A viscoelastic constitutive relationship derived from a solid mechanics viewpoint using the curvilinear form of Hooke's law as opposed to an Eulerian context was developed. This approach has the advantage of obtaining the deformation relative to the original configuration as opposed to the current one. A constitutive model was derived for viscoelastic fluids where the stress depends on the history of the first spatial gradient of the displacement. A material (convected) coordinate system and body tensor mathematics were invoked to formulate the viscoelastic relationship. Complicated integration of a Jaumann or convected rate derivative was not required. Development of the model and its theory are given here in part A.