The staggered structure of bone - Mechanical behaviour and toughening mechanism

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 to enhance the stress level in the soft matrix that operates protein deformation. The

nanosized super­strong mineral crystals allow the protein and the protein–mineral interface to have an opportunity to enhance their

strengths without fracturing the crystals. 11

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The optimal design is to raise the strength of the protein­mineral interface, together with that of the soft matrix, to a highest possible

∫ ≤ S ρ

¿ /

m

level, without breaking the crystals. The condition is ; since large aspect ratios are needed to compensate for the

S =S ¿

P

S ≥(30 40)S

softness of the matrix, there is also . Taking the yield strength of protein to be around 20­50 MPa, the

m P

mineral strength according to these conditions must be of the orders of GPa, which is near the theoretical strength of the mineral:

that’s why it is important that the HA crystals are at the nanoscale (size effect).

We can also understand the toughening mechanism with a strain amplification, provided by the staggered arrangement: the strain in

the soft matrix is magnified over the composite strain by the aspect ratio of the mineral crystals, allowing the protein to fully deform

and dissipate energy at the microstructural level without inducing large deformation on the composite level.

ON THE RELATIONSHIP BETWEEN THE DYNAMIC BEHAVIOUR AND NANOSCALE STAGGERED

STRUCTURE OF THE BONES

Failure of the bone occurs often under dynamic, and not static, conditions. Systematic finite element analysis was performed on the

dynamic response of nanoscale bone structures under dynamic loading. It was found that for a mixed mineral volume fraction and

unit cell area, there exists a nanoscale staggered structure at some specific feature size and layout which exhibits the fastest

attenuation of stress waves.

INTRODUCTION

Bone, a composite primarily consisting of mineral (calcium phosphate or hydroxyapatite) and protein (collagen) in a hierarchical

architecture, is a robust support for the whole human body, but also protects its various organs from external impacts.

The mechanical properties of the bone are believed to be dictated by its hierarchical structure of seven levels, made of strong and

stiff mineral and weak and soft protein. whole bone; osteons (100 µm) with Haversian lamellae and canals; fiber

arrays (7 µm); collagen fiber (+ fibrils); collagen fibril (300 nm) with enzymatic cross links; MCF made of collagen molecules (280 nm)

and HA nanocrystals (1­10 nm).

A staggered arrangement was suggested for the nanoscale structure: collagen matrix embedded with nano­sized HA crystals.

Importantly, this structure was experimentally observed.

Sizes, shapes, layouts and composition are closely related to their functionalities, for example, mechanical support and protection,

which require superior toughness and strength. In addition, biocompatibility and biodegradability also put stringent requirements for

12

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their components, such as proteins and mineral crystals. Bones provide extraordinary fracture toughness, strength as well as the

ability to resist flaws.

Computer simulations as finite element methods have also been used to study the mechanical behaviour of bones by considering

either single or multi­hierarchy.

Natural materials adopt a fascinating structure, which becomes insensitive to flaws at the nanoscale: this tolerance allows the bone

to be able to survive at extraordinarily high stress levels. However, it should be recognized that an important functionality of the bone

is to protect various fragile organs inside it, such as brain and marrow, from external impact. Hence, a fast stress wave attenuation of

the bone under dynamic conditions becomes vitally important. The stress wave attenuation arises from the wave refraction and

reflection at the interfaces of the hard mineral and soft protein portion. The structure with the fastest stress wave attenuation is the

one which is able to most efficiently damp the dynamic or impact load exerted to the structure.

FEM analysis was used to investigate the stress wave traveling across different staggered structures. It is shown that the staggered

structure is able to reduce the magnitude of the stress wave compared to the layered structure. There exists an optimal structure with

some specific size and layout that can lead to the fastest attenuation.

FEM MODELING AND VALIDATION

Qwamizadeh et al., 2015 validate their numerical procedure by using the solution of the layered structure and comparing it with

analytical solutions from literature.

On the left vertical side (x=0) a step load is applied: subsequently, the wave starts to travel within the structure. Both components

σ 0

are assumed to be linear elastic. A good agreement between the FEM results and the analytical model was found.

RESULTS AND DISCUSSION

The volume fraction of the mineral portion in the structure can be calculated from the unit cell:

ld

ϕ= (l+a)(b+d )

For the bone nanostructure, the mineral volume fraction is still currently unknown (2015). The value of 42% has been widely used

and experimentally verified for the whole bone. Since there are seven hierarchical levels, the volume fraction at the nanoscale level in

general should be greater than that for the whole bone. In addition, the length of the tropocollagen is about 280 nm; the thickness of

the HA platelets is in the range of 2­10 nm and their length from 15 to 150 nm; the horizontal distance between the plates was

reported to be on the same order as the thickness (2­4,5 nm).

In the simulations, the structure (layered and staggered), , the unit cell area, the mineral plate thickness d and length l have been

Φ

varied in order to perform parametric studies.

E = 130 GPa, E = 1 GPa, = 0,28, = 2 g/cm , = 1 g/cm .

3 3

ν ρ ρ

m c m c 13

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Why does the bone take a staggered structure rather than a layered structure? The figure shows the wave profiles for two different

points at different distances from the impact surface with respect to time when passing through the layered structure (a), the collagen

portion (b) and the mineral portion (c) of the staggered structure. The stress wave profiles passing through the layered structure are

similar at different distances from the impact surface. However, the transition times are different, causing the waves to start at

different times. Thus, for a larger distance, the wave needs more time to reach that position. For the staggered structures, the stress

peaks are not the same in the mineral and the collagen components. Clearly, the magnitude of the stress in the mineral portion is

greater than that in the collagen zone. 14

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To analyze the stress wave attenuation, Qwamizadeh et al. impose a unit pulse load on the vertical left side of the structure (x = 0,

impact surface). The duration of the unit load is taken to be the time that the wave needs to pass one unit cell. Then, they plot the

average stress of unit cell with respect to the distance from the impact surface to study the attenuation. The average stress in the unit

cell can be found based on the current stresses in its elements:

n

∑ σ A

ei ei

i=1

(σ ) =( )

ave t n

j ∑ A ei t j

i=1

Finally, the maximum average stress for a particular unit cell during the time history is used as the average stress within the unit cell.

The decrease rate of stress over the distance is used as a measure for stress attenuation. For the incidence of the elastic wave in the

x direction, the stress in the layered structure is greater than that in all staggered arrangements with the same and unit cell area.

Φ

This may provide an explanation on why the staggered structure is taken by nature for load­bearing biological materials. Among all

these staggered structures, the stress attenuation levels are different. It should be noted that the cases with thickness larger than 10

nm is geometrically not allowed for this mineral volume fraction. 15

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Along the y direction, the stress wave attenuation in all the staggered arrangements is again much faster than that in the layered

structure. Among the staggered structures, the stress attenuation levels are different from those of the counterparts along the x

direction: there is a specific structure for each mineral and unit cell area which leads to the fastest attenuation of the wave. With a

Φ

higher , the fastest attenuation of wave demands a larger mi