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 micro­length 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. One

1

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 stress­strain 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 extra­fibrillar protein matrix, and

separation between MCFs under loading was observed in experiment.

One typical feature observed in MCF array is the high aspect ratio (lenght over thickness). Extra­fibrillar 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 extra­fibrillar protein matrix. The focus is to

investigate the effects of the dimension of MCF and the failure energy in extra­fibrillar protein matrix on the overall mechanical

performance.

MATERIALS AND METHODS

At a particular length scale, it is reasonable to simplify the internal structure ad a regular two­dimensional MCF array in protein. The

numerical model consists of a staggered arrangement of MCFs, and extra­fibrillar protein (thickness of 1 nm) is included between

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

2 https://en.wikipedia.org/wiki/Toughness 1

MICROMECHANICS

fibrils. Three­point 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 elasto­plastic material, and the stress­strain relationship is simplified with a bi­linear response.

The intermolecular slip between HA platelets and tropocollagen starts at the yield (Y) point, and the saw­tooth shaped curve is due to

this continuous process. The behaviour of MCFs is assumed as isotropic with = 0,28.

ν

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.

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. 2

MICROMECHANICS 3

MICROMECHANICS Damaged protein

corresponding to points a­f 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. 4

MICROMECHANICS

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. 5

MICROMECHANICS

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. 6

MICROMECHANICS

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. 7

MICROMECHANICS

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 = L/h (high values for long reinforcing particles). The

ρ

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

8

MICROMECHANICS

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 stress­free short side. This tension­shear 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

2τdx= dxh

∂x

τ

σ x x

( )=2

HA 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:

1

τ σ

=2 HAm ρ

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

C

μ

collagen has tangential stiffness .

c 9

MICROMECHANICS

σ L 1−ϕ

HAm 2 γ h

+ C

2 E ϕ

HA

ϵ =

Macro L

4(1−ϕ)

1 1

= +

́ 2 2

E E

ϕ

μ ρ

ϕ HA

c

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

́

E E

/

magnitude. Therefore, the ratio increases with , rapidly for small values and with lower slope for higher value. An

ρ

HA ́

E=ϕ E E

+(1−ϕ)

asymptotic value is obtained for very high (rule of mixture): (the second term is negligible because

ρ HA C

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 = 60 is consistent with these numerical

ρ

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 BONE­LIKE MATERIALS (GAO, 2005, CAP. 4) 10

MICROMECHANICS

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

bone­like 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 bone­like 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 of protein (corresponding to

τ p p

the stress required for domain unfolding), the protein–mineral interface strength S and the limiting strength of the mineral crystals

int

S ; denotes the effective strain to which the soft matrix can deform before failure. It is quite obvious that the toughness of bio­

Θ

m P

composites should increase with the volume fraction of protein (1− ): the more protein, the more material absorbs and dissipates

Φ

fracture energy. The effective strain , as a measure of the deformation range of protein, is a key parameter for fracture energy of

Θ P

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 =S , then should include not only domain

Θ

int p P

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