 In this lecture, we will learn about nanomaterials in nature specifically for a natural bond. We earlier we have talked about the multi-scale hierarchy, so we will see how this hierarchy is already present in many of the natural objects. So in this case, we will learn about how the structures develop at a nano level or at a molecular level and how do they go about making some ultra structures or finally, a bulk structure. So in this case, in this particular lecture, we learn how the nanomaterials in nature, how they are devised in terms of forming some basic fundamental units, how do they go on to forming certain structures which are uniform and then forming a higher level of scale such as starting from hydroxyapatite or collagen to a fiber and then to a lamellae and then to a sort of a macro structure and then ultimately forming the spongy bond. So talking about natural bond, generally all the calcified tissues they have some inorganic component that is hydroxyapatite, it is a basic fundamental unit of C F 5, P O 4 3, O H and each of those two units are occupied in a unit cell. So it gives rise to C A 10, P O 4 whole 6, O H twice, apart from that it also has a protein collagenous content and certain other organic phases which are present on those calcified tissues and again it is very, very essential to learn how the characterization of bond, how it depends on the length scale of observation because if you are talking about molecular scale, whether we are talking about microstructural scale, we are talking about macroscale or we are talking about the bulk scale or the actual structure of the bond. So we can see, we can see how those properties, it can be mechanical, it can be nutritional, it can be biological, it can be chemical, how it is affected by the structure which provides a certain functionality to the bond. So as we see in natural bond, we have certain components of either some organic components, some inorganic components and inorganic component we have basically hydroxyapatite which is present in all the calcified tissues apart from collagen and it is also essential to see how this hydroxyapatite is spread into collagen both intra and intra fibrillous regions. So that will give a very peculiar structure at certain length scales and again the characterization of these is also very critical. So we will also learn about how we can determine the elastic constant or rheological properties of bond. So that when a certain stress is applied on to it, how it can relax or how it can distribute the stress. So those things become very critical and they depend much on the way the structure of that particular bond is designed. So learning about the compact cortical bone and as we said earlier that there are 4 levels of hierarchy. So in first case we are talking about a basic molecular construct and that comprises collagen, triple helical structure which is also called tropo collagen or the arrangement of hydroxyapatite within there. So collagen and half or hydroxyapatite they form the basic molecular construct and as we said hydroxyapatite has been defined as a structure of C A 5, P O 4 whole thrice, O H and 2 such units are there in a unit cell. So it becomes C A 10, P O 4 whole 6 and O H twice. So collagen and hydroxyapatite there are basic fundamental blocks to form the basic structure or molecular structure. So that is the first level of hierarchy. Then this half starts spreading both in the entire and inter fibrillous region of collagen and that forms the second level of construct which is called ultrastructural. So we have molecular structure, then we have ultrastructure and now coming to the third component all these fibrils form fiber or fiber bundles to give rise to a lamella type of a structure and that is again called microstructural. So starting from molecular we have gone to ultrastructure and that has gone to a microstructure and then finally all these lamellar units they form secondary osteons or haversian systems which are called bone tissue structure which is at a structural level or bulk structure. So we can see we have four levels of hierarchy. In first level we see a basic display of collagen and hydroxyapatite to give rise to a molecular structure and how this half is being spread on to the collagen fiber that gives rise to ultrastructure. Then we go on to forming those as fibers, fiber bundles or lamellae and then that is called a microstructural entity and how these things go on to evolve a osteon or haversian system to give rise to a entire structure which is called a bulk structure or the bone tissue structure. So we can see in compact cortical bone how these four levels of hierarchy are defined and these are inherently present that has been designed by nature. So these are inherently present in nature and these all have different way of providing certain property to the bone. So let us see how these entities are basically being developed. So initially we have the crystals of apetite. So we can see there are certain crystals of apetite which are couple of angstroms. So it starts at approximately apetite mineral crystal. So we can see there are certain apetite mineral crystal which has spread intra and inter the collagen fibers. So we can see this apetite crystals they are approximately 200 to 400 angstroms long and all these are inter laid on to certain collagen fibers. These are certain fibers. So they are present both in the inter and the intra. So we have apetite and collagen and these are from the basic molecular unit and from that the arrangement of this gives rise to certain concentric lamina. So we can see there are certain concentric lamina