Physical material properties of foam - concepts and equations re: compression

Mattress Talk General Mattresses Questions
1

I have been struggling to find resources re: engineering of bedding, specifically regarding latex, and foams writ large. I don’t fully understand why we can’t treat foam layers as springs. I.e. (Hookean) springs in series have an equivalent spring rate = 1 / Keq = 1/k1 + 1/k2 + … , such that combining springs in series always yields a lower equivalent spring rate and combined springs in series will compress equivalently regardless of which end they are compressed from. Foams seem to behave differently. I.e. firm foam over soft foam feels differently if the firm side or the soft side is laid on (unlike as, mentioned, a coil spring system, which can be flipped and would feel equivalent).

My best guess is that this is a result of the non-Hookean (non linear stress-strain response) of foams. It seems from looking at some papers and articles that foams tend to have 3 compression phases: Hookean, non-Hookean (or at least less-Hookean/lower K, i.e. lesser slope, curving upwards), and then Hookean again, resuming a K/slope similar to phase 1. I am wondering if the surface tension of the foam material plays a role in limiting point elasticity and thereby force transfer to lower layers (if you cut the foam into a fine grid, would it behave more like an ideal spring?). It seems that the compression modulus of foams tops out at values (2.6) that are just equivalent to a Hookean spring, meaning most foams, in the 25% to 65% compression range where this is measured, are regressive in spring rate and less supportive than a helical spring of equivalent initial K, and the most supportive foams only match a helical spring.

These are basic concepts and yet it’s been surprisingly hard to find info on the ideas and equations that would predict the physical properties (ILD/resilience/hysteresis/etc.) of combinations of foams of known properties in a known arrangement. I am most concerned with understanding the effective ILD/Keq at various depths, i.e. the stress-strain curve. The actual maths are probably PDE’s or some complicated series expansion but I’m hoping there is a simplified version for a basic understanding, as with the high school physics version of the spring equation I gave above. If an engineer is designing a bed, they must have some equation or guideline by which to model the result of combining different foams based on the order, thickness, and ILD (or other properties) of each layer, and I want to know what it is. Otherwise, it is impossible to design a mattress save for trial and error.

I am hoping that the site moderators or someone else knowledgeable in bedding or engineering can explain the math/physics, or point me to resources such as papers, trade organizations, textbooks, engineer friends, etc., so I can find my answer. Thanks!

2

BMG:

In case you haven’t come across it yet, you may enjoy diving into this Journal of Solid Mechanics paper, Modeling of Compression Curves of Flexible Polyurethane Foam with Variable Density, Chemical Formulations and Strain Rates, and specifically the references contained within. Lots of good stuff there that may help point you in the direction you desire.

3

[quote=“BMG post=91460”]I have been struggling to find resources re: engineering of bedding, specifically regarding latex, and foams writ large. I don’t fully understand why we can’t treat foam layers as springs. I.e. (Hookean) springs in series have an equivalent spring rate = 1 / Keq = 1/k1 + 1/k2 + … , such that combining springs in series always yields a lower equivalent spring rate and combined springs in series will compress equivalently regardless of which end they are compressed from. Foams seem to behave differently. I.e. firm foam over soft foam feels differently if the firm side or the soft side is laid on (unlike as, mentioned, a coil spring system, which can be flipped and would feel equivalent).

As you guessed the problem is that although mechanical springs respond quite lineraly unfortunately foams (poly, latex, memory, etc) do not.

My best guess is that this is a result of the non-Hookean (non linear stress-strain response) of foams. It seems from looking at some papers and articles that foams tend to have 3 compression phases: Hookean, non-Hookean (or at least less-Hookean/lower K, i.e. lesser slope, curving upwards), and then Hookean again, resuming a K/slope similar to phase 1. I am wondering if the surface tension of the foam material plays a role in limiting point elasticity and thereby force transfer to lower layers (if you cut the foam into a fine grid, would it behave more like an ideal spring?). It seems that the compression modulus of foams tops out at values (2.6) that are just equivalent to a Hookean spring, meaning most foams, in the 25% to 65% compression range where this is measured, are regressive in spring rate and less supportive than a helical spring of equivalent initial K, and the most supportive foams only match a helical spring.

Yes, tension is a big part of explaining why the foam layers don’t act like normal springs. It also explains why thinner layers act differently from thicker layers of the same firmness (ILD).

These are basic concepts and yet it’s been surprisingly hard to find info on the ideas and equations that would predict the physical properties (ILD/resilience/hysteresis/etc.) of combinations of foams of known properties in a known arrangement. I am most concerned with understanding the effective ILD/Keq at various depths, i.e. the stress-strain curve. The actual maths are probably PDE’s or some complicated series expansion but I’m hoping there is a simplified version for a basic understanding, as with the high school physics version of the spring equation I gave above. If an engineer is designing a bed, they must have some equation or guideline by which to model the result of combining different foams based on the order, thickness, and ILD (or other properties) of each layer, and I want to know what it is. Otherwise, it is impossible to design a mattress save for trial and error.

You’re not going to find much of this out on the internet. Mattress engineering is extremely specialized and much of the knowledge is highly proprietary since it’s a competitive advantage in the marketplace (many reasons for this). I’ve done a bunch of graphing of latex foam response and posted them on here but much of the knowledge needs to basically be reverse engineered via empirical testing.

I am hoping that the site moderators or someone else knowledgeable in bedding or engineering can explain the math/physics, or point me to resources such as papers, trade organizations, textbooks, engineer friends, etc., so I can find my answer. Thanks![/quote]

Comments in red above.

