Another way of looking at this is:

Relative to the biker who is travelling on the bike, the wheel is not rotating around its contact point on the road, but its axle, as one would intuitively expect. Therefore while the top of the tyre is doing twice the speed of the vehicle relative to someone standing stationary on the side of the road, the top of the tyre is only doing the same speed as the vehicle with respect to the biker or the bike.

As far as force and stress goes:

Lets say a 2kg lump of rubber is doing 180km/h

And it hit something and stops completely in say 100ms (complete guess) then you can work out the deceleration needed to do that from

V2 = V1 + a T

From that you can work out the force exerted by the rubber on what ever it hits which is simple F= m a

So that would give you

a = 180 /(3.6 *0.1) = 500 m/s2

thus force = 500 * 2 = 1000N which equivalent to roughly 100kg

If the rubber happens to hit edge-on then that force is absorbed by a small area lets guess and say 2 cm x 1 cm = .0002 m2

So the stress exerted would be 1000/0.0002 = 5 MPa which should not cause steel to fail

Just a quick thumbsuck

Of course it all depends on lots of different things (eg the calc's above assume a loose piece of rubber flying through the air that stops when it hits something, not a flap of rubber hanging off the wheel, that is being driven by the bike's engine which obviously means you need to include the bike's torque and momentum into the calc)

Which ever way you look at it, this could get very nasty for anyone nearby those flying bits of rubber.