The dissertation presents theoretical models for processes involving continuous failure and displacement of material via contact with a rigid object. Focus is mainly on processes relating to material-wheel interaction, which are central in a number of engineering applications (e.g., vehicle mobility and metal rolling). The analysis considers indentation, transient rolling, and steady-state rolling on an elastic-perfectly plastic or a rigid-perfectly plastic cohesive-frictional material. Mechanics-based models are developed using two separate approaches. The first is based on comprehensive numerical simulation using the Finite Element Method (FEM), which enables rigorous analysis of the three-dimensional deformation occurring for narrow wheels. The second approach is analytic and formulated by considering the entire process of deformation as a sequence of incipient plastic flow problems. Using the theoretical models, the relationships between dimensionless variables are quantitatively assessed. It is further shown that the predictions display reasonable agreement with experimental data from two types of small-scale indentation and rolling tests: one aimed at measuring the force-penetration relationship for indenting and rolling wheels and the second type concentrated on measuring the incremental displacement field at wheel midplane using Particle Image Velocimetry (PIV). The models and experiments provide insights into how indentation and rolling processes are influenced by three-dimensional effects, non-associativity, localization, and the presence of displaced material.