Quote by skazis
I got confused thinking about cause and effect. If force is applied to a car due to collision, it deforms car. The longer is the time of a collision, the smaller is average force applied. Longer time is achieved by crumple zone deformation, which is affected by force. I find here circular argumentation. Could someone clarify for me this?

Yes, the deformation depends on the force and the force depends on the deformation. Many problems are like this in structural engineering  even static problems can usually only be solved if one knows the properties of the materials involved. It would be particularly difficult to find a closedform solution to your crumple zone problem since it has the additional complexities of being 1) dynamic and 2) inelastic. That's why such problems are not solved analytically. Rather, they are solved using methods that march forward in time in incremental steps, numerically satisfying the equation of motion along with the applicable forcedeformation relationships (dependent on material properties) at each increment in time. Such methods can be terminated once a steadystate is reached (i.e. the collision is complete), for example.
As mentioned by jambaugh, the acceleration of the driver would be the key parameter. One would seek to design the crumple zone so as to minimize the acceleration that the driver experiences, at the expense of the "headlights" and hood, etc.
Quote by skazis
Energy absorbed by car equals its kinetic energy (assuming headon collision and that both cars stop after), so work done by average force on the car is
[itex]F_{aver}d = \frac{1}{2}mv^2[/itex]
where d is the deformation of the car.
From 2nd Newton's law the average force is equal to
[itex]F_{aver} = m\frac{\Delta v}{\Delta t}[/itex]
Are these 2 average forces the same?

Looks fine to me