If the velocity is very slow, solve the model in Static Structural. Solving in Dynamics adds a term to the equation, which is the inertial forces of mass x acceleration. If the acceleration is zero, the force is zero and the equation is like a statics equation.
Maybe the reason you are solving in Explicit Dynamics is because it makes computing self contact easy or it makes element erosion easy. If the issue is the solution time is very long due to the slow velocity, then the answer is to increase the velocity. As long as the kinetic energy remains a very small fraction of the total energy, the result will be quasi-static.