Static Structural is the correct solver. You are using a nonlinear material and large deflection (I hope). Show your analysis settings.
Mesh convergence is typically done by solving once with a coarse element size, then manually using smaller elements, especially near the high stress regions and obtaining another value of the important result, such as Force vs Displacement. This is done several times and the the Force as a function of element size is put on a chart to see if it is changing.
You don't have a sharp corner because you used a Tangent Modulus of 1.25 GPa. The authors used a Tangent Modulus of 0. I know that because they used the words Perfectly Plastic in the text.
I think you have made the correct setting for displacement since the initial slope of the stress-strain curve matches the paper quite well.
They didn't say Explicit and they did say nonlinear mechanical.
That is wrong. See #3 above.
Good luck!