In contrast to the engineering material, living structures show the ability to grow and adapt thereform, shape, and microstructure when it is subjected to the mechanical loading. The phenomenonof growth when it is subjected to mechanical loading conditions on the theoretical and the computationallevel is formulated in this report. The biomechanics problems like bone remodeling, hipreplacement, wound healing are the fields of application of the growth. Growth is defined as theadded mass which can occur due to hyperplasia (cell division), hypertrophy(cell enlargement), secretion(discharge)of hard tissue matrix, and the accretion at external or internal surfaces. There canbe a negative growth in the biological structure this can occur due to cell death, cell shrinkage or resorption.
Remodeling can be defined as the process which involves changes in material properties.This changes are typically adaptive and can be brought into account by alternations in modulus,or internal structure, strength, or density. Our tissues never stop changing and growing, for nomatter, it is for good or bad. One of the standard problem in the context of biomechanics is howthe gene of one dimentional in nature transfer the information to the three dimentional structure.In the present work we will first review the fundamental definition in the context of biomechanicsand then the existing formulation present for the continuum level theory. Then we apply the opensystem formulation presented1.
1 Basic forms of growthGrowth is a generic term in which mass of the body changes over time. In biology, growth is a fundamentalaspect of life. In realization of biological processes such as cell division, morphogenesisdevelopment, maintenance, cancer, aging, all these processes come under growth. One of the basic1challenges in modern biology is to understand how the role of genetic code in transforming thecell into fully mature organisms and how these organisms manage to regulate through growth andremodeling.
Growth also appears in some physical processes. For example, epitaxial growth, a thinlayer of crystal is produced by depositing raw material on an existing crystalline substrate. Unlikein the biological growth here boundary line of the inter-phase in the physical growth, a process is aline of discontinuity with no material properties. The swelling of gels is another example arguablythe closest non-biological process that mimics growth as it is non-diffusive and occurs in the bulkof the material. It can be used as a physical analog to gain insight into the role of mechanics inbiological pattern formation.
1.1.1 ClassificationGrowth fulfills the many purposes associated with different qualitatively different processes. Classicallygrowth process is a divide by the way growth alter a body, by changing its volume, or bychanging its properties or relative rearrangement of the material points.• Growth. The word growth itself represent a change in mass. It is commonly understood asthe increase in mass, but the concept can be extended to mass reduction and shrink. Changes inmass can be by changes in mass at constant volume as densification of bone or change in mass ata constant density as a development of the soft tissue or in both ways found in developing bone.
Mathematically, the theory of growth must account for the changes in mass at the boundary asmass flux and accumulation of mass as a source term. 1• Remodeling. In the process of aging, it is known that the tissues may become stiffer or softer.Remodeling related to the evolution of the material properties such as stiffness, fiber orientation,fiber strength and so forth without changes in mass. Remodelling related to the typical changes inmicrostructure that determine the overall behavior of tissue. For example, the typical compositionin many animals is composite of collagen fibers and within an elastin matrix. Since the elastin fiberdoes not change over many years but collagen fiber changes continuously depend on the biologicaland mechanical stimuli. 1• Morphogenesis.
In the initial stage of life when baby born, many cell divisions occur andthe major reorganization and cell differentiation leads to major structural changes in the organism.Thus, this process of shape changes is called as the morphogenesis. The weak tissues attach to theother organism due to adhesive stimulus and overall shape of the body changes. 1Classification of the growth process is also done in quantifying mode such as tip growth, accre2tive growth, volumetric growth.1.1.2 Relative GrowthExtensive studies show that children and puppies are cute and lovable.
Cuteness factor whichdefends a person due to a relative size of body and facial feature of child versus adults. In the earlyembryonic period, the size of a skull as compared to other parts of the body is larger 2 and lengthof hand till knee, but as a child started to grow, then some parts of the body grow in slow rateand the other parts of the body grow in a fast rate. If take the example of the head then it growsrelatively slower and legs relatively faster. This type of growth is called as allotropic growth whichin turn known as the relative growth.1.2 Continuum Models for Growth AnalysisIn the classical approach of the thermodynamics, the amount of mass is always conserved no matterhow the system is accelerated, deformed or moved.
