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The mechanics of deformable cones

Discussion in 'Biomechanics, Sports and Foot orthoses' started by Simon Spooner, Apr 10, 2009.

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    This is something I've been thinking about for a while, perhaps some of you can help me.

    Lets say we have a cone made of a material sitting on a bench such that the apex of the cone is at the top (see cone.jpeg attached). If we walk across this cone the pressure between the foot and the apex of the cone would increase as more body weight is loaded on to the cone. The pressure at this interface would be high because the contact area between the foot and the apex of the cone is very small and pressure = force/ contact area.

    Now lets say the cone deformed under the foots loading when we walked over it so that it resembled the picture in the cone deformed.JPEG (attached), the pressure should now be lower because the surface area in contact with the foot is greater.

    So the more force we apply to the deformable cone the greater the contact area- right? Is the relationship between load and increase in contact surface area likely to result in pressure remaining the same regardless of loading? Or is it possible to make a cone that results in decreased pressure the more it is loaded i.e. a non-linear stress/ strain cone? The material that the cone is made out of would obviously be significant, but what about the geometry of the cone itself?

    The pictures I have supplied showed the deformed cone basically as the first cone chopped in half, this is obviously an over-simplification because if the cone had deformed in this way its density would have increased too- also influencing contact interface pressure. In reality the cone would "bulge". This is where my brain starts to hurt.

    The more astute of you should see where my head is going with this. Can anyone help?

    Attached Files:

  2. Little Sesamoid

    Little Sesamoid Active Member

    Hi Simon,

    Interesting post!!

    The material of the cone is indeed very important. There is the hardness and the elasticity of it that will determine whether or not it will deform at all. If you were to step on an aluminium cone for instance, it would deform but a little, a rubber cone would deform much more. Hence for the aluminium cone your pressure would certainly not remain the same with the increase of force (because the surface area remains relatively unchanged), for the rubber cone it is possible that the pressure would remain roughly the same.

    Theoretically I guess it is possible to have pressure decrease with increasing force, but you would have to get a material soft enough for the surface area to quadruple when the force is doubled. Is there any such material??
    What gets really complicated is the fact that the proportionality constant between stress and strain is a fourth order tensor, and with such rubbery materials the higher order terms (which can be neglected in harder materials) can no longer be ignored. Thus one cannot assume that Hooke's law is obeyed for rubbery materials. And yes indeed it will bulge, but before we get to that, it's already complicated enough!!

    hope this helps!!

  3. Simon:

    I can see where you are going with this and it is a good mental exercise. I don't know if you can solve this problem without some complex modeling that would likely be best accomplished using finite element analysis.

    However, to simplify the problem, let's look at a few examples. If the cone was made of a pillow foam, for example, the low stiffness of the material would allow the foot of a normal size adult to totally compress the cone shaped "pillow" to the point that the whole foot was in contact with the foam with possibly the central, most-compressed portion having slightly increased contact pressures due to there being a greater thickness of compressed material at the apex than at the edges of the cone.

    In a rigid, steel cone. The increased stiffness of the steel versus the foam material for the cone would cause very high pressures on the foot when the apex of the cone contacted the plantar foot due to the steel cone deforming very little to body weight from the foot with the likely result being injury (if not skewering) of the foot by the steel cone from the very high contact pressures at the apex of the cone.

    Now, let's say we had a more dense foam such as a cone made of neoprene rubber (i.e. Spenco insole material). The intermediate stiffness of the neoprene would allow less deformation per Newton of force applied when compared to the more compliant pillow foam but would certainly deform much more than the steel cone. At initial contact with the foot, the material would compress fairly easily which, in turn, would widen the surface area of contact between the foot and the neoprene cone which, would in turn, decrease the pressure, but only if the force required to deform the cone was constant (which it wouldn't). Therefore, this is where the math (or in the UK "the maths") becomes very complex since one would need to know, for the material being modeled, how much the material would compress under increasing plantar foot load and whether that deformation would widen the surface area of contact with the plantar foot sufficiently to decrease the contact pressure that would otherwise be increased by the increasing plantar load.

    Like I said, get out the finite element analysis software, put some extra caffeine into the water dishes of the hamsters running their treadmills inside your computer, and this interesting question will likely be solved in a fairly straight forward fashion.:drinks
  4. That's why the cone shape is important. A cone with a relatively wide base to height ratio may help, so geometry is important- right.:drinks
  5. I guess that ultimately I'll have to do this (need a few more hours in each day), but rather than randomly make cones of varying geometry I was trying to get my head around the secrets of cone geometry (if there are any). On that note is there any advantage of a cone say over a triangular, square, pentagonal or hexagonal based pyramid- something tells me hexagons are good because they maximise surface area when placed together, but I'm not sure it's relevant?
  6. Little Sesamoid

