Doubt about friction force It's probably a very basic question, but I am somewhat stuck with it: Let's consider some object at rest and some force acting on it, trying to put it to motion. Then, there will be a static friction force which stays equal to the applied force, until some critical force $\mu_s F_N$ is reached. At this point the object starts moving and feels a kinetic (or sliding) friction force which is approximately constant and given by $\mu_k F_N$, where $\mu_k<\mu_s$. So the friction force looks like in the following graph: http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
However, there is also a rolling friction force, which is given by $\mu_r F_N$ with $\mu_r<\mu_k$. Again, if I try to put an object into motion, it will first be subject to a static friction force until some critical force is reached, the object starts rolling and the friction force drops to a lower value $\mu_r F_N$.
Now, the question is, is the critical force at which the object starts to move dependent on whether sliding or rolling will ensue? If it is, is the critical force required for initiating a rolling movement equal to zero? If not, how can it be calculated?
Intuitively, it is easier to make things roll than make them slide, so not only are the friction forces different when the object is already moving, but also the critical force required to make them start moving is different (lower for rolling than for sliding).
But on the other hand, the static friction coefficient is said to be only dependent on the pair of materials in question (say, steel on steel or rubber on asphalt) and on nothing else.
I understand that there is most probably some very complex physics going on and there are better models than $F_{fr} = \mu F_N$ for the friction force, but I still wonder, if this question can be answered using this simple model?
 A: 
Now, the question is, is the critical force at which the object starts
to move dependent on whether sliding or rolling will ensue?

What you call "rolling friction" is more properly referred to as "rolling resistance". Unlike kinetic friction, which is energy loss due to the relative motion (slipping/sliding) between surfaces, rolling resistance is related to energy loss due to inelastic behavior of the materials of the contacting surfaces (rolling object plus surface) when the materials are  compressed and uncompressed while in contact with one another. The compression force acting opposite the direction of motion is greater than the decompression force acting in the direction of motion, for a net force opposing motion.
The coefficient of rolling resistance is generally much smaller than the coefficient of kinetic friction which, in turn, is generally somewhat less than the coefficient of static friction. See the Wikipedia article on Rolling Resistance.
That said, which type of "friction" is applicable depends on what type of "moving" is involved. There is "moving" without rotation (translational motion), "moving" without sliding (rotational motion), or a combination of the two (rotation with slipping).
If the object is not capable of rolling, for example, a block, then "move" means relative motion between the object and the surface below, i.e., strictly translational motion. In that case, the critical force is the maximum possible static friction, or $\mu_{s}N$, as discussed in the Hyperphysics link. In this case rolling resistance does not apply.
If the object is capable of rolling (i.e.,a  circular object), then you need to distinguish between the initiation of rolling and the maintaining of rolling.
To initiate rolling without slipping requires a static friction force. That force will match the pushing or pulling force on the object, or the force the object exerts on the surface due to torque applied to the object, up until the maximum possible static friction force is reached when slipping occurs and friction becomes kinetic. If the applied force or torque does not result in the maximum static friction being exceeded, then pure rolling begins almost instantaneously. Opposing the pure rolling is rolling resistance. But since the rolling resistance force is so much less than the static friction force during acceleration, the net force is essentially the static friction force. If the maximum static friction force is exceeded, slipping occurs and friction opposing motion becomes kinetic. However, again rolling resistance is much less than kinetic friction so the net force opposing motion is essentially the kinetic friction force.
So when does rolling resistance play an important role? When there is pure rolling with no external forces acting in the direction of motion, then rolling resistance is one of the primary reasons a rolling object will slow down. The others are air resistance, bearing friction, etc. An example is the slowing of a bicycle or car that is coasting.
Hope this helps.
A: Your two scenarios are categorically different and cannot be compared as you do:

*

*In the sliding-block scenario, static friction is replaced by kinetic friction.

*But in the rolling-wheel scenario, static friction appears simulatenously with rolling friction (which is more appropriately called rolling resistance).

