Mixture Theory Modeling of Multiphasic Deformation
In this dissertation, the bio-mechanical response of a ber reinforced solid matrix (soft tissue) has been formulated. A constant magnetic eld eects has been incorporated in the binary mixture of uid and porous solid. The governing dynamics involved in the multiphasic deformation was based upon the loading imposed at the rigid bony interface. The uid ow through the cartilage network depends upon the rate of applied compression as well as strain-dependent permeability of the soft tissues. The components of the mixture were assumed intrinsically incompressible; however, in the derivation of governing dynamics, visco-elastic behavior of the solid and an interstitial uid were developed. The continuum mixture theory approach is employed in modeling solid deformation and local uid pressure. In deriving the governing dynamics, strain-dependent permeability has been incorporated in the governing equations of binary mixture. The governing nonlinear coupled system of partial dierential equations was developed for the solid deformation and uid pressure, in the presence of Lorentz forces. In the case of permeability dependent ow, a numerical solution is computed, whereas, an exact solution is provided for constant permeability case. Graphical results highlight the in uence of various physical parameters both on the solid displacement and fluid pressure.
In the second problem, the mechanical response of a radially constrained elastic porous shell during the passage of charged uid has been formulated . The motion of uid as well as solid deformation were based upon the rate of applied compression at the inner radius of the shell. A nonlinear diusion equation applicable to plana and radial geometries was developed for the porosity along with informal integral boundary conditions on both the extremities. An equation for solid deformation is derived in the form of an integral equation. The governing system of equations is solved numerically for the transient case, whereas, an exact solution is provided for the steady-state problem. In the case of linear permeability, an excellent agreement is noticed between both the solutions. The comparison of the uid ow through the planar, cylindrical, and spherical shell is used in exploring the process of uid ow aected by the geometrical constraint. Graphical results highlight the inuence of dierent physical parameters on the porosity and solid displacement. Moreover, a detailed analysis of the uid ow through a thick and thin wall elastic porous shell is also presented.
Finally, for the same geometry as was in the second problem, the mathematical model describing visco-elastic behavior of an elastic porous shell during passage of non-Newtonian uids was developed. In formulating the ow behavior, powerlaw model was used in the constitutive equations of the mixture theory. The dominant mechanism of the uid ow was considered outwardly directed when loading imposed at the inner radius of the shell. The outer boundary of the shell is considered as rigid mesh which oers negligible resistance for the passage of uids. The general system of equations is derived for the porosity and solid deformation both for planar and radial geometries. The governing system of equations is solved analytically for steady-state case, whereas, numerical solution is computed for the transient problem. The signicance of power-law index on the porosity and solid displacement is presented graphically.