Poissons Ratios

Poisson s ratio is defined as the ratio of the change in a sample s width per unit width to the change in the length per unit length, as a result of straining a plastic sample, Examples of Poisson s ratios of fluoropol5miers are illustrated in Table 3.45. 

Poisson s ratio v is the ratio of transverse contraction to longitudinal extension in the stretching direction. For the purposes of visualization only, picture a tough rubbery sheet, push a dull thin rod into the sheet, and you have transverse contraction and longitudinal extension. In a practical mode, Poisson s ratio is about 0.33 for many solid plastics and just under 0.5 for elastomers. 

The transverse strain -Etransverse is defined as negative, and the longitudinal strain Eiongitudinai is defined as positive so that v has a positive value. 

Poisson s ratio can be viewed, in simple terms, as directly related to the modulus of elasticity E and inversely related to the shear modulus G  

Poisson s ratios shown in Table 15.1 are for isotropic materials that have identical stmctirre and properties in all directions. Fibers, especially polymer fibers, are anisotropic and their Poisson s ratios deviate from the values reported in Table 15.1. 

The bulk modulus (B) is a measure of resistance to the brrlk stress or hydrostatic press, and it can be defined as  

Since the engineering compression ratio is dimensionless, the unit of bulk modulus is the same as the hydrostatic pressure, i.e., pascal. 

For isotropic materials, the bulk modulus has a simple relationship with tensile modulus and Passion s ratio  

Sources. Boresi, A.R, et. sH., Advanced Mechanics cfMatehalSy Fifth Edition, John Wiley Sons, Ltd., 1993. Ward, I.M., et. Mechanical Properties of Solid Polymers,l mA tt on.,  

The Poisson s ratio, v, is defined as —eyyjcxx = Szzl xx- Using the last two equations as before to express in terms of e y, we can write 

For plastics the ratio falls within the range of 0 to 0.5. With a 0 ration there is no reduction in diameter or contraction laterally during the 

Plastic Poisson s ratio Shear modulus MPa Shear stress MPa 

The negative sign is to ensure that the numerical value of Poisson s ratio is positive. For isotropic materials, the theoretical value of u is Most metals show values in the range 0.25-0.35. Some materials, counterintuitively, have a negative Poisson s ratio, and get thicker when under tension. They are discussed later in this chapter. Representative values of Poisson s ratio are given in Table 10.1. 

Most materials have o between 0.0 and 0.5. Cork is close to 0.0, polysilicon is around 0.22, single crystal silicon is around 0.28, most steels are around 0.3, and rubber is almost 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson s ratio of exactly 0.5. Some materials, mostly polymer foams, have a negative Poisson s ratio if these auxetic materials are stretched in one direction, they become thicker in perpendicular directions. 

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The Poisson ratio 0 < k < 1/2 is given. By the first Korn inequality, we have... 

Material Tensile modulus, GPa Tensile strength, yield, MPa Flexural strength, MPa Poissons ratio Fracture toughness, MPaVm... 

Strained set of lattice parameters and calculating the stress from the peak shifts, taking into account the angle of the detected sets of planes relative to the surface (see discussion above). If the assumed unstrained lattice parameters are incorrect not all peaks will give the same values. It should be borne in mind that, because of stoichiometry or impurity effects, modified surface films often have unstrained lattice parameters that are different from the same materials in the bulk form. In addition, thin film mechanical properties (Young s modulus and Poisson ratio) can differ from those of bulk materials. Where pronounced texture and stress are present simultaneously analysis can be particularly difficult. 

The modulus term in this equation can be obtained in the same way as in the previous example. However, the difference in this case is the term V. For elastic materials this is called Poissons Ratio and is the ratio of the transverse strain to the axial strain (See Appendix C). For any particular metal this is a constant, generally in the range 0.28 to 0.35. For plastics V is not a constant. It is dependent on time, temperature, stress, etc and so it is often given the alternative names of Creep Contraction Ratio or Lateral Strain Ratio. There is very little published information on the creep contraction ratio for plastics but generally it varies from about 0.33 for hard plastics (such as acrylic) to almost 0.5 for elastomers. Some typical values are given in Table 2.1 but do remember that these may change in specific loading situations. 

Example 2.4 A glass bottle of sparkling water has an acetal cap as shown in Fig. 2.14. If the carbonation pressure is 375 kN/m, estimate the deflection at the centre of the cap after 1 month. The value of Poissons ratio for acetal may be taken as 0.33. 

