predicting compressive stress on fibrous wool structures

I have searched for reference and handbook information (yes, google too) describing the load behavior of randomly tangled wool-like fiber matte. The
interest is to design a wool-like material for elastic constant and stiffness depending on the materials properties, volume fraction and geometry (fiber thickness, cross section, length) of the fibers involved. Materials will be inorganic fibers and not polymers or natural fibers. Fibers will likely be curled or twisted and not straight. That should not constrain the governing equations. I have no idea what governing equations there may be. I expect there are empirical relations to use? There must be some engineering work from ancient subjects like pillow making, steel wool, filters or packaging. Fiber books don't cover this topic. No materials books or chapters in them address such topics. If there is a clue out there among you to resolve this, and you have read this far, then let me add further refinements. Say I want to make a fibrous matte much like Fiberfrax or the like. How might the mechanical compressibility change if a heat treatment allows fiber interconnects to form (sintering). Adhesives or sintering can ultimately make the matte into a rigid structure. I wish to control stiffness or the spring constant but maximize the compressed volume ratio. How would the properties change if I substituted sapphire fibers for silica glass for example? A book, web site, name of subject area or good keyword is all I ask.
Prof. K
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You have a fundamental problem, in that fibre mats themselves have no intrinsic strength. They are all resilient and easily deformed. It is only by mechanically bonding the fibres together that any web strength is achieved. This may involve mechanical looping, as in needle mat, where chopped strands are held in a veil support, or chemical bonding as typified by emulsion or powder-bound chopped strand mat, or typical insulation fibre blankets. Another approach is to make a paper-like material of dispersed fibres.
You ask about the use of sapphire fibres. I presume that you have done a Google search for "Saffil"?
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Terry Harper
http://www.terry.harper.btinternet.co.uk /
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You understand the problem. Thank You. Saffil is like Nextel which is one of my materials. I should revisit Saffil since the material might be engineered to a greater degree than their commercial products. Thanks for activating that neuron that opened that interconnect. I wish to take advantage of the resiliency and predictably increase the strength until the resiliency is reduced to my threshold. I had not heard of powder-bound support, but it sounds like electrostatic binding. I wish not to count on veil support. Instead, since I plan to avoid chopped fibers, approach a similar state while including the coiling of fibers . I seek 'loft' which is a term used in down (as in goose) which is another material I wish to be biomimetic with.
Dr. K

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Saffil
Powder binders are usually thermoplastic resins, used in chopped strand mat and continuous strand mat to bind the fibre bundles together. Emulsion binders are applied in aqueous emulsion, and have additional constituents to modify the properties, like drape, softness, and so on. Usually PVAlcohol is the base. In Insulation fibres, the binder used tended to be phenolic but I don't know what they use these days.
Have you read J Gilbert Mohr's book, "Fiber Glass", Van Nostrand Reinhold, ISBN 0-442-25447-4 ? My copy is dated 1978.
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Terry Harper
http://www.terry.harper.btinternet.co.uk /
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William Kaukler wrote:

about microstructure and material properties, and how little we actually know about it.
Your problem points it out quite nicely.
I came to this conclusion in studying the effects of porosity on materials behavior.
First I concluded that we really didn't know what the heck porosity was.
I used open cell, closed cell and steel wool (actually plastic or metal coil sink scrubber) materials to illustrate the point.
Secondly, I concluded that we often had no useful micromechanical or microproperty theories...... and I used the same materials to illustrate the point.
Dear Dr. K..... do you agree with this short summary?
Dr. Ashby hit the nail on the head in discussing mechanical applications as: 1) Tension (Or compression) 2) Bending 3) Torsion
In the systems of loose fibers such as steel wool, the mechanisms of bending and perhaps even torsion dominate the micromechanical behavior. This gives great softness or high compliance because of "lever" like effects.
Eventually you can fill the voids enough that the compression response of the matrix becomes dominant.
Or you can cross bond or cross link the filaments enough that eventually the compression or tension modes of the fibers become dominant.
Theory sucks in understanding this.... and most other porosity effects, except for the trivial.
Jim Buch
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Yes, I fully concur. I might add it seems fair to assume that all the stresses from fibers in different orientations can be summed by superposition. The only problem is quantitatively identifying the various fiber attributes of angle, bending, bridging, torsion, cantilever etc. Then the various contributions can be summed. Certainly someone has done this years ago? Perhaps there are some empirical ensemble equations? How about twisting up some piano wire into a ball and measuring the spring compression constant and then repeating with a double diameter wire of the same volume? Then double the volume fraction. Will it be double the stiffness? Who knows? I am interested in curly long fibers and not straight chopped fibers of some short length. The latter may be more tractable. I don't wish to address the complication of particulate binders of varying strength and composition. Should I continue an investigation in this; what journal would it best go to? Now, if you have an answer for that, tell me what journal do I need to search for related papers for the literature search? The answer to that is what I need whether I publish or not.
Prof. K

