The mechanical behavior of cortical actin cytoskeleton, a network of cross-linked semiflexible actin filaments (F-actin), is of interest to biologists and engineers since it plays a key role in many fundamental cellular properties and processes, such ascell shape and motility. The dynamics of flexible filaments in low-Reynolds number viscous shear flow has also gained much interest in the last decades in a wide variety of applications involving biological systems like DNA, polymers and proteins. In this dissertation, two numerical algorithms based on the Immersed Boundary Method (IBM) are proposed and implemented for the study of the actin cortex and the dynamics of semiflexible filaments immersed in low-Reynolds flows.

A new, modified, and more computationally efficient version of the IBM that combines the Coarse-Graining Method (CGM) with IBM is developed for modeling of inextensible filaments in shear flow at low Reynolds numbers. The various two-dimensional orbit regimes of flexible filaments are studied and the results of the proposed method are validated with theoretical results and previous works, numerical and experimental, showing excellent agreement. They are subsequently used to develop a prediction model using Artificial Neural Networks (ANN) to effectively forecast the orbit regime of a filament in shear flow with different parameters.

An extension of the traditional IBM to include a stochastic stress tensor is also proposed in order to model the thermal fluctuations in the cytoplasmic fluid surrounding the actin cortex. The theoretical values for time-averaged contraction for a single inextensible filament under hydrodynamic thermal fluctuations are verified through numerical simulation. The mechanical behavior of the actin cortex and its elasticity when subjected to shear flow is investigated, illustrating a stiffening of the cross-linked network with increasing strain under shear flow, as other experimental and numerical studies have shown. By implementing the proposed extension of the IBM, the behavior and interaction of passive F-actin under thermal fluctuations is also studied in the current work, where a trend of filaments to spread out is observed.

The diffusion of liquid and gas through porous solids is of considerable technological interest and has been investigated for decades in a wide spectrum of disciplines encompassing chemical, civil, mechanical, and petroleum engineering. Porous solids of interest are made of either natural materials (e.g., soil, sand) or man-made materials (e.g., industrial filters, membranes). In both cases, liquids (e.g., water, crude oil) and gases (e.g., air, oxygen, natural gas) are driven through the voids in the porous solid by naturally or artificially induced pressure. Nafion⃝R is an important example of a well-characterized man-made porous medium due to its extensive use in proton- exchange membrane fuel cells. Here, while the fuel cell is in operation, a mixture of air and water diffuses through the pores of a Nafion⃝R membrane. The efficiency of the fuel cell is affected by the variation in water concentration. In addition, high water concentration has been experimen- tally shown to cause substantial volumetric deformation (swelling) of the membrane, which may compromise the integrity of the device.

In this dissertation, a continuum approach for modeling diffusion of fluid through a porous elastic solid is proposed. All balance laws are formulated relative to the frame of a macroscopic solid resulting from the homogenization of the dry solid and the voids. When modeling only liquid diffusion through the macroscopic solid, the displacement of the macroscopic solid and the liquid volume fraction are chosen to characterize the state of the porous medium, and Fick's law is used as the governing equation for liquid flow. When modeling multiphase diffusion through the macroscopic solid, the displacement of the solid, the gas pressure and the liquid saturation are chosen as state variables, and both fluid diffusions are assumed to follow Darcy's law. Both single phase and multiphase diffusion models are implemented in the finite element method, and tested with various loading conditions on different types of materials. Numerical simulation results are presented to show the predictive capability of the two models.

Cell motility, and in particular crawling, is a complex process, with many as yet unknown details (\textit{e.g.}, sources of viscosity in the cell, precise structure of focal adhesions). However, various models of the motility of cells are emerging. These models concentrate on different aspects of cellular behavior, from the motion of a single cell itself, to -taxis behaviors, to population models. Models of single cells seem to vary significantly in their intended scope and the level of detail included.

Most single cells are far too large and complex to be globally amenable to fine-scale modeling. At the same time, cells are subject to external and internal influences that are connected to their fine-scale structure. This work presents a continuum model, including the use of a continuum theory of surface growth, that will predict the crawling motion of the cell, with consideration made for the appropriate fine-scale dependencies.

This research addresses several modeling aspects. At the continuum level, the relationships between force and displacement in the bulk of the cell are modeled using the balance laws developed in continuum mechanics. Allowances are made for the treatment of and interaction between multiple protein species, as well as for the addition of various terms into the balance laws (\textit{e.g.}, stresses generated by protein interactions). Various assumptions regarding the nature of cell crawling itself and its modeling are discussed. For instance, the extension of the lamellipod/detachment of the cell is viewed as a growth/resorption process. The model is derived without reference to dimensionality.

The second component of the presentation concerns the numerical implementation of the cell motility model. This is accomplished using finite elements, with special features (\textit{i.e.}, ALE, discontinuous elements) being used to handle certain stages of the motility. In particular, the growth assumption used to model the crawling motility is represented using ALE, while the strong discontinuities that arise out of the growth model are represented using the discontinuous elements. Results from representative finite element simulations are shown to illustrate the modeling capabilities.

Shape memory alloys have found diverse applications in several engineering

systems including biomedical devices and thermal actuators. This

is due to their superelastic and shape-memory behavior, which occur as a result of

solid-solid transformations from a parent phase to several variants of the product

phases. The most commonly used shape-memory alloy is a nearly equiatomic

NiTi alloy known as Nitinol. Much research has been devoted to modeling

polycrystalline Nitinol under various thermomechanical loading conditions. As a result,

several phenomenological and micromechanics-based models have been proposed

to characterize the complex behavior of Nitinol in both monocrystalline and

textured polycrystalline form.

In this work, a multiscale thermomechanical model for Nitinol is developed

that takes into account the temperature-dependent multivariant phase transformations

at the single-crystal level and the interaction between various crystals in a textured

polycrystalline aggregate. The single-crystal thermomechanical model

is relevant to both thermal loading and mechanical loading at high strain-rates.

The coupled thermomechanical problem is solved using a monolithic approach

in a finite-element framework. Specializing this model to isothermal conditions leads to a

temperature-independent mechanical response, which is suitable for quasistatic

mechanical loading. Most models in literature account only for isothermal stress-induced

phase transformations between the austenite and multivariant

martensite phases in Nitinol. In this work, such a constitutive model

is extended to include the formation of intermediate multivariant

rhombohedral phase as well. In order to model the macroscopic

response of polycrystalline Nitinol, first a statistics-based method is developed

to determine the optimum size of Representative Volume Element (RVE) meant

for solving the microscale problem. The macroscale constitutive response is

then derived through computational homogenization of this RVE response.

A finite element-on-finite element architecture is employed to solve

this multiscale problem accurately. Representative numerical

simulations are performed in order to validate the modeling approach with

several experiments on thin-walled tubes.