New insights into long-bone biomechanics: are limb safety factors invariable across mammalian species?

The most common function of limb bones is to provide stiff levers acting against muscles and gravity; however, a general mechanical description is not yet available. This research attempts such a description by modeling the bone's intrinsic biomechanics through elastic stability of solid long cylinders considered in non-critical, transient and critical mechanical regimes distinguished conventionally through maximal resisting elastic strains. The non-critical regime controls bones' adaptation through the safety factor (bone strength related to the peak functional stress) S2. This is ensured by bone-diameter (d=1/3+beta) and bone-length (l=1/3-beta) scaling exponents generally following from compressive-stress constraints. Prange's index (0<beta<<1) known from long-bone allometry is related to the components of bone-stress tensor. The tensor-stress components depend weakly on body size, whereas the overall peak limb-compressive stress in running animals remains almost weight-independent. The transient regime (1<S<2) activated in animal vigorous activity determines elastic stability of slightly curved limb bones by avoiding critical-stress bending via non-critical torsion and critical torsion via moderate bending. A physical description of the transient regime suggests a united mechanical pattern. Established under most general consideration, the scaling rules for peak strains, forces, momenta, and stresses challenge locomotor patterns distinguished in small mammals and birds, lizards, primates and non-primate mammals. Taking into account that all scaling rules are limited by S=1 associated with critical regime, reliable estimates for critical body masses are obtained for living elephants and extinct dinosaurs. Our study of the variable limb safety factor provides evidence that land-dwelling and land-moving giants are biomechanically accommodated to the peak bending and torsion functional stresses, respectively.