which basically generate of this and then these are fibrous entities and these are approximately couple of microns may be say 2 to 10 micrometers and these are nothing but the ultra structural microstructural. So the arrangement of this, arrangement of apetite and collagen is called ultra structural and then it goes on to forming these fibers which are microstructural and again all these fibrous entities go on to forming certain osteons and eventually they go on to forming haversian systems, haversian canals or haversian systems. So this is again more at a micro level microstructural level and then we see we get finally, a spongy bone kind of a structure. So we have all the cartilages we have a spongy bone. So we can see that a spongy bone kind of a structure is being generated and then we have a compact bone and in this region we see the compare how the osteons and the haversian systems. So we have this osteons and this haversian system, haversian canals. So all these lamellas are being organized in a very nice channels of making haversian, haversian canals and finally, we get a entire spongy bone and a compact bone structure with periosteum. So we can see that initially we start with apetite and collagen is type of a structure which is a basic molecular unit approximately 2 to 4 inch long of apetite crystals. They are both present in the intra and the inter fibrillous region of collagen and that gives rise to our ultra structural entity that forms a basic. The basic molecular units comprise ultra structural units and these going to forming certain fibres or lamellas or fibro less kind of entities which are again at the micro structural level and all these fibres going to forming osteons which eventually form haversian canals and this haversian canals are again at the micro structural regime to finally, give a bone which is spongy or the compact bone which is at the bulk structural level. So that is how we can see the overall how the hierarchy is basically classified into 4 different regimes starting from the fundamental unit of apetite and collagen. Then going on to ultra structural regime of these arrangement of apetite on to the collagen is fibre. Then how this collagen is fibre are properly linked to form fibres or lamell type of a structure and then eventually form osteons or the haversian systems at micro structural level and finally, give rise to the bone structure. So basically we can see the fundamental unit of hydroxyapatite hydroxyapatite is a apetite is a hexagonal unit cell with space group of p 6 3 by m. It is a lattice parameter of a 9.880 angstrom and c of 6.418 angstroms. So essentially it has 2 units of c 5 p o 4 3 o h and 2 such units are present molecular units are present in a each unit cell. So that is the reason it becomes c 8 10 p o 4 6 o h twice and it has a small crystallite size of generate forms a small crystallites to size of 2 by 20 by 40 nanometer that is 20 by 200 by 400 angstroms and because it is a nano crystalline in nature. So, the x-ray diffraction of the bone shows very very considerable line broadening and because of line broadening it is very hard to identify which other elements are present in this particular structure. So hydroxyapatite it is the basic fundamental structural unit and again because of the presence of this calcium and phosphate in this hydroxyapatite which is nothing but the inorganic component of the bone. It is led to the development of certain calcium phosphate based bio ceramics because that is the main mineral block which is the c e by p ratio 1.67. So that is the motive behind that is the basic impetus behind initiating new calcium based bio ceramics as a potential body bone implant materials. So, we can see hydroxyapatite it is a hexagonal unit cell with a space group of p 6 3 by m with the lattice parameters of 9.88 and c of 6.418 angstrom and it contains 2 molecular units of c 5 p o 4 whole thrice o h and it is a very small crystallite size to the order of 2 by 20 by 40 nanometers. So, it has a very forms very nanometer regime apetite crystals and because of that the x-ray diffraction of the bone it shows very very high line broadening because of the nanocrystallinity of this inorganic component of the bone and that makes the identification very very difficult because there is so much line broadening the background is generally very very high for this particular ceramic component and then, but this but because of presence of calcium and phosphate those have led to the development of calcium based bio ceramic which is now the major field of research in the biomedical industry. So, that is the basic fundamental unit and we can see the structure of hydroxyapatite looks more like this. So, there are 2 blocks of c f i p o 4 thrice o h. So, in a unit cell will get c a 10 p o 4 6 o h twice. So, we see there are total of 10 c a atoms. So, we have this orange one is the calcium that is what we are seeing here. So, we have 10 of such calcium atoms in a unit cell and then there are 6 of phosphorous those are the nothing but the yellow atoms which you see here those are the phosphorous and then we have oxygen. So, we can see there are total of 26 oxygen o 4. So, 24 plus 2 26 such oxygen which are present in a unit cell and then we have hydrogen which is at the corners apparently that hydrogen is present as o h and the total of out of total number of sites only half of them only 50 percent of such sites are being occupied by the o h. So, in this particular case we have o h twice. So, only 2 of the o h components are present in this case and that is occupying only the 50 percent of the available sites for the o h. So, we