Introduction

In the first part of this dissertation, we considered how the kinetic energy of an incoming mass is imparted and distributed to the shock mitigating foam. We assumed that the footprint of the incoming mass, that is, its impacting area, was the same size as the foam slab. But, what happens when the incoming mass footprint is substantially smaller? It would seem that this situation occurs more frequently than the latter. First, we need to mentally separate the behavior of the foam under the inbound mass from that everywhere else. What we would like to do, but cannot, is to assume the foam under the inbound mass to be a series of isolated springs and dashpots, totally unconnected to each other and to the rest of the foam structure. If we could assume this, then the analysis would be easy. The foam under the inbound mass would behave exactly as described in Part 1. And the foam away from the inbound mass would be totally unaffected by the presence of the inbound mass. At best, however, we can assume that the foam at some far distance, albeit undefined here, is unaffected by the presence of the inbound mass. And that the foam immediately under the geometric center of the inbound mass area is described by the discourse in Part 1. And, unfortunately, everywhere else, the presence of the inbound mass affects and is affected by the foam cell structure. So, let’s see how to go about analyzing this problem.

The Net, Envisioned

Consider an elastic fishing net with its edges rigidly fastened. Call it an elastic mesh with, say, square elements. An inbound mass, impacting the mesh, will act to stretch all elements of the net, with the possible exception of the area directly under the mass. The mesh free of the mass contact area is stretched in what is known as “plane strain” fashion. That is, if you look perpendicularly at the surface of the mesh, all the elements will still look rectangular. But from the side, the elements will stretch between the edges and the mass contact area. (The arithmetic for plane strain deformation is given in J.L. Throne, Technology of Thermoforming, Hanser/Gardner, 1996.) In other words, the mesh is deformed everywhere by the incoming mass impact. The elastic mesh material response is given not in terms of compressive stress-strain but in terms of tensile stress-strain. Again, the higher the modulus of the mesh material becomes, the greater the material resistance to incoming kinetic energy and the lower the deformation of the elastic mesh will be when the incoming mass kinetic energy has been completely converted to elastic potential energy in the mesh.

Extending the Net Concept

Certainly if we replace the elastic mesh with a monolithic very thin rubber sheet that has been inscribed with square elements, we should not expect to see any difference between the way in which the sheet and the elastic mesh deforms. Now, let us increase the sheet thickness. The tensile load bearing characteristic of a polymer, or any material, for that matter, is proportional to the cross-sectional area of the specimen. In other words, tensile stress is the applied force per unit area. As we increase the thickness, we increase the “unit area,” as it were. Therefore, the thicker the rubber sheet becomes, the lower the deformation of the sheet will be under the same inbound kinetic energy. Hold this thought as we continue to explore the nature of the rubbery sheet.

Now, consider the sheet to be foamed, very lightly, at first. As seen earlier, foaming reduces the effective modulus. Since the tensile modulus is one of the design criteria for the material resistance to tensile load, one would expect the tensile modulus of a foam to decrease in proportion to the square of the density, as with the compressive modulus. And it does. Therefore, foaming increases the extent of deflection under constant inbound kinetic energy

What about Thick, Highly Foamed Sheet?

So far, we have considered only the tensile aspect of the sheet. And that’s because we assumed the sheet to be thin and only lightly foamed. In actuality, the mechanical behavior of thin, low-density foam, such as polyethylene furniture wrap, is very well characterized by the plane strain model. It’s when the thickness becomes large, say, greater than about 3 mm, that a second mechanism begins to influence the mechanical response to the applied load. Why is this?

First, the foam directly under the applied load is compressed in the fashion described in Part 1. Recall that the polymer properties become very important if the applied stress collapses the foam to an extent greater than about 65%. If the foam is thick enough, compressive levels do not exceed about 65% and the rules described in Part 1 apply, but again, only directly under the applied load. Then, at the edge of the applied load, the foam is undergoing a transition between compression and tension. The edge of the applied load represents one of the “fastening points” for tension in the foam. The other is the far edge of the foam. We’ll discuss the nature of fastening at the far edge of the foam momentarily. The tension is applied here to the top of the foam (assuming that the incoming load is impacting the foam from above). The tension at the other end is usually applied to the underside of the foam (assuming again that the other end is, in fact, fastened to something). As the foam is stretched under applied load, the tensile load is therefore applied through the foam, from the top edge on one “end” to the bottom edge on the other “end.” This causes a shearing effect through the foam. In other words, in order to accommodate the applied load, the foam cell walls are not compressed or stretched, but are bent by differential forces parallel to the foam surface. Consider the simple example of pushing on the top of a cleaning sponge while the underside is against a solid surface. This is the shearing action.

