Investigating the mechanical dynamics of bacterial motility has led to a deeper understanding of the behaviors and lifecycle of many bacterial species. We discuss chain driven sliding motility where the bacteria maintain connections between daughter cells following division, resulting in long chains that expand across the viscous substrate. These chains grow exponentially, suggesting the chain tips may accelerate to very fast speeds. We devise multiple mathematical frameworks encapsulating the key physical dynamics and interactions to investigate the dynamics of bacterial chains and the biological implications of this motility. Our first framework, the rigid rod model, provides a set of equations describing the chain growth dynamics. Analysis of these equations reveals the stress maintaining cell-cell linkages increases unsustainably at an exponential rate. We devise a perturbation analysis of the rigid rod model in order to predict the critical stress associated with mechanical failure of these linkages. A phenomenological population model reveals that repeated chain breakages limit the expansion of the entire population to linear growth. Through experimental observation and computer simulations, we identify two key mechanical instabilities that emerge in growing bacterial chains. The first is sharp localized kinking that leads to the chain breakage mentioned above. In the second dynamic, the chain buckles due to compressive drag forces resulting in the emergence of large curvatures throughout the chain. We devise a continuum mechanics framework to examine the curvature dynamics in the growing chain. Through linear stability analysis of the rigid rod model and the continuum mechanics framework, we predict the dominant instability dynamic based on the physical properties of the chain and its environment. We use rigid rod model simulations to investigate the biological implications of these dynamics. Lastly, we introduce a number of methods that extend the rigid rod model to allow for the investigation of interacting chains. We consider methods that implement forces due to the entanglement of cell body appendages as well as collision dynamics. In total these models provide generic frameworks for investigating mechanical dynamics of growing bacterial chains. Our models provide testable predictions and suggest biological motivations for the typical behaviors that are observed in these cell chains. / Doctor of Philosophy / Motility is crucial in the life of many bacterial species. Effective motility allows bacteria to obtain nutrients and avoid dangerous hazards. Since motility is such an important part of bacterial survival, understanding bacterial motility has strong implications in bacterial control and utilization. We consider a motility in which the bacteria move by forming long, often straight chains of many cell bodies that expand across the surface. This is known as chain-mediated sliding motility and can allow the bacteria to move at very high speeds. We present multiple physics based mathematical frameworks that provide the tools to investigate chain-mediated sliding motility. These frameworks are generic and can be applied to study any bacterial species that use chain growth as a means for motility. Using these tools, we learn the speed at which these chains can expand is limited by the mechanical strength of the linkages connecting adjacent cells with in the chain. This limitation means the chains will repeatedly break into shorter chains, a pattern that limits the speed at which the entire bacterial population can expand. Additionally, we discover two interesting behaviors exhibited by these bacterial chains, one in which the chain kinks before breaking into two shorter chains, and a second in which the chain buckles, resulting in curved chains. We apply our mathematical frameworks to determine how the physical conditions dictate which of these two behaviors will emerge and learn the chains may curve and bend as a means to avoid breaking. Lastly we introduce additional methods that extend these frameworks to allow for investigating the behavior of the bacteria when multiple chains interact with one another. The mathematical frameworks we present allow for investigation into the specific mechanical properties that make chain growth possible as well as the mechanics that limit its efficiency. The models also give insight into the biological impact of this motility, suggesting how it affects the growth-coupled spreading of an entire bacterial population.