Original Full Length ArticleCrack propagation in bone on the scale of mineralized collagen fibrils: role of polymers with sacrificial bonds and hidden length
Introduction
The human bone structure is a hierarchical composite of collagen and hydroxyapatite (HA) with several mechanisms to resist fracture at various scales [20], [27], [32].These size scales relate to the characteristic structural dimensions in bone, which vary from twisted peptide chains at the nanometer scale to the (secondary) osteon (haversian) structures, which are several hundred micrometers in size. The hierarchical structure at the intermediate scales includes (i) hydroxyapatite-impregnated twisted collagen fibrils at the scale of tens of nanometers; (ii) collagen fibers that are typically a micrometer in diameter and (iii) the lamellar structure of collagen fibers at several micrometer dimensions. The combination of this complex geometry and unique blending of material properties provides bone with remarkable levels of strength and toughness [10], [11].
In this paper, we focus on the mechanical response of a single mineralized collagen fibril sliding on a polymeric layer that includes sacrificial bonds and hidden length (SBHL) systems [36]. The fibril utilizes the breakage of sacrificial bonds and the release of hidden length to dissipate energy while being stretched [14]. This process introduces a microscopic mechanism for fracture resistance [12]. Our primary focus is investigating the effect of the polymeric glue material on the basic characteristics of crack propagation such as critical crack size, stable crack growth speed and energy dissipation.
The basic structure and operation mechanism of the SBHL system is shown in Fig. 1. The assembled glue molecule may include more than one polymer chain with sacrificial bonds forming within the chain itself, crosslinking the different chains and connecting the chains to the collagen fibrils. The large scale separation of the collagen fibrils is resisted by an array of parallel gel molecules as shown in Fig. 1(a). As long as the bond is intact, it shields parts of the polymer length from contributing to the end-to-end distance. This corresponds to a reduction in the chain entropy (the possible number of configurations resulting in the same end-to-end distance) and a corresponding increase in the initial stiffness of the polymer chain. After the sacrificial bond is broken, the shielded loop unfolds and significant energy is dissipated in reducing the chain entropy as it straightens out.
Previous theoretical models describing the mechanical behavior of bone glue polymers [e.g. [10], [13], [25]] have implemented the worm-like chain model [8] as an approximation for the AFM experimental curves. We adopt this model here as well. The more flexible the glue polymers are, the better this approximation will be. Nonetheless, further work is required to constrain the force displacement relation of single polymer molecules in the bone glue along its whole deformation history.
The existence of SBHL systems is incorporated in the worm-like-chain model by introducing a dynamical variable: the available length [10]. This available length is the difference between the polymer contour length and the sum of the length of the hidden loops that have not been unfolded yet. The rate dependence of the SBHL system is modeled using the transition state theory [4], [25]. In this paper we will implement the rate and displacement model developed by Lieou et al. [25] as the constitutive law for the polymeric layer with SBHL system.
The primary component of the human bone structure is mineralized collagen fibril. Buehler [6] developed a model for the mineralized fibril in nascent bone in which collagen is represented by tropocollagen molecules, cross-linked by hydrogen bonds, and the mineral plates are hydroxyapatite (HA) crystals forming in the gap regions between the collagen fibrils. The stiffness of collagen fibrils depends on the mineralization percentage. With aging, bone properties degrade [e.g. 47]. The mineralization percentage decreases and both the stiffness and the peak strength of the fibrils are reduced. We will also study the influence of mineralization on fracture properties of the mineralized fibril-polymer system.
We developed a coarse grained model for the mineralized collagen fibril with polymeric glue. The fibril is modeled using a one-dimensional mass-spring system. The stiffness of the springs is calculated using the fibril geometric properties and the stress strain behavior computed from Beuhler [6]. The polymeric layer is modeled using the constitutive description of Lieou et al. [25]. The system is integrated in the quasistatic limit which is appropriate for exploring nucleation characteristics and stable crack growth speeds. Depending on the polymer density, the system may fail by the breakage of the collagen fibril and not the detachment of the polymers. In this limit we use a fully dynamic approach to track these instabilities. This failure mode is relevant for understanding the deterioration of bone quality with age since the ability of bone cells to produce the polymeric glue decreases with age. We have also investigated the properties of the SBHL system since it has been shown previously that these molecular bonding provide a small scale energy dissipation mechanism and hence contribute to fracture toughness [36]. There are still insufficient experimental data about the geometrical properties of these systems and the effect of this mechanism on crack resistance. Hence, we pursue in this paper a parametric study to explore their relative contribution on fracture processes.
