why bones are tough · embedded in protein matrix (b) simplified tension–shear chain model...

A.Ashok Reddy Why bones are tough although consisting of brittle materials?

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Why bones are tough??? Bone is a very interesting material in our body. All bones put together called

skeleton. To say that no part of exist without bones is not an exaggeration. When a child is born his/her skeleton consists of about 276 bones, but as he/she continues to mature, by the time he/she reaches adult age, number reduces to 206. This is because in the process of maturing he/she looses some bones. Skeleton (Bones) gives a shape to the body. Bones, along with muscles, holds up our body in position, and also gives protection to our body. Try to imagine our body without skeleton in it; I think it will be a scary thing. Although extensive studies and research is done to understand the behavior, structure and etc.. about bones some of the important properties about it especially origin of toughness is not perfectly understood. This paper is an attempt to put all the recent developments and details about bone and reasons for its superior strength.

1. Introduction Bone is a living, growing tissue, can be viewed as highly porous material. In generic sense bone refers to a family of materials [11], all of which are composites of mineral crystals (Calcium phosphate) and protein matrix (Collagen). Along with these two it also consists of water, vessels, cells, minerals etc... It has highly complex structure; from macroscopic to microscopic length scales they exhibit seven orders of hierarchical structures. It is a very tough and stiff material. It is considered as nanocomposite of the mineral and protein matrix. Mineral element is very hard, but has very poor energy absorbing energy, where as protein is very soft but has very high energy absorbing energy. Ideally mineral is qualified as stiff material for its high stiffness, where as protein is qualified as tougher material for its high fracture energy. But nature takes these two and makes wonderful material like bone which is stiffer and tougher. Important thing to notice here is that bone does not behave as general composite made out of two materials, its properties are far different from its constituents [7]. So what makes this possible? Answer to this is attributed to nanostructure of bone. So let us first look at the structure of bone briefly.

2. Structure Bone has various structural arrangements at different length scales. These are chosen by nature for various functions of bones, such as for structural support, protection, storage of healing cells and etc. [3]. There is no specific technique to explore the structure of bone at different length scales. It has different components in its complex structure at different length scales. Understanding of structure of bone (At different length scales) becomes important when we are trying to understand its mechanical properties. Some of the structures at different length scales are [3],

1. The macrostructure: cancellous and cortical bone 2. The microstructure (From 10 to 500 mm): Haversian systems,

osteons, single trabeculae 3. The sub-microstructure (1–10 mm): lamellae 4. The nanostructure (From a few hundred nanometers to 1 mm):

fibrillar collagen and embedded mineral


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5. The sub-nanostructure (Below a few hundred nanometers): molecular structure of constituent elements, such as mineral, collagen, and non-collagenous organic proteins.

These hierarchical structures are highly irregular organizations. But these are optimized, with preferred orientations of components making bone to be heterogeneous and anisotropic.

Figure.1:- Hierarchical structural organization of bone [3] As we are interested in the nanostructure of bone let us look at only nanostructure and sub-nanostructure of bone.

2.1 Nanostructure Structures, which can be seen at this scale, are collagen fibers surrounded by mineral elements [3].

2.2 Sub-Nanostructure This length scale consists of three main materials [3]; they are mineral crystals (Apatite), collagens, and non-collagenous organic proteins. Mineral crystals in bone are discrete and discontinuous, owing to their occurrence within discrete spaces within collagen fibrils. These are plate shaped crystals. These crystals grow with their C axes parallel to longitudinal axes of collagen fibril (Growth in specific crystalline orientation), but their growth is limited due to limited available space within collagen fibril. Approximate dimensions of these crystal plates are 50nm (Length), 25nm (Width), and 2-3nm (Thickness). The main organic constituent of protein matrix is Type I collagen. Proteins such as phosphoproteins (Osteopontin, sialoprotein, osteonectin, and osteocalcin) will constitute to non-collagenous organic proteins, which regulate the size, orientation, and crystal habit of mineral deposit.


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Figure.2:- A schematic diagram illustrating the assembly of collagen fibrils and fibers and bone mineral crystals [3]

2.2.1 Minerals in Bone Mineral are inorganic elements [14], suggesting that they are not produced by plants and animals. For proper functioning of the body our body needs different mineral along with vitamins. Among so many minerals that body requires calcium and phosphorus are important as they serve as building blocks of bones (For teeth also).

