Mechanical behaviour of single mineralized collagen fibril using finite element simulation coupled to quasi-brittle damage law

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Mechanical behaviour of single mineralized collagen fibril using finite element simulation coupled to

quasi-brittle damage law

Abdelwahed Barkaoui, Awad Bettamer, Ridha Hambli

To cite this version:

Abdelwahed Barkaoui, Awad Bettamer, Ridha Hambli. Mechanical behaviour of single mineralized

collagen fibril using finite element simulation coupled to quasi-brittle damage law. European Congress

on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), Sep 2011, Vienna,

Austria, September 10-14, 2012, Austria. pp.9. �hal-00788758�

Vienna, Austria, September 10-14, 2012

MECHANICAL BEHAVIOR OF SINGLE MINERALIZED COLLAGEN FIBRIL USING FINITE ELEMENT SIMULATION COUPLED TO

QUASI-BRITTLE DAMAGE LAW

Barkaoui A ¹ , Bettamer A ² and Hambli R ³

1,2,3 PRISME Laboratory, EA4229, University of Orleans Polytech’ Orlans, 8, Rue Lonard de Vinci 45072 Orlans, France

1 abdelwahed.barkaoui@univ-orleans.fr

2 abettamer2003@yahoo.com

3 ridha.hambli@univ-orleans.fr

Keywords: Bone, mineralized collagen fibril, finite element simulation, quasi-brittle damage law

Abstract. Numerical modeling of biocomposite at different length scales will provide improved understanding of the mechanical behavior of structures such as bone, and also guide the devel- opment of multiscale mechanical models. In this study, a new three-dimensional model based on finite elements method (FEM) was achieved to model the hierarchical structure- compounds properties relationship at the sub-structure scale (minera-lized collagen fibrils) and investigate its nanomechanical equivalents properties Furthermore, fracture strength/strain of fibril model was different from those reported for collagenous tissues of higher hierarchical levels, indicat- ing the importance of obtaining these properties at the fibrillar level for multiscale modeling.

Here, the results of finite element simulations coupled to quasi-brittle damage law were reported

to probe the mechanical response of mineralized collagen fibril subjected to uniaxial tension.

Barkaoui A, Bettamer A and Hambli R

1 INTRODUCTION

Bone is a mineralized biocomposite material which serves, among its functions, the support of the other tissues in the body. The origin of the mechanical properties of bone is in relation, with structural organization over different length scales and with the properties of elementary constituents [1, 2, 3]. Mineralized collagen fibril (MCF) is one such constituent that act as an organic framework in all bone material to play a significant role in determining the mechani- cal response of bone [4]. MCF is distinct building blocks for bone material and performs an important mechanical function.

Several researchers reviewed in detail the composition and structure of bone at the nanostruc- tural level [5, 6, 7, 8]. There are also various models proposed in literature for modeling bone at the nanoscale. Experimental works were performed to investigate the ultrastructure of mineral- ized collagen. Some authors focused on the mechanical properties of individual collagen fibrils [4, 9, 10, 11]. In addition, mechanical models for mineralized collagen fibrils have been devel- oped by several authors in order to estimate the mechanical properties of mineralized collagen fibrils and bone tissues [12, 13] and to model the 3D orthotropic elastic properties of a single collagen fibril [14]. Nikolov and Raabe [15] proposed a homogenization method to model the elastic properties of bone at the level of mineralized collagen fibrils from the staggered arrange- ment of collagen molecules up to an array of parallel mineralized fibrils. Jaeger and Fratzl [16]

proposed a model of mineralized fibrils with a staggered arrangement of mineral particles dis- tributed unequally in the gap and overlap zones of collagen fibrils. This geometric model has served as the basis for almost all the FEM models proposed for modeling of bone at nanoscale.

