Fundamental toughening landscape in soft–hard composites: Insights from a minimal framework

Soft–hard composite strategy is a highly general yet powerful approach to overcome the inherent trade-off between strength and toughness in material design. However, the underlying toughening mechanisms, veiled by nonlinearities and complex network interactions, remains unclear. Here, we employ a three-dimensional soft–hard composite (SH- com ) framework by arranging randomly distributed linear-elastic soft and hard elements to explore the toughening mechanisms of soft–hard composites, while shielding the influence of complex nonlinearities and network architectures. Key features observed in soft–hard composites, including mechanical hysteresis, sacrificial bond-driven toughening, and brittle-to-ductile (BTD) transitions, are successfully reproduced, suggesting that the simplest model captures the essence of toughening in soft–hard composites. Visualization of internal fracture reveals distinct fracture patterns associated with the BTD transition, while numerical and theoretical analyses elucidate its mechanical origins. Furthermore, we identify an optimal toughening composition governed by a unified scaling relation linked to the fracture toughness ratio between soft and hard components. A fundamental toughening phase diagram is also established in terms of strength and toughness. This work sheds light on the underlying toughening landscape of soft–hard composite systems.