2.2.1.1 Calcium:- Calcium plays an important role in formation of bones. Long term deficiency of calcium will lead to the problems like osteoporosis in which bones become too brittle that they loose their strength and density, and breaks easily.

2.2.1.2 Phosphorus- This mineral also involved in bone formation. Deficiency of this will lead to malformation of bones and pain [16].

So from above descriptions it can be established that bone is a nanocomposite (Fig.3) of proteins and mineral crystals, and its basic building blocks spring-up at nanoscale. These building blocks are best described as plate like mineral crystals embedded in protein matrix.

3 Toughness Toughness of a material is defined as its ability to absorb impacts and resistance it offers to effects like fracture [5], when it is suddenly stressed. It is the amount of energy that a material can absorb before breaking. This can be found by finding the area (i.e., by taking the integral) underneath the stress-strain curve, and in general called as modulus of toughness. Figure.4:- Toughness of material

Figure.3:- Bone is made of plate-like crystals (2-4 nm thick, up to 100 nm long) embedded in a (collagen-rich) protein matrix [1,2,7]


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Origin of toughness of bone is still mysterious. Thompson et al [5,6] try to explain by saying that collagen (The main organic constituent of the bone) has sacrificial bonds are responsible for toughness of the bones. These sacrificial bonds break upon loading without significant damage to the bone, and some of these will reform when the force applied is removed, allowing bone to repair itself to some extent. Bone’s toughness attributed to the mechanical work required to overcome these bonds. We will look at these aspects of the bone in subsequent discussion.

4. Mechanical properties of Bones With decades of work to understand nature of the bone, now we are able to

explain reasons for some of the mechanical properties of the bones like stiffness, but reasons for toughness is some what not perfectly understood, although researchers speculated in different ways about it. Higher strength and toughness of the bone hinges on its nanoscale structure, and also viscoelastic properties of the bone. Bone is as tough as mineral and as stiff as protein. Some of their mechanical properties of individual elements of bone are shown in Table 1

Table.1:- Mechanical properties of bone and its constituents [7] It is very clear from the above values that although stiffness of the bone is less than its constituents, it is quite surprising that its strength and toughness are much higher than even mineral (Has these values higher than protein). If we follow basic composite theories, the properties of bone should be in between the properties of its constituents, but experiments revealed that it is not the case, as we can clearly see that except for stiffness properties of bone is exceeding even that of mineral. It fascinates a person how wonderfully nature choose structure and materials to form stuff like bones. It urges us to look more closely at the bone. Researchers worked in that direction and came out with some models to explain these things. Figure.3 shows the nanostructure of bone where in mineral platelets are arranged in a staggered pattern, which is very important aspect to notice.

4.1 Stiffness The table also indicates that although protein content in bone is sometimes more than and comparable to that of mineral, which is 1000 times harder former, bone. For a two-phase linearly elastic composite, for given volume fraction the upper bound to the stiffness is given by Voigt model [8,9], which states that

2211 VEVEEC += Similarly the lower bound defined by Reuss [8] model is,

2

2

1

11EV

EV

EC

+=


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Where CE is stiffness of composite, E1 is stiffness of first material, E2 is stiffness of second material. V1is volume fraction of first material, and V2is volume fraction of second material.

Figure.5:- Schematic representation of (a) Iso-strain and (b) Iso-stress models [8]

So if we go with these models to calculate bone stiffness, the values we get from

Voigt model ranges from 20-45 GPa, where as values from Reuss ranges from 0.08-0.18 GPa. Experimental values from Table.1 are clearly approaching values suggested by Voigt model. This indicates that both materials in bone are aligned with the longitudinal direction of bone, in which direction most of loads (Compressive and tensile) are expected to be applied. So this much loads obviously not taken by the mineral, because it is too brittle, and cannot sustain high stress. So protein is assumed to play an important role for bone to achieve higher stiffness. This is so because protein matrix acts as a soft wrap around mineral platelets, homogenizes the stress distribution in bone, and hence protects mineral platelets from peak stresses because of applied loads.

Developments in the fields of microscopy and etc., such as high-voltage electron microscopy (HVEM) and small angle X-ray scattering method (SAXS) made it possible to study the internal structure of bones. Studies had shown that shape of mineral crystals is highly anisotropic, owing to platelets of 3nm thick and up to 100nm long. At this nano level the mechanical properties of bones will depend on the precise arrangement of the mineralized crystals within collagen fibrils. It is quite clear that in basic structure of bone mineral crystals have large aspect ratio (Length to thickness ratio) approximately of the order of 30, and aligns, parallel (Preferably) to the longitudinal axis of the bone, in collagen matrix. The suggested structure of bones is “mineral crystals in a staggered fashion in collagen matrix”; this was consistent with previous studies and the actual arrangement in bone.

