  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.
