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Research Papers

Modeling of Stiffness and Strength of Bone at Nanoscale

[+] Author and Article Information
Diab W. Abueidda

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Mechanical Engineering Building,
1206 W Green Street,
Urbana, IL 61801
e-mail: diababueidda@gmail.com

Fereshteh A. Sabet

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Mechanical Engineering Building,
1206 W Green Street,
Urbana, IL 61801
e-mail: fereshteh.sabet@gmail.com

Iwona M. Jasiuk

Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Mechanical Engineering Building,
1206 W Green Street,
Urbana, IL 61801
e-mail: ijasiuk@illinois.edu

1Corresponding author.

Manuscript received December 5, 2016; final manuscript received March 16, 2017; published online April 6, 2017. Assoc. Editor: David Corr.

J Biomech Eng 139(5), 051006 (Apr 06, 2017) (10 pages) Paper No: BIO-16-1496; doi: 10.1115/1.4036314 History: Received December 05, 2016; Revised March 16, 2017

Two distinct geometrical models of bone at the nanoscale (collagen fibril and mineral platelets) are analyzed computationally. In the first model (model I), minerals are periodically distributed in a staggered manner in a collagen matrix while in the second model (model II), minerals form continuous layers outside the collagen fibril. Elastic modulus and strength of bone at the nanoscale, represented by these two models under longitudinal tensile loading, are studied using a finite element (FE) software abaqus. The analysis employs a traction-separation law (cohesive surface modeling) at various interfaces in the models to account for interfacial delaminations. Plane stress, plane strain, and axisymmetric versions of the two models are considered. Model II is found to have a higher stiffness than model I for all cases. For strength, the two models alternate the superiority of performance depending on the inputs and assumptions used. For model II, the axisymmetric case gives higher results than the plane stress and plane strain cases while an opposite trend is observed for model I. For axisymmetric case, model II shows greater strength and stiffness compared to model I. The collagen–mineral arrangement of bone at nanoscale forms a basic building block of bone. Thus, knowledge of its mechanical properties is of high scientific and clinical interests.

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Figures

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Fig. 1

Geometric representations of the two models considered in this study: (a) a staggered arrangement of HA inside collagen fibrils (image is taken from Ref. [34]) and (b) the HA minerals residing outside collagen fibrils (image is taken from Ref. [17])

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Fig. 2

Schematic of Jäger and Fratzl model (model I) [34]: (a) plane stress/strain cases and (b) axisymmetric case

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Fig. 3

Model for bone at the nanoscale assuming minerals lie outside collagen fibrils (model II). The model is investigated under several arrangements and assumptions: (a) plane stress/strain and no matrix, (b) axisymmetric and no matrix, (c) plane stress/strain and matrix, and (d) axisymmetric and matrix is used to separate the multiple sentences.

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Fig. 4

Schematic illustration of traction-separation law: (a) pure normal deformation (opening mode) and (b) pure tangential deformation (sliding mode)

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Fig. 5

Stress–strain curves of model I (uniaxial longitudinal tensile loading) at different fracture energies under (a) plane stress, (b) plane strain, and (c) axisymmetric assumption. The values in the legends are in J/m2. The strength of the interface is 64 MPa.

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Fig. 6

Effect of fracture energy on the strength of model I for uniaxial tensile loading. The strength of the interface is 64 MPa.

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Fig. 7

Von Mises stress contours for model I: (a) plane strain, (b) plane stress, and (c) axisymmetric cases under uniaxial tensile loading. The strength of the interface is 64 MPa while the fracture energy is 0.2 J/m2 and the applied strain is 2%. The unit of the stress values is GPa.

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Fig. 8

Stress–strain curves of model II. The interfacial fracture energy is 0.2 J/m2 while the strength of the interface is 64 MPa.

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Fig. 9

Longitudinal tensile elastic modulus of model II under different geometrical conditions. The strength of the interface is 64 MPa, and the fracture energy is 0.2 J/m2.

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Fig. 10

Effect of fracture energy on the longitudinal tensile strength of model II. The strength of the interface is 64 MPa.

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Fig. 11

Effect of the interfacial strength on the longitudinal tensile strength of model II. The interfacial fracture energy is 0.2 J/m2.

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Fig. 12

Von Mises stress contours for model II: (a) plane stress with matrix, (b) plane stress without matrix, (c) axisymmetric with matrix, and (d) axisymmetric without matrix. The strength of the interface is 64 MPa while the fracture energy is 0.2 J/m2 and the applied strain is 0.5%. The unit of the stress values is GPa.

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Fig. 13

Comparison between model I and model II. The strength of the interface is 64 MPa while the fracture energy is 0.2 J/m2.

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