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Poisons ratio is defined as negative of the lateral strain divided by the longitudinal strain when a loading is applied in the longitudinal direction. It theoretical value ranges from -1 to 0.5 based on thermodynamic considerations of strain energy in the theory of elasticity .
Materials with Negative Poison's Ratio:
In practical examples all materials which undergo or which show lateral contraction upon expansion/extension or which show lateral expansion/extension upon compression have positive Poison ratio values ranging from 0 to 0.5 . For example, the Poisson ratios for various materials are approximately zero for cork, 0.14 _ 0.17 for SiC, 0.26 for Al2O3, 0.33 for aluminum, 0.42 for silicon, 0.45 for lead, and nearly 0.5 for rubbers and soft biological tissues [3-6].
The materials which show behavior opposite to this i.e. lateral expansion/extension upon expansion/extension and lateral contraction upon compression exhibit negative Poison ratio values ranging from -1 to 0. The materials with negative Poison's ratio are only theoretical permissible with only few exceptions. Negative Poisons ratio has not yet been practically observed, and still unknown to a certain extent. However, Love presented a single example of cubic ''single crystal'' pyrite with a negative Poison's ratio of 0.14 and suggested that the effect may result from a twinned crystal . Recently, it was indicated that negative Poisson's ratios may be a feature of cubic metals [8,9]. In addition, Lakes obtained a type of foam material with a negative Poisson's ratio by designing cellular structures  and he pointed out that the effect of a negative Poisson's ratio may play an important role in the fracture toughness of the materials [10,11]. In particular, the man-made hierarchical laminates with a chevron structure were recently predicted to have a negative Poisson's ratio in a specified direction, the physical mechanism of which was illustrated by a hinged framework that unfolds under tension [12, 13], whereas the natural hierarchical laminates were deduced to rarely have such a characteristic [14,15].
"Recently, the invention of negative Poisson's ratio foams was reported [16-18]. The polymer foams exhibited negative Poisson's ratios as small as -0.7, and values to -0.8 have been observed in metal foams. In the theory of elasticity, it is well known that some solutions for stress or displacement fields are dependent on Poisson's ratio. However, since most engineering materials have Poisson's ratios very close to 0.3, such dependence has received comparatively little attention. With the advent of new materials which exhibit controllable negative values of substantial magnitude, the effect of Poisson's ratio on stress and displacement fields deserves re-examination within the context of design." .
Behavior of the material with negative Poisson's Ratio:
Compression: on compression they when the longitudinal strain is negative they show the contraction laterally.
Stretching: On the longitudinal stretch these materials show the expansion in lateral direction.
So that in both cases Poisson's ratio becomes negative
There might be two reasons for deformation in such materials at nano scale.
By tensile deformation mechanisms.
By the existence of micro-rotational degrees of freedom.
For foam and chiral honeycombtype structures:
these are the structure which always have compacted type structure specially in hexagonal foam arrangment fibres so that when loaded they try to open up like in the honeycomb and that the hexagones to form the square geometry(foam case) by them and in such a way thatshow latterally expansion or shrinking behavior.
As shown in the following images:
structue of open cell polymer foam structure of chiral honey comb
For lattice structures: 
We applied uniaxial compressive stress using instron machine on the square lattice of circular holes. These holes represent the voids in the microstructure.
Initially as shown in fig (a) we have circular voids and no any sort of deformations:
After compression loading internal voids deform in the form of ellipse in such a way so that some are in horizontal deforming and some are in the verticle deformation that this is the behavior which allows the
-ve Poisson's effect to bieng compressed in longitudnal as well as shrinkage into latteral directions.
Fig (b,c) are showing the resulting changes.
