I am working on a project where I must find the maximum principal stresses on an object subjected to impact loading condition. I know the velocity of the object (the one which will hit the test object) and its weight. My questions are:

  1. For analysis should I apply force or pressure to the area of impact?
  2. How do I calculate the force/pressure?

1 Answer 1


Your problem is ill-posed as stated.

However, a zeroth-order estimate of the impact force can be calculated using a one-dimensional balance of energy. The kinetic energy of the projectile is $$ \text{KE} = \tfrac{1}{2} m v^2 $$ where $m$ is the mass of the projectile and $v$ is its velocity.

The strain energy of the target, assuming linear elasticity, is $$ \text{U} = \tfrac{1}{2} \sigma \varepsilon V $$ where $\sigma$ is the stress in the target, $\varepsilon$ is the strain in the target, and $V$ is the volume of the target.

Equation the two, we have $$ m v^2 = \sigma \varepsilon V = \frac{F}{A} \varepsilon A L = F \varepsilon L $$ where $F$ is the impact force, $A$ is the impact area and $L$ is the length of the target normal to the impact direction.

The impact force estimate is, therefore, $$ F = \frac{m v^2}{\varepsilon L} = \frac{m v^2}{d} $$ where $d$ is the depth of penetration.

If, instead, you would like to use the pressure, $$ p = \sigma = \frac{m v^2}{\varepsilon A L} = \frac{m v^2}{A d} $$ Either can be used, but note that the force has to be applied over the contact area.

The problem is ill-posed because, even if you know $A$, you don't know $\varepsilon$ and $L$. People usually assume that $L$ is the thickness of the object (which may be infinite in some cases) and that $\varepsilon$ is small (e.g., 0.001). On the other hand, you may know a value of $d$ and get a reasonable estimate that way.

The only accurate way to compute the impact forces is to do a full-physics dynamic simulation.

  • $\begingroup$ You correctly said the OP's problem is ill-posed, but this "approximate" solution may be wrong by orders of magnitude for several reasons (too long for a comment, and I'm not going to post an answer that just explains why another answer is wrong, except for the last sentence: either the OP needs to explain the problem situation in more detail, or do a dynamic analysis to model the impact itself. $\endgroup$
    – alephzero
    Jun 20, 2018 at 8:23
  • $\begingroup$ @alephzero: As I said, it's a zeroth order approach that, strangely enough, is used widely by engineers. I don't have the time to get into details at this time and will greatly appreciate it if you could edit my answer to add your list of reasons why the accuracy can be poor. Or at least add the list as a set of comments that I can incorporate into the answer later. $\endgroup$ Jun 20, 2018 at 20:32
  • $\begingroup$ thank you for your answer, indeed the question is ill-informed, sorry about that. I am trying to do stress analysis where a moving object collides with a static object. working on a dynamic simulation would have been better but i need an answer by static analysis. I still do have a question, if I assume the impact time, will I be able to calculate the Force from it? Using this formula F=mv/t $\endgroup$ Jun 23, 2018 at 1:17
  • $\begingroup$ @raihannirvik: In many cases the failure of the target is caused not by the impact but by internal tension created by the interaction of elasto-plastic waves. So a simplistic approach will give you some number which may have no relationship with reality. There are no shortcuts. To get a feel for the problem, even when both bodies are rigid, see ruina.tam.cornell.edu/research/topics/collision_mechanics/… . $\endgroup$ Jun 23, 2018 at 22:38

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