Poisson’s ratio, defined

For an isotropic body, there are only two stress components that are independent of the other. This means that while different loadings and strains can be imposed, the material constants relating stress and strain are not all unique. There are six common elastic constants (Table 6.1), including Poisson s ratio, defined as the ratio of the lateral strain accompanying a longitudinal strain, V = -eiilsn- Since only two are unique for an isotropic body, each elastic constant can be expressed as a function of any other pair these expressions are tabulated in Table 6.1 in terms of Poisson s ratio (Mott and Roland, 2009). 

For a polymer film possessing orthorhombic symmetry, there are three lateral compliances 5i2, and S2i, which relate to the six Poisson s ratios defined in Section 8.2.2. 

The difference between the bounds defined by the simple models can be large, so that more advanced theories are needed to predict the transverse modulus of unidirectional composites from the constituent properties and fiber volume fractions (1). The Halpia-Tsai equations (50) provide one example of these advanced theories ia which the rule of mixtures expressions for the extensional modulus and Poisson s ratio are complemented by the equation... 

One final point. We earlier defined Poisson s ratio as the negative of the lateral shrinkage strain to the tensile strain. This quantity, Poisson s ratio, is also an elastic constant, so we have four elastic constants E, G, K and v. In a moment when we give data for the elastic constants we list data only for . For many materials it is useful to know that... 

Contact mechanics deals with the deformation of solids in contact. Consider two elastic bodies, shown schematically in Fig. 3, of radii of curvature R[ and Rt, Young s moduli E and E2, and Poisson s ratios and V2. Define... 

For isotropic materials, certain relations between the engineering constants must be satisfied. For example, the shear modulus is defined in terms of the elastic modulus, E, and Poisson s ratio, v, as... 

Lekhnitskii defines the coefficients of mutual influence and the Poisson s ratios with subscripts that are reversed from the present notation. The coefficients of mutual influence are not named very effectively because the Poisson s ratios could also be called coefficients of mutual influence. Instead, the rijjj and ri y are more appropriately called by the functional name shear-exitension coupling coefficients. 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

For a calculation of d. see R- H. Fowler. Statistical Thermodynamics. Second Edition, Cambridge University Press. 1956. p. 127. In Section 1.5a of Chapter 1 we defined the compressibility and cautioned that this compressibility can be applied rigorously only for gases, liquids, and isotropic solids. For anisotropic solids where the effect of pressure on the volume would not be the same in the three perpendicular directions, more sophisticated relationships are required. Poisson s ratio is the ratio of the strain of the transverse contraction to the strain of the parallel elongation when a rod is stretched by forces applied at the end of the rod in parallel with its axis. 

Tensile stress-strain tests give another elastic constant, called Poisson s ratio, v. Poisson s ratio is defined for very small elongations as the decrease in width of the specimen per unit initial width divided by the increase in. length per unit initial length on the application of a tensile load ... 

Generally our material will be compressible and zxx zyyj2 = ezz/2 and we have to introduce another material parameter, Poisson s ratio, v. Poisson s ratio is defined as the ratio of the contractile to the tensile strain, i.e. v = zyy exx. Equation (2.4) now becomes ... 

Stress is equal to the force per unit area, and strain or elongation is the extension per unit length. For an isotopic solid, i.e., one having the same properties regardless of direction, the strain is defined by Poisson s ratio, V = y /j which is the change in thickness (lateral contraction) to the change in length. 

The minus sign in Eq. (5.9) is to account for the fact that Ad as defined above is usually negative. Thus, Poisson s ratio is normally a positive quantity, though there is nothing that prevents it from having a negative value. Eor constant volume deformations (such as in polymeric elastomers), v = 0.5, but for most metals, Poisson s ratio varies between 0.25 and 0.35. Values of Poisson s ratio for selected materials are presented in Appendix 7. 

T Total stress tensor V microslip, defined by Eq. (2.120) Poisson s ratio... 

At large W/b9 the stress path, i.e.9 the ratio of 01/02 as 02 increases, is calculable (21), provided that the Poisson s ratio along the path is known. Since Poisson s ratio (if indeed it can be defined for finite strain) varies from about 0.4 typically to about 0.5 at yield, the stress path is not radial, i.e.9 01/02 is not constant. This situation is illustrated schematically in Figure 2. Since accurate values for Poisson s ratio of the polymers tested are not available, the path dependence of our results will have to await further investigation. 

The elastic properties of a body in axial tension can be characterized using Poisson s ratio vp, which is defined as the negative ratio of the transverse strain to the longitudinal strain. 

Poisson s ratio v is defined from the ratio of transverse and longitudinal strains in tension ... 

Near the contact, the vertical arrows at the dashed contour schematically represent the surface forces which cause an additional deformation of the elastic sphere thus increasing the contact radius from aH (Hertz) to aJKR (JKR). The contact radius for the JKR model is a function of the external load, the work of adhesion, the radius of the contacting sphere (or the reduced radii of the contacting spheres, if two spheres are in contact) and the elastic constant K (a combination of the Young s moduli and the Poisson s ratios of the contacting materials), defined as... 

For Si cantilevers, the material properties (Young s modulus and Poisson s ratio) are known in any crystal orientation [7]. By contrast, the material properties of the SisN4 cantilevers are not well defined and may vary significantly [8] due to differences of the CVD processes [9, 10]. For instance, Young s moduli and Poisson s ratios of Si3N4 cantilevers in the range of 120—200 GPa and 0.22—0.27, respectively, have been reported. 

Conventionally elastic properties of solids are described using moduli called Young s modulus, E, shear modulus, G, bulk modulus, K and the Poisson s ratio, v (due to the fact that mechanical properties were studied more extensively by engineers).Young s modulus is defined by the relation... 

Similar parameters may be defined for plane strain problems. The previous results have shown that the solution of crack problems in FGMs is not very sensitive to the Poisson s ratio. Thus, v is assumed to be constant throughout the medium. In this study it is further assumed that in the graded materials the variations in En, E22 and G12 are proportional. Referring to... 

Poisson s ratio v is defined by Equation 11.5 for an isotropic (unoriented) specimen. It describes the effect of the application of a deformation (strain) in one direction (i.e., along the x axis) on the dimensions of the specimen along the other two directions (i.e., the y and z axes) perpendicular to the direction of the applied deformation. The fractional change of volume dV/V of the specimen is given by Equation 11.6 in terms of the strains dex, d y and d z along... 

Equations 11.8 and 11.9 are isomorphous to equations 9.9 and 9.10 which define the storage and loss components of the complex dielectric constant . Similar equations are also used to define the complex bulk modulus B, the complex shear modulus G, and the complex Poisson s ratio v, in terms of their elastic and viscous components. The physical mechanism giving rise to the viscous portion of the mechanical properties is often called "damping" or "internal friction". It has important implications for the performance of materials [8-15],... 

Figure 20.2. Effects of platelet orientation relative to the direction of deformation, predicted by using the model of Brune and