D = external drill pipe diameter in ft D. = internal drill pipe diameter in ft V = steel Poisson ratio... 

Figure 4-325. Overburden gradient and Poisson ratio (a) variation of the overburden gradient with depth (b) variation of the Poisson ratio with depth. (Courtesy Editions Technip.)...

Gqb overburden gradient in psi/ft Gp = pore pressure gradient in psi/ft K = coefficient related to Poisson ratio... 

The static Poisson ratio is determined in a triaxial cell. The dynamic Poisson ratio is calculated with the sonic compressional and shear wave velocities. They usually are different. 

Compute the fracturation pressure gradient and fracturation pressure at 8,460 ft assuming a Poisson ratio of 0.4. 

The load or stress has another effect on the creep behavior of most plastics. The volume of isotropic or amorphous plastic increases as it is stretched unless it has a Poisson ratio of 0.50. At least part of this increase in volume manifests itself as an increase in free volume and a simultaneous decrease in viscosity. This decrease in turn shifts the retardation times to being shorter. 

To illustrate the correct approach, consider applications in which a material is used in sheet form, as in automotive body panels, and suppose that the service requirements are for stiffness and strength in flexure. First imagine four panels with identical dimensions that were manufactured from the four materials given in Table 3-1. Their flexural stiffnesses and strengths depend directly on the respective material s modulus and strength. All the other factors are shared in common with the other materials, there being no significantly different Poisson ratios. Thus, the relative panel properties are identical with the relative material properties illustrated in Fig. 3-3. Obviously, the metal panels will be stiffer and... 

Because there is a large literature, we restrict ourselves to an interesting example, useful for understanding the axial support of thin mirrors. Consider a thin circular plate of radius a and thickness h, with elastic constant E and Poissons ratio v. Let this plate be axially loaded by gravity and assume we will support this plate against this load by N supports. This is shown in Fig. 2. 

Couman W.J., Heikens D., and Sjoerdsma S.D., Dilatometric investigation of deformation mechanism in polystyrene-polyethylene block copolymer blend Correlation between Poisson ratio and adhesion. 

Sjoerdsma S.D., Bleijenberg A.C.A.M., and Heikens D., The Poisson ratio of polymer blend, effects of adhesion and correlation with the Kemer packed grain model. Polymer, 22, 619, 1981. 

An attempt has been made by Spiering et al. [39,40] to relate the magnitude of the interaction parameter F(x) as derived from experiment to the elastic interaction between HS and LS ions via an image pressure [47]. To this end, the metal atoms, inclusive of their immediate environments, in the HS and LS state are considered as incompressible spheres of radius /"h and Tl, respectively. The spheres are embedded in an homogeneous isotropic elastic medium, representing the crystal, which is characterized by specific values of the bulk modulus K and Poisson ratio a where 0 < a < 0.5. The change of molecular volume A Fas determined by X-ray diffraction may be related to the volume difference Ar = Ph — of the hard spheres by ... 

The contact force between two particles is now determined by only five parameters normal and tangential spring stiffness kn and kt, the coefficient of normal and tangential restitution e and et, and the friction coefficient /if. In principle, kn and k, are related to the Young modulus and Poisson ratio of the solid material however, in practice their value must be chosen much smaller, otherwise the time step of the integration needs to become unpractically small. The values for kn and k, are thus mainly determined by computational efficiency and not by the material properties. More on this point is given in the Section III.B.7 on efficiency issues. So, finally we are left with three collision parameters e, et, and which are typical for the type of particle to be modeled. 

The mechanical properties of the inorganic nanotubes have only been investigated to a relatively small extent. The Young s modulus of multiwall BN nanotubes was measured using the vibrational method within a TEM (17) and was found to be 1.2 TPa, which is comparable to the values measured for carbon nanotubes. The Young s modulus of the b-P nanotubes was calculated (88a). The observed value, 300 GPa, is some 25% of the Young s modulus of carbon nanotubes. The Poisson ratio of b-P nanotubes was calculated to be 0.25 in this work. 

In order to proceed with the evaluation of the time-dependent Poisson ratio v(0, both sets of relaxation behaviour are required. Now from Chapter 2 we know the Poisson ratio is the ratio of the contractile to the tensile strain and that for an incompressible fluid the Poisson ratio v = 0.5. Suppose we were able to apply a step deformation as we did for a shear stress relaxation experiment. The derivation then follows the same course as that to Equation (4.69) ... 

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