Force to close a

Yes you want non-linear.  Large Deformation and possibly Elasto-plastic material

Here is the difference in small displacement and Large displacement (small on left)

Here is how you do the 1.4mm deflection

Here is how you prevent infinite stress at the edge constraint... with a small region made rigid.

Here is the measure used to obtain the force.

Here are the Analysis definitions.  If using elasto-plastic material then Plasticity will have a check mark.

Plotting the measure vs. time

Final results.  If stress is above yield then you should use ELASTO-PLASTIC material.  Nonlinear material + non-linear geometry.  Otherwise force is generally predicted too high.  

Confirmation?  Run this model with found force (48N in this case) in X direction as a load in place of the enforced displacement.  Confirm max displacement about 1.4mm in X direction.

BONUS - I used a Mapped Mesh (not totally necessary but results are so smooooth!)