Strain energy density maximization principle for material design and the reflection in trans-scale continuum theory

Traditional efforts in the design of damage-tolerant structural materials were largely exercises in optimizing the combination of strength and ductility. However, the simultaneous consideration of these two conflicting mechanical indices, improving one inevitably sacrifices the other, makes the design extremely complex and difficult, due to the dilemma of choosing between them. Here, physically guided by the energy variational expression in trans-scale continuum mechanics theory, we propose a general mechanics principle for material design that involving only one index: towards strong and tough material the strain energy density limit (w) should be maximized, i.e., strain energy density maximization principle, referred to as w_max principle. It aims to guide the attainment of exceptional comprehensive mechanical properties, while circumventing the dual-index dilemma by employing a singular index w. Extensive experimental data analyses prove that (i) the maximum w_max always exists, at a critical dimension of characteristic microstructure d_c,micro, and (ii) w can effectively index strength-ductility synergy and the w_max is conjugated with both high strength and high ductility, verifying the validity of w_max principle. The universality, practicality and downward compatibility are also examined. The d_c,micro approaches twice the span of strain gradient region around internal boundary, suggesting that the microstructure state with w_max is the critical state with strongest strain gradient. Importantly, the w improvement as a function of the characteristic size of either microstructure or deformation field can be well captured by strain gradient theory, confirming the consistence between the w_max principle, experimental results and trans-scale continuum theories. This principle opens up a new design concept for advanced structural materials from the perspective of microstructure-w-mechanical properties relationship.