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Effect of material and structural factors on fracture behaviour of mineralised collagen microfibril using finite element simulation

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Bone is a multiscale heterogeneous material and its principal function is to support the body structure and to resist mechanical loads without fracturing. Numerical modelling of biocomposites at different length scales provides an improved
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    Effect of material and structural factors on fracture behaviour of mineralized collagen microfibril using finite element simulation Abdelwahed Barkaoui   SYMME laboratory, Université de Savoie, BP80439, F74944 Annecy-le-Vieux, France Ridha Hambli PRISME laboratory, EA4229, Université d’Orléans, 8 Rue Léonard de Vinci 45072 Orléans, France João Manuel R. S. Tavares    Instituto de Engenharia Mecânica e Gestão Industrial, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Corresponding author: Professor João Manuel R. S. Tavares Departamento de Engenharia Mecânica Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias 4200-465 Porto Portugal Phone: +351 225 081 487, Fax: +351 225 081 445 Email: tavares@fe.up.pt, url: www.fe.up.pt/~tavares     Effect of material and structural factors on fracture behaviour of mineralized collagen microfibril using finite element simulation ABSTRACT Bone is a multiscale heterogeneous material and its principal function is to support the body structure and to resist mechanical loads without fracturing. Numerical modelling of biocomposites at different length scales provides an improved understanding of the mechanical behaviour of structures such as bone, and also guides the development of multiscale mechanical models. Here, a three-dimensional nano-scale model of mineralized collagen microfibril based on the finite element method was employed to investigate the effect of material and structural factors on the mechanical equivalent of fracture properties. Fracture stress and damping capacity as functions of the number of cross links were obtained under tensile loading conditions for different densities and Young’s modulus of the mineral phase. The results show that the number of cross-links and the density of mineral as well as Young’s modulus of mineral have an important influence on the strength of mineralized collagen microfibrils which in turn clarify the bone fracture on a macro-scale. Keywords: nano-scale model; cross-links; finite element method; fracture mechanical properties    1.   I NTRODUCTION   Bone is a multiscale material with a very complicated hierarchical structure. This hierarchical structure has different scales or levels, specific interactions between these levels and a highly complex architecture in order to fulfil bone biological and mechanical functions (Barkaoui and Hambli, 2011; Sergey, 2010). Katz et al. and Feng et al. (Katz et al.,1987; Feng et al., 2010) divide the hierarchical structure into five levels that have been widely accepted in the scientific community: (i) a Nano structural level (ranging from a few nanometres to several hundred nanometres) - bone at this level can be considered as a multi-phase nano-composite material consisting of an organic phase (32 - 44% of bone volume), an inorganic phase (33 - 43% of bone volume), and water (15 - 25% of bone volume); (ii) a Sub-micro-structural level, also called a single lamella level (spanning from one to a few microns); (iii) a Micro-structural level (from tens to hundreds of microns), or a single osteon and an interstitial lamella level; (iv) a Meso-structural level (from several hundred microns to several millimetres), or the cortical bone level; and finally, (v) a Macro-structural level, or whole bone level (several millimetres to several centimetres, depending on the species). Bone strength is governed by the characteristics of collagen, including the collagen cross-links that have an important role in the reinforcement of bone strength. The biomechanical effects of collagen depend largely on the cross-linking (Knott and Bailey, 1998; Viguet-Carrin, 2006). The strength and stability during maturation of the microfibrils are attained by the development of intermolecular cross-links (Stančíková et al., 1999) . Experimental evidence (Allen et al., 2008; Wu et al., 2003; Boxberger and Vashishth, 2004) has shown that collagen cross-linking in bone tissue significantly influences its deformation and failure behaviour. Additionally, experiments in vitro  (Wu et al., 2003; Boxberger and Vashishth, 2004) and in vivo  (Viguet-Carrin, 2006a; Allen et al., 2008; Wu et al., 2003) revealed that an increases in the number of cross-links is associated with the enhancement of certain mechanical properties (strength and stiffness) and a reduction in others (energy absorption). There are two types of cross-links :  enzymatically and non-enzymatically (Seigmund et al., 2008). Considering the macroscopic response of bone, enzymatic cross-linking has been related to improving mechanical properties (Banse et al., 2002) whereas non-enzymatic cross-linking prevents energy absorption by micro damaged formations and may accelerate brittle fracturing (Vashishth et al., 2004; Tang et al., 2007; Nyman et al., 2007; Vashishth, 2007). Natural cross-linking gives collagen a high tensile strength and proteolytic resistance (Friess, 1998).    