4.1.1 Tension-shear chain model Jager and Fratzl [18] worked on staggered arrangement of bone and developed a simple model to estimate the stiffness of bone. With this work Gao et al [1,2,7]further


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explored bone and proposed a one dimensional composite model to study the nanoscale effective stiffness of bone. His figure is shown in Figure.6

Figure.6:- A model of bone (a) A schematic diagram of staggered mineral inclusions embedded in protein matrix (b) Simplified tension–shear chain model showing load transfer mechanism in bones, tensile regions of protein are eliminated to emphasize the load transfer within the composite structure.(c) The free body diagram of a mineral crystal [1,2,7]

Mineral crystals are hard by nature and have large aspect ratios compared to soft protein matrix. Tensile zone in protein matrix near the ends of mineral crystals are assumed to be carrying no applied load. By and large the load transfer is accomplished by high shear zones of protein matrix. Most of load is taken by the hard mineral crystals. Fig. 2(b) best explains this, where in tensile zones in protein matrix are eliminated and protein matrix in between mineral crystals is responsible for load transfer across the mineral crystals, while these mineral crystals are taking most of the load applied. As the path of load transfer is simplified to a one dimensional serial spring system consisting of protein matrix and mineral crystals, which are responsible for load transfer through shear and tension respectively, this model is named as tension-shear chain model.

Fig. 6 shows the stress distribution over the entire length of the mineral elements. From the assumptions it is clear that at the both ends of the mineral crystal stress developed is zero. From figure it is clear that if the length of the platelets is very large than thickness, i.e. if the aspect ratio is large the force applied is going to be distributed over large shear region, inducing negligible stress in protein. It is assumed that stress distribution along the length of the mineral platelets is linear. So average and maximum stresses can be formulated as

pm ρτσ = , and pρτσ = Where mσ is the maximum tensile stress,

σ is the average tensile stress,


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pτ is the shear stress of protein, and ρ (=L/h) is the aspect ratio of mineral platelets, L and h are length and thickness of the platelets.

It is clear from these that as the aspect ratio increases tensile stress of mineral is more with other conditions remain same. So tensile-shear chain model best describes the mechanism of load transfer in bones. How nature able to maintain stiffness of bone close to that of mineral platelets in spite of higher volume fraction of protein matrix in it? This is explained now. As explained previously as protein does not carry any tensile load the effective tensile stress of the bone is given by

mσσ Φ= , WhereΦ is the volume fraction of the mineral.

In similar way effective strain of the bone due to shear and tensile deformations of protein and mineral respectively is given by

Lhpm ΦΦ−+∆

=)1(2ε

ε , m

mm E

L2σ

=∆ , p

pp µ

τε = .

Where m∆ and pε are tensile elongation and shear strain of the mineral and protein respectively,

pµ is shear modulus, and

mE is the Young’s modulus of mineral platelets. So from all this effective stiffness of bone can be established as

mP EE Φ+

ΦΦ−

=1)1(41

22ρµ

Results from this analysis are well agreement with the calculations from finite element method, when the aspect ratios are much deviated from unity, and can be used for analytical descriptions of bones. The widely used Mori-Tanaka method of engineering composite theory lags behind in comparison with analysis of Gao et al [7], for it is not considered the staggered arrangement of mineral crystals in protein matrix.

Figure.7:- Comparison of the Young’s modulus of biocomposites predicted by the tension–shear chain model, by FEM calculations and by the Mori–Tanaka method. The results show that the simple chain model gives predictions in reasonable agreement with FEM calculations. In contrast, the Mori–Tanaka method significantly underestimates the composite stiffness. E, Ep and Em are the Young’s modulus of the composite, the protein and the mineral. Em/EP = 1000 and Φ = 45% (bone) [7]


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So this proves that mineral platelets are in reality arranged in staggered pattern, other wise bone would have been very softer with definite fracture planes as volume fraction of protein is more. The above analysis indicates that large aspect ratio of mineral elements compensates for low stiffness due to protein through the product 2ρµ p , i.e. an increase in one order of mineral aspect ratio will compensate two orders of reduction in stiffness of bone due to protein. Comparisons are shown in Figure.7. Hence large aspect ratio and staggered arrangements of the mineral platelets are key factors for bone to achieve high stiffness (Approaching upper limit defined by Voigt model).

