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.

自相似的分形是从人体肌肉骨骼系统内的生物材料中广泛获得的，其粘弹性行为可以用分数阶导数来描述。但是，现有的粘弹性模型忽略了生物材料的分形结构与其分数阶时间响应之间的内部相关性。我们进一步扩展了分形超细胞（FHC）粘弹性理论来研究这种时空相关性。首先将FHC单元与其他材料单元和弹簧-阻尼粘弹性模型进行了比较，从而突出了其离散和分形特性。为了证明FHC的效用，分别从人体软骨，肌腱和肌肉横断面提取了树状，梯状和三角形的FHC。进一步讨论了FHC元素的对偶性和对称性，在此过程中，对偶变换的操作产生了新型的FHC元素，而FHC的对称性破坏改变了其分数阶粘弹性响应。因此，建立了FHC的交错模式与其流变幂律次序之间的相关性，并且多级FHC元件的粘弹性行为在腱层结构的宏观和纳米水平上都非常适合应力松弛实验。 。FHC元件为理解衰老或疾病期间生物组织的结构变性与其功能变化之间的联系提供了理论基础。FHC的对称破坏改变了其分数阶粘弹性响应。因此，建立了FHC的交错模式与其流变幂律次序之间的相关性，并且多级FHC元件的粘弹性行为在腱层结构的宏观和纳米水平上都非常适合应力松弛实验。 。FHC元件为理解衰老或疾病期间生物组织的结构变性与其功能变化之间的联系提供了理论基础。FHC的对称破坏改变了其分数阶粘弹性响应。因此，建立了FHC的交错模式与其流变幂律次序之间的相关性，并且多级FHC元件的粘弹性行为在腱层结构的宏观和纳米水平上都非常适合应力松弛实验。 。FHC元件为理解衰老或疾病期间生物组织的结构变性与其功能变化之间的联系提供了理论基础。并且在应力松弛实验中，多层FHC元件的粘弹性行为在腱层的宏观和纳米层面上都非常合适。FHC元件为理解衰老或疾病期间生物组织的结构变性与其功能变化之间的联系提供了理论基础。并且在应力松弛实验中，多层FHC元件的粘弹性行为在腱层的宏观和纳米层面上都非常合适。FHC元件为理解衰老或疾病期间生物组织的结构变性与其功能变化之间的联系提供了理论基础。