can see the hexagonal unit cell its p 6 3 by m that is the basic space group which is for the hydroxyapatite and there are total of 10 calcium atoms total of 6 phosphorous and there are total of 26 oxygen and 2 hydrogen which are present in a unit cell of hydroxyapatite. And to understand how the collagen triple helix is basically formed we need to learn about how the amino acids going to forming polypeptides and how they eventually turn out to be proteins. And in the case of collagen triple helix we have 3 alpha change which are the left handed screw they coil to form tropo collagen which is the triple helix structure and these tropo collagen eventually form protofibrals. So, for the collagen triple helix we have to start with the fundamental amino acids how the polypeptide bonds the hydrogen and the nitrogen bonds how do they basically emerge to form finally proteins. And how this alpha change coil to form tropo collagen and this tropo collagen eventually form protofibrals in the bone structure. So, let us see how this collagen structure is basically developed starting from the change of this alpha helix structure. So, we can see that alpha helix is the left handed helix. So, we can see it has a certain repeat unit. So, it has a repeat unit of approximately 9.3 angstroms. So, we can see it has certain structure of. So, it has a residue of around 3.1 angstrom and it has a it is a repeat length of 3.1 angstrom and again it has a residue which is which is of length around 3.1 angstrom and it basically comes as a residue. Residue is again the repeat point where the same point appears again. So, that is approximately 9.3 angstroms. So, three such units are there in a repeat unit. So, this is for the alpha helix structure alpha helix left handed structure and after that once we start this is the alpha left handed helix helix and once it is being opened up and coiled with the right handed helix. So, we have one left handed helix and then one right handed helix. So, they basically so, because of its opening the overall residue reduces from 3.1 to around 2.8. So, the overall residue is approximately 2.86 angstroms, but the repeat length the repeat length because of the presence of two type of a structure repeat length is now the 10 times it becomes instead of here it was 3 times. So, the repeat length is approximately 3 times in the alpha helix which increases to 10 times. So, in this case we have overall. So, repeat length it is in decreases from 3.1 to 2.86, but the overall repeat length overall repeat length is basically increased by 10 times. So, it is the residue which is decreasing from 3.1 to 2.86, but the repeat length is now increased to around 10 times. So, that becomes 28.6 angstrom in the twisted right handed helix. So, we start with alpha helix we try to open it and then the residue is decreased by 2.86, but the overall repeat length is now increased from 9.3 angstrom to 28.6 angstrom in the twisted right handed helix and then these are threaded to form. So, three such units come together and then they form this tropocollagen. So, there are three such units which go on to form this particular tropocollagen which is very very complicated. So, three alpha helices they form a threaded. So, three such alpha helices they get threaded to form this tropocollagen structure. So, initially we start with alpha helix which is a left handed screw which is a residue of around 3.1 and repeat length of 9.3 angstrom and then it go because of its uncoiling the residue is basically decreased to 2.86 angstrom with a repeat length of around 28.6 and this is approximately 10 times that of earlier case. So, 10 times of this residue. So, in this case it has gone from 9.3 angstrom to 28.6 angstrom and three such alpha helices they are now threaded to form this tropocollagen structure. So, that is what we see at the basic fundamental unit. So, this we have now hydroxyapatite which is p 63 by m and now we have this tropocollagen and now these chains the chain of tropocollagen and the crystals of apetite they are present in both intra and the inter collagenous region to form the ultra structural unit. So, we had fundamental block of hydroxyapatite and this tropocollagen and dispersion of hydroxyapatite in this particular tropocollagen is basically giving rise to the second level of hierarchy that is the ultra structure. So, this hap collagen organization. So, it is controlled by a bonding at the molecular level. So, we can see the basic units of collagen and hydroxyapatite they their arrangement is getting controlled by the bonding at the molecular level and now here in this what is happening is the collagen structure can also change during the formation because the collagen is getting replicated from the alpha helices. So, during the initial stage of its formation its structure can be very very different and again when it is just forming its physical properties or the mechanical properties are very very inferior. So, that has been observed by bone pathology and that can have a weak mechanical properties as well. So, those inferior physical properties relate directly to the mechanical properties and it might take very long time to heal if certain stresses are applied to it. So, that structure is very very important in terms of dictating the overall properties of the bone. So, for fundamental unit itself is very very feeble or very very weak it will lead to such changes in the bulk property as well. So, that is also very very critical at this stage that hap collagen organization is completed. So, we get a very proper fundamental block for its repeat to form microstructure or the final bone structure