Design and Optimization of Nano-Bio-Inspired Hierarchical or   Nano-Grained Materials
  In this letter a mathematical model to design nano-bio-inspired hierarchical materials is proposed. An optimization procedure is also presented. Simple formulas describing the dependence of strength, fracture toughness and stiffness on the considered size-scale are derived, taking into account the toughening biomechanisms. Furthermore, regarding nano-grained materials the optimal grain size is deduced: incidentally, it explains and quantitatively predicts the deviation from the well-known Hall-Petch regime. In contrast with the common credence, this deviation does not arise at a universal value of grain size but it is strongly dependent on the mechanical properties of the mixture. 
Nicola M. Pugno