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jabbrer on about microstructure and material properties, and how little we actually know about it.

as:
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The Journal of the Textile Institute or Textile Research Journal might have some relevant papers. See http://www.texi.org/pub.htm or http://www.textileresearchjournal.com/ respectively.
I did some research on the fibres projecting from the surface of textiles, many years ago. The stress-strain curve is roughly exponential as one engages more and more projecting fibres, but I stopped as soon as the body of the textile began to compress, which is the point at which you wish to begin.
Keep friction in mind. A lot of what happens depends on friction at the points of contact between fibres. In some circumstances, tacking the fibres at the points of contact, by sintering or with another material, makes very little difference to the compressive properties, if friction was high to begin with.
The stiffness of the assembly is a very rapidly increasing function of the packing density, because the average strut length decreases.
You might get some useful clues from "Particle Packing Characeristics" by Randall M. German, ISBN 0-918404-83-5.
There was a paper in the Journal of the American Ceramic Society in the last few years (sorry, I have lost the reference) that gave a powder compression analogue of the general gas equation. There might be some clues in that.
I think you have an intersting area to work in :)
Cheers
Alan Walker
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From your message text I presume you want to design for compressive elastic modulus or compressive yield stress for a ceramic felt?
The people that make fiberglass insulation know a lot about how fiber/fiber bonds affect elastic recovery from compression ("Loft Recovery"). It is important that they be able to compress insulation for shipping but have it recover its fluffiness for installation.
Glues are usually applied during manufacture to lock fiber/fiber junctions together thereby maintaining the microscopic geometric arrangement of the felt's fibers. When such a felt is compressed, it will return to its original state if the bonds don't fail or the fibers don't permanently bend or break.
It is no surprise that there is no ab-initio theory for predicting the large deformation mechanical behavior of felts. Indeed there is not even any reasonable way to describe the geometric details of felts.
You say that the fibers will probably be bent. How much are they bent? What about corkscrew curves? These are very difficult geometries to describe quantitatively. If we've no tools to describe the geometric details of an arrangement of fibers how can we hope to describe the mechanical properties?
Fortunately there are a few things which are probably true about the mechanical properties of felts:
If the fibers themselves are elastic (ie. recover their original state when deforming stresses are removed) and the contact points between fibers are either locked (ie. fibers cannot slide over each other) or frictionless (ie. contact points have no mechanical consequence), then the felt will be elastic because its geometry cannot be fundamentally altered.
If any of the conditions in the preceeding paragraph are not met, the felt will be inelastic.
Felts thus fall into two categories, elastic and non-elastic.
Sources of inelasticity:
Plastic deformation or fracture of fibers; these are obvious. Some practical examples follow:
It is often found that heat treatment of ceramic fiber felts results in drastic changes in elastic recovery because crystal growth can severely weaken fibers. The weakened fibers break during deformation resulting non-recoverable deformation (the felt's geometry is altered during deformation).
I suppose it is possible that fiber/fiber sintering might increase elasticity by locking fibers together; however, I fear that fiber weakening by crystal growth might dominate.
If full elastic recovery is desired fiber/fiber junctions are usually bonded with some kind of adhesive as in the case of fiberglass for insulation.
Clearly, if stresses at bonded fiber/fiber junctions exceed a certain level the bonds fail and fibers slide with respect to each other resulting in a changed felt geometry and subsequent recovery.