Behaviour example (in general):
For example leading to the general case we take an example of polytetrafloroethylene 
Structure of particles when polymer dispersion coalescing of fibers and merging of dispersal particles forming node particle structure. Fig (1) during compression. Fig (2)
Showing normal condition of fibril particles and the dispersal particles formation in result of the particulate nodes. Fig (3) Extension. Fig (4)
"Fig 1 shows a high-magnification picture of one such node comprising the original spherical polymer dispersion particles. These particles are approximately 0.3 pm in diameter. In this fully extended form the fibrils are approximately 20 pm long and 500 8, wide. Under the action of low compressive loads, applied transverse to the fiber direction, the fibers are seen to coalesce together (figure 5). Also, the granular structure of the nodes becomes less distinct. At higher magnifications the cross fibers seen in figure 2, which are not present in the fully extended form are seen to be due to fibers coalescing together and occasionally crossing from one cluster to another . It can also clearly be seen in this picture that individual condensate particles are also coalescing together. It is possible to fully compress this microstructure, eliminating nearly all porosity and all indication of microstructure from SEM examination. As will be shown below, under uni axial tension in the fiber direction this process is completely reversed and full details of the structure reemerges"
"Studies of the sheet material reveal very similar structural features. Figure 3 again shows the fibrillar connection of the anisotropic particles. Again, at higher magnifications, in figure 4, these nodes are seen to be clusters of the original dispersion particles. As can be seen in figure 3 the particles and fibers lie at different orientations. This is a feature of the partially expanded structure"
Synthesis of the material with negative poisons ratio:
These materials do not exist naturally but synthesized in a fibrous arrangement way. The properties these materials get are due to focus upon there fiber structure than there composition. We use the soft lithography technique to make such structural materials. . It is based on the different types of printed cells like shown in the figure:
D:\engg works\self\4th semester\mom31\proj 2\Untitled.png
Transparent polymer film deposited upon the Si, Cr or Au metal.
The high accuracy printer creates the fiber structure upon this model.
There are then two possibilities before modeling:
Electroplating to give deep penetration of fiber pattern into the metal
Making of the mold cavity for molding purposes
After printing depending upon the polymer material thus we can use following techniques:
Micro transfer molding
Micro molding in capillaries
Finally the mold is filled by capillarity in UV light to make above shown structural negative Poisson's ratio material.
Synthesis in 3-D: 
Many times we require the 3D structure of -ve Poisson's ratio .following Figure illustrates 3D method.
" We made a PDMS Membrane that has a NPR pattern"
"Second, the PDMS membrane was coated with gold by thermal evaporation and wrapped around a glass cylinder"
"Nickel was electroplated on the gold layer"
"Freestanding, metallic NPR structure was obtained by dissolving the PDMS using TBAF and removing the cylinder"
So it was the 3D extension of the planar 2D -ve Poisson's ratio material sheet.
Materials with negative Poisson ratio are special materials; they are different from ordinary material and deceive the common intuition, since they expand laterally when stretched longitudinally. This particular behavior is achieved when cell type re entrant structure are formed, their chain structures are unfolded when they are stretched. 
In true sense many materials occur which can give rise to negative Poisson ratio, ranging from naturally occurring polymers to some crystalline materials and synthetic polymers.
During early stage of theory of elasticity, it was believed (Navier, Poisson) that isotropic material can be described as having a single elastic ratio, during those days they suggested that Â¼ is Poisson ratio for all the isotropic materials, according to experimental results and theoretical analysis for some materials . This inconstant theory was based on assumption of having central force interaction among the atoms of symmetric lattice. Since non central forces are present in some structures, so this theory was failed with discovery of some materials with P.R of 1/3. Classical theory of elasticity is now used to explain elastic constants for isotropic materials and Poisson ratio is also explained with the help of same theory. According to classical theory of elasticity, Poisson ratio varies from -1 to 0.5 for isotropic materials in 3d and from -1 to 1 in 2d. 
To satisfy equilibrium equations, non central forces in a structure must be balanced by a moment. The non balanced forces which do not depend upon distance between atoms, and exists between any two separate atoms will produce a deformation that is non homogenous locally or in other words that is non affine. While in affine deformation particles are displaced according to uniformly distributed strain.