Jaeger and Fratzl [16] used this model to explore the effect of the mineral volume fraction and thickness as well as the distance of the HA platelets on the longitudinal elastic modulus, maxi- mum elastic strain, and maximum elastic stress (strength) of the MCF. Kotha and Guzelsu [17]

extended the Jaeger-Fratzl model [16] to investigate the effect of interphase and bonding on elastic properties of bone. Ji and Gao [18] used the Jaeger-Fratzl geometry [16] using analyti- cal formulation and a FEM analysis to obtain the transversely isotropic elastic constants of the MCF as a function of mineral aspect ratio. Yuan et al. [19] used a FEM analysis to predict the elastic properties of a mineralized collagen fibril both in 2D and 3D and verified their computa- tional results with experimental data obtained by synchrotron X-ray diffraction. They improved the shear lag model [16] by incorporating more structural features of the mineralized collagen fibril. They extended their 2D FEM analysis to a 3D geometry. To our knowledge, this is the first 3D mechanics model of a mineralized collagen fibril in which the staggered arrangement of HA crystal within the collagen phase is consi-dered. Molecular dynamics simulations (MD model) [20] have been developed to investigate the mechanical response under uniaxiale tension of individual mineralized collagen fibril. The results showed that the deformation and failure mechanisms of a collagen fibril are strongly influenced by its length and width as well as cross- linking density which, in turn, indicates the size dependence of failure mechanical properties of collagen fibrils.

Several parametric studies were done to investigate the effects of collagen Young’s mod- ulus, mineral Young’s modulus, mineral volume fraction, distance between Hydroxyapatite (HA)platelets, and thickness of HA crystals on the apparent modulus of the MCF. In this study, we expand upon previous models dealing with bone ultrastructure modeling in three aspects: (i) Creation of realistic 3D FE model to represent the structure of the mineralized col-lagen fibril with three constituents (mineral, mineralized collagen microfibril and cross-links); (ii) investi- gation of the fibril deformation behavior under tension (iii) validation of the model based on

2

experimental data from literature (stress-strain curve). The proposed 3D FE model of miner- alized collagen fibril enables the bottom-up investigation of structure/property rela-tionships in human bone. Such model can be applied to study the effects of biochemical de-tails related to the collagen, the mineral or the cross-links components on the strength of human bone.

2 METHOD AND MATERIALS

2.1 Mineralized collagen fibril composition

The elementary components of the mineralized collagen fibril can be distinguished as follow:

(a) Mineralized collagen microfibril: The microfibril is a helical assembly of five TC molecules (rotational symmetry of order 5), which are offset one another with apparent periodicity of 67 nm. This periodical length is denoted by the letter D and it’s used as a primary reference scale in describing the structural levels. The helical length of a collagen molecule is 4.34 D ≈ 291 nm and the discrete gap (hole zone) is 0.66 D ≈ 44 nm ( = 35 nm in some other references) between two consecutive TC molecules type1 in a strand. These gaps in bone are the sites of nucleation for hydroxyapatite crystals (the mineral component of bone tissue) to be deposited [21, 22, 23]

(b) Mineral: Bone mineral is composed of poorly crystalline hydroxyapatite (HA). HA is cal- cium phosphate [Ca10(PO4)6(OH)2] and recent studies using transmission electron microscopy (TEM) and scanning small-angle X-ray scattering (SAXS) have shown that HA is plate-like in shape. Fratzl et al. [24] determined the most probable size of the particle to be 15-200 nm long, 10-80 nm wide and 2-5 nm thick.

(c) Cross-links: They join the TC molecules. Cross-linking is either enzymatically or non- enzymatically mediated [25]. Cross-linking of TC molecules plays a critical role in bone mi- crofibrils, fibrils and fiber connectivity [26].The formation of these cross-links is mediated by the enzyme lysine oxidase [27]. It has been reported that the formation of intermolecular cova- lent cross-links has significant effects on material properties (strength and brittleness) [20], and mechanical behavior [27].