A similar situation holds for fiber junctions locked by static friction or electrostatic forces; above a certain stress fibers will slide with respect to each other resulting in hysteresis effects. This is the case for felts used for piano string hammers.
Many people will attest to the effectiveness of hair conditioners in easing combing out a rat's nest of hair. Similarly, surface chemistry effects can drastically alter the mechanical properties of a felt. A felted wool shirt feels different on a very humid day. These are everyday examples of how a felt's mechanical properties can be altered by surface chemistry effects.
The deformation and recovery behavior of a felt may depend strongly on its past or present environment.
Elastic Felt descriptive parameters: For small enough deformations, felts made from elastic fibers behave elastically; ie. deformation is recoverable. It might be more precise to say that an elastic response to deformation can be approached asymptotically. I include this hedge because a fresh felt might not be in its lowest energy configuration and small periodic mechanical deformations may drive it towards an equilibrium configuration.
It is important to understand how felts scale dimensionally. That is, given a particular felt structure, how do the felt properties change if all dimensions are changed linearly? For example, what would happen to a felt's elastic modulus if all spacial dimensions are doubled?
Let's consider a simple case: a one dimensional array of fibers of modulus Eo, like a bunch of parallel cables in tension. The cables are x units apart and are of diameter d. A load F on a area of L*L is shared between N=(L/x)*(L/x) cables. The stress on an individual cable, So, is:
So=F/(N*(Pi*r*r)=F*(x*x)/((L*L)*Pi*r*r)
The strain is
e=So/Eo=(F/(L*L)/Eo)/(Pi*(r/x)*(r/x))
The nominal stress S is F/(L*L) and the nominal modulus is:
E=S/e=Eo*(Pi*(r/x)*(r/x))
The important scaling parameter in this simple case is the square of the fiber radius to spacing ratio: (r/x)^2
It shows that the elastic properties of the cable assembly is independent of magnification; the modulus stays the same if all dimensions are changed equally.
Other important cases for fiber geometry include "S" shaped fibers (see Therory of Elasticity, Timeshenko & Goodier, 3rd ed. art.33, McGraw Hill, 1970) vertically loaded in the plane of the S.
In this case a new dimension can be introduced, R, the curvature of the "S"; it turns out that the dimensional grouping that dictates the effective modulus of a parallel assembly of squiggley cables x units apart is proportional to:
Eo*(r/R)^2*(r/x)^2
Finally we look at fibers shaped like axially loaded helical springs (see www.clc.tno.nl/projects/recent/spring.html for a nice description of spring theory). In this case the spring's fiber radius is r, the spacing between springs is x, the spring's loops are radius R, and the loops are t units apart along the spring's axis (like inches per turn); the appropriate dimensional grouping is:
Eo*(t/R)*(r/R)^2*(r/x)^2
We therefore expect that a felt's elastic modulus might be described by a linear combination of terms involving the fiber modulus times a nested hierarchy of dimensionless groups, (r/x), (r/R), and (t/R) that describe the felt's geometry, like Eo*(r/x)^2*(a +(r/R)^2*(b+(t/R)*(...))))
Note that in all cases the felt's effective modulus is proportional to the fiber modulus, so holding structure constant and doubling fiber modulus should double felt modulus. decreasing fiber spacing by a factor of 2 quadruples modulus, etc...
Disclaimer: The above discussion pertaining to dimensional groups describing the modulus of felts is entirely hypothetical and untested; i.e. I made it up. I believe it to be reasonable and based on well established mechanical principles. It assumes one dimensional loading and does not consider the response of a spring loaded off-axis for example.
I look forward to hearing critical commentary.
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A thesis by T.J. Lahey at U. Waterloo has a good discussion of the mechanical properties of fabrics.
http://etd.uwaterloo.ca/etd/tjlahey2002.pdf
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