4.1.2 Viscoelasticity of Bone It is well known that protein matrix in bone exhibits significant viscoelasticity. By saying this we mean is that it has time dependent stress-strain behavior. Experiments [7,18] have shown volume fraction of the protein matrix in bone is about 55-60 %. But stress relaxation curves (Graph between stress and time) are found to be similar to that of demineralized bones [7]. This shows that there is major contribution from protein matrix to viscoelasticity of bone. Viscoelasticity of any material helps it to dissipate fracture energy under dynamic loads, and increases the area under stress-strain diagrams significantly.

4.1.3 Insensitive to flaws-Critical length scale Mineral platelets must sustain large tensile stress without fracture, and the protein

matrix and protein/mineral interface must sustain large shear deformation to dissipate fracture energy, in order to ensure optimum strength and toughness of the bone. So tensile strength of mineral platelets is important factor in enhancing fracture toughness of the bone. But these mineral platelets are too brittle, and easily undergo fracture. Nature’s design of bone against these constraints is studied by Gao et al [1,2,7]. Theoretical strength of mineral platelets indicates that it can sustain that much mechanical stress without fracture. So a perfect defect free crystal will take mechanical stress equal to theoretical strength ( thσ ) of the mineral element. But reality is not so. Take that (For analysis) there exists a thumb-nail flaw on mineral surface, by this we mean that these are proteins that are trapped in the process of biomineralization within the mineral crystals. This consideration is indeed true because of protein matrix’s low modulus. From Griffith’s criterion (was based on the observation that glass is not internally homogeneous but rather contains pores) fracture strength of this crack-induced mineral platelet is given by

ψασ mf

m E= , hEm

γψ =

Here γ is surface energy of mineral, h is its thickness, and

α is a parameter which depends on the geometry of the crack; here it is approximately π (for a crack depth equals to half of the thickness of the mineral platelet).


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Figure.8:- A length scale for optimized fracture strength in mineral platelets. (a) A schematic diagram of mineral platelet with a surface crack. (b) Comparison of the fracture strength of a cracked mineral platelet calculated from the Griffith criterion with the strength of a perfect, defect-free crystal [1,2,7]

Rearranging the terms in the above equation for fracture strength of the mineral

leads to 22

th

mEh

σγ

α≈∗

Where ∗h is the critical length scale below which fracture strength of cracked and un-cracked mineral platelet is same. A rough estimate of this critical length scale for =γ 1J/m2, 30mth E=σ , Em=100GPa will give a value around 30nm for a half-cracked platelet, which indicates that nature chosen this nano size for the platelets in order to optimize on the fracture strength of bone. So whenever thickness of the mineral platelets crosses this length scale, the fracture strength of bone depends on structural size and hence fracture strength is governed by the conventional engineering concepts, which describes strength of composite from pre-existent flaws, and propagation of flaws leading to the failure of composite under stress concentration at crack tip.

On the other hand, strength of a perfect mineral platelet is maintained in spite of flaws, if the size of mineral lies below the critical length scale. And failure is governed by theoretical strength rather than by the Griffith criterion and the material becomes insensitive to pre-existing flaws. From all these we can make the following postulate for bone fracture strength: The nanometer size of the mineral crystals in bone may have been selected to ensure optimum fracture strength and maximum tolerance of flaws. So the length scale 2

thmE σγ is an inherent property of the bone which measures the zone size of fracture process, and as the thickness of the mineral reaches this length scale bone material becomes insensitive to the flaws.

Fracture of solids is a nonlinear process involving breakage of atomic bonds. To explain the failure mechanism in nano materials like bone Gao et al used VIB (virtual internal bond) method. This model incorporates an atomic force law in to constitutive model for nano materials to model failure mechanism. Near crack tip, this model could describe the fracture process as a localization zone. It is based on the so-called Cauchy –Born rule which is a multi scale assumption about how the motion of atoms can be


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related to continuum deformation measures. According to this atoms, in a crystal subjected to a homogeneous deformation, move according to a single mapping from the undeformed to the deformed configurations. Equating the strain energy density function to potential energy stored in atomic bonds due to bond stretching gives the linkage between micro structural description of a crystalline material and the corresponding continuum constitutive properties. The VIB model assumes that each bond can be described by a potential energy function, which depends on bond length. It has been demonstrated that the onset of fracture predicted by the VIB model is not determined solely by the choice of bond potential, but also by the state of deformation in the localization zone.