The remainder of the paper is organized as follows: In Section II we introduce our model for numerical simulation as well as its discretization. Then, the material properties of collagen fibrils and the polymer system are discussed. In Section III, we describe the numerical method and the integration scheme. In Section IV, the results of our simulations are presented demonstrating the effect of different properties of SBHL system, polymer density and mineralization ratios. We discuss the implications of our simplified model on bone fracture in Section V.
Section snippets
Kinetic model
In this section, we introduce the basic elements of the coarse grained model for the mineralized collagen fibril and the polymeric layer. We consider a single fibril, idealized as 1-D array of masses and nonlinear springs, sliding on a viscoplastic polymeric layer. As the fibril is pulled, the motion is resisted by the interfacial forces provided by the polymeric system. Detachment of polymer end bonds and failure of collagen fibril are expected as limit states.
Results of simulations
We numerically integrate the equations of motion of the different blocks coupled with the rate and displacement model of the polymeric interface using Newmark’s integration method and a predictor-corrector scheme. The detailed numerical approach is provided in Appendix A.2. Here we describe some of the quantitative predictions of our model for the different characteristics of crack nucleation and propagation in the fibril-polymer system.
Discussions
Problems involving dynamics of cohesively held interfaces arise broadly in biological [39 and references therein], engineering [[3], [37], [44], [45]], and geophysical [[24], [34]; and references therein] applications. Common to all of these applications are fundamental physical processes involving deformation, rupture nucleation, propagation and arrest. In strongly nonlinear problems, like dynamic fracture, small scale instabilities can lead to large scale system fragilities and it is
Acknowledgments
The authors are grateful to two anonymous reviewers for their thoughtful and constructive comments which greatly enhanced the manuscript. The authors are also grateful to Jean Carlson, Paul Hansma, Charles Lieou and Darin Peetz for constructive discussions on SBHL systems. This research is supported by the Department of Civil and Environmental Engineering (CEE Innovation grant) at the University of Illinois (Urbana-Champaign).
References (47)
Mathematical theory of equilibrium cracks in brittle fracture
Adv Appl Mech
(1962)- et al.
Merger of structure and material in nacre and bone – Perspectives on de novo biomimetic materials
Prog Mater Sci
(2009) - et al.
Sacrificial bonds and hidden length: unraveling molecular mesostructures in tough materials
Biophys J
(2006) - et al.
Effect of aging on the transverse toughness of human cortical bone: evaluation by R-curves
J Mech Behav Biomed Mater
(2011) - et al.
Mechanics of forced unfolding of proteins
Acta Biomater
(2009) - et al.
A damage-based cohesive model for simulating fatigue crack growth
Int J Solids Struct
(2009) - et al.
Multiscale modeling of bone fracture using cohesive finite elements
Eng Fract Mech
(2013) - et al.
Cohesive finite element modeling of age-related toughness loss in human cortical bone
J Biomech
(2006) - et al.
Anisotropy of age-related toughness loss in human cortical bone: a finite element study
J Biomech
(2007) - et al.
Crack growth resistance in cortical bone: concept of microcrack toughening
J Biomech
(1997)
Fracture length scales in human cortical bone: the necessity of nonlinear fracture models
Biomaterials
Changes in the stiffness, strength, and toughness of human cortical bone with age
Bone
Molecular energy dissipation in nanoscale networks of dentin matrix protein 1 is strongly dependent on ion valence
Nanotechnology
Experimental determination of the fracture toughness via microscratch tests: application to polymers, ceramics, and metals
J Mater Res
Models for the specific adhesion of cells to cells
Science
Nonequilibrium thermodynamics of driven amorphous materials. II.Effective-temperature theory
Phys Rev E
Molecular nanomechanics of nascent bone: fibrillar toughening by mineralization
Nanotechnology
Gene mapping with recombinant inbreds in Maize
Genetics
Entropic elasticity of lambda-phage DNA
Science
Properties of earthquakes generated by fault dynamics
Phys Rev A
Dynamics of polymer molecules with sacrificial bond and hidden length systems: towards a physically-based mesoscopic constitutive law
PLoS One
Sacrificial bonds and hidden length dissipate energy as mineralized fibrils separate during bone fracture
Nat Mater
Nanoscale ion mediated networks in bone: osteopontin can repeatedly dissipate large amounts of energy
Nano Lett
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