Bacteriophage T4 is one of the most common and complex of the tailed viruses that infect host bacteria using an intriguing contractile tail assembly. Despite extensive progress in resolving the structure of T4, the dynamics of the injection machinery remains largely unknown. This paper contributes a first model of the injection machinery that is driven by elastic energy stored in a structure known as the sheath. The sheath is composed of helical strands of protein that suddenly collapse from an energetic, extended conformation prior to infection to a relaxed, contracted conformation during infection. We employ Kirchhoff rod theory to simulate the nonlinear dynamics of a single protein strand coupled to a model for the remainder of the virus, including the coupled translation and rotation of the head (capsid), neck, and tail tube. Doing so provides an important building block toward the future goal of modeling the entire sheath structure which is composed of six interacting helical protein strands. The resulting numerical model exposes fundamental features of the injection machinery including the time scale and energetics of the infection process, the nonlinear conformational change experienced by the sheath, and the contribution of hydrodynamic drag on the head (capsid).