The size of the fracture localization zone is correlated with the fracture energy and the virtual bond potential of the VIB model. During the fracture simulation using a VIB-based finite element method (VIB-FEM), crack growth and initiation are represented by a separation of two adjacent nodes at the crack tip, and the localization zone of fracture is represented by one overstretched sheet of mesh. In contrast to a conventional FEM calculation where the mesh size is a numerical concept which is selected only to achieve a desired computational accuracy, the mesh size of VIB-FEM has specific physical meaning (directly related to the J-integral and the fracture energy of the materials). This is an important difference between VIB-FEM and conventional FEM calculations.

Figure.9:- The VIB-FEM calculated stress near the critical thickness for optimum fracture strength of a mineral platelet with a thumbnail surface crack. The color map of normal stress yyσ obtained from a 3D FEM simulation based on the VIBmodel. At large thicknesses (h/h*=20,200), the stress concentration at the crack tip significantly reduces the fracture strength f

mσ from the theoretical strength thσ . Near the critical thickness h., the stress concentration vanishes and the strength approaches the theoretical strength [1,7] Figure.9 shows calculations for stress field for a mineral platelet with flaw existing on its surface, which is loaded to close to failure. Form figure.9 it is clear that as the thickness of the platelet is reaching critical length scale, the stress field becoming uniform and reaches the theoretical strength of the platelet at critical length. These results are beyond comprehension of the conventional concept of stress concentration at macroscopic level. This means that as the material becoming insensitive to the existent flaws surface energy required for crack growth will no longer released in the form of strain energy at the crack tip.


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4.1.4 Fracture Energy of Bone Based on tension-shear chain model Gao et al [2,7]developed a simple estimate of the fracture energy of the biocomposite. Consider a crack as shown in Figure.10. For simplicity assume that near crack tip dissipation of fracture energy is concentrated within a strip of localized deformation along the prospective crack path. For this model fracture energy can be calculated as ∫= εεσ dwJC )(

Here w is the width of localization strip, and )(εσσ = is the continuum constitutive law for biocomposite. To evaluate the integral, we make use of simple tension-shear chain model to account for the large deformation of protein within the fracture process zone. Mineral crystals are assumed to be strong enough to maintain their integrity in the process of composite fracture. So width w of the process localization zone should be larger than and proportional to the length L of the mineral. So if we write this as Lw ξ= , 1≥ξ . Inserting the appropriate equations previously formulated into the integral will lead to

∫ ∫ ∫Φ−+∆Φ== .)1(21

ppmmC dLddLJ ετξσξεσξ

Figure.10:- Estimation of fracture energy of biocomposites. The fracture energy dissipation is assumed to concentrate within a strip of localized deformation with width w. The tension-shear chain model is used to estimate the fracture energy in the localization strip. The stress and strain relation of material within the localization strip is assumed to obey a cohesive law [2,7]

In which first and second terms corresponds to the contributions from mineral and protein phases to the fracture energy of the biocomposite respectively. Out of these two first term is expected to be negligible in comparison to second one. So that leads to ( )∫ ∗Φ−=Φ−= f

pPpPC LdLJ ετξετξ 1)1(

where fpε is the effective shear strain of protein before fracture, and


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∗pτ is the shear stress in the protein, which can be expressed as

),,min( int ρστττ fm

ffpp =

∗

where fpσ is associated with domain unfolding, f

intτ is the strength of the interface, and f

mσ is the tensile strength of the mineral. Substitution of all these parameters will lead to simple estimate of the fracture energy of the biocomposite; which is given by ),,min()1( int ρσττεξ f

mff

pfpC LJ Φ−=

By seeing at the above equation one can easily tell that toughness of the biocomposite will increase with the volume fraction of the protein )1( Φ− , and length of the mineral L, effective strain of protein before fracture f

pε , and effective stress of the

protein ∗pτ , which is bounded by on three parameters. In the above mentioned parameters

volume fraction of protein is quite clear, in the sense that more volume fraction of protein more the energy dissipation of the biocomposite. As far as mineral length is considered it sets the intrinsic length scale at the crack tip for the strain localization. Longer the mineral element more is the delocalization of the crack-tip deformation, hence more is the fracture energy. Under the application of the stress domains of proteins starts unfolding, so naturally this allow the protein to under go large deformations before fracture. Slippage of the protein along the mineral length further increases this effective strain. Stress parameter is obtained by the lower bound of protein strength, mineral strength and interface strength.