and hap is being observed both intra and the interfibrillary regions with the collagen. And now what is happening in this particular case now we have proper mix of hap and collagen. So, here what we can do we can model the elastic properties based on the weight of each component. So, if you know we have certain content of hydroxyapatite and certain content of collagen now we can identify what can be the elastic properties of the structure elastic properties by weight of or volume of this structure. So, till now once we had only hap or once we had only collagen and the dispersion was still under progress. So, in this case once they have organized now we know exact content of hydroxyapatite and exact content of collagen and from here we can really model what will be the elastic properties of this particular organization at microstructural level. And from there on we go on to forming lamellar type of a structure. So, we know that these generation of this appetite on the collagen structures is are forming certain fibres of fibrolyer type of composites. And these eventually form fibre bundles and these fibre bundles can be layered. So, these fibre bundles can get layered to form certain lamellar. So, we can see that all the fibre components are forming certain lamellar. So, these lamellar structures can either go as circular lamellar units. So, it will it can form circular lamellar units which form secondary osteons or haversian systems and this is present mainly in the mature human bone. So, this circular lamellar units can go on to forming osteons and haversian systems which are predominant in the mature human bone. These are also go on to forming straight lamellar units. So, it can also form straight lamellar units and which forms plexiform bones and these are observed in quadruped animals such as cats or rabbits. So, we can see that the lamellar can be distinguish into two parts circular lamellar units which go on to forming haversian systems or osteons which are predominant in the mature human bone or these are also these can also go on to forming straight lamellar units which form plexiforms in the quadruped animals such as cats or rabbits or higher than that. And now in this case we have developed the entire structure. So, we have formed the fibro less units these have finally formed the lamellar units and almost to the lamellar we can identify what is the overall mechanical performance or what is the mechanical response of this particular unit or that particular structure. So, now we have the composite available to us and we can analyze this particular composite of how its response will be to a certain external stimuli. It can be elastic properties it can also be the rheological properties in terms. So, we can identify what is happening at the macroscopic level. In the previous case we knew the microstructural arrangement of it. So, we could weigh its properties the elastic properties by the composition of hydroxpirate and collagen. In this case we know exactly how these units are being circulated and what is the overall system or the overall structure which is being developed either as circular lamellar or as straight lamellars. So, now we have the entire composite available to us and now we can model its elastic properties that is the beauty of this structural the generation or the development of this structural units. So, now we can model the elastic properties of this tissue which will render macroscopic properties to the board. So, that is very nice flow of properties that we start from basic structural units. Those go on to forming ultra structural units and ultra structural units we know what is their distribution. So, we can identify certain properties at that level, but in this case we can identify how these layers are now arranged as lamellar units. And now because it is a lamellar we know how is the overall arrangement of these osteons or haversian systems or maybe as plexiforms. So, now we have a composite available to us and then we can model the elastic properties of the tissue. So, it can mimic or it can tell us about the macroscopic properties of the bone because this is the unit which is eventually forming the bone. So, this is the complexity which is associated with the bone hierarchy. So, we start with the basic fundamental blocks of hydroxpirate and collagen. So, we know we have certain crystals of hydroxpirate which are 2 by 20 by 40 nanometer. We have collagenous fiber, tropo collagen and these go on to forming certain fibrous or lamellar kind of structure and these eventually form osteons. So, we can see the osteon construction like this. So, we have haversian canal out here and these osteons how do they form the circular units or circular haversian osteons. So, we can see that part out here. These are nothing but the osteons of a compact bone. So, we can see how these units are now arranged, how this lamellar, the circular lamellar are now arranged to form this entire construct of structure of bone and now in this case when we have everything available to us we can eventually form get the elastic properties as well as the mechanical properties. It can be either the stress response or the creep response we can also be observed. So, we can see that we have periosteum, spokesman canal, we have haversian canals, we have osteons, osteocytes, we have osteocytes, osteons even the entire haversian system is now haversian canals and haversian system is now available to us to completely form the spongy bone. So, once we have this entire construct ready we can now play with it and find the mechanical response of this entire structure. So, that is the