Elastic Modulus of Mechanical Model for Mineralized Collagen Fibrils

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1 Yapeng Sun et al.: Model for mineralized collagen fibrils Original Journal of Hard Tissue Biology 25[1] (2016) The Hard Tissue Biology Network Association Printed in Japan, All rights reserved. CODEN-JHTBFF, ISSN Elastic Modulus of Mechanical Model for Mineralized Collagen Fibrils Yapeng Sun 1), Yanqiang Liu 1) and Xufeng Niu 2,3) 1) School of Mechanical Engineering and Automation, Beihang University, Beijing, China 2) Key Laboratory for Biomechanics and Mechanobiology of Ministry of Education, School of Biological Science and Medical Engineering, Beihang University, Beijing, China 3) Research Institute of Beihang University in Shenzhen, Shenzhen, China (Accepted for publication, October 12, 2015) Abstract: Bone is a biomaterial with excellent mechanical properties. Because of its hierarchical structure, bone combines the stiffness and toughness of its components properly. Mineralized collagen fibril is nanocomposite, which plays the foundational role in the hierarchical structure of bone. In this paper, we derive the effects of aspect ratio of mineral platelets and collagen distributed in the longitudinal and transverse direction between adjacent mineral platelets in different arrangements. The Results show that the elastic modulus of mineralized collagen fibrils will increase with the volume fraction and aspect ratio of platelets, but decrease with number of sub unit in representative volume element (RVE) of fibrils. But when the volume fraction of mineral is low, the RVE with more sub unit could obtain a higher elastic modulus by adjusting the distribution of collagen matrix. Key words: Modeling, Biocomposites, Mechanical properties Introduction Mineralized collagen fibril is nanocomposite in bone, which is composed of hydroxyapatite crystal, collagen and other organic 1,2). The hydroxyapatite crystal has a higher stiffness while the collagen has a higher toughness 3,4). Mineralized collagen fibrils are the basic level of hierarchy. With level increased, lamellae, Harversian system and cortical bone could be observed in bone 1,5). Though varied in function and sharp 6), bone shows outstanding mechanical properties according to experimental test for it combines the stiffness and toughness of its component properly 7-9). That characteristic attracts many researchers to investigate what we can learn from its hierarchy. Since mineralized collagen fibrils are the basic elements in hierarchical structure 10), it is indispensable to understand the structure mechanism of mineralized collagen fibrils. According to previous studies, collagen molecule forms collagen fibril in a specific arrangement, in which the gap zone and overlap zone can be observed 1). The gap zone refers to the hole between adjacent molecules. The gap zone is about 35 nm Correspondence to: Dr. Yanqiang Liu, School of Mechanical Engineering and Automation, Beihang University, No. 37 XueYuan Road, Haidian District, Beijing , China; Phone: ; Fax: ; liuyanqiang@buaa.edu.cn and Dr. Xufeng Niu, School of Biological Science and Medical Engineering, Beihang University, No. 37 XueYuan Road, Haidian District, Beijing , China; Phone: ; Fax; ; nxf@buaa.edu.cn 75 and the overlap zone is about 32 nm in the longitudinal direction 11). The hydroxyapatite crystal nucleated inside the gap region originally and extended into the overlap region, grew into a plateshaped crystals eventually 12, 13). That is to say, the distribution of crystal is decided by collagen fibrils. In the mature bone, the size of mineral platelet is about nm, the transverse distance between adjacent mineral platelets is 4-5 nm 12), and the volume fraction of the mineral is about 45-55% 7,14). The main component of the collagen matrix is Type I collagen and the size of collagen is about Φ nm 11). The Young s modulus of collagen is about 1-3 GPa and the Young s modulus of mineral platelet is about GPa 1,9,15,16). Although many studies had focused on the interfacial interactions between mineral platelet and collagen matrix 17,18), the mechanism of stress transfer between them is still unclear. Jäger and Fratzl 10) proposed a stagger model in which the mineral platelet was considered as rigid and the stress in collagen phase was divided into four parts. They proved that the shear stress in the collagen phase between the overlapping mineral platelets plays a key role in this model. Ji and Gao 19) proposed a tension shear model to describe the load transfer among platelets. When the mineralized collagen fibrils were elongated, the mineral platelets were under tension stress and the collagen matrix was under shear stress. Zhang et al. 20) yielded the mechanical properties of mineralized collagen fibrils based on the tension shear model,

