The staggered structure of bone - mechanical behaviour and toughening mechanism

POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Finite Element Methods

It It is becoming more precise There is much difficulty in

choosing real loading history

It enables to estimate the

strains wherever one likes The bone is still assumed

mechanically homogeneous

and isotropic

It has the possibility to

include many kind of

external forces The model has to be

validated with experiments

POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: assumptions

Quasi-static

Quasi-static analysis

Three-point bending

• Staggered arrangement of MCFs in extra-fibrillar protein matrix

• MCF modelled as elasto-plastic material

• Bilinear stress-strain relationship

• Cohesive Zone Models

•

Dynamic analysis

Staggered arrangement at the nanoscale: nanosized HA crystals

• in tropocollagen matrix

Both components assumed as linear elastic

• Comparison between a layered and staggered structures

• A step load is applied on the left vertical side

• POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Quasi-static analysis POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Dynamic analysis

EHA = 130 GPa

ETC = 1 GPa

ν = 0,28

ρHA = 2 g/cm3

ρTC = 1 g/cm3 POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Parametric studies

Quasi-static

Quasi-static analysis

Lenght of MCFs

• Thickness of MCFs

• Cohesive law of the extra-fibrillar protein matrix

•

Dynamic analysis

Structure of a single MCF

• Volume fraction of HA

• Thickness of HA

• POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughening mechanisms POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results

Decrease rate of stress over the distance along x-direction

POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results

Decrease rate of stress over the distance along y-direction

POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results

a, b: normal and shear stress attenuation along x-direction;

c, d: normal and shear stress attenuation along y-direction.

Unit pulse stress of 1 MPa applied on the vertical left side of the structures. The volume fraction of HA

is fixed. POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: results

a: x-direction; b: y-direction. POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Mechanical behaviour: conclusions

Increasing

protein failure

energy in the

interface

direction (// to

fibril axis)

Increasing Enhanced

aspect Staggered

mechanical

ratio of structure

performance

MCFs Nanoscopic

HA crystals in

order to avoid

size effect

(see also

toughness) POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughness: TSC model

• Anisotropic tissue

• Axial loading: HA particles

• Shear loading: TC matrix

• TC and HA perfectly

bonded

• Elasticity

• Length of HA distributes

shear stress along a wide

area, lowering the axial

stress

The collagen matrix acts as a protective deformable shell

surrounding the reinforcing particles. Homogeneous distribution

of stress is achieved and HA crystals are shielded against stress

concentrations. POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughness: TSC model POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughness: TSC model

●

= 60 also from experimental

observations. POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughness: crack in a staggered structure

W = fracture energy

w = width of the localized deformation strip

τp = shear stress in the plastically deforming TC

Sp = yield strength of protein

Sint = protein–mineral interface strength

Sm = limiting strength of the mineral crystals

ΘP = effective strain to which the matrix can deform before failure POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughness: crack in a staggered structure

●

Conditions to raise the strength of the protein and of the

interface at the highest level without breaking HA crystals.

20-50 MPa: the mineral strength must be of the order of GPa

theoretical strength of HA: nano-sized crystals are needed.

POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughness: molecular model

Only nascent bone

 Loading is applied by displacing a thin layer of particles

 Strain rate: 7,6*10^(-8) per integration step

 POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics

Toughness: molecular model

CF The

Deformation

● CF MCF

Deformation starts

Deformation starts at 6,7% tensile

at 5% tensile strain strain

The tissue strain The tissue

(applied strain) is (applied) and TC

always larger than strain remain much

the strain within TC closer during

molecules deformation

POLITECNICO DI MILANO

Nadia Paderno – 851112 - Micromechanics