Growth time of static friction

I have a doubt about the best model of friction to use ina dynamic simulation. The most popular model, the Coulomb-Morin's friction, is defined by:

$F =\cases{ -F_d \cdot sign(v) & with$v \neq 0$and$F_d$known real constant\\ F_s & with$v = 0$and$F_s \in \mathbb{R} \; \big\vert \; -F_{s,max} \leq F_s \leq F_{s,max}$}$

This model leads to a instantaneous growth of the static friction: at $v=0$, the force assumes immediately one value in the interval $[-F_{s,max},F_{s,max}]$. This fact causes me some doubts: is the Coulomb's model the most "realistic" one to model the static friction? I know that there are several other models, but the way they deal with the static friction is nearly the same: a discontinuity at $v=0$ or a very steep growth. The finite growth time is often introduced for computational purposes, since this allows to obtain a continuous model. This finite growth time is to consider:

• only a trick to make the function suitable for computer implementation;
• a way to model an experimentally observed effect.

I think that, in the real world, the value of friction should rise very rapidly but with a finite growth time; however, I am not enough expert in tribology to make conclusions.

Thank you in advance for the help.