Key to enhance first two parameters can be attributed to Ca2+ induced sacrificial bonds. Figure.11 shows the mechanism of the sacrificial bonds. As sacrificial bonds formed by Ca2+ ions starts breaking one by one, will allow protein in between the mineral platelets to under go deformation. This will result in long flat tail of force-extension curve with saw toothed undulation. Under optimum conditions three governing strengths will be same, i.e. ρσττ f

mff

p == int . If this condition were to occur there should be

optimum mineral aspect ratio which will be given by fp

fm τσρ =∗ .for a mineral

obeying Griffith’s criterion it is already shown that hEmf

m γασ 2= , so if we substitute this in the above equation will allow us to estimate the optimum aspect ratio of the

mineral element.hEm

fp

fp

fm γα

ττσ

ρ21

==∗ . This suggests that optimum aspect ratio is

inversely proportional to the square root of the mineral element thickness. In bone mineral crystals have usual aspect ratio of the order of 30-40, and thickness ranging over few nanometers. These values are some what close to values estimated from above scale law. As maximum aspect ratio required to compensate on the low modulus of the protein, mineral strength should be ff

m int)4030( τσ −≥ . In biological materials, the organic molecules and the inorganic mineral crystals are locally polarized and the interface strength is dominated by electrostatic interactions.


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Figure.11:- A schematic illustration of protein modules deforming between mineral platelets in the biological nanostructure and the inherent force-extension relation of protein with cross-linking mechanism of Ca++ formed sacrificial bonds. The sacrificial bonds are formed by Ca++ ions linking negatively charged functional groups along the peptide chain. The sacrificial bonds effectively convert the usual entropic elasticity behaviors of biopolymers to one that resembles metal plasticity. The long flat tail and the saw tooth undulation of protein deformation are due to breaking of sacrificial bonds and protein unfolding in the modules instead of molecular backbone [2,4,5,6,7]

Taking shear strength of the protein to be around (20–50) MPa, we can immediately estimate the mineral strength needs to be on the order of a few GPa, which is near the theoretical strength of mineral. This reveals the reasons why nature has chosen mineral crystals at nanometer scale. High strength, flaw tolerant mineral crystals are crucial to maintain a significant effective stress in protein which, together with large shear deformation inside protein and along the protein–mineral interface, gives high fracture energy.

4.1.5 Role of protein in toughness of bone The fact that biomaterials like bone contains more protein content than the required for mineral crystal deposition indicates that protein plays an important role in various features of bone, beginning from bio-mineralization to higher toughness of bone to the applied loads. Protein matrix act as an organizer of mineral platelets among them in bone at various length scales. Protein matrix induces distinguished properties to composite like bone, which are in drastic contrast with properties from calculations based on conventional composite theories. Protein acts as barrier to crack propagation, source


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of micro-crack initiator to homogenize the stress distribution, and fracture energy dissipater in bones. Since mineral elements in bone are too brittle they are expected to have only a linear stress-strain curve, end of which shows that when failure load occurs mineral fails suddenly without any prior indication. But protein is expected to have flatter one because of domain unfolding under load application. The arrangement of minerals in protein matrix is highly complicated, but we tried to explain that with tension-shear chain model. Similarly behavior of bone under load, crack propagation, etc... are highly complicated. Some attempts were made to explain reasons to these through simple models. Some of the models are presented below. Gao et al [7] used a notched specimen in which protein layer is sandwiched between two mineral platelets, to understand the phenomena when a crack enters protein layer from mineral elements. Figure.12 gives the idea about the arrangement for the experiment.

Figure.13:- Plot of stress concentration near the crack tip versus the thickness of a protein layer at the critical condition of imminent crack propagation. Oσ is average stress far ahead of the crack tip [7]

Figure.12:- A model to study the phenomena when a crack propagates into the protein layer-A notched specimen in which a protein layer is sandwiched between two mineral platelets [7]


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As shown in figure.12 crack propagates through the mineral and then falls in to the protein layer, load is applied on the top side of the specimen. Calculations are done for the stress at the critical point of crack propagation. In experiment [7] thickness and stiffness of the protein matrix are varied to understand the dependability of stress concentration on these parameters. Results of the study are presented in the Figure.13. The plot clearly indicates that stress concentration decreases very rapidly with increase in thickness of the protein matrix. It also indicating that when thickness of the protein is 20nm and stiffness is about 1000 times smaller than the mineral stress concentration disappears completely. This shows that whenever favorable circumstances occur protein layer will trap the crack and arrests its propagation, and hence failure is confined to a local zone. This model thus explains the response of proteins to cracks.

