The staggered structure of bone - Mechanical behaviour and toughening mechanism

Micromechanics

Mechanical behaviour of staggered array of mineralized collagen fibrils in protein matrix: effects of fibril dimensions and failure energy in protein matrix

Introduction

The constituents at the nano- and micro-length scale play a critical role in determining the mechanical performance of biological composites. The mechanical behaviour of MCF array depends on the fibrils dimensions and on the intrinsic failure energy in extra-fibrillar protein matrix (≠ tropocollagen?).

Bone: combination of high stiffness and toughness. Stiffness is the rigidity of an object — the extent to which it resists deformation in response to an applied force. Toughness is the ability of a material to absorb energy and plastically deform without fracturing. One¹ definition of material toughness is the amount of energy per unit volume that a material can absorb before rupturing. It is also defined as a material's resistance to fracture when stressed. It is the energy of mechanical deformation per unit volume prior to fracture. The area under the stress-strain curve is called toughness.²

The structure of bone is made of seven hierarchical levels. One of the building blocks identified is mineralized collagen fibrils, consisting in hydroxyapatite platelets and tropocollagen protein matrix. MCFs are connected through extra-fibrillar protein matrix, and separation between MCFs under loading was observed in experiment.

One typical feature observed in MCF array is the high aspect ratio (length over thickness). Extra-fibrillar protein matrix plays an important role in energy dissipation.

Due to the complexity of the internal structure, and to technical difficulties, deformation and failure mechanisms have not been well understood. Alternatively, numerical methods have been widely used to investigate the mechanical behaviour (as finite element simulation).

In this study, the attention was focused on the staggered array of MCFs embedded in extra-fibrillar protein matrix. The focus is to investigate the effects of the dimension of MCF and the failure energy in extra-fibrillar protein matrix on the overall mechanical performance.

Materials and methods

At a particular length scale, it is reasonable to simplify the internal structure as a regular two-dimensional MCF array in protein. The numerical model consists of a staggered arrangement of MCFs, and extra-fibrillar protein (thickness of 1 nm) is included between fibrils. Three-point bending was applied on the model. The MCF dimensions were systematically changed to study the effects of length, thickness and aspect ratio on the mechanical behaviour.

MCFs were modelled as elasto-plastic material, and the stress-strain relationship is simplified with a bi-linear response.

The intermolecular slip between HA platelets and tropocollagen starts at the yield (Y) point, and the saw-tooth shaped curve is due to this continuous process. The behaviour of MCFs is assumed as isotropic with ν = 0,28.

To simulate the material behaviour and failure process of the matrix, cohesive zone models were used. The material is linearly elastic until the maximum strength, then it is followed by a linear evolution of material softening until complete damage.

• Cohesive Law A: Similar material softening slopes in the interface (s = along the fibril axis) and thickness (t) directions.
• Cohesive Law B: Slower material softening rate in the interface (s) direction.
• Cohesive Law C: Slower material softening rate in the thickness (t) direction.

¹ https://en.wikipedia.org/wiki/Stiffness

² https://en.wikipedia.org/wiki/Toughness

Results and discussion

High tensile stresses are generally observed at the middle of MCFs. Throughout the loading process, the highest stress is normally located around the sharp notch or around the tips of damaged protein.

The MCF array deforms elastically at the beginning. Negligible amount of damaged protein is observed when the displacement is less than 200 nm. With the increase of the damage zone in protein, the stiffness starts to decrease. The force continues to increase up to the peak force, then drops gradually.

Damaged protein corresponding to points a-f in the force displacement curves (length = 2000 nm, thickness = 100 nm).

The initial stiffness with CLA and CLB are almost identical; the curves start to differentiate at the displacement of 260 nm. Higher maximum force and lower protein failure rate are observed with CLB. Small differences are observed between CLA and CLC. Therefore, the cohesive energy analysis indicates that the maximum force of MCF array can be enhanced considerably if the protein failure energy in the interface direction (s) is increased. On the other hand, the protein failure energy in the thickness direction has a minimal influence on the mechanical behaviour.

The longer the MCFs, the higher the maximum force. The displacement at the maximum force also increases with increasing length. The fraction of damaged protein is very similar changing the lengths.

The maximum force increases with reducing MCF thickness, and the displacement is also higher. An increase in the protein failure rate with increasing thickness is observed.

Less tortuous damaged protein patterns and delayed damage progress are observed in those having thinner MCFs. Therefore, the reduction of thickness is a way to sustain a high load bearing capacity.

Both the max force and energy dissipation increase with increasing protein failure energy in the interface direction. The improvement is greater for the array with longer or thinner fibrils. The mechanical performance of the array can be improved increasing the failure energy in the matrix in the interface directions. Increased maximum force and energy dissipation can be achieved either by increasing the MCFs length, or reducing the thickness. Both the max force and energy dissipation thus increase with increasing aspect ratio.

More noticeable difference in the fraction of damaged protein is also observed in the result of different MCF thicknesses compared to the results of different lengths. These results suggest that the thickness has a greater influence on the damage of protein and energy dissipation.

During early stage of mineralization, HA platelets situated in the gap areas between collagen phases in fibrils can grow and affect the overall mechanical behaviour. However, some studies claimed that HA platelets are also present outside the fibrils: the computational model has to be further improved. MCFs acts as effective deflectors to delay the growth of damaged protein.