4

Thanks, that’s a thoughtful contribution. I saw that paper a few days ago when I was googling late at night, but I need to look at it more closely. The idea of the Maxwell arm and the polymeric mathematical model is interesting and may set me onto some good concepts, but I believe that paper dealt with single layers rather than the combination of layers I am after. Section 4 is “POLYMERIC MAHEMATICAL MODEL (PFM) FOR FOAM MATERIALS WITH SINGLE LAYER.” As I said though, I could look at it more closely (I am however leery of a paper that doesn’t spell “mathematical” correctly - must not have had a thorough peer review, granted though that English may not be their first language).

5

Thanks for your feedback. I would have thought equations for stress-strain curves of foam combinations would be common knowledge in physics/engineering and something I could find in a text book, but it’s looking like that may not be the case. I would think that if you know the curve for each layer, then it wouldn’t be that hard to model how they act in combinations. A couple people I spoke to seemed to think that in industry it is more a matter of experimentation than using predictive math, but I don’t have enough data yet to draw that conclusion, as I’m just beginning this enquiry. Can you link to your testing?

6

Keep in mind the response curves may actually also change based on thickness of materials and also materials are not perfect so the ILDs are ranges (ILD will vary across the layer itself). It’s definitely complex math involved to model it correctly (I suspect you’re right that the mattress industry uses experimentation more than modelling for development purposes but hard to say for sure). The other complication is that the human body isn’t flat either and may also have varying density across the body as well. Lots to think about when doing actual mattress engineering. I started down that route but it quickly became clear you could spend alot of time/money modeling/experimenting instead of just using trial and error to triangulate in on what works best for you to get a good night’s sleep.

7

Here’s the whole thread: https://forum.mattressunderground.com/t/simmons-beautyrest-mattress-rebuild

8

I have been reading your thread; captivating. I am about halfway through (I’ll finish it later) and have to see how the story turns out, but so far I have come to a lot of the same conclusions you have, and identify with your persistence and need to tinker. I wanted to yell out early on “Zoning! You need zones, and custom zones, not the predetermined ones, because they never fit everyone perfectly, and, as you noticed, small changes can make a big difference. Having the zone be 1cm off can mean the difference between pleasure and pain.” I was going to mention Flobeds and suggest you cut or have them cut custom sizes.

I also believe that, unless you want to firm up your latex mattress, a stretchy case is important to let the foam expand. Using an adjustable air bed as a base is a smart idea. The same thing can more or less be accomplished by swapping cores in an all latex mattress, but obviously is more of a hassle, and probably won’t let you dial in as precisely. I also like the idea of buying a foam saw.

At this point, I am coming to the conclusion that all mattresses need to be adjustable to ideally fit a person (and not up/down adjustable like most adjustable bases do, but adjustable zoning). You can get close with the right set up, at least for one sleeping position, but the idea that any consistent spring rate can perfectly adapt to your body is a lie, let alone do it in different positions. I wonder how a human body would have to look, for each sleeping position, for a non-zoned mattress to perfectly conform and support - is that even a possible body? A mattress really has to be bespoke to fit you well, not ito of overall firmness but zoning, and you probably need 2 sides (like a split king) for each person: one for back/stomach and one for side sleeping if you are a combination sleeper, or combination user (side sleep but sit/lie in bed on your back to work or relax).

The interesting question to me right now is the best way to accomplish this: you can use pillows/bolsters instead of zones, have bespoke zones in your mattress, or have an adjustable base of some kind, either one that is motorized and has a lumbar/shoulder position that can raise/lower (although it probably will not be a perfect size/latitude) or a Euro-slat or -tile-style adjustable base, of which there are many. Right now I am exploring various Euro-slat/tile designs and am keen to see if it makes more sense to do the zoning in the base or in the comfort layer, or to do both.

Haven’t gotten to the latex stress/strain graphs yet.

PS If you ever need more pressure relief and want more of a pillow-top feel, you can try a shredded latex topper (or a convoluted topper, which would be halfway in plushness between solid and shredded. It’s a mess, but some, like the Flobeds shredded topper, have baffles that can be individually adjusted with fill (the whole side zips open), so you can do some micro-zoning in the pillow-top layer).

9

I agree with pretty much everything you said above. Alot of people could definitely make use of flexible zoning for the best sleeping experience possible but since the turn of the century the major brands seem to be more interested in volume than customer satisfaction or product quality. Very thankful to have great small companies around like Flobeds/SleepEZ/APM/etc to help people build what they really want. Again I think with some adjustable zoning you can get pretty close for most people to handle multiple sleeping positions (although side vs stomach needs are a often at odds with each other). Rest Bed (https://restperformance.com/) has an adjustable air bed with multiple chambers I looked at to be able to handle fine tuning alignment adjustments but besides being ungodly expensive I’d be worried about getting parts for it if something ever broke or they went out of business. So for now as you’ve noticed I decided to go with a very common air base and custom latex foam zoning for making mine adjustable. Right now the comfort layer is perfect but the zones still aren’t right on my side (need firmer in the lower back / waist and softer at the legs/feet to change the hip angle) so that’s what I’m working to fix. As you mentioned the standard zones don’t work for me so that’s where the foam saw will come in handy. Biggest pain honestly has just been getting stuff across borders during Covid but hopefully that’s coming to an end soon and I can get this design finished the way I want it to perform.