Low-density thermoplastic foams are frequently used as energy-absorbing foams. That is, kinetic energy from an incoming mass is mostly dissipated in the foam, resulting in little, if any, throwback or reverse propulsion of the mass from the foam. Typically, energy absorption is described in terms of the area under the foam stress-strain curve. The typical stress-strain curve for a low-density foam is depicted as having three general components:
At low compression, say, less than about 5%, the foam acts as a Hookean elastic spring. That is, the extent of compression, ε, is directly proportional to the applied stress, σ , with the proportionality, Ef, being the modulus of the foam: σ = Ef ε.
For essentially all foams, the modulus of the foam is directly related to the modulus of the polymer, Ep, and the ratio of squares of the foam density, Ep, and polymer density, ρP: Ef = Ep (ρf / ρo)².
At high compression, say, greater than about 65%, the foam cells have collapsed to such an extent that the strut supports are interfering with one another. Now the resistance to further compression is mostly related to the compressive character of the polymer itself and not so much to the compressive nature of the foam.
In between, from about 5% to about 65% compression, the compressive nature of the foam is given as a combination of the bending and/or buckling characteristics of the strut supports and the compressive nature of the gas in the foam cells.
First, consider that the impact loading does not compress the foam more than 65% or so, or into the region where the stress-strain curve begins to turn up. Then, consider that compression is not the idealized uniform birdcage strut bending pictured in most textbooks on the subject. Instead, compression usually occurs only in the impact region of the incoming mass and then usually in a plane essentially parallel to the foam surface in that region. The latter is sometimes referred to as “conjugated cell collapse.” Planar compression usually begins with bending and folding of the thinner and therefore weaker cell walls. If there is a preponderance of weak cells in a given plane, the initial compression will be centered there. Elsewhere, the foam cells will not seem compressed at all. Basically, the struts will bend and fold until further compression requires compression of the polymer itself. Then the incoming mass will force weaker cell walls in other portions of the foam to begin to collapse.
Of course, as the mass continues into the foam, conjugated cell collapse will occur in more and more planes parallel to the foam surface. The general shape of the stress-strain curve in this region tends to be nearly linear with the load-bearing characteristics of the foam gradually increasing as the foam is compressed. The region from about 25% compression to about 65% compression is sometimes referred to as the “comfort zone,” and the shape of the stress-strain curve in this region is most important in seating and bedding.
It is somewhat surprising to learn that stress-strain curves for most polymeric foams have very similar shape, with the primary difference being the effect of foam density on the foam resistance to incoming mass. Thermoplastic foams have stress-strain curves that are very similar to those of thermosetting foams, such as urethanes and silicones. And open-celled foams have stress-strain curves that are similar but not identical to closed-cell foams. The energy absorption of open-celled foams is the subject of a subsequent paper in this area.
Energy Absorption in Foams
So the question is: How is the energy dissipated from the incoming mass? Consider two scenarios. In the first, the area or footprint of the mass is essentially the same as the area of the foam. This allows the foam to be compressed essentially uniformly across its surface. In the second, the footprint of the mass is substantially smaller than the area of the foam. Therefore the foam is compressed locally under the mass while the foam at some distance from the mass is essentially uncompressed. Then consider the nature of the impact. Again there are two scenarios. In the first, the impact is a singular event. That is, the mass impacts the foam once, and comes to rest atop the foam. In the second, the impact is part of a multiple event. In other words, the mass impacts the foam over and over. For purpose of this discussion, consider that the repetition is periodic and that the amount of energy to be dissipated is the same each time. In other words, consider vibrational impact as an example of this case. Of course, there are other types of multiple impacts, such as a mass that rebounds, then re-impacts, only to rebound again. And again and again.
Finally consider the classic case of an incoming mass impacting the foam surface, compressing the foam, and coming to rest atop the foam without rebounding free of the surface.
And then consider the general energy absorption picture insofar as the foam is concerned. As noted above, the key to foam energy absorption is the stress-strain curve. As the incoming mass impacts the foam surface, compression begins and the foam resistance is dictated by following up the stress-strain curve. The foam continues to compress until the mass velocity reaches zero. In technical terms, the foam has compressed to ε dynamic. The extent of stress at this condition is σ dynamic. In other words, the kinetic energy of the mass has been converted into heat and potential energy in the foam.
This is obviously a metastable state, since the foam has substantial elastic character and the “foam spring has been compressed,” so to speak. Now the elastic character of the foam pushes against the now-stationary mass, reversing its direction. Consider the following two scenarios.
If the foam were completely elastic, the mass would be propelled or rebounded from the foam with the same kinetic energy it had when it first contacted the foam. The foam, in essence, would resemble a bed of perfect springs. On the other hand, if the foam were completely dissipative, the foam would remain compressed by the mass. The foam, in essence, would resemble a bed of Play-DohTM.
Quite obviously, no foam is either a bed of perfect springs or perfect Play-Doh. Some, hopefully, most of the incoming kinetic energy is dissipated but typically, some of the kinetic energy is returned to the mass. Certainly the greater the elastic modulus of the foam, the greater the “spring effect” of the foam will be. Again, since the elastic modulus of a foam is the product of the modulus of the polymer and the square of the foam density, the “spring effect” is diminished by using a low-modulus polymer such as EVA or LDPE, or by foaming the polymer to a lowered density.
