Does the contact area affect friction forces? I recall studying a law of friction some years ago, in engineering school. All I remember is that when first approximation was taken, the popular $f=k \times N$ was derived.
Why would such an approximation be deemed so useful as to be taught as "engineering truth"? What is the proof that contact area does not significantly affect friction force?
EDIT:
So, the book is about designing a friction clutch. This device consists of a number of z disc pairs, one connected to the input shaft and the other disc from the pair connected to the output shaft.
If I understand the formula and meanings of the variables correctly, the maximum torque the device can transmit is proportional to $F \times z$ where F is the force pushing the disc package together.
 A: Of course the simple linear relationship F=k*N is just a crude approximation.
The major force in friction of mainly flat areas is the van der Waals force between the molecules in the two layers being close to each others, which also are affected by the deeper layers in the materials.
The amount of friction also depends on how long the two areas have been in contact, their temperature and a myriad of other factors, with plenty of ongoing research.
A: It affects and it does not in different cases
Case 1: When contact area do not affects the frictional force
You know that frictional force $f$ is $$f = N*\mu$$ where $\mu$ is the frictional coefficient of the surface and $N$ is the normal force acting on the surface. Now look at the image below

both the blocks have same mass in $fig(1)$ and $fig(2)$ but the contact area is different let the block 1 have the are $Am^2$ and block 2 have the are $2Am^2$, in the both the images there is equal frictional coefficient that is $\mu$
Now force exerted by the first block per square meter is $$N_1 =\frac{mg}{Am^2}$$and the force exerted by the second block is $$N_2 = \frac{mg}{2Am^2}$$so since both the blocks have same frictional coefficient $\mu$ so frictional force between block 1 and the bottom plank is$$f_1 = N_1*\mu*a_1$$ here $a_1$ is the area of the contact between the blocks, so$$f_1=\frac{mg}{Am^2}*\mu*Am^2\implies mg\mu$$ and the contact force between block 2 and bottom plank is$$f_2 = N_2*\mu*a_2$$ here $a_2$ is the area of the contact between the blocks, so$$f_2=\frac{mg}{2Am^2}*\mu*2Am^2\implies mg\mu$$$$\implies f_1=f_2$$ so you see here there is no effect of contact area on the frictional force.
Case 2:When contact area do affects the frictional force
i don't if this is a correct example or not but here is goes 
take example of books kept in shelf standing vertically side by side and are tightly packed so if you will try to remove a big book with a greater surface are you will have to apply more force as compared to in removing smaller books with smaller surface area
Hope you got it
A: For what they call "dry friction" (no sticking, viscous lubricants, etc) indeed, the empirical evidence indicates that among many possible factors (contact area, sliding velocity, etc) only load matters. This is known as Amonton's law of friction, and, I agree, it seems very counterintuitive. But being an empirical fact (see e.g. the reference below) it does not need theoretical proofs; what it needs is an explanation, and as far I can tel, there is no satisfactory explanation yet (one hypothesis, as John Rennie, mentions is that the actual contact area grows proportionally with load). Here is one relatively recent review of the topic: https://pubs.acs.org/doi/10.1021/jp036362l.
A: Well I'm not aware of any well publicized change in the standard teaching that dry static friction is area independent, and due to the microscopic contacting as described above by John Rennie.   However when it comes to clutch design, there are other factors to consider. It is desirable to have very smooth increase in friction with increased contact pressure (macro), and this dictates a macro surface area, that is hugely greater than the real micro contact area. So that the making and breaking of these contacts is not detectible at the individual contact level.  The actual contacts tend to weld, in the case of metal on metal, and the friction force itself is the shearing of these micro welds (when sliding friction occurs.)    The other major concern as regards clutch design, is the removal of heat; and that feeds into the calculation of suitable macro areas, so that temperature rise is minimized.    This is even more important, in the case of a brake shoe, where energy dissipation as heat, is the end aim of the system.
