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# Poisson's ratio10

## Poisson's ratio

(OP)
Hello,

I'm looking for informations about the poisson' ratio, deeper thant the basic definiton (ratio between transverse and longitudinal deformation).

Where does such a property have its origin ?

A fully amorphous material such as glass has a poisson' ratio around .3, as have a fully cristallized material such as common steel.

Some foams have  negative PR...

Why ? What is mainly driving this ratio ?

Thank you.

### RE: Poisson's ratio

Although I have not tried to confirm this, my opinion is that one controlling factor is shear strength of the material.  This is particularly evident in the foam you mentioned.  If shearing occurs before "bulging", then the Poisson's ratio gets out of whack.  Rate of loading also throws it off.  For pseudo-homogeneous materials (concrete), Poisson's ratio is lower than you might expect (0.15), whereas for unconfined soil, it is comparable to steel (0.3 to 0.4).

I realize I haven't answered your question, just thought I'd throw a few more thoughts in the mix.

### RE: Poisson's ratio

The Poisson's ratio is related to the ability of a material to change it's volume when elastically deformed under a nonuniform stress distribution.
The physically maximum possible value is 0.5, that corresponds to a material that doesn't change it's volume under deformation (many rubbers have values of PR>0.499).
Up to now I thought the physically minimum possible value was 0: this value indicates a material that may freely change it's volume under deformation (a tension specimen with PR=0 would not develop a reduction in the cross  section, that is normally found in real materials to contrast the increase in volume associated with the tension).
If you find materials with negative PR, I would guess this corresponds to a condition where the cross section of a tension specimen increases instead of decreasing, but this sounds quite odd.

prex
motori@xcalcs.com
XcalcS
Online tools for structural design

### RE: Poisson's ratio

Prex made a good observation about the negative PR, but odd as it is there are some foam substances and composite laminates that have been developed that do exert this strange phenomenon. I have also been advised that certain Laminates can also be seen to have PR's in excess of 1 (some materials getting above three), the effects of an internal lattice structure compound the standard tension/necking and compression/expansion effects of the material and produce a "substantial" change in internal volume. The application of a foam with a negative PR is reputed to be in void fillers where the expansion under tension could be put to good effect. Unfortunately I can not remember the name of the material or its developer.

### RE: Poisson's ratio

6
A novel foam structure is presented, which exhibits a negative Poisson's ratio. Such a material expands laterally when stretched, in contrast to ordinary materials.

Virtually all common materials undergo a transverse contraction when stretched in one direction and a transverse expansion when compressed. The magnitude of this transverse deformation is governed by a material property known as Poisson's ratio. Poisson's ratio is defined as minus the transverse strain divided by the axial strain in the direction of stretching force. Since ordinary materials contract laterally when stretched and expand laterally when compressed, Poisson's ratio for such materials is positive. Poisson's ratios for various materials are approximately 0.5 for rubbers and for soft biological tissues, 0.45 for lead, 0.33 for aluminum, 0.27 for common steels, 0.1 to 0.4 for cellular solids such as typical polymer foams, and nearly zero for cork.

Negative Poisson's ratios are theoretically permissible but have not, with few exceptions, been observed in real materials. Specifically, in an isotropic material (a material which does not have a preferred orientation) the allowable range of Poisson's ratio is from -1.0 to +0.5, based on thermodynamic considerations of strain energy in the theory of elasticity. It is believed by many that materials with negative values of Poisson's ratio are unknown; however Love presents a single example of cubic 'single crystal' pyrite as having a Poisson's ratio of -0.14; he suggests the effect may result from a twinned crystal. Analysis of the tensorial elastic constants of anisotropic single crystal cadmium suggests Poisson's ratio may attain negative values in some directions. Anisotropic, macroscopic two-dimensional flexible models of certain honeycomb structures (not materials) have exhibited negative Poisson's ratios in some directions. These known examples of negative Poisson's ratios all depend on the presence of a high degree of anisotropy; the effect only occurs in some directions and may be dominated by coupling between stretching force and shear deformation. The materials described in the following, by contrast, need not be anisotropic.

