The constitutive relation of self-healing hierarchical fiber bundle materials

Biomaterials have the fascinating self-healing property, and improve their mechanical behaviors through a hierarchical architecture. For example, collagen fibrils, as the major constituent of fibrous biological materials (e.g., tendon and bone), are structured in a hierarchical bundle. In this paper, in the light of the Weibull distribution and Daniels’ theory, the constitutive relation of fiber bundle material is first established. Based on the Weibull distribution, the self-healing effect is studied for fiber bundle materials, and the constitutive relation of self-healing fiber bundle materials is then developed. Together with Daniels’ theory, the hierarchical effect is also studied for fiber bundle materials, and the constitutive relation of hierarchical fiber bundle material is therefore founded. Finally, typical mechanical properties such as strength, rigidity and toughness are evaluated for consecutively hierarchical fiber bundle materials, which are in good agreement with the references if available. Our results show that there exist the thresholds of healing rate pertaining to strength and toughness, only greater than which the corresponding mechanical properties benefit from the hierarchical architecture while rigidity is insensitive to the self-healing effect. The current work indicates that, knowing the constitutive relation of other biomaterials, the disparate mechanical properties can be studied in a same manner as those for self-healing hierarchical fiber bundle materials.