For negative Poisson ratio we need three main phenomena to occur in a solid structure: non-affine deformation, non central force interaction, or chiral structure.
Structure and forces in an N.P.R material:
Honey comb structures in which chains of molecules are arranged in hexagonal cellular manner have occurred to have Poisson ratio of 1.
In the fig. a honey comb re-entrant structure is shown: made up of bendable ligaments. If this structure is stretched, ligaments BD and AC will straighten and the space between A and B will increase hence dilatation is produced so we get a negative Poisson ratio. While points C and D have their previous distance between them. A non homogenous deformation is produced in ligaments (non affin). The structure is an orthotropic one. The deformation of ligaments results in unfolding or expanding the cells. These unfolding have great impact on stress strain behavior of material when sufficiently higher loads are applied: both stress strain begin to show a linear relation for a re-entrant foam structure, which was non linear before. (1)Re-entrant honeycomb cells as shown in Fig. gives a negative Poisson's ratio to structure (4).
Another type of material structure with negative Poisson ratio is shown in fig below. When it is stretched, ligaments which bond particles will be stretched giving rise to a dilatation. (1)PoissonStrucF7.gif
Negative Poisson's ratio composites: Negative Poisson ratio has been discovered in some of composite materials. In composites negative Poisson ratio is mainly influenced by angle and arrangement of fibers i.e. staking sequence of fibers. N.P.R can be achieved by applying load perpendicular to fiber.
Properties of materials exhibiting "Negative Poisson Ratio":
Properties of materials with negative Poisson ratio depend upon the degrees of rotational freedom, deformation kinematics of non-affin, starting materials, processing time, temperature, presence of humidity, volumetric compression, relative density and cell size and on similar things like this. For negative Poisson's ratios the non-affine kinematics are seen to be essential. The materials with negative Poisson ratio usually have following properties:
Toughness and density:
Negative Poisson ratio enhances the toughness of the materials due to volumetric compression occurred in these materials like foam etc which in turn increases the density of that material.
E =2G/(1 - ï®ï€ ).
So the toughness for given shear modulus does not diverge, but for ï®ï‚®ï€ -1, G becomes large in comparison with E.
Deformation and tear resistance:
Negative Poisson ratio increases energy absorbing ability which avoids fracturing of that material. It increases the volume strain energy to about 100% which in turn decreases the deformation strain energy to about 44%, so it increases the deformation ability of material so instead of fracture material deforms so it increases material ability to resist the propagation of crack.
Effect on flexural rigidity:
It enhances the flexural rigidity the ability of non rigid materials to deform.
The indentation resistance increases so load bearing ability of material increases, which enhances its ability to resist stress concentration to larger extent.
Indentation rigidity is
P/w = E/2a (1- ï®2)
E = Young's modulus,
P = pressure
w = indentation
Material with value of "-1" Poisson ratio will be difficult to indent although if we are using compliant type of material
These materials show greater dilatation because of their ability to easily change volume. Volumetric compression ratio is expressed as:
= âˆš[âˆš + sin/2 + Î¦)]-3
Sound absorption ability:
Dynamic analysis of materials exhibiting negative Poisson ratio shows that their sound absorbing ability is greater than materials with positive Poisson ratio and lower ability to reflect sound. It lowers cut off frequencies.
Examples of Materials with Negative Poisson's Ratio
Gore-Tex was invented by Wilbert L. Gore and patented on 18 March 1980. It is famous as a waterproof fabric. It is a granular form of the polymer polytetrafloroethylene, which is also used in making Teflon. The molecular structure of Gore-Tex consists of islands of the chemical bonded with each other through a network of thin fibers. The space between the islands allows air to pass through making the material breathable but the gaps are too small (9 billion every square inch) for water droplets to cross making Gore-Tex waterproof as well. For these properties, Gore-Tex is used for insulation and sealing in almost every type of industry.
