With rising demand to develop versatile composite materials for modern applications, computer-based solutions to arrive at the material characteristics are now preferred. This necessity is due to the large cost involved for the nanoadditives, as well as elaborate and precise setups required for detailed load tests or experiments. Even with the numerical simulations, considerable information has to arrive from experiments for these highly nonlinear viscoelastic materials. Stress accumulation and relaxation are important aspects that need prediction before considering composites for applications. Being a relatively new engineering domain, the significant information to use commercial codes like Ansys®, MSC®, or Abaqus® are now scattered in literature and the solvers themselves are still evolving. Our attempt through this work is to provide a comprehensive basic set of information on the same, while using Ansys® for arriving at the material characteristics of Thermoplastic Polyurethane-Carbon Nanotube (TPU-CNT) polymer composite. A simulation procedure to study stress relaxation behavior with Maxwell’s Prony relaxation parameters is detailed, suitability of incorporating basic hyperelastic models, such as Mooney–Rivlin and Ogden available in the solver, is compared, and the influence of coefficient of friction (COF) in the numerical simulations is investigated. On appropriate validation with available experimental results, we found that hyper-viscoelastic model is best suited for TPU-CNT with maximum error as low as 5% during stress relaxation phase (for 1800 s), in comparison with 15% and 25% for viscoelastic and hyperelastic models. During the short loading phase of the material, none of the models are accurate. The two-parameter first-order equation-based Mooney–Rivlin model fed with uniaxial load test information was satisfactory for low strain predictions in comparison with higher-order Ogden model. COF is found to significantly affect the solution, and a value of about 0.3 was found suitable for the present 0.05% TPU-CNT composite. Further, we present the transient stress contours to show how the stress relaxation within the composite material would be predicted with each model.