In addition to the elastic effect of the foam, the compressibility of the gas in the cell structure must also be considered. Assume for the moment that the foam is closed cell. And assume that the incoming mass has already buckled or bent the supporting struts. The buckling or bending of the supporting struts acts to reduce the individual cell volume. Since the air in the individual cells is compressible, the result is an increase in cell gas pressure. If the compression is isothermal, that is, the gas and polymer temperatures remain constant, the cell gas pressure is inversely proportional to the cell volume: Pl / Po – (Vl / Vo)-1.
Now, as the cell is compressed, the internal pressure increases, thus offering resistance to the incoming mass. This resistance is in addition to the polymer elastic resistance. Extensive technical studies have shown that the easiest way to consider the effects of both cell gas pressure and polymer elasticity is to simply add the effects together:
Foam resistance = Polymer elasticity + cell gas pressure
Again, polymer elasticity dominates the early portion of compression whereas cell gas pressure influences the later portion of compression. Now technically, polymer elasticity is reversible and cell gas compression can be reversible. The above discussion assumes that the polymer and gas temperatures remain constant. In certain cases, such as very high-speed impact, the impacting energy is dissipated as heat, raising both polymer and gas temperatures. The gas temperature is raised adiabatically, and the cell gas pressure is no longer inversely proportional to the cell volume to the first power. More importantly, since energy is dissipated as heat, the amount of energy remaining in the foam to rebound the mass is diminished. And the more energy that is dissipated as heat, the less energy remains to rebound the mass. In other words, the key feature to achieving Play-Doh-like dissipation is heat absorption by the gas and the polymer.
So, how does the polymer absorb heat? Two general ways. First, through energy interchange with the adiabatically compressed cell gas. It is admitted, however, that heat transfer between a quiescent gas and a polymer is notoriously poor. The second way is through localized bending or buckling of the polymer struts. Recall the old experiment of rapidly stretching a rubber band and quickly placing it against your lip to feel the heat? This is a similar situation. Molecular stretching due to folding, bending, or buckling results in internal heat generation in the polymer. Heat dissipation in polymers is extremely poor, since the primary mode of energy transfer is conduction from the folding or bending source. This means that the bent or folded area remains “hot” for awhile after impact. This also means that the elastic characteristic of the polymer can be mitigated by this.
Consider what the local heating does to the polymer morphology. Low-density foam cell walls are films that are biaxially oriented as much as four to five times (4X to 5X). When the film is bent elastically, it recovers elastically. However, when the film is heated locally, the film is “de-oriented” locally. If the energy absorbed is sufficiently high, the film may actually melt locally, then cool into a new, unoriented state. And for crystalline polymers, the new state may be a different crystalline morphology. This localized softening or melting effect happens primarily at compression levels far beyond the 5% elastic condition discussed earlier. And this effect is also one of the primary causes of “compression set,” where the foam simply does not recover its full thickness after the mass has been removed.
Furthermore, if the foam is subjected to repeated impacts, as in vibrational applications, the local heat is never fully dissipated and thermal damage and de-orientation can continue throughout the foam cell structure.
[It should be noted parenthetically, however, that insofar as I can tell, the type of permanent damage described above has not been observed or photographed. For those seeking confirmation of this type of thermal damage, I suggest careful 30X examination of singularly impacted "crystal" or unmodified polystyrene foam cell walls, using crossed polarizers. Localized softening and recooling should yield much lower strain fields than what can be observed in the surrounding biaxially oriented film.]
Energy Balance in Shock Mitigation
Okay, now the foam is compressed to its maximum. The total kinetic energy of the incoming mass is now zero, with the energy having been transferred to the foam. Part of the energy has been converted into heat, through adiabatic compression of the cell gas and internal heating of the polymer at the to-be-permanent folds, bends or buckles in the cell walls. This energy is dissipative, or unrecoverable. Whatever energy is left must be elastic or completely recoverable. This energy, such as the springiness of the elastically bent cell walls and the compressibility of the cell gas, pushes back against the now-still mass. Certainly, the mass itself provides a static stress against the foam, given by the weight of the mass divided by its footprint area. Thus, the energy balance for the inbound mass is given as:
Incoming kinetic energy of mass = Heat dissipated to gas and polymer + Static stress of the mass + Kinetic energy of rebounded mass.
It is apparent that if the mass is not rebounded from the foam, viz, the Play-Doh example, all the inbound energy must be converted to dissipative heat or must act to simply support the static mass. Therefore, the rebound or kinetic energy imparted to the mass must be related to the elastic potential energy stored in the foam, given as: Rebound energy = Elasticity of cell walls + Differential gas pressure.
Nature of Cell Gas Compression
Before continuing, consider the importance of cell gas pressure on energy absorption. It is well known that the cell gas pressure of fresh foam at room temperature is sub-atmospheric. This is because foam stops expanding when the polymer and cell gas is quite warm, near the polymer melting or glass transition temperature and the cell gas pressure is about one atmosphere. And as the foam cools, the cell gas pressure drops below atmospheric. There follows a gas diffusional interchange, with air diffusing into the cell structure while the foaming gas diffuses out. Typically, cell gas pressures of mature foams are greater than one atmosphere. Certainly cell gas pressures greater than one atmosphere will support the static load better than cell gas pressures less than one atmosphere.
So far, we’ve considered energy distribution of an inbound mass that occupies the same footprint as the foam. In the next section, we consider what might happen if the foam is substantially larger than the footprint of the inbound mass.