I am modelling the crashing of a train with a deformable crumple zone on the front. I am assuming that the crumple zone deforms from the front back, so I am essentially modelling it as if the crumple zone passes through the wall so a cross section of the crumple zone is touching the wall at each point, and at each point the wall exerts a force of the train that depends of the area of the cross section of the crumple zone and the speed of the train at that point in time. My question is, how do I express the force on the train as a function of the speed of the train and the area of the cross section of the crumple zone that is touching the wall at a given time? I have a feeling that the force on the train should be proportional to the area of the cross section if you assume that the material that the crumple zone is made of is uniform, but I don't know how it should relate to the speed of the train.
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You can determine the average force experienced by the train during the collision by applying the work energy theorem which states that the net work done on an object equals its change in kinetic energy. Assuming the velocity of the train is $v$ just prior to impact, and that the train comes to a stop due to the collision, you would have
$$F_{ave}d=\frac{mv^2}{2}$$
where $d$ is basically the stopping distance of the train, which would be the length of the crumple zone. This average force does not depend on the cross sectional area of the crumple zone. The average stress or pressure on the front of the train would depend on that area. So you would need information on the cross sectional area to calculate the average stress.
Hope this helps.
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$\begingroup$ What if I want to calculate the instantaneous force on the train at a given point in time? $\endgroup$ Commented Nov 12, 2019 at 14:29
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$\begingroup$ @windingnumberone The work energy principle will only give you the average force over the stopping distance. In order to determine the instantaneous force on the train, you would need a way to develop a graph of force vs time over the stopping distance. $\endgroup$– Bob DCommented Nov 12, 2019 at 15:19
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$\begingroup$ I meant to ask if, given that I know the cross-sectional area of the crumple zone at every point, and I know the speed of the train at every point in time, and I know all the relevant properties of the material that makes up the crumple zone, can I determine the instantaneous force on the train? $\endgroup$ Commented Nov 12, 2019 at 15:34
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$\begingroup$ @windingnumberone I'll get back to you on this $\endgroup$– Bob DCommented Nov 12, 2019 at 15:53
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$\begingroup$ @windingnumberone What you might consider is modeling the crumpling of the front of the train as the compression of an ideal spring. Of course when material crumples energy is dissipated as heat, whereas when the spring is compressed energy is stored as elastic potential energy. But in either case, application of the work energy theorem would still apply. The benefit of using the spring model it gives you force varying linearly with the compression distance, something you can't conclude from the work energy theorem alone. $\endgroup$– Bob DCommented Nov 12, 2019 at 19:01