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Can anyone tell me how to go about this problem?

There is a square shaped laminate (polyethylene fiber and epoxy resin composite) that is constrained on one side. A gun is fired and the bullet hits the laminate. What kind of stresses result in the laminate? How should I go about their calculation? When I say stresses, I mean the stress tensor components $\sigma_{i,j}$. Also, can the strain tensor components $\epsilon_{i,j}$ be calculated?

Textbooks and other sources don't show or tell how the stress components are measured. Also, in the above problem, the mass and the velocity of the bullet are known.

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    $\begingroup$ Terminal ballistics might be of interest to you. $\endgroup$
    – grfrazee
    Apr 5 '16 at 12:55
  • $\begingroup$ @grfrazee- what about the stress and strain tensors? $\endgroup$
    – nihal
    Apr 5 '16 at 13:06
  • $\begingroup$ That I can't help you with, unfortunately. $\endgroup$
    – grfrazee
    Apr 5 '16 at 13:20
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Expanding on the comment by @grfrazee, it isn't as simple as with a static load. Several non-linear parameters change the response of the composite component.

  • Extremely high strain rate increases the observed brittleness and decreases the strain-at-failure and strain-at-rupture of the impacted materials.
  • There is a non-obvious limitation on penetration depth of projectiles in materials, though for a thin plate this is may be irrelevant.
  • Contact surfaces are constantly changing with time.
  • Velocity is changing with time, though this may be a trivial decrease depending on relative energy and thickness.
  • Force exerted is changing with time.
  • Momentum and energy of the bullet and composite are changing with time.
  • The composite may partially rupture, changing the local geometry, and its effects on stress and strain. The state of the geometry is also changing with time.

Assuming you've accounted for the non-linear change in material properties, the rest could be modeled carefully using a non-linear dynamic structural finite element simulation. However it must be stressed that you will almost certainly need a dynamic mesh and careful application of boundary conditions. Once the simulation is complete, the simulation software should be able to output the stress and strain of the material in each direction, from which you could rebuild the stress and strain tensors.

An alternative to this approach is to explore @grfrazee's answer and try to locate simpler models for deformation. It may be that a simple tensor model does not exist, in which case you would have to create your own from even simpler, 1D or 2D models.

An experimental alternative, as pointed out by @grfrazee, is to shoot things and see what happens. Doing so is a bit more tricky than that statement might imply. Some possibilities for tracking live material deformation characteristics would include attaching strain gages to the component in a carefully determined configuration, or use a laser interferometer, or even perform the test inside a real-time x-ray machine. For the latter test, you would probably want to do something like dope the fibers with osmium tetroxide or a similar compound containing a similarly high-atomic-number element for contrast between the composite components.

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  • $\begingroup$ Or, better yet, shoot stuff and see what happens! $\endgroup$
    – grfrazee
    Apr 5 '16 at 13:21
  • $\begingroup$ Good point. I'm mostly a theorist and modeller, so I forget the obvious experimental answers. I'll add that. $\endgroup$
    – wwarriner
    Apr 5 '16 at 13:49
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An additional complication is the behaviour of the bullet itself on impact, which is not only difficult to model but can vary a lot according to the construction of the bullet, it's geometry and mass, and the angle and energy of impact.

For example a low-velocity, unjacketed lead bullet might squash against the impact surface with little or no penetration, which is a reasonable approximation of an inelastic collision and so the best modelling approach might be to consider its momentum and here it is certainly possible, although not trivial, to construct fairly credible finite element model based on dynamic stress and strain. Indeed you may even be able to get meaningful order of magnitude numbers with manual calculations.

On the other hand a high-velocity, armour-piercing bullet might well pass through cleanly and transfer relatively little energy to the target.

A third case is that fragmentation or tumbling of the bullet or target. In high energy impacts you may even observe fluid-type behaviour.

In addition to the actual contact surface between the bullet and the target there will also be shock waves that propagate through the target, which can cause high stresses and failures (eg. spalling) and can eject material from the back side of the target even if the bullet itself does not penetrate it.

In general the big difficulty with modeling this sort of situation is that you are dealing with strain rates far outside the assumptions underlying the material properties and equations used for static and most dynamic modelling.

To put it another way firing a bullet at a target is a fundamentally different condition from pushing it through a target slowly regardless of how much force is applied. Indeed in practical impact testing it is more usual to think in terms of energy than stress and strain.

In fact when it gets right down to real world applications standards for body armour will usually specify resistance to specific rounds (in terms of caliber, propellant and bullet type) at a specific range. Simply because there isn't a measurable physical property which is generally descriptive of this sort of performance.

You also need to think carefully about what you are trying to determine. In the context of armour something which generates a wide cone of debris and fragmentation on impact may actually be far worse than just allowing the bullet to pass through intact.

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