very nice thing about the difference in hierarchy. So, the properties of hydroxyapatite with collagen will be very very different. Once they go on to forming a fibrillous kind of a structure or a lamellar kind of a structure then the construct will be very very different. We can find properties at that level because we know the percentages of the haps and the collagen and once they come together to form osteons the overall outlay of this osteons either as circular or as a straight lamellage. We can identify what will be the response of the bone because now these are the units the way they are arranged they will they will give rise to certain gives and takes at interfaces as well as the property the individual properties and how they are arranged at that level they they mimic the nature of the bone. So, now we can get the bulk properties of the bone via this construct and apparently the way all the individual components are present we know that bone is a multifunctional entity we are running we jog a lot. So, bone undergoes much more mechanical stress it undergoes fatigue it also has to give some yielding. So, that the some stress can be easily dispersed at the same time all the blood cells have to be supplied certain nutrients. So, now we can see the different complexity in terms of supplying the nutrients supplying the gases to those cells also being able to accommodate all those cells without letting them die. So, incorporating certain stress distribution having some porosity to absorb some shock. So, these are the multifunctional entity which is being incurred by the bone on a daily basis. So, we can see the stiffness it arises because of the composite structure of the mineral part. So, we have micro crystals which are present in the bone and again the proteins which also absorb certain shock. So, we have stiffness which is comprising out of the loading. So, loading is being taken by the micro crystals of hydroxyapatite and protein takes the shock. So, in entirety we get a stiffness arising because of the composite structure of mineral micro crystals of hydroxyapatite and collagenous protein fibers. So, that is one component of the bone functionality other than that it can also undergo certain slow creep. So, for allowing the creep. So, that the stress can be distributed easily we need to allow slipping. So, the slip can also occur at the cement lines between the osteons. So, we have certain osteons. So, slip can occur between those osteons. So, slip can easily occur between the osteons and within the osteons as well to give rise to certain slip that can also accommodate discharge of certain stress. And again the toughness can also come as because of the weak interface. The interfaces are very very strong then it can even crack the bone that part we do not want because we do not want to damage the bone. So, the nature has devised or nature has designed the bone in a manner that it can also incorporate certain slipping at the cement lines. And that occurs because of the weak interfaces and in the process we can also absorb some shock. And again the lacunae or the ellipsoid pores are present to provide space for the osteocytes. If there is no place for the osteocytes then basically there would not be any living entity in the bone. So, if there are certain osteocytes they have to be supplied certain nutrient they have to be supplied certain gases or some oxidizing atmosphere. So, that they can absorb the nutrients and they can survive and they can reconstruct the bone as and when required. If we do not have any space available for them then eventually the bone will start getting dead and that part will again hamper the functionality of the bone. So, the way it has been designed that there are certain lacunae or ellipsoid pores which will provide space for this osteocytes and will make the life living for this bone cells. Again bone cells at this level of scale they also permit bone tissue to remodel its structure. So, in order to remodel we also need to have some space we need to have some regime to allow its reconstruction to its remodeling at the same time should also have access to getting certain nutrients. So, all those things are allowed by the bone cells at this level of scale to permit the remodeling of structure in response to the stresses. So, certain stresses are being applied it should be able to remodel itself like in by slipping at certain regimes by allowing the collagen to stretch by allowing the protein to impart certain toughness to it. So, all these things require remodeling and that is again permitted because of the complex structure of the bone. And then we have haversian canals or cylindrical pores which are present to contain blood vessels because the all the tissues in the bone they also need nourishment. So, nourishment is being provided by the haversian canals which are nothing but the cylindrical pores which contain the blood vessels. Then we also have fine channels or canalically they also assist by pumping the nutrients. So, once a certain mechanical stresses are being applied because of certain physical activity it can be running jogging or an impact or even walking. So, they basically get nutrition by the pumping of this mechanical stress through this channels and that is being taken incurred by the canalically. And again the pore structure is designed in such a manner that it is able to adapt to the mechanical stresses. So, we can see there are so many components of the bone how complex they are and how they can be organized together to give a multifunctionality to the bone. So, stiffness we can see stiffness is arising because of the composite