Foams with negative Poisson's ratios were produced from conventional low density open-cell polymer foams by causing the ribs of each cell to permanently protrude inward, resulting in a re-entrant structure. A polyester foam  was used as a starting material and was found to have a density of 0.03 gm/cubic cm, a Young's modulus of 71 kPa (10 psi), a cell size of 1.2 mm, and a Poisson's ratio of 0.4. The method used to create the re-entrant structure is as follows. Specimens of conventional foam were compressed triaxially, i.e. in three orthogonal directions, and were placed in a mold. The mold was heated to a temperature slightly above the softening temperature of the foam material, 163 deg.C to 171 deg.C in this case. The mold was then cooled to room temperature and the foam was extracted. Specimens which were given a permanent volumetric compression of a factor of 1.4 to a factor of 4 during this transformation process were found to exhibit negative Poisson's ratios. For example, a foam subjected to a permanent volumetric compression of a factor of two had a Young's modulus of 72 kPa, and a Poisson's ratio of -0.7. Polyester foams of similar structure and properties but different cell sizes (0.3 mm, 0.4 mm, 2.5 mm) transformed by the above procedure were also found to exhibit negative Poisson's ratios. Reticulated metal foams were transformed by the alternate procedure of plastically deforming the material at room temperature. Permanent compressions were performed sequentially in each of three orthogonal directions. Foams transformed in this way were also found to exhibit re-entrant structures.

### RE: Poisson's ratio

And now back to simple explanations....

Imagine a cube of pure rubber 3" x 3" x 3".  Place it on the table in front of you and and press down on the top with a flat plate.  As you push down, the sides bulge out.  The amount of bulge is determined by Poisson's ratio.  For rubber, which is essentially incompressible, for every unit volume you push down, an equal volume squishes to the outside, and since it can go in two directions, the ratio is 50% as described by prex and eran above.

Now consider a block of steel of the same size and that you are strong enough to compress it like you did the rubber.  The steel is compressible, so it doesn't bulge as much for every unit volume you compress it, so Poisson's ratio is lower for that material.

Now go back to the block of rubber and imagine it constrained between two parallel plates so it can only bulge out two sides.  You can guess that it would be harder to push down the rubber.  This is why Poisson's ratio must be considered in some deflection calculations, where a material is constrained by it's shape.  The constraints are going to affect how the material deflects.

Now imagine a block of open cell foam, like that gray packing foam that electronics are sometimes protected with.  This stuff is very compressible, enough so that you can imagine that the sides of the cube might not bulge at all when compressed.  That would mean a Poisson's ratio of zero.  It is even possible to imagine that the sides might collapse inward as you compress the foam, and that foam would have a negative Poisson's ratio and would be constructed as eran describes above.

Thanks for the good discussion everyone!

tcampbe1

### RE: Poisson's ratio

Gortex (registered trademark) is a material with negative Poisson's ratio

### RE: Poisson's ratio

(OP)
Hi,

Thank you for all these answers.... :)
BUT I think we didn't get it : we are dealing with common things, known by everyone studying mechanical properties of materials :)

I'm looking for a Physical explanation of PR : why such a propety ? Okay, we can observe that a compressed material (in a specified direction) usually expand (if not constrained) in the perpendicular directions, but where does PR have its fundaments ?

For example, Young's modulus is due to the bonds between atoms. Considering a lattice structure (let say CC) of atoms, we stretch it, atoms tends to resist to this stretch due to attraction forces at their scale, like a spring. Great.

Do I have to look to the thermodynamical side ?

Regarding all this discussion and the informations that I have, I think that PR is one of the most important material parameter, but one of the less defined (in physical side) :))

Thanks again.

### RE: Poisson's ratio

When a solid particle is stressed in compression, there is a vertical component of the stress and horizontal components in all lateral directions.  As you align axially with the load, the shear stress particle interaction is greatest, but because solid materials are, on the whole, granular materials, the particle interaction occurs at various angles.  This allows a creation of the horizontal components of stress, since granular particle "A" might impinge on granular particle "B" at an angle greater than 0 degrees from the vertical.  This is what creates an orthogonal stress distribution, which in most solids, is considered to be a triaxial state of stress.  Thus, theoretically, if you push down on something, you must transfer stress to a lateral direction as well.  Using this premise, as shear strength approaches 0, the PR might go negative, as the excessive deformation changes the state of stress in the material.

In my humble opinion, the reason that PR for soils or other granular substances is as high as high strength solid materials, is that the soil shape and granular texture mimicks, in the physical sense, the molecular interaction as well.

### RE: Poisson's ratio

I don't think PR is so important. Unless you are concerned with something that is critically related to the deformational behaviour of a component, you may neglect the influence of PR.
Concerning stresses, the influence of PR is always quite limited. Moreover, as stresses due to a nonzero PR are deformation related quantities (secondary stresses as defined by ASME VIII Div.2), you will get a safe result for ductile materials (practically all materials of structural importance) if you take PR=0.
Concerning the physical interpretation I cannot be very precise on that, but it seems to me that the relation of PR to the change in volume is highlighting. If you compress a specimen, this will tend to expand laterally to avoid the change in volume (and this is again because of the atom to atom relations). For some reason that I don't know (and this is indeed what you are looking for) some materials are 'stiffer' than others to that change in volume.

prex
motori@xcalcs.com
http://www.xcalcs.com
Online tools for structural design

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