Nano-scale failure properties of the mineralized collagen microfibril were studied here and the mechanical and structural parameters of all phases analysed. The cross-linking is one of the most important structural elements of type I collagen in mineralized tissues that provides the fibrillar structure and contributes to various mechanical properties, such as tensile strength, fracture toughness and viscoelasticity (Elham and Iwona, 2012). Fritsch et al., (Fritsch et al., 2009) used the multiscale micromechanics elastic theory in an elasto-plastic analysis to predict the strength of cortical bone. These authors found that the failure of bone material initiates at the nano-scale by a ductile sliding of hydroxyapatite (HA) crystals along layered water films, followed by a rupture of the collagen cross-links. Siegmund et al., (Seigmund et al., 2008) proposed a model that addresses the effect of collagen cross-linking on the mechanical behaviour of a mineralized collagen fibril. Buehler (Buehler, 2008) extended a Molecular Dynamics Model of a single collagen molecule to an individual collagen fibril to obtain its mechanical response under uniaxial tension. The results showed that the deformation and failure mechanisms of a collagen fibril are strongly influenced by the cross-linking density. Barkaoui and Hambli (Barkaoui and Hambli, 2011) proposed a three-dimensional (3D) finite element method (FEM) model to represent the structure of the mineralized collagen microfibril with three constituents: mineral, tropocollagen (TC) molecules and cross-links. This model was used to study the elastic and mechanical failure behaviour of mineralized collagen microfibril under a varying number of cross-links, based on an array of five collagen molecules that are cross-linked together by spring elements and embedded in a mineral matrix. In this study, the 3B FEM model proposed by Barkaoui and Hambli (2011, 2012) was enhanced to allow dapping and fracture stress calculations, which were used to study the failure of mineralized collagen microfibril as well as the biomechanical effect of the constituent properties on the fracture behaviour. The enhanced 3D FEM model of mineralized collagen microfibril proved to be a good solution for bottom-up investigations on structure-property relationships in human bone. 2.   M ETHODS AND TOOLS  2.1.   Mineralized collagen microfibril The existence of sub-structures in collagen fibrils has been a topic of extensive debate. Recent studies have suggested the presence of microfibrils in fibrils (Yang et al., 2012). Experimental    work conducted by Orgel, Fratzl and others have proved that all collagen-based tissues are organized into hierarchical structures, where the lowest hierarchical level consists of triple helical collagen molecules (Fratzl, 2008; Orgel et al., 2006; Orgel et al., 2001) and the multiscale structure is defined as TC/fibrils/fibres. Also, other authors have observed a longitudinal microfibrillar structure with a width of 4 - 8 nm (Habelitz et al., 2002; Baselt et al., 1993). Three-dimensional image reconstructions of 36 nm-diameter corneal collagen fibrils also showed a 4 nm repeat in a transverse section that was related to the microfibrillar structure (Holmes et al., 2001). Using X-ray diffraction culminating in an electron density map, Orgel et al. (Orgel et al., 2001) suggested the presence of right-handed super-twisted microfibrillar structures in collagen fibrils. The microfibril is a helical assembly of five TC molecules that offset one another with an apparent periodicity of 67 nm. These five molecules create a cylindrical formation with a diameter of 3.5-4 nm however its length is unknown. Smith (Smith, 1968) suggested that each microfibril consists of exactly five molecules in a generic circular cross section (Figure 1). Lee et al., (Lee et al., 1996) suggested that collagen microfibrils have a quasi-hexagonal structure. Barkaoui and Hambli (Barkaoui and Hambli, 2011) also used the cylindrical representation of Smith's microfibril model (Figure 1) to develop their 3D finite element model of microfibrils. This choice was based on the fact that the mechanical behaviour of microfibrils under tensile load depends mainly on the area of the cross section, while the shape has no influence under tensile load functions. [Insert Figure 1 about here] As aforementioned, bone is a complex composite material consisting mainly of collagen (TC molecules) and mineral (HA crystals). Figure 2 illustrates that collagen is a very resistant material that give bone its toughness. The Figure also depicts that the mineral HA is very rigid, which gives bone its rigidity and resistance to fracture. As such, the mechanical properties of bone depend on the characteristics of its basic components and the interaction between them (Li et al., 2003; Walsh and Guzelsu, 1994). [Insert Figure 2 about here] The fundamental components of the microfibril can be distinguished as follows:
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