structure of the hydroxyapatite crystal and the protein creep also can be accommodated by the slipping at cement lines. Toughness also is being incurred by the cement lines at the weak interfaces. Then we have lacunae which will provide space for the osteocytes so that it can provide the nutrients and it can allow the living of those bone cells. And then bone cells they are required because they want to remodel the structure in case of any stresses are applied to it. Then we have haversian canals they also contain blood vessels for providing the nourishment to the cells. We have canalically or very fine channels which allow the pumping of this nutrients through this channels so that it can reach the cells. Then we have a particularly designed pore structure to allow the repression to the mechanical stresses and that is why we see we have all those lacunae or the slip lines between those osteons and how we can supply the nutrients we have canalically or the pore structure and then we have blood vessels which can provide the nutrients. So, all these things are so complicated and well so connected at the at this level to allow the multifunctionality of the bone. So, that is what makes the structure very important at different line scales. Unless we have a smaller entity smaller pore they cannot generate a bigger pore and how these canals or this how this lamellas are devised so that they can yield as and when required in response to the mechanical forces or in response to the slipping in response to certain other shock. So, that is what we see that multifunctionality of the bone is highly dependent on the structure which is predominant at different line scale. It can be very fine like canalically to pump the nutrients it can be much bigger. So, at haversian canal systems to contain the blood vessels. So, the same porosity at different length scales it can it can render different properties. So, we can see canalically can pump nutrients haversian canals can contain blood vessels for providing the nourishing to the nourishment to the tissues. And again the pore structure itself can allow adaptation to mechanical stress. At the same time the cement lines which are again nothing but the porosity they can also yield slipping as and when a mechanical stress is applied by acting as a weak interface. So, that is the overall functionality of the bone which is so very so essential for devising the multifunctionality to the bone. Bone the elastic properties of bone and it treated by its components and bone is viscoelastic in nature. It means that the bone gives response because of the bonding between the atoms or the molecules that is the elasticity part. Viscosity or the viscoelasticity comes because of the diffusion of atoms at that particular level. So, diffusion of atoms or molecules gives rise to viscous nature of the bone and the elasticity comes because of the strong or the mineral part of the bone. So, once we have a mineral component that gives elasticity and some molecular or amorphous nature or liquid type nature gives rise to some viscous type of response. That is because of the diffusion of atoms and elasticity is because of the immediate response with because of the bonding between the atoms and bone is a viscoelastic entity. So, learning that elastic response or the viscoelastic response will tell us how the bone will perform when certain stresses are being incorporated or how it will respond to certain external stimuli. There are many properties of bone which have been studied because by doing certain experimentation. So, Causes static strain can give rise to stress strain curve. It can give rise to the evaluation of Young's modulus. We can also identify Poisson's ratio by the response of the transverse component with respect to the longitudinal component. And then we can also identify yield stress, yield strength, fracture stress or fracture strength. And other than that if we apply a certain elastic or plastic strain in tension, compression, torsion, shear and fatigue, we can again identify certain properties of the bone. So, the bone is a very complicated structure because of its viscoelastic nature. And elastic properties will tell about the overall performance or understanding how it will behave when any external stress or external stimuli is being applied on to it. We can either have Causes static strain to identify certain static properties or elastic plastic strain to identify its viscoelastic properties. So, that will make the structure very complicated. So, there are certain ways which we are relating the bone to. So, basically it is the hydroxybride part which will give rise to the elastic part of the bone. So, again we can, bone can be considered either as a transversely isotropic or orthotropic because of the hexagonal symmetry of the appetite, which is the basic load bearing component of the bone. So, there is much relation or much more profound relationship between the microstructure and the mechanical properties. So, the manner in which we have the microstructure or how these appetite and collagen are arranged at a microstructural level that will dictate how the mechanical properties will basically evolve. So, bone is can be considered either as transversely isotropic because of its hexagonal symmetry. And when it is considered hexagonal, it will require only 5 elastic compliances or stiffnesses to describe its elastic properties. But in cases when it is not so, when it is orthotropic symmetry will require 9 such components and by devising them that way, it will help us understand the resorption and remodeling. Because that is very well known that once a