A more complex simulation model (Figure.14) is proposed to explore the role of proteins in bone. In this model several mineral platelets are glued by protein layers to form multi-layered specimen, and a lateral load is applied on its two sides. Outcomes of this model are shown in Figure.15

Figure.14:- Simulation model of multiple cracks nucleation in a multi-layered biocomposite specimen [7]

Figure.14:- Results of simulation.(a)The color map of the normal stress in the loading direction; (b)The color map of the shear stress showing the high shear regions between the mineral platelets [7]


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Color maps shown in Figure.14 indicate initiation of multiple cracks in the tensile zones of protein near the long ends of the mineral platelets. This multiple cracking pattern is consistent with recent experiments [7]. So it concluded that the micro-cracks could redistribute and homogenize stress concentration. The crack pattern revealed by this VIB simulation is similar to that observed in experiments [7]. The micro-cracks can delocalize damage and dissipate fracture energy. Studies [6,7] on the molecular mechanics of protein demonstrated that protein could dissipate fracture energy by its saw-tooth like force-extension relationship. We also note that there are high shear regions between the long sides of the mineral platelets. These results are consistent with the TSC model proposed earlier. So the simulation confirms the importance in the protein, as the load transfer between minerals is mostly through the shear of the protein layers between the long sides of the mineral elements. Micro-cracks themselves will not cause any significant damage to the bone, unlike in conventional composite systems where initiation of micro-cracks leads to the loss of stiffness and toughness of the composites according to damage theory (This correlates the stress level to no of cycles). So the stiffness and toughness of bone hinges on the tensile strength and shear strength of the mineral and protein respectively. According to studies on bone micro-crack damage in non-crucial zones is beneficial for animals.

4.1.6 Role of protein-mineral interface In the analyses as far we have gone through we have assumed, implicitly, that protein-mineral interface is perfectly bonded. But in reality it is not so owing to so many reasons like imperfections in mineral and protein, deposition conditions, etc… Some times failure of bone can be caused by interfacial slippage of protein along the length of mineral elements. In such cases knowledge of interface becomes extremely crucial. Hence it can be said that protein-mineral interface also plays important role in bone toughness mechanism. To demonstrate the effect of interface nature on the toughness of bone an impact simulation [7] is conducted for a nano scale biocomposite with weak interface between protein and mineral element. Here we assume that strength of mineral is greater than strength of protein, which is greater than interface strength. Simulation model was shown in Figure.16, in which volume concentration of the protein is much higher than the previous simulation. An impact load is applied on the left side of the upper half of the model by a rigid- wall and a symmetric boundary condition is applied on the right side of the model (to simulate bulk material).

Multiple cracks at interface greatly depend on the modular ratios of protein and mineral. Subsequent figure explains the debonding that occurs in model due to impact load for various modular ratios of the constituents. It also shows that for modular ratio greater than 400 (close to reality) no cracks are observed. Proteins deform by gradual unfolding of their domain structures. It can take a large amount of deformation before the primary structure of protein, the peptide backbone, is directly stretched. So the hierarchical structures of proteins are best suited for absorbing dissipating fracture energy. Fracture energy is defined as the area under the stress–strain curve. Figure shows that enhancement of the composite (bone) fracture strength can be achieved by enhancing the fracture energy of the protein molecules (even if we keep the shear strength of protein as constant-illustrated by the embedded figure).


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Figure.16:-Simulation of deformation and interfacial failure of a multi-layered biocomposite specimen under various impact conditions. (a) The simulation model; (b) Ep/Em=1/400, no cracks are observed; (c) Ep/Em=1/200, cracks are nucleated; (d) Ep/Em=1/20, multiple crack nucleation and propagation occur at protein–mineral interfaces. The impacter is a rigid wall acting on the left side of the upper half part of the material. The right part is subject to a symmetric boundary condition to simulate a bulk material [7]

Figure.17:- Fracture strength of biocomposites; (a) Force-displacement curves of protein with different fracture energy (Range of deformation of protein is varied under constant cohesive strength). Embedded figure-composite failure under the same external load for different force–displacement curves of protein, corresponding to (b) curve 1, (c) curve 2 and (d) curve 3 [7]


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Previous studies suggested that for high toughness of bone both large effective stress and large effective strain are needed. But the usual entropic elasticity of biopolymers involves relatively small effective stress. How to solve this paradox? Solution to this problem in bone has been provided by mechanism of “sacrificial bonds”, which involves linkage of Ca++ ions with negatively charged functional groups along the peptide chain. These are quite strong bonds, whose strength reaching up to 30% of covalent bonds in peptide backbone. These bonds binds the functional groups along different segments of protein and along the protein-mineral interface, thus increases the effective stress in protein which in turn ensures high fracture energy for bone. Added to this these sacrificial bonds allow protein deformation and interface slipping to occur simultaneously under similar stress levels, which leads to maximum amount of fracture energy. There is still another way for the protein to dissipate fracture energy; it is through viscoelesticity.

