Self-similar fractals are widely obtained from biomaterials within the human musculoskeletal system, and their viscoelastic behaviours can be described by fractional-order derivatives. However, existing viscoelastic models neglect the internal correlation between the fractal structure of biomaterials and their fractional-order temporal responses. We further expanded the fractal hyper-cell (FHC) viscoelasticity theory to investigate this spatio-temporal correlation. The FHC element was first compared with other material elements and spring–dashpot viscoelastic models, thereby highlighting its discrete and fractal nature. To demonstrate the utility of an FHC, tree-like, ladder-like and triangle-like FHCs were abstracted from human cartilage, tendons and muscle cross-sections, respectively. The duality and symmetry of the FHC element were further discussed, where operating the duality transformation generated new types of FHC elements, and the symmetry breaking of an FHC altered its fractional-order viscoelastic responses. Thus, the correlations between the staggering patterns of FHCs and their rheological power-law orders were established, and the viscoelastic behaviour of the multi-level FHC elements fitted well in stress relaxation experiments at both the macro- and nano-levels of the tendon hierarchy. The FHC element provides a theoretical basis for understanding the connections between structural degeneration of bio-tissues during ageing or disease and their functional changes.

Although various phenomena have confirmed that surface geometry has an impact on surface energy at micro/nano scales, determining the surface energy on micro/nano curved surfaces remains a challenge. In this paper, based on Lennard-Jones (L-J) pair potential, we study the geometrical effect on surface energy with the homogenization hypothesis. The surface energy is expressed as a function of local principle curvatures. The accuracy of curvature-based surface energy is confirmed by comparing surface energy on flat surface with experimental results. Furthermore, the surface energy for spherical geometry is investigated and verified by the numerical experiment with errors within 5%. The results show that (i) the surface energy will decrease on a convex surface and increase on a concave surface with the increasing of scales, and tend to the value on flat surface; (ii) the effect of curvatures will be obvious and exceed 5% when spherical radius becomes smaller than 5 nm; (iii) the surface energy varies with curvatures on sinusoidal surfaces, and the normalized surface energy relates with the ratio of wave height to wavelength. The curvature-based surface energy offers new insights into the geometrical and scales effect at micro/nano scales, which provides a theoretical direction for designing NEMS/MEMS.