stress is applied to a bone, it will try to resorb or it will try to generate itself or remodel itself. So, that it can take that stress in a better manner or a better fashion and where there is no stress being incorporated by the bone, that part of the bone will start dissolving itself. So, that is the beauty of the bone that it can remodel itself or it can take the stress and it can regenerate itself. So, for that it is very essential that we learn how these elastic constants or how this elastic or the mechanical properties or the elastic properties are very much essential for the development of bone. So, when we consider it transversely isotropic, we can define it by using 5 constants and that arises because of the hexagonal symmetry of the bone. So, we can see the properties along 1 and 2 they turn out to be the same. So, in that particular case, we can see that the overall compliance matrix can be given the first of all the dependence of i j. So, the overall compliance matrix can be given as c 1 1, c 1 2, c 1 0 0 0 and we have c 1 2 and c 2 2 is now similar as c 1 1 and c 2 3 also is now similar as c 1 3. So, that is the reduction of the elastic constant out here, the compliance constant out the constant out here and now we get we have c 4 4. Now, again c 2 2 is this c 5 5 is similar as c 4 4. So, in that case we get. So, we can see we get c 1 1, c 1 2, now c 1 3 and c 2 3 are the same. So, we can get them a similar entity, c 2 2 is also similar as c 1 1. So, that is the dependence we get, then c 1 2, c 1 3, c 3 3, c 4 4 and c 5 5 is similar as c 4 4 and c 6 6 and here we can see that totally of 6 constants, but c 6 6 can be given as a dependence of half of c 1 1 minus c 1 2. So, in total totality we have now 5 constants which are required for defining the transversely isotropic crystal. So, because of the hexagonal symmetry of the hexagonal symmetry, we can define them by a only 5 constants. So, that is that is how we can see out here, but what happens once we are stretching it might lead to certain porosity and once it is leading to certain porosity, then obviously the bone cannot be defined by a transversely isotropic or via hexagonal symmetry. So, in that case it can also change the properties along the radial in the in the tangential direction. So, because of that in such as in such as in plexiform lamellar bone, plexiform lamellar bone or because of porosity these dependencies are no more valid. So, in that case we have to define them via orthotropic symmetry. So, in that case we do not have a dependencies of c 1 c 1 3 being equal to c 2 3. So, now this additionally certain terms arise. So, c i j now becomes c 1 1 c 1 2 c 1 3 now c 2 2 is not equal to c 2 3 or c 1 1. So, c 2 2 is different in c 1 1 is different and then our c 2 3 also is different in c 1 3 which was not. So, in the earlier case. So, we can see and now our c 4 4 is also different from c 5 5. So, that is the dependence what we get in this particular case that our c 2 3 and c 1 3 they are not similar and c 1 2 and c 2 2 they are again not similar and c 4 4 and c 5 5 also not similar. So, in that case once we have a non transversely isotropic property because of generation of porosity or because of plexiform lamellar kind of a structure these are no more uniform along the 1 and 2 direction. So, now we need to also provide the values of properties along those directions. So, in that case c 1 c 1 3 and c 2 3 they are not the same and c 4 4 c 5 5 also are not the same c 1 1 and c 2 2 also are not the not the same. So, now we need total of 9 constraints to define the overall elastic properties of this orthotropic structure and so bone is generally orthotropic in both directions and that is how we can define the mechanical properties along these two directions. Now, coming to the visco elastic properties of the bone. So, basically there is no time temperature superposition to obtain its properties because the way in which the time and temperature will behave is very very different. So, we cannot impose by increase in the temperature that will be the time response of the bone that is not so. So, we need to define them via Kelvin work model that can portray the creep properties. We can also define them via Maxwell model that will provide the stress relaxation of the bone. So, when we are applying certain stress how that stress is getting reduced that can be defined via these two models and again these are only two element models. So, in Kelvin work model or the Maxwell model we only have one of each the resistance and the capacitance part. So, that is how we define them. So, in Kelvin work model we have everything in parallel. So, we have this resistance and then this capacitance in parallel whereas, in Maxwell model we have both of them in series. So, we have this resistance and this capacitance in series and from there we apply the stress, but these models may not be sometimes enough to define the properties of bone. So, three elements or more models or more elements will be representing the bone response in a much better manner and will be near to the actual system. So, let us now see how the response will occur for the Maxwell and the Kelvin work model. So, in the Kelvin work model what we are doing we are defining the creep compliance to it and we have this entities in parallel. So, we can see we have a resistance and a capacitance with certain stress of being applied to it. So, in this case we can see the resistance part can be k and this capacitance part can be n. So, in this case what we are seeing is the stress is now being sheared between the two components or the strain is same and sigma is nothing