2 J.Hard Tissue Biology Vol. 25(1):75-80, 2016 in which the platelet was arranged randomly. Yuan et al. 15) built finite element models of 2-D and 3-D staggered model, gave the influence of material properties and structure parameters on the mineralization. Lei et al 21) yielded the up and down bounds of elastic properties of stairwise model, which was modified from the tension shear model, according to the principle of minimum potential energy and minimum complementary energy. Zhang and To 22) demonstrated the damping characteristics of hierarchical staggered structure and proved that the structure of hard tissue and Hierarchy had an effect on damping characteristics. According the observation of the mineralized collagen fibrils, a simplified model of this composite is developed in this paper, in which a new parameter is introduced to demonstrate the effect of Fig. 3 (a) Schematic illustration of three partitions of shear zone. (b) matrix lie in axial space between adjacent platelets. The distance Shear stress produced by a longitudinal elongation between adjacent mineral platelets in both longitudinal and transverse directions are introduced. The structure parameters According to the tension shear model, the stress is transferred including aspect ratio and volume fraction of mineral platelet, and by shear strain of matrix between platelets. The tensile strain of the arrangement of platelets is also considered in this paper. collagen and the shear strain of mineral platelet are assumed to be zero when the RVE is under load. The shear stress of matrix is Materials and Methods closely related to the size of shear zone between adjacent platelets. The sub unit of mineralized collagen fibril is composed of The shear zone on each side of platelet could be divided into mineral platelet and collagen matrix, as showed in Fig. 1. The three parts, as showed in Fig. 3, based on the difference between length of platelet is ι, the width is d. The axial distance between the gap zone and the overlap zone. The shear stress in each part is adjacent platelets is a, the lateral distance is b. Both mineral assumed as constant and have no influence on other regions. When platelet and collagen matrix are considered as isotropic materials, the mineralized collagen fibrils under tensile stress in the and the relation between them is assumed as ideal bonding. longitudinal direction (x), the elongation of the collagen matrix lie in axial space between adjacent platelets is assumed as.the shear stress of collagen matrix in region I, II and III could be yielded according to the relative displacement of platelets. These shear stress are Fig. 1 A sub unit of mineralized collagen fibril (2) According the observation of the mineralized collagen fibrils 6,7), the arrangement is established as Fig. 2. Mineralized collagen fibril is formed by ordered sub unit, the platelets are (3) arranged in collagen matrix, and each sub unit is shifted up by a distance e. This offset distance e makes the projection of platelets in row i coincident with row (i + n) (n 2) in transverse direction. (4) Thus the mineralized collagen fibril could be represented by a representative volume element (RVE) composed of n sub units. Where G m is the shear modulus of collagen matrix, n is the relative The relation between e and n is displacement of platelet in row i and row (i + l ), s is the number e = ι + a n of platelets vacated in region III. They can be expressed as (1) (5) Fig. 2 A schematic description of RVE that is indicated by the black dashed line E m Where and are the Young s modulus and the Poisson s V m ratio of collagen matrix, respectively. The transverse distance of region III was decided by s, and the model will be similar with staggered model 10) proposed by Jäger if a < e, According to the type of shear zone on sides of platelet, the normal stress in the platelet is given separately over three regions in each half platelet 76