The reason is that rolling won't happen without static friction. A sliding wheel that experiences kinetic friction is not rolling. By definition, rolling means that the periphery motion follows the surface (it is often term rolling without slippin to make this point clearer).
An ideal wheel experiences no kinetic friction or resistances of any other kind while rolling. It only experiences static friction at the contact point which is kept stationary (doesn't move during the moment of contact - if it did move, then we would have sliding, and we wouldn't call it rolling). But the wheel has inertial motion and "topples over" this stationary contact point, which lifts the contact point out of contact and places a new contact point.
So, static friction is a prerequisite, a necessacity, for rolling to take place.
Under non-ideal circumstances, we might see

*

*axle and bearing frictions,

*soft wheel or soft surface (rubber tire expanding and compression or sand beach deforming, which consumes energy via work done on displaced particles)

*"bulging up" of material in front of the wheel (think of driving on a sandy beach where the wheel constantly has to "lift itself out" of the hole is is digging for itself)

*a slightly deformed wheel causing non-centered normal forces introducing counteracting torques

etc. All such effects absorb energy that is removed from the kinetic energy of the motion. All these effects are gathered under one headline: rolling resistance or rolling friction. So, essentially, rolling friction is not a friction at all but rather a collection of all the non-ideal factors that come into play during rolling in real life. And of that reason you can't expect to fully behave just like kinetic friction, i.e. don't expect constancy nor linearity of the force, don't expect it to replace static friction etc.
A: 
Now, the question is, is the critical force at which the object starts
to move dependent on whether sliding or rolling will ensue? If it is,
is the critical force required for initiating a rolling movement equal
to zero? If not, how can it be calculated?

We have been discussing this in the context of what the minimum force a man must exert to initiate motion of a car by pushing on it in neutral on a flat horizontal surface. Also we ask why, in real life experience, does it usually seem harder to initiate that motion than it is to maintain that motion once it occurs?
Although we have considered the possible contribution of rolling resistance, as defined by the product of the coefficient of rolling resistance and the load on the wheel, or $f_{rr}=C_{rr}N$, it appears it is not applicable to the initiation of rolling. Rolling resistance is defined as the force that resists motion while rolling, as opposed to the force that resists the initiation of rolling motion. Testing for the coefficient of rolling resistance of tires is typically determined by a rotating"drum test". For a description and discussion of the test, see here: https://ascelibrary.org/doi/full/10.1061/%28ASCE%29TE.1943-5436.0000673
So it appears that the only external force acting on the car to oppose the  initiation of motion pushing is the static friction force that acts backwards on the wheel in response to the pushing force, and which enables the wheel to roll without slipping. Normally we consider that this force only has to overcome the rotational inertia of the wheel in order to achieve the necessary angular acceleration, on the assumption that wheel bearing friction is negligible. But John Rennie, in his answer to this post: How much force needed to push a car on neutral? suggested the reason it is harder to get a car rolling than it is to keep it rolling is "Car bearings are designed to maintain a thin film of oil when they're moving, but when the car is stationary this film collapses and the friction rises considerably."
In researching bearing friction I found his comment corroborated at this site: https://www.linearmotiontips.com/faq-what-is-stick-slip/#:~:text=Stick%2Dslip%2C%20also%20referred%20to,transition%20from%20standstill%20to%20motion. It makes the following statement:
"With recirculating bearings, the initiation of motion draws lubrication into the contact area between the two surfaces, reducing surface-to-surface contact and causing friction forces to drop. As the speed increases, the lubricant film increases and friction is further reduced."
In conclusion, it appears that, in addition to the tires rotational inertia, static friction between the tire and road also needs to overcome static bearing friction to initiate rolling motion. But that bearing friction drops once rotation occurs.
With regard to quantifying the initial resistance to motion John suggests, in his above answer, placing the car on a ramp and jacking the ramp up until the car starts rolling. The force acting down on the car when it starts rolling is then $mg\sin\theta$ where $\theta$ is the angle of the ramp with the horizontal. That force would then approximate the threshold force opposing the initiation of motion.
Hope this helps.
