<p>Models of human musculoskeletal tissue are the missing component needed to make significant advances in clinical orthopedics. Developing these models requires an in-depth knowledge of techniques and procedures which are rarely considered or taught in universities.  Essential skills like deriving the foundational physics and the constitutive theory from first principles are the building blocks which will deliver future models or ligaments, tendons, and muscles. This thesis presents the first step in a journey to understand these modeling techniques in order to move toward developing a model of human tissue. The material utilized in the experiments</p> <p>was recipe of Ecoflex 00-20 which represents an idealized example of a large deformation viscoelastic solid. The Finite Strain Viscoelastic constitutive law was derived from first principles, uni-axial tension experiments provided the raw data, and the constitutive laws were fit to the data. The derived model outperformed most of the models with some exceptions described in the results. These results justify the development of more complicated models and experiments such as modeling surface field data and considering more complicated material properties.</p>

Collagenous tissues such as ligaments and tendons are viscoelastic materials. They exhibit a slow continuous increase in strain over time, or creep, when subjected to a constant stress and a slow continuous decrease in stress over time, or stress relaxation, when subjected to a constant strain. Moreover, the loading and unloading stress-strain curves are different when the tissues are subjected to cyclic loading, showing hysteresis and softening phenomena. The micro-structural origin of the viscoelasticity of these tissues is still unknown and the subject of debate among experts in biomechanics. Therefore, formulating viscoelastic models by accounting for the mechanical contributions of the structural components of these tissues can help in understanding the genesis of viscoelasticity. A nonlinear viscoelastic modeling framework has been developed to describe the elastic and viscoelastic properties of ligaments and tendons by considering their main structural components, the collagen fibers and proteoglycan-rich matrix. The mathematical models derived within this framework can illustrate the tensile behavior, stress relaxation and creep by as suming that the collagen fibers are elastic and the surrounding proteoglycan-rich matrix is viscoelastic. The collagen fibers are represented by linear elastic springs that are engaged to support load at different values of the tissue's strain according to a Weibull distribution function. The mechanical contribution of the matrix is introduced via a Maxwell-type viscoelastic element arranged in parallel with the collagen fibers. According to the proposed mathematical framework, both the collagen fibers and the proteoglycan-rich matrix are responsible for resisting tensile loads. However, the collagen fibers play a significant role in creep while the proteoglycan-rich matrix has a dominant role in stress relaxation. The model parameters that define the stress relaxation and strain stiffening phenomena are estimated by using published experimental on rabbit medial collateral ligaments and are then used to predict creep. The above modeling framework has been also extended to capture the in uence of preconditioning on the mechanical properties of ligaments and tendons. The stress softening and decrease in hysteresis that are observed during successive loading cycles in preconditioning are assumed to be determined by a decrease in the elastic properties of the collagen fibers and proteoglycan-rich matrix. Preliminary data collected on stress relaxation and preconditioning on rat medial collateral ligaments by collaborators are used to evaluate the model parameters and analyze its predictions. The elastic and viscoelastic properties of single collagen fibers are studied by formulating a nonlinear viscoelastic framework by accounting for their main components: microfibrils, cross-links and proteoglycan-rich matrix. The model illustrates tensile behavior and stress relaxation of a single collagen fiber by assuming that the microfibrils and the cross-links are elastic and the surrounding proteoglycan-rich matrix is viscoelastic. The mechanical contribution of the microfibrils is included via a linear elastic spring while the cross-links are represented by linear elastic springs that progressively fail at different values of the tissue's strain according to an exponential distribution function. The matrix is defined by linear dashpots arranged in parallel with each single spring that represents an individual cross-link. The viscous properties of the matrix associated with the unbroken and broken cross-links are assumed to have different values. In the model formulation, the microfibrils and the cross-links are assumed to determine the elastic response of the fibers while the proteoglycan-rich matrix determines the stress relaxation. Microfibrils, cross-links and the proteoglycan-rich matrix are responsible for resisting the loading force during tensile behavior. Experimental data collected by performing incremental stress relaxation tests by other investigators on reconstituted rat tail tendons are used to estimate the parameters in the model and evaluate its performance. / Ph. D.