3 (6) Where c is the smaller value of e and, y is the decimal part of. Since the stress applied on platelet is centrally symmetric, the normal stress in another half part can be expr essed as: (7) The elongation of platelet is (8) Where E p is the Young s modulus of mineral platelet. The normal stress of RVE can be obtained by adding the normal stress σ p in platelets, which lie in the end face of RVE. The Young s modulus of mineralized collagen fibril is yield as (9) Results The modulus ratio between mineral platelets to collagen matrix is defined as E p /E m = 100, according to the experimental results 4, 13). The Poisson s ratio of mineral platelets and collagen matrix are taken to be v p = v m = 0.3. In order to demonstrate the effect of structural parameters, the sub unit is characterized by three auxiliary parameters: (10) Where ρ is the aspect ratio of mineral platelets, t is the distribution factor of collagen matrix, of mineral in sub unit. The Voigt model is also introduced to reveal the upper bound of the Young s modulus of composite. However, the structure of composite is not considered in Voigt model, only the volume fraction of platelets and material property are essential parameters 21,23,24). The Voigt model can be expressed as Yapeng Sun et al.: Model for mineralized collagen fibrils Voigt bound with increasing aspect ratio ρ. The figure also indicates that models of RVE including fewer sub units always achieve higher modulus with the same parameters. Fig. 4 (b) plots E voigt = E p φ p + E m ( l- φ ) p is the volume fraction is assumed as constant and decided only by relative displacement (11) the effective Young s modulus of composite as a function of the platelet volume fraction for t = 10, ρ= 15. The Young s modulus increases with increasing but the trends is obviously different from Fig. 4 (a). The difference between Voigt value and predict values would reaches its peak at about increasing = 60%. That is to say, would be a method to enhance the stiffness of composite, but the effect is remarkable only in the case that platelet volume fraction is above 60%. Fig. 5 (a) and 5 (b) plot the effective Young s modulus of RVE with respect to the distribution factor of matrix for two different volume fraction of mineral = 90%, 15%, espectively. The modulus of RVE with large volume fraction of mineral will increase with factor t, and the figure indicates that the modulus of composite made up of fewer sub unit is higher than that made up of more sub units. However, it is clear that the figure in Fig. 5 (b) show a different feature, which increase first then decrease. The effective Young s modulus could reach a peak by adjusting factor. That peak implies that there is an optimal factor t for RVE. It is worth noting that the peak for composite with different platelets are obtained on different factor t and the RVE with more sub unit demonstrate a higher peak modulus than that with fewer sub units. That indicates for composite with litter volume fraction of mineral could perform a high modulus by adjusting arrangement η and factor t. Discussion In this paper, a new parameter is introduced to calculate the effect of collagen distribution on the elastic modulus of mineralized collagen fibrils. The distance between in adjacent mineral platelet is ignore in original tension shear model, but its influence is remarkable shown in this paper. The effect of aspect ratio and volume fraction of mineral platelets are analyzed in different arrangement. The platelet and matrix in composite are considered as isotropic and ideal coherence between them. The shear stress in each region of platelets and the tensile stress of matrix and the shear stress of platelet are ignored. The accuracy of this model is reduced by those assumption. When the RVE contains two sub units (n = 2), the force and moment are balance. But the moment balance of mineral platelets can t be archived any more, if there are more units in RVE (n > 2), that affects the elastic properties of composite. That may explains why the RVE with fewer sub unit always perform a higher Fig. 4 (a) plots the effective Young s modulus of composite as a elastic modulus. function of the platelet aspect ratio ρfor t = 5, = 45%. The The effect of the collagen matrix lie in the lateral space between Young s modulus monotonically increases until approaching the adjacent mineral platelets has been analyzed thoroughly in the 77

4 J.Hard Tissue Biology Vol. 25(1):75-80, 2016 Fig. 4 Variations of the Young s modulus of RVE with aspect ratio ρand volume fraction p Fig. 5 Variations of the Young s modulus of the composite with t for the two different volume fraction of mineral Φ p = 90%, 15%, respectively former studies based on tension shear model, but the effect of collagen matrix lie in the axial space between adjacent mineral fraction is relatively low, the effective improvement for the stiffness of the composite element can be achieved by adjusting the lateral platelets is seldom considered. Besides the aspect ratio and space and axial space of mineralized platelets and raising the volume fraction of mineral Φ p, the distribution of matrix also number of sub unit in RVE. have a significant effect on the elastic modulus of composite. Typically, a RVE formed by more sub units performs a lower modulus than that formed by fewer sub unit. However, for the Acknowledgment This research was financially supported by the National RVE with low volume fraction Φ p the modulus of composite Natural Science Foundation of China (No ), the National could achieve a higher value by adopting structure including more sub units and adjusting the distribution of matrix. In conclusion, the results indicate that the increasing aspect ratio and volume fraction of mineral platelet will enhance the stiffness of mineralized collagen fibrils, and the Young s modulus of fibrils will approach to the upper limit with the aspect ratio Science & Technology Pillar Program of China ( No. 2012BAI18B01), the Fundamental Research Funds for the Central Universities (No. YWF-15-YG-021), the 111 Project (No. B13003), and the International Joint Research Center of Aerospace Biotechnology and Medical Engineering, Ministry of Science and Technology of China. increases. The results also demonstrate that, when the volume 78

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6 J.Hard Tissue Biology Vol. 25(1):75-80,