but sigma of k component plus sigma of n component. So, that is what we are able to see here in the Kelvin work model. So, eventually we can see is that the strain how the strain develops one with the how with the increasing time we are getting the strain to release the stresses or how the stress is being released with the time. So, we have sigma n p this is also called the creep compliance model. So, once we are applied a certain constant load how the stress strain basically starts increasing how the stress basically starts decreasing with time and again we have one more model. So, we can see once we so once we have this other model of Maxwell system or Maxwell model is also called stress relaxation model. So, in this case what we have we have the system in series. So, we have this k component and this capacitance component in series. So, your stress k component n component. So, in this case what is happening is the overall stress is being sheared. So, sigma so the stress is similar the strain is being sheared. So, sigma is equal to and this is being equal. So, sigma equal to sigma k to sigma n in the Maxwell model and again approximately it is the overall it can also be represented by. So, if you apply certain frequency that is omega is applied frequency and delta is the phase angle that is the phase lag between the stress and strain. So, with this we can see that the overall strain response with respect to time will appear out to be a linear dependence and then with respect to the time or stress we get a decrease. So, that is sigma eventually dies out with time. So, Maxwell model is very good for showing the stress relaxation and Kelvin model is very good for the creep compliance. So, we can see that in the Kelvin work model we have the resistance and the capacitance in parallel. So, in this case the strain is similar in both the components and that may not be so in the real cases, but the strain is now being sheared between those two components it is being k and n. Whereas in Maxwell model we have stress is the similar in all the components where the strain is the combination of the resistance and the capacitance and that is what we are seeing here in both the cases. This is nothing but the rheological response of the bone and that can arise because the resistance can be given mainly from the hydroxyapatite this is a mineral part and then n can be can come mainly from the protein or the collagenous component. Apart from that the overall construct or the mechanical structure can also lead to much more stiffening the stiffness part and this is the compliance part which can also come because the slip lines. So, it can be again from the lamellae. So, lamellae or the overall stiffening part can come out as a resistance whereas the complying part can come out or the capacitance part can come out from the slip lines between the osteons, the porosity or the protein collagenous type of a structure. So, we can see that the bone structure can be easily represented by these two components of resistance and capacitance to give out a final remodeling or the final reconstruct of the structure to yield the rheological response with respect to certain stress. So, once you apply certain stress how the strain develops or how the reconstruction or remodeling of the bone occurs with respect to time can be easily represented by the Kelvin model where we see the creep compliance or it can also be represented by the Maxwell model where we can see how the stress relaxation can occur in that particular case. So, that tells the hierarchy part that is the multi-scale hierarchy which is inherent in the bone that how the hydroxyapatite and collagen come together to give out certain properties elastic properties and how those elastic properties are being utilized and when a more lamellae structure or a lamellae or a osteone structure is formed to give out the bulk properties or the bulk property to the bone structure. So, in summary we can see that the bone hierarchy is very very complicated at molecular level we can see the construct of collagen and hydroxyapatite eventually this collagen and hydroxyapatite they organize to form ultra structure. So, from molecular structure we go to ultra structure then these go on to forming fiber, fiber bundles or lamellae type of structure there is nothing but the microstructure and now these go on to forming osteones and haversian systems to finally yield the bone tissue structure and at this level we can identify the rheological response via Maxwell model or via Kelvin work model work model and also we can also identify the mechanical properties by making them transversely isotropic or orthotropic in nature in that particular manner. So, that is what we see the elastic properties can either be isotropic or those can again be orthotropic and once it has hexagon symmetry we can reduce the number of constraints and because of the presence of this lamellae type of a structure linear type of a structure that can or presence of porosity that can basically make the structure orthotropic and further we can define the overall rheological response via Kelvin work model where we can define the creep relaxation or the creep compliance or it can also be represented by the Maxwell model where we can learn about the stress relaxation in a much nicer fashion and these are the two element models but we can always increase the number of elements to more than to and learn about the overall rheology of this bone. So, that we can learn about the response mechanical response of these structures with respect to certain stress or loading with this I end my lecture here. Thank you.