This dissertation establishes a mathematical framework for analyzing a viscoelastic model that displays thixotropic behavior as a model parameter gets very small. The model is the partially extending strand convection model, originally derived for polymeric melts that have long strands that get in the way of fully retracting. A Newtonian solvent is added. The uniaxial and equibiaxial extensional flows are studied using combined asymptotic analysis and numerical simulations. An initial value problem with a prescribed elongational stress is solved in the limit of large relaxation time. This gives rise to multiple time scales. If the initial stress is less than a critical value, the initial elastic elongation is followed by settling to an unyielded state at the slow time scale. If the initial stress is larger than the critical value, then yielding ensues. The extensional flows produce delayed yielding and hysteresis, both associated with thixotropy in complex fluids.