(d) Different non-collagenous organic molecules: These are predominantly lipids and pro- teins which regulate HA mineralization, probably by proteins supporting or inhibit-ing min- eralization, possibly also by lipids [28]. In the organic matrix, in addition to type-I collagen, there are hundreds of non-collagenous proteins [29]. However, in this study, because of their small volume fraction (altogether ¡10% of total organic com-ponents) and lack of evidence of a direct relationship with mechanical properties, these secondary proteins are typically ignored in deformation studies [30].

(e) Water: Water is the third major component in bone and occupies about 10-25% of the

bone mass [30]. Recent theoretical and experimental works highlighted the important role of

water for the failure properties of bone by a mechanism which consists in glueing together the

collagen and mineral phases [31].

Barkaoui A, Bettamer A and Hambli R

3 3D FE model of mineralized collagen fibril

In this section we present the 3D finite elements geometric model of mineralized collagen fibril adopted in this study and the mechanical quasi-brittle damage law used to investigate the mechanical behavior of fibril.

In this modeling a symmetrical and repetitive portion of the collagen fibril is considered. The dimensions of the square is ¡ (1/20) of the diameter of a mineralized collagen fibril, where the representation of a cylindrical shape by a square shape is acceptable (Figure 1). The microfi- brils have cylinders form. These microfibrils are regularly distributed in the inorganic matrix (mineral) of the mineralized collagen fibril.

Figure 1: 3D FE model of mineralized collagen fibril

In this study, we consider a three-dimensional model with a square shape (3D EVR) of the mineralized collagen fibril described above, for simulation by the FE method using the ABAQUS software. The model consists of approximately 26,000 linear tetrahedral elements type C3D4. For boundary conditions, some assumptions were taken into considerations: (i) the mineralized collagen fibril retains its rectangular elastic during loading, (ii) there is a constant movement in the axial direction of the upper and lower surfaces, in because of the periodicity along this direction and (iii) a uniform displacement in the radial direction for all nodes on the outer surface of the mineralized collagen fibril. View the lack of information and complexity of modeling, sledding in the interface between the two phases (microfibrils and mineral) has been neglected. The mineralized collagen fibril was tested in tension under tensile loading.

The fibril is considered a quasi-brittle material; consequently, a constitutive law coupled with a quasi-brittle damage has been used to study its mechanical behavior.

4

3.1 Isotropic damage law

To model the progressive development of crack propagation of bone through the decrease in its elastic stiffness, a homogenized measure of damage is introduced, which in the simplest case is represented by a scalar D. The stress-strain relation of elasticity based damage mechanics is expressed by Lemaitre, 1985 [32]:

σ _ij = (1 − D)C _ijkl ε _kl (1)

Where D denote the damage variable (repetition), σ _ij the Cauchy stress components, ε _kl and the linear strains and C _ijkl are the components of elasticity tensor.

In the case of anisotropic directions the effective stress will expressed in terms of the Matrix M(C) equation (1) within the assumption of homogenized measure of isotropic damage variable:

 

 

 

 

 

 

 

 

 



σ ₁₁ σ ₂₂ σ 33

σ ₁₂ σ ₁₃ σ 23

 

 

 

 

 

 

 

 

 



= (1 − D)



 

 

 

 



C ₁₁₁₁ C ₁₁₂₂ C ₁₁₃₃ 0 0 0 C ₂₂₂₂ C ₂₂₃₃ 0 0 0 C 3333 0 0 0

sym C ₁₂₁₂ 0 0

C ₁₃₁₃ 0 C 2323



 

 

 

 



 

 

 

 

 

 

 

 

 



ε ₁₁ ε ₂₂ ε 33

ε ₁₂ ε ₁₃ ε 23

 

 

 

 

 

 

 

 

 



(2)

Where

 

 

 

 

 

 

 

 



 

 

 

 

 

 

 

 

C ₁₁₁₁ = E ₁ (1 − ν ₂₃ ν ₃₂ )γ C ₂₂₂₂ = E ₂ (1 − ν ₁₃ ν ₃₁ )γ C 3333 = E 3 (1 − ν 12 ν 21 )γ