This is a material which is primarily used in blast protection although it also has various other uses. It was developed by Auxetic Technologies Ltd. As auxetic materials expand under an applied force, the Zetix fabric gets thicker in the region where a force is applied, becoming denser and thereby providing greater resistance to deformation and failure. It has proved capable of enduring multiple explosions and was one of five finalist technologies at the Global Security Challenge. The British government is a major client of Zetix fabrics. Zetix combines 1 part high-performance auxetic material with 100 parts of low quality fabric making it considerably cheaper than other explosion protection materials. Apart from the defense industry it is also used in natural disaster protection, medical sutures and road safety gear.
Some forms of Bucky paper have been experimentally shown to display auxetic behavior. This property could prove to be very useful in advancement of the auxetic industry, say researchers. Bucky paper is made from nanotubes of carbon woven into sheets. Carbon nanotubes are tiny fibrils of carbon that are only one atom thick. These tubes are stacked up to form fibers of carbon called multi-walled nanotubes (MWNTs) which are made into slurry and dried to produce Bucky paper. The slurry consists of both multi-walled and single-walled nanotubes (SWNTs). It is the proportion of MWNTs that dictates the Poisson's ratio of Bucky paper. Researchers discovered the auxetic property of Bucky paper when the Poisson's ratio went from 0.06 to -0.2 by increasing the amount of MWNTs. This provided ground for the auspicious possibility of being able to adjust the Poisson's ratio of an auxetic material according to the intended purpose. Bucky paper made of both SWNTs and MWNTs has proved to be stronger, tougher and more durable than that made from any one component alone. The adjustability of the Poisson's ratio means that Bucky paper can have a host of diverse applications ranging from use in composite materials, gaskets, stress/strain sensors and even artificial muscles.
Negative poisons ratio materials have extensive properties because of their unusual behaviors. One of the reasons for interest in materials of unusual mechanical properties comes from the fact that they can be used as matrices to form composites with other materials of other required properties, e.g. electric, magnetic, etc. These new types of materials have a lot of potential applications to Defense such as personal protective equipment's (e.g., protective clothing, body armor, bullet-proof vest, etc) and others (e.g., "smart" sensors, sonar, panels etc). .
These new types of materials were named auxetic by Evans . They can be also used for future work it is necessary to collaborate with researchers from textile, chemical & biological areas to explore the potential applications for protecting military personnel from injury, or chemical and biological attacks .
NPR materials offer a new direction for achieving unusual and improved mechanical performance . Studies, with support from NASA and Boeing, have demonstrated enhancements in mechanical properties for auxetic materials, including shear resistance , indentation resistance , and fracture toughness .
Artificial blood vessel is a typical example for the medical application. If the blood vessels made of conventional material, it tends to undergo a decrease in wall thickness as the vessel opens up in response to a pulse of blood flowing through it .
There are also used as composites. Auxetic reinforcing fibers should enhance fracture resistance of composites. It is well known that the interface between matrix and fibers is the weakest part of a composite material .
The counterintuitive property of auxetic materials, namely, lateral expansion (compression) under longitudinal tensile (compression) loads, is essential from the point of view of modern technology . Many applications for auxetic materials have been designed in various fields of human activity, from vascular implants, strain sensors, shock and sound absorbers, "press-fit" fasteners, gaskets and air filters, to fillings for highway joints .
Auxetic materials are also used for defense clothing and armor suits. Auxetic materials is useful to manufacture better body armours,because auxetic body armor could give the same safeguard but thinner, lighter, and conform better to the synclastic double curvatures of the human body. Also, the convex shapes are more appropriate than saddle shapes for sandwich panels for aircraft or automobiles .
Another promising application area is using auxetic polymers to make bullet-proof helmets or vests more resilient to knocks and shrapnel. When an auxetic helmet suffers an impact from one direction, material should flow in from other directions to compensate for the impact .
They are also used for making the sensors in making fastener and rivets. Also used in Energy Absorption Material for the Protection of Airborne Cargo Drops . Negative Poisson ratio materials still need research in their field .they are the materials of the future.