Here is a paragraph on Johnson-Cook Strength material model from the ANSYS Help.
Use this model to represent the strength behavior of materials, typically metals, subjected to large strains, high strain rates and high temperatures. Such behavior might arise in problems of intense impulsive loading due to high velocity impact.
This model is only available in Explicit Dynamics, which is the solver to use for very high speed events.
When you say you want to "simulate the tensile testing process at various strain rates", what strain rates are you talking about?