C ₁₁₂₂ = E ₁ (ν ₂₁ + ν ₃₁ ν ₂₃ )γ = E ₂ (ν ₁₂ + ν ₃₂ ν ₁₃ )γ C ₁₁₃₃ = E ₁ (ν ₃₁ + ν ₂₁ ν ₃₂ )γ = E ₃ (ν ₁₃ + ν ₁₂ ν ₂₃ )γ C 2233 = E 2 (ν 32 + ν 12 ν 31 )γ = E 3 (ν 23 + ν 21 ν 13 )γ C ₁₂₁₂ = G ₁₂

C ₁₃₁₃ = G ₁₃ C 2323 = G 23

(3)

And γ defined as:

γ = ( 1

1 − ν ₁₂ ν ₂₁ − ν ₂₃ ν ₃₂ − ν ₁₂ ν ₂₁ − ν ₃₁ ν ₁₃ − 2ν ₂₁ ν ₃₂ ν ₁₃ (4) The restrictions on the elastic constants due to material stability are:

 

 

 

 

 

 

 



C ₁₁₁₁ , C ₂₂₂₂ , C ₃₃₃₃ , C ₁₂₁₂ , C ₁₃₁₃ , C ₂₃₂₃ > 0,

| C ₁₁₂₂ | < (C ₁₁₁₁ C ₂₂₂₂ ) ^1/2 ,

| C ₁₁₃₃ | < (C ₁₁₁₁ C ₃₃₃₃ ) ^1/2 ,

| C ₂₂₃₃ | < (C ₂₂₂₂ C ₃₃₃₃ ) ^1/2 , det(C) > 0

(5)

3.2 Isotropic quasi-brittle damage law

Quasi-brittle damage growth depends on a damage loading function in terms of the strain

components. The growth of the damage variable is controlled by the damage threshold para-

meter k, which is defined as the maximum of the equivalent strain measure ε _eq reached during

the load history [33]:

Barkaoui A, Bettamer A and Hambli R

k = max(ε _eq ) (6)

The loading function of damage is expressed by Mazars and Pijaudier-Cabot, 1996 [33]:

f (ε _eq , ε ₀ ) = ε _eq − max(k − ε ₀ ) (7) Where ε ₀ is the initial value of k when damage starts. If the loading function f is negative, damage does not develop. During monotonic loading, the parameter k grows (it coincides with ε eq and during unloading and reloading it remains constant:

f ≤ 0: No damage growth and the material behaviour is elastic. f ≥ 0: Damage growth and reduction of the stiffness.

Combining experimental law and the dependence of damage growth to the hydrostatic pres- sure, a quasi-brittle damage law can be expressed by Ridha et al. 2012 [34]:

 

 

 



D = 0 ; ε eq ≤ ε 0

D = ^{( ε} _ε

^eq

f

) _n

; ε ₀ < ε _eq < ε _f D = 1 ; ε eq ≥ ε f

(8) Where ε _f denotes the strain at fracture given by:

{ ε _f = ε ^T _f ; (T ension)

ε _f = ε ^c _f ; (Compression) (9) Where ε ^T _f and ε ^C _f are the tensile strain and compressive strain at fracture respectively.

4 RESULTS

Here we present the result of numerical simulation of the fibril (Figure 2) shows the stress- strain curve, at Young’s modulus of mineral is E _m = 114 GPa, Poisson’s ratio ν _m = 0.27, Young’s modulus of microfibril E _f = 1 GPa [21], and the Poisson’s ratio ν _f = 0.28 [21].

Figure 2: Stress-strain curve on mineralized collagen fibril

Fracture of the fibril consists of two steps: Stepe1: Fracture of immature mineral Step 2:

Frac-ture of mature mineral in Gaps. Good agreement is obtained between predicted and experi- mental results in the small strain response based on microelectromechanical systems platform to test partially hydrated collagen fibrils under uniaxial tension [11].

6

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