Impact, momentum, demand and capacity

[Jpcgandre]

I'm struggling with trying to find how conceptually need demand and capacity to be conciliated during impact.

In particular, I find two different formulas in papers and websites dealing with impact.
In all cases, they state that there are two forces acting on objects on impact, where they disagree is what are those forces.

One set states that these are gravity and the impact force (see https://www.wired.com/2014/07/how-do-you-estimate-impact-force/, https://www.wired.com/2011/09/can-bird-poop-crack-a-windshield/ and https://www.wired.com/2009/01/im-iron-man-no-im-not/, http://www.civil.northwestern.edu/people/bazant/PDFs/Papers/476.pdf),
whereas the second set states that these are the capacity force ($F_{capacity}$) and the impact force ($F_{i}$) (see https://www.researchgate.net/publication/358267183_The_Total_Collapse_of_the_Twin_Towers_What_It_Would_Have_Taken_to_Prevent_It_Once_Collapse_Was_Initiated):

$$ F_{net}dt = (F_{i} - mg)dt = Δp $$
$$ F_{net}dt = (F_{i} - F_{capacity})dt = Δp $$

There is an obvious difference: one uses gravity whereas the other uses the capacity force (which may be multiple times larger than the former). Not only this but fundamentally in the former set there's no explicit way to account for the (resistance) capacity to determine the impact force.

Although the second set explicitly accounts for the (resistance) capacity I have doubts this is correct since the capacity will limit the demand (ie the impact force) simply because the demand cannot be larger than the capacity (ie if the capacity isn't larger than the impact force then the impact force will be capped by the capacity value). So it appears to me that determining the demand from the impulse=change of momentum should not involve the capacity.

Having said that, I'm struggling with the aftermath of the latter hypothesis. If capacity does not appear in the demand equation, then from the velocities profile (obtained from conservation of momentum) one can get the displacement increments. At the same time from the impulse=change of momentum, one can get the impact force.

I'm struggling to see how these pair of displacement increments and impact forces can be consistent with any arbitrary material constitutive model, ie if I plug in the displacement increment in this model I bet I won't get the impact force that I derived from the impulse=change of momentum. The same applies to the displacement increment: if I plug in the impact force to the model. So it appears that demand should indeed be determined considering explicitly the capacity. But how?
Thank you.

A follow-up question.
Scenario: Object A and object B are aligned vertically and the former is positioned below B. Object B is released and begins a course of collision with object A, which is rigidly attached to Earth.

If the capacity of object A is lower than the demand, then the velocity during impact of the mass formed by object B and the fractured part of object A will be larger than the case where the capacity of object A is higher than the demand. This can be obtained from the conservation of momentum by considering that the fractured part of object A will concentrate most of the after impact velocity while the remaining part of object A will have a much smaller velocity.

Is this reasoning OK?

 

[berkeman]

Please post links to what you have been reading about this question (required), and please use LaTeX to post your thoughts on the math equations you are asking about (see the LaTeX Guide link in the lower left of the Edit window). Thanks.

 

[Jpcgandre]

Hi Berkeman, I gave it a shot, but not sure if Latex is showing up correctly... I see the "code" and not the intended result.
Added links.

 

[berkeman]

Thanks for that, @Jpcgandre

Thread is closed temporarily for Moderation...

 

[berkeman]

After a Mentor discussion, the thread is provisionally reopened.

 

[Jpcgandre]

Thanks, having though a litlle bit about this I think Eq. (2) in my original post is wrong.
For the scenario I defined at the end of my original post:
Using Eq. (1), one gets from the $Impulse = Change of momentum$ theorem:

1) Before the pressure wave reaches Earth:

Net force on object B:
$$F_{i} - m_{B}*g$$
Impulse = Change of momentum
$$F_{i} - m_{B}*g = \frac {d(m_{B}*v_{B})} {dt}$$

Net force on object A:
$$-F_{i} - m_{A}*g$$
Impulse = Change of momentum
$$-F_{i} - m_{A}*g = \frac {d(m_{A}*v_{A})} {dt}$$

Applying the conservation of total momentum:
$$m_{A}*v_{A}|t_{i} + m_{B}*v_{B}|t_{i}= m_{A}*v_{A}|t_{i+1} + m_{B}*v_{B}|t_{i+1}$$
$$-(m_{A}*v_{A}|t_{i+1} - m_{A}*v_{A}|t_{i}) = m_{B}*v_{B}|t_{i+1} - m_{B}*v_{B}|t_{i}$$
So:
$$-(-F_{i} - m_{A}*g)*dt = (F_{i} - m_{B}*g)*dt$$
$$m_{A}*g = -m_{B}*g$$
Which doesn't make sense so I think that before the pressure wave reaches Earth the total momentum is not
conserved!

After the pressure wave reaches Earth, total momentum is conserved by the extra term that appears in the Fnet of object A account for the reaction on the ground which is equal to $m_{A}*g + m_{B}*g$ as predicted by the conservation of total momentum.

Of course $F_{i}$ is given by the minimum of the resistances of objects A and B.
Is the above correct?