and methodological basis for the multi-scale strength optimization of composite structures.