The role of the interfaces and cross-links on the mechanical behavior of mineralized collagen fibrils. A numerical approach

Abstract

Mechanical properties of bone tissue are highly dependent on its hierarchical structure. The presence of microcracks and diffuse damage in lamellar bone is correlated with the failure of the collagen-mineral interface in mineralized collagen fibrils (MCF). The main goal of this work is to evaluate the mechanical behavior of the interfaces and quantify the stiffness loss of the MCF associated with different failure mechanisms, under controlled in-plane displacement. Additionally, we aim to study the role of the cross-links on the fibril mechanical response, beyond the interface failure. Inter- and intra-microfibrilar cross-links are analyzed. In order to address the first issue, a detailed representative volume of the MCF is analyzed by means of the finite element method, under the assumption of plane strain and periodic boundary conditions. In this model the interfaces between constituents are modeled with an exponential cohesive law. Enzymatic cross-links, located at the molecular terminals connecting each 4D ($D=67\text{nm}$) staggered molecules, are represented by non-linear springs. Three in-plane controlled deformations are applied. The results of this work provide the anisotropic stiffness loss of the tissue involved in the different failure mechanisms at the nano-scale length. The initiation of microcracks and the presence of damage zones are compatible with the failure mechanisms observed at interfaces. Interface failure entails a progressive stiffness loss, bringing a non-linear behavior of bone. The strength obtained for the longitudinal maximum deformation is more than 20 times the transverse strength and 3.5 times the shear strength. The quantification of the reduction percentage in the elastic moduli and the shear stiffness when the fibril is damaged, has a potential application in improving failure criteria based on degradation of elastic constants. When longitudinal elongation is applied, the mechanical contribution of the cross-links in delaying the failure initiation of the interface is shown. Likewise, results of this work confirm the scarce influence of the cross-links in the strain range analyzed. Additionally, a three-dimensional numerical model of several microfibrils is defined with the aim of analyzing the mechanical relevance of inter- and intra-microfibrilar cross-links, beyond the interface failure. Results confirm that cross-links transfer the load when strain increases, being highlighted the mechanical competence of the trivalent cross-links.