As this conservation of the mass found valid inmany application there exist problems in biomechanical and the chemomechanical systems whereconservation principle is a mere definition and not applicable. Both the systems biomechanics andthe chemo-mechanics falls in the category of the open system, where mass interaction to the systemis not only through a generation of mass due to source term as well as due to the mass influx oroutflux. Thus the newly added mass influence balancing equations for the quantities like mass,momentum, energy, and entropy.The theory of open system is closely related to the ‘theory of mixture’ as given by Truesdelland Toupin 3 . In Theory of mixture, the individual constituent can exchange the mass with theother parts while the mass of the overall mixture will remain constant. The fundamental assumptionfollowed in the theory is that individual particle superimposed particularly on each spatialcoordinate so the moment of the one constituents will be compensated by other. While in manyapplications it seems reasonable to make attention on only one constituent. This system which canexchange mass, momentum, and energy with the outside world comes in the category of the opensystem thermodynamics.
The lines of taught given by Mougin in open system defined as the permeable,diathermal, deformable membrane which can exchange the mass through outside world.Cowin and Hegedus 4 give the first model in the context of open system thermodynamics underthe name of ‘Theory of adaptive elasticity.’ In this adaptive elasticity theory mass addition to the3system is considered through the volume source term only. Epstein and Mougin gave the ‘Theoryof volumetric growth,’ herein the mass addition through flux through boundaries of the system includedwith the volumetric generation term. Kulh and Steinmann 5, 6 give numerical realizationof the open system mechanics problem using finite element method.1.
3 Open System Thermodynamics1.3.1 Mass and Volume Specific Views1.3.2 Spatial and Material Settings1.4 Material Force Method1.4.1 Healing as an application of aterial force method1.
5 Motivation1.6 Thesis Outline4Chapter 2Bone structure and propertiesBone is solid structural element part of the body which constitutes the whole skeleton structure.With specific structural and the material properties of the solid, bone protect the small tissue whileundergoing the mechanical forces and moment by maintaining a shape of the body. For the engineeringperspective, bone is non-homogeneous, an anisotropic and viscoelastic material.
Like mostbiological tissues, it can adapt its structure according to a type of loading condition. Moreoverit exhibits wide range of morphology depending on the porosity of structure. Bone exists in twobasic types depending on their relative densities or the volume fraction of the solids: cortical andcancellous 7.
The core spongy cancellous part is surrounded by the dense outer shell. the denseouter shell is known as the cortical bone or compact bone. Cortical bone has a volume fraction ofsolid more than 70% of overall bone. In the cancellous (or trabecular or spongy ) bone is porousand contain less than the 70 of solid volume fraction. The distribution of the cortical bone and thecancellous bone varies from bone to bone and also from person to person. At the molecular level,bone represents the true composite material 8 . Bone consists of 65% mineral (HA), 35% organicsmatrix (mostly collagen fibers), water, cells, and vessels. The mineral content is impure mainly HAin the form of small crystals with the shape of needles, plates, and rods located within and betweenthe organic matrix.
The organic matrix consists of 90% collagen and 10 of various non-collagenousproteins 9. Bone can also be categorized architecture and arrangement of collagen fiber on intotwo types: woven and lamellar bone. Bone grows rapidly and typically found at younger ages andduring fracture healing process. The collagen fiber of oven bone is comparative randomly orientedand the loosely packed in contrast to the lamellar bone. Lamellar bone is more regularly arrangedand the growth rate is not as fast as in woven bone.
The fibers of collagen and associated calcium5phosphate are presents in the form of sheets, known as lamellae. The mineral content of lamellarbone is usually lower than the woven bone 8. Lamellar bones are also categories as the primarylamellar bone and the secondary lamellar bone.