    Little Sesamoid Active Member

    Geometry is indeed important, the only thing is that due to the sheer complexity of it there are no analytical solutions governing the mechanics for the compression of a non-cyllindrical object that I know of.
    In fact the poisson's ratio for many objects is still unknown as there are so many factors to take into account (von Mises stress, shear stress etc).
    As for the shape of the base, if you are packing several hexagon-based conical objects together you would have to take into consideration the ballistic-type energy transfer between the "cones". So packing them closely would probably not be the best way to go about it...would be my guess. The more room they have to "spread out" the less axial stress they will have to overcome.
    An analytical solution, in my opinion would be nearly impossible (far too many variables to consider) and a numerical solution would probably be best!!
    This is just my opinion and I'm certainly not the last say in the field.
    Hope this helps!
  7. It might help if we know who you are.
  8. Little Sesamoid

    Little Sesamoid Active Member

    Im the wife of little sesamoid (who is a Podiatrist), im a physicist, with a keen interest in the field of Podiatry.
  9. And your name is:..... don't be shy, you're obviously a shining star... My name is Simon, but my friends call me Spooner, Spoonz, Dog (long story), Doc (derivative) etc. I've lost track of the names my enemies have for me.

    As a physicist, I suspect you have a great deal to offer. Not too many people here know much about cones under load ;)

    "If I was the wife of an acrobat...
    Could of had it all...Could of had it all..."
    Suede: "the living dead"

    And that's not a song I give out at the drop of a hat- tune, sad tune, but tune, great tune none the less. http://www.youtube.com/watch?v=IX7n6kLRjPQ
    Welcome to the show; only begs the question, will the real little sesamoid please stand up? And reveal himself? ;):drinks

    When I get time, I'll run it through the FEA software and see what we get- unless you can beat me to it?
    Last edited: Apr 10, 2009
  10. Wife of Little Sesamoid:

    Welcome to Podiatry Arena! We would love to have a real name to match your obvious knowledge in physics and mechanics. Your knowledge is very valuable to many of us that are trying to convert podiatric biomechanics from a clinical field of undefinable joint positions, foot deformities, and normal and abnormal motion to more of a mainstream science where we use commonly accepted biomechanics, engineering and physics concepts and terminology to describe the kinetics and kinematics of the human body during weightbearing activities.

    I am excited that someone with your academic background has taken an interest in contributing to this site. Along with your name, could you please tell us a little about your research background or fields of interest in physics? You may become a very popular person here on Podiatry Arena. From your responses so far in this thread, you are, honestly, a like a breath of fresh air to me!:drinks
  11. Little Sesamoid

    Little Sesamoid Active Member

    Thank you both for the warm welcomes!
    My name is Maya, I'm a postgrad doing research in astrophysics...which is a far cry from podiatry but my husband and I regularly discuss podiatric biomechanics from a physical point of view.
    We were both contributing to this thread using his log-in name. I don't have an understanding at all about the podiatric side which is where my husband steps in, but as for the physics/mechanics side if I can contribute in any way I'd be glad to!!!
  12. Maya:

    Hmmmmm.........astrophysics and podiatry......are there any black holes close by that we could drop off a few of our infamous podiatrists into so that they would never been seen or heard of again???

    Welcome Maya......now, who is your husband? Bet those dinner conversations are very interesting!!:drinks
  13. Little Sesamoid

    Little Sesamoid Active Member

    No...i'm afraid there aren't any close enough that i know of!!

    My husband's a recent graduate working in Australia.... He would like to be known as little sesamoid. :D
  14. Simon:

    If you were modeling a single cone structure using finite element analysis (FEA), then using a simple cone would likely be the best to model due due to the lack of edges as would be present in, for example, a pyramid-shaped or hexagonal-shaped structure. The edges of a pyramid or hexagon would likely create greater pressures along those edges as the deformable cone was compressed, which may complicate analysis.

    However, if multiple cones were modeled, using the pyramid or hexagon shape would allow close packing of the cones so that no dead space between cones would exist. It all depends on what your purposes and goals are in your experiment.

    Here's an article on FEA of Foam Mattresses which is not too far off from what you may be looking for.

    If you don't want to get too complicated by using FEA, then a relatively simple way to do this is to take foam objects of different densities and different shapes (e.g. cone, sphere, cube) and then compress them step by step onto a pressure mat to see how the load vs pressure vs distribution of pressure changed with increasing deformation of the foam object. Even though the pressure mat would be a flat surface, using it versus the FEA software would likely give you a more real-life idea of how the forces and pressures change during the gradual compression of a "deformable cone". Optimization of cone materials to achieve the load-deformation characteristics that you desire could then be further fine-tuned by using FEA.

    Hope you and your family have a nice Easter, good buddy!:drinks
  15. efuller

    efuller MVP

    Hi all,

    Interesting discussion. The only thing I can think of to add is making the cone air filled, like a blow up toy. Force applied to a small area, like the tip of the cone will have a much greater pressure and deform it more than a force on the side. So, with an air filled cone, if you were to step on it, the deformation would be much greater with lower forces and as the pressure increased inside of the cone, there would be less deformation.

    And now I'm going to hop around in my backyard on a fine Easter morning. Happy hunting.


    Eric Fuller

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