5. Future Mechanics theories on the stiffness, hardness, strength and toughness of biomaterials can be expected to play an important role in the development of bio-inspired multi-functional and hierarchical materials in the coming decades. Further exploring this materials may allow us to develope composite like bone in laboratory, from two materials which are lacking in specific qualities.

6. References 1. Gao, H., Ji, B., JKager, I.L., Arzt, E., Fratzl, P., 2003. Materials become

insensitive to flaws at nanoscale: lessons from nature. Proc. Natl. Acad. Sci. USA. 100, 5597–5600.

2. Gao, H., Ji, B., Buehler, M.J., Yao, H., 2004. Flaw tolerant bulk and surface nanostructures of biological systems. Mechan. Chem. Biosystems, 1, 37–52.

3. Rho, J.Y., Kuhn-Spearing, L., Zioupos, P., 1998. Mechanical properties and the hierarchical structure of bone. Med. Eng. Phys. 20, 92–102.

4. Thompson, J.B., Kindt, J.H., Drake, B., Hansma, H.G., Morse, D.E., Hansma, P.K., 2001. Bone indentation recovery time correlates with bond reforming time. Nature 414, 773–776.John currey, 2001.

5. Sacrificial bonds heal bone (letter). Nature 414, 699. 6. Smith, B.L., Schaeffer, T.E., Viani, M., Thompson, J.B., Frederick, N.A., Kindt,

J., Belcher, A., Stucky, G.D., Morse, D.E., Hansma, P.K., 1999. Molecular mechanistic origin of the toughness of natural adhesive, fibres and composites. Nature 399, 761–763.

7. Ji, B., Gao, H., 2004. Mechanical properties of nanostructure of biological materials. J. Mech. Phy. Solids, 1-28.

8. Lakes, R.S., 2001. Extreme damping in compliant composites with a negative-stiffness phase. Philosophical magazine letters, 81 (2), 95-100.

9. http://ceaspub.eas.asu.edu/concrete/elasticity2_95/sld049.htm 10. http://www.risoe.dk/afm/personal/bsqr/fracture.htm 11. http://www.medes.fr/Eristo/Osteoporosis/BonePhysiology.html 12. http://www.engin.umich.edu/class/bme456/bonestructure/bonestructure.html 13. http://www.medify.com/pat_info/osteoporosis/htm/typesofbone.html 14. Book-Comparing Behavior: Studying Man Studying Animals


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15. http://kidshealth.org/kid/ill_injure/aches/broken_bones.html 16. http://www.cdc.gov/powerfulbones/boneup/hardfacts.html 17. http://www.trinity.edu/rblyston/bone/intro2.htm 18. JKager, I., Fratzl, P., 2000. Mineralized collagen fibrils: a mechanical model with

a staggered arrangement of mineral particles. Biophys. J. 79, 1737–1746.

6.1 Additional References 1. Gao, H., Klein, P., 1998. Numerical simulation of crack growth in an isotropic

solid with randomized internal cohesive bonds. J. Mech. Phys. Solids 46, 187–218 2. Zioupos, P., 1998. Recent developments in the study of failure of solid

biomaterials and bone: ‘fracture’ and ‘pre-fracture’ toughness. J. Mater. Sci. C6, 33-40.

3. Zude, F, Jae, R, Seung, H, Israel, Z., 2000. Orientation and loading dependence of fracture toughness in cortical bone. J.Mater. Sci. C11, 41-46.

4. Turner, C, H., 2002. Biomechanics of Bone: Determinants of Skeletal Fragility and Bone Quality (Review article). Oste. Int. 13, 97-104.

5. Norman, T.L, Vashishth, D., Burr, D.B., 1995. Fracture toughness of human bone under tension. J. Biomech.28, 309–320.

why bones are tough · embedded in protein matrix (b) simplified tension–shear chain model...

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