Small cavities connected by their tubular canals(canaliculi) are found throughout the both woven and the lamellar bone. Entrapped bone cells (osteocytes)occupy the lacunae and their long cytoplasmic processes occupy canaliculi, respectively9.2.1 Mechanical properties of cortical boneIn the human skeleton, cortical bone represents 80 % by mass and only contain 20 % volume of thetotal bone structure. It is solid contact tissue constituting the diaphysis of the long bones and outershell of the epiphyses and metaphyses. Cortical bone with an enormous tensile strength and elasticmodulus in a longitudinal direction than in transverse direction thus Cortical bone is anisotropic.
Moreover Cortical bone is stronger in compression than tension. 8 measured the elastic propertiesof cortical bone. They estimated the elastic moduli along the longitudinal direction around 20- 22Gpa and 12-14 around the transverse direction. In spite of anisotropy, the transversely isotropicand orthotropic behavior is shown by the cortical bone. 9. Elastic properties of the cortical bone,measured using mechanical tensile testing is reported in a range of 17.5 – 19 Gpa 10.
However, thisvalues of elastic modulus are depended on ethnicity, age, and sex of the person.11 reported thestress-strain behavior of human femoral cortical, they observed that there are three distinct regionsin the stress-strain curve. In the initial region, the stress-strain curve shows the linear relation with17 Gpa. In the intermediate region, the bone exhibits the non- linear elastoplastic behavior; thisregion is characteries by bone yielding with the yield strength value around 110Mpa.
The finalregion exhibits linear plastic behavior with strain hardening modulus of 0.9Gpa. The bone wasfound to have fractured when tensile stress was 128 MPa, and the corresponding tensile strain was0.026 11. The study also shows that the elastic modulus values and the strength values depend onthe strain rate, which indicates the viscoelastic properties of bone 11.
Other investigations basedon mechanical loading also shows that stress-strain behavior of bone dependent on the orientationof bone concerning a direction of loading.62.2 Mechanical properties of cancellous boneDepending on the mode of loading the elastic properties of cancellous or trabecular bone varies extensively.
The stress-strain curve of cancellous bone under compressive load contains an initial learelastic region, followed by a constant plateau region of almost constant stress and finally increasingthe stiff region culminating in the fracture 7, 11. The material yielding tends to occur when thetrabecular begin to fracture. Mechanical yield property of cancellous bone varies significantly withanatomical location 12.
In contrast to the cortical bone, cancellous bone fractures abruptly undertensile force loading condition and behave like the brittle material behavior. Cancellous bonehas more energy absorption capacity in the case of compressive loading as compared to the tensileloading . The material properties and stress-strain relationship depend on the mode of loading aswell as the apparent density. It is reported that cancellous bone shows the behavior similar to theother mechanical cellular structure like polymeric foam 13. The apparent or relative densities areequivalent to the volume fraction of solids in the cancellous bone, which can be calculated from thecancellous bone density and volume of trabeculae (solid shell wall).
Low relative density showsthe open-pored structure while high value of the relative density shows the closed network of thecell. In 7 gibson suggested that Young modulus for the open system varies with the square ofdensity and the enclosed system elastic modulus vary with the cube density. Carter and hayes in1977 14 predict the transformation of the rod-like structure to a plate-like structure at the relativedensity of 0.20. A series of experiments have been done to find out the empirical relations betweenbone density and the elastic modulus15, 12. Different equations were deduced with the sameempirical relationships: E = C?D. Where C varies in the range 3000 – 30000, whereas the value ofD ranged between 1.
14 and 3.2., depending on the location of bone 12. In the recent experimentalstudy by 12 is suggested that for human body strain-based criteria for trabecular bone may bemore simple and statically more powerful.
They tested the cylindrical specimens of human trabecularbone taken from a different atomic site under both uniaxial tensile and compressive loadsubsequently proved that yield strains could be considered uniform within a single site despitesubstantial variation in yield stress and elastic modulus.7Chapter 3Different Modeling AlgorithemsIn the past, very massive amount of works of literature is based on the stability and the uniquenessof the suggested models. Many cases in attribution to the finite element formulation shows theill-posedness of the underline continuous problem. So we will make the clear distinction betweencontinuous problem, the time discrete problem and the fully discrete finite element problem toclarify the notion of stability and uniqueness.