So, there we have it. Compression of the foam under the incoming load. Tensile stretching of the foam surfaces at or near the edges. And shearing forces on the foam between. Oh, and there is one more factor that we need to include. We are assuming at this point that the foam underside is unencumbered and can freely. Later we consider what happens when the foam underside is rigidly in contact with a solid substrate.

So, how do we characterize the foam response to incoming load? Again, we look at potential elastic and dissipative modes in the foam. These are the same as described in Part 1. Energy is converted to heat by cell compression and by internal heating of the polymer. And energy is elastically stored in the spring action of unfolded, unbuckled cell walls and in the P-V-T characteristics of the cell gas. However, we need to keep certain aspects of the stretching behavior of the foam in mind. For example, uniaxially stretching of foam does not necessarily decrease cell volume, and may, in fact, increase it. Therefore the P-V-T cell gas effect may be absent. Now tensile distortion, buckling and folding of the cell walls do take place and as a result, we should expect some internal heating and energy dissipation. Shearing of the foam cells again may not effect individual cell volume and therefore P-V-T effects may not be significant. But, again, shear distortion, buckling and folding of cell walls do take place and again, some internal heating and energy dissipation does take place. However, from all but the highest stretching cases, the tensile and shear dissipative characteristics should be small. This means that for all intents, the net/mesh/membrane characteristic of foam sheet should be more elastic than dissipative. In short, one should expect the incoming mass to rebound from a foam sheet as fully as if it were solid.

The Role of the Substrate

Aha, you say, but what happens if the foam sheet is rigidly supported? Or if it’s really, really thick? First, it is apparent that the underside of the foam is no longer under tension and it therefore represents the neutral tensile axis plane. The top surface is still divided into the region of compression directly under the incoming load and the tensile region stretching from the edge of the load to the edge of the foam, which we are still assuming to be rigidly affixed (to something). Unlike the earlier case, it is uncertain whether plane strain can be assumed. In fact, from the simple observation described earlier for the net/mesh/membrane deformation, it appears that rectangular elements scribed on the undeformed foam surface show substantially more distortion in the region or the load than in regions far from the load. One apparent reason for this must focus on the compressible nature of the foam itself, particularly when rigidified from the underside, either by a rigid support or simply because the foam is very thick. In other words, the tensile stretching of the foam at the upper surface is compromised by compression of the foam beneath.

This complex interaction makes detailed technical analysis difficult. But a heuristic analysis goes something like this. The maximum amount of energy dissipation in the foam is determined by assuming all the foam is compressed to the same level as that directly under the incoming load. In other words, consider the mass footprint to be the same size as the foam sheet. The minimum amount of energy dissipation assumes that the underside of the foam is unsupported, as detailed above. In fact, in the limit, one can assume that the mass footprint is negligible compared with the surface area of the now-very thin sheet. As a result, this yields the maximum stored kinetic energy in the foam. The real case is somewhere in between. Certainly as the mass footprint increases in size as compared to the foam surface area, the stored kinetic energy in the foam will decrease and the foam will behave more like a uniformly compressed foam.

What about Edge Supports?

In some cases, the foam edges are not adhered or fastened to other rigid structures. In other cases, the foam edges are free to distort under incoming loads. How does edge support effect the performance of the foam under incoming load?

Well, in the heuristic analyses considered so far, we’ve assumed that the edges of the foam have been rigidly fixed. As a result, for the net/mesh/membrane stretching under load, we noted that the membrane was stretched between two fixed points, the edge of the foam and the edge of the incoming load. Certainly if the edge of the foam is not fixed, neither the top of the foam or the underside of the foam can be under tension. As a result, the foam is simply folded inward as the load impacts it. Obviously this is a useless situation. But what about rigidly supported or very thick foam?