Slide 02

Slide 03

The protein matrix can absorb part of the energy otherwise available for fracture propagation in brittle material. The collagen matrix acts as a protective deformable shell surrounding the reinforcing particles; these latter are shielded against stress concentrations as the stress are more homogeneously distributed within the particles. The ceramic component is represented by rectangular platelets of length L and width h for which a dimensionless aspect ratio is defined ρ = L/h (high values for long reinforcing particles). The typical characteristic length of the particles falls in the range 100 nm and 500 nm; the typical characteristic width is about the order of few nanometres: the typical values of the aspect ratio is therefore between 10 and 100. The tissue is anisotropic: it can be expected that the mechanical response along the axial direction (long axis of the particles) is different than that exhibited along the transverse direction.

Axial loading is mainly borne by the reinforcing particles (HA) and that tensile load is transmitted from one particle to the next one through the shear stress in the collagen matrix; this latter being perfectly bonded to the lateral surfaces of HA crystals. This model implies that the tensile stress acting in the collagen matrix in the interspace between two adjacent particles (along the short side) is negligible and therefore that reinforcing particles has a stress-free short side. This tension-shear chain model (TSC) can be represented by a mechanical system with a series of two elastic elements, the HA (tensile stress) and the collagen (shear stress). The significant length of the ceramic particles allows to distribute the shear stress (from the collagen) along a wide area, thus lowering the axial stress.

The overall elastic modulus can be obtained as the ratio between the macroscopic strain and the macroscopic uniaxial stress. On the reinforcing particle the shear stress acting on the lateral surfaces and the axial stress acting on the cross sections are considered. As an approximation, the axial stress is linearly distributed over the length.

∂σ_HA/∂x = 2τ/h

σ_HA(x) = 2τx/h

At the centre of the particle the axial stress achieves its maximum value. Considering the average particle stress and introducing the aspect ratio, the relationship between the average particle stress and the shear stress in the collagen is obtained:

τ = σ_{HA m} / 2ρ

The average macroscopic stress can be obtained as the average value of the tensile stress in the two phases of the composite. Further, the average macroscopic strain can be estimated by considering the summation of strain in the two phases. The component γ due to HA crystals is obtained by integrating the strain field within them, while the component due to the shear strain in the collagen has tangential stiffness μ_c.

ε_Macro = σ_{HA m}L / 4E_HA + γ_Ch / 2

1/Ē = 1 / E_HAΦ + 4(1 − Φ) / ρ²μ_cΦ

The macroscopic elastic modulus will depend on the aspect ratio; in particular, the higher the aspect ratio, the higher the elastic modulus. The mismatch between the elastic modulus of the ceramic phase and that of the collagen can be as high as three orders of magnitude. Therefore, the ratio Ē/E_HA increases with ρ, rapidly for small values and with lower slope for higher value. An asymptotic value is obtained for very high ρ (rule of mixture): Ē = ΦE_HA + (1 − Φ)E_C (the second term is negligible because of the three order of magnitude).

From experimental data, the bone tissue is characterized by: volumetric fraction of about 45% of HA; ratio between the macroscopic elastic modulus and that of HA (100 GPa) of about 0,2; from the graph, one can infer that ρ = 60 is consistent with these numerical data, and this value is also validated from experimental observation. Therefore, the TSC model is good.

Application of fracture mechanics concepts to hierarchical biomechanics of bone and bone-like materials (Gao, 2005, cap. 4)

We have discussed that the staggered alignment and large aspect ratios of mineral crystals play an essential role in creating a stiff biocomposite in spite of a high volume fraction of the soft matrix. The mineral crystals thus provide the required structural rigidity for bone-like materials. However, a rigid structure by definition does not deform much and is usually brittle. How does nature build toughness into the structure? To estimate the fracture energy of the staggered nanostructure of bone-like materials, consider a crack growing in an infinite medium made of the staggered biocomposite as shown.

The deformation in the protein layers is assumed to remain uniform; another hypothesis is that HA crystals remain intact during fracture.

W = w ∫ σ(ε)dε

Where w (bigger than L) is the width of the localization strip and W the fracture energy. At the composite level, crack propagation occurs by pulling the hard crystals out of the soft matrix. The fracture energy becomes: being the shear stress in the plastically deforming soft matrix which is limited by the yield strength S_p of protein (corresponding to the stress required for domain unfolding), the protein–mineral interface strength S_int and the limiting strength of the mineral crystals S_m; Θ_P denotes the effective strain to which the soft matrix can deform before failure. It is quite obvious that the toughness of bio-composites should increase with the volume fraction of protein (1 − Φ): the more protein, the more material absorbs and dissipates fracture energy. The effective strain Θ_P, as a measure of the deformation range of protein, is a key parameter for fracture energy of the biocomposite. The hierarchical structures of proteins are ideally suited for absorbing and dissipating fracture energy. If the mineral crystals are strong enough to remain intact during the deformation and S_int = S_p, then Θ_P should include not only domain unfolding of protein molecules but also slipping along the protein–mineral interface. Therefore, it will be advantageous to let the interface have the same strength as protein to maximize the deformation range of the soft matrix (plus slipping along the interface).

In order to achieve maximum toughness, large deformation alone is not sufficient as it is the area under the stress–strain curve which defines the fracture energy. It is also important