The steady flow of incompressible elasto-viscoplastic liquids through a planar expansion-contraction is investigated. A novel constitutive model is employed to describe the mechanical behavior of the flowing liquids. Numerical solutions of the constitutive and conservation equations were obtained via a finite element method to investigate the role of elasticity, yield stress, and inertia. The fields of velocity, stress, elastic strain, and rate of strain were obtained for different combinations of the governing parameters. It was observed that these fields, as well as the shape and position of the yield surface, are all strong functions of elasticity, yield stress, and inertia. The trends observed agree well with previous numerical and visualization results available in the literature. The present work offers a detailed study on the effects of elasticity, presenting, in particular, the fields of elastic strain.