The top surface of the foam is usually connected to the underside by cell walls. As the top begins to follow the impacting load downward, the tension is applied first to the top surface or skin, then to the cell walls that make up the cut edges that are at right angles to the top surface. The tensile strength of the cut edge is related to the modulus of the polymer and the amount of polymer in the cut edge. The tensile strength of foam polymers is usually assumed to be proportional to the density ratio to the 3/2-power (not the 2-power for elastic modulus). Now since the top surface of a foam usually has a skin, the tensile strength of the surface is usually higher than that of the cut edges. This means that the cut edges will stretch at least as much if not more than the top surface. The result of the stretching is a combination of shear and compression on the foam at or near the top layer and the corner of the foam slab. Again the general effect of compression is one of dissipation rather than elasticity. As a result, energy absorption of the inbound mass should always be greater with foams with free surfaces than with foams with fixed surfaces.

There is one more aspect of shock mitigation to consider. Consider the fact that at high impact energy, the inbound load generates an energy wave that propagates through the foam at a sonic velocity. Although the wave is dissipated to some degree by the non-homogeneous nature of the structure, viz, cells and cell walls, it propagates to the reverse side of the foam slab, where it is reflected back into the foam. The returning wave acts as a reinforcing agent to the foam cell structure itself, thereby momentarily increasing the foam resistance to the inbound load. It also results in an enhanced localized heating and consequently localized failure of the cell wall. In higher density foams, the effect is dramatic, with the foam exhibiting a remarkable ductile-to-brittle failure layer that clearly demarks the specific plane where the sonic wave is locally reinforced. Furthermore, this sonic reinforcement is used to an advantage in military structures such as tanks, where laminate structures of differing sonic velocities are used to attenuate incoming small-arms fire.

Shock Mitigation for Open-Celled Foams

It is well-documented that the stress-strain curve of an open-celled foam is very similar to that for a closed-cell foam, particularly at high impacting speeds. Why is this? After all, there should be no closed cell gas compressibility factor, right? This assumption depends entirely on the degree of “open-celledness.” Consider the foam to generally be closed except for a hole in each of the cell walls. When the cell is compressed by the incoming mass, the cell gas must flow through the holes to the surrounding cells. Of course if all the cells in a given area are being compressed to the same degree, the cell gas pressure in each cell equals that in the neighboring cells and so there is no flow. This would generally be the case if the mass footprint is the same as the area of the foam. If the mass footprint is much smaller than the area of the foam, then cell gas can flow from the more compressed region to the less compressed region. It is surprising that cell gas flow under this type of impact is perpendicular to the impacting foam surface and in a direction opposite that of the incoming mass. To confirm this, you can sprinkle a fine film of talc on the surface of the foam prior to impact. Upon impact, the ejected cell gas will blow talc from the surface at a short distance from the edges of the impacting mass.

The rate of cell gas flow depends entirely on the size of the hole in the cell wall. If the hole is very small, the cell gas flow is very low. As a result, the cell gas is compressed essentially as if there are no holes! Even if the holes are relatively large, there should still be some measure of cell gas compression. As discussed earlier, any cell gas compression provides kinetic energy dissipation. And of course, the internal energy dissipation in the polymer cell walls is the same, whether or not the cells are open. In other words, as the holes in the cell walls become larger and larger, the adiabatic energy dissipation effect of the cell gas slowly diminishes. The true effect of open cell foams is noted when the mass becomes static, since the auxiliary cell gas “pillow” support is absent.

Summary

So, what have we learned about shock mitigating foams? The compressive stress-strain curve teaches us the maximum amount of energy that can be transferred to the foam from the inbound mass. The extent of energy dissipation depends on the internal energy of the polymer as it is bend, folded or buckled by the force of the incoming mass, and on the adiabatically compressible nature of the cell gas. For masses with footprints smaller than the foam, the surface of the foam that is impacted by the mass is put under tension. Energy absorption depends on the thickness of the foam, whether the foam is supported on the reverse side, and whether the foam edge is fixed or free. And open cell foams show energy absorption curves that are similar to those of closed cell foams.

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