The polymers are integral part of vast number of products used in day to day life due to their anomalous viscoelastic behaviour. The remarkable flow behaviour exhibited by the polymeric fluids including rod climbing, extrudate swell, tube-less siphon, viscoelastic jet, elastic recoil and sharkskin instability is attributed to the complex microstructures in the polymeric liquids that arise due to the interactions of long chain polymer molecules with each other and with the surrounding fluid particles. The significance of polymer in transportation, packaging, pharmaceutical, chemical, biomedical, textiles, food and polymer processing industries highlights the requirement to comprehend the complex rheology of polymeric fluids. First, we investigate the flow features exhibited by different shear thinning vis-coelastic fluids in rectangular cavities over a wide range of depth to width ratio. We have developed a viscoelastic flow solver in order to perform numerical simulations of highly elastic flow of viscoelastic fluids. In particular, we discuss the simulations of flows of constant viscosity Boger and shear thinning viscoelastic fluids in the complex flow problems using different constitutive equations. The effects of elasticity and inertia on the flow behaviour of two shear thinning vis-coelastic fluids modeled using Giesekus and linear PTT constitutive equations in rectangular cavities is studied. The size of the primary eddies and critical aspect ratio over which the corner eddies merge to yield a second primary eddy in deep cavities is discussed. We demonstrate that the flow in the shallow and deep cavities can be characterized using Weissenberg number, defined based on the shear rate, and Deborah number, specified based on the convective time scale, respectively. The study of flow in driven cavities is important in understanding of the mixing process during synthesis of blends and composites. Next, we study two phase polymeric flow in confined geometries. Nowadays, polymer processing industries prefer to develop newer polymer with the desired material properties mechanically by mixing and blending of different polymer components instead of chemically synthesizing fresh polymer. The microstructure of blends and emulsions following drop deformation, breakup and coalescence during mixing determines its macroscopic interfacial rheology. We developed a two phase viscoelastic flow solver using volume conserving sharp interface volume-of-fluid (VOF) method for studying the dynamics of single droplet subjected to the complex flow fields. We investigated the effects of drop and matrix viscoelasticity on the motion and deformation of a droplet suspended in a fully developed channel flow. The flow behaviour exhibited by Newtonian-Newtonian, viscoelastic-Newtonian, Newtonian-viscoelastic and viscoelastic-viscoelastic drop-matrix systems is presented. The difference in the drop dynamics due to presence of constant viscosity Boger fluid and shear thinning viscoelastic fluid is represented using FENE-CR and linear PTT constitutive equations, respectively. The presence of shear thinning viscoelastic fluid either in the drop or the matrix phase suppresses the drop deformation due to stronger influence of matrix viscoelasticity as compared to the drop elasticity. The shear thinning viscoelastic drop-matrix system further restricts the drop deformation and it displays non-monotonic de-formation. The constant viscosity Boger fluid droplet curbs the drop deformation and exhibits flow dynamics identical to the shear thinning viscoelastic droplet, thus indicating that the nature of the drop viscoelasticity has little influence on the flow behaviour. The matrix viscoelasticity due to Boger fluid increases drop deformation and displays non-monotonic deformation. The drop deformation is further enhanced in the case of Boger fluid in viscoelastic drop-matrix system. Interestingly, the pressure drop due to the presence of viscoelastic drop in a Newtonian matrix is lower than the single phase flow of Newtonian fluid. We also discuss the effects of inertia, surface tension, drop to matrix viscosity ratio and the drop size on these drop-matrix systems. Finally, we investigate the emulsion rheology by studying the motion of a droplet in the square lid driven cavity flow. The viscoelastic effects due to constant viscosity Boger fluid and shear thinning viscoelastic fluid are illustrated using FENECR and Giesekus rheological relations, respectively. The presence of viscoelasticity either in drop or matrix phase boosts the drop deformation with the drop viscoelasticity displaying intense deformation. The drop dynamics due to the droplet viscoelasticity is observed to be independent of the nature of vis-coelastic fluid. The shear thinning viscoelastic matrix has a stronger influence on the drop deformation and orientation compared to the Boger fluid matrix. The different blood components, cells and many materials of industrial importance are viscoelastic in nature. Thus, the present study has significant applications in medical diagnostics, drug delivery, manufacturing and processing industries, study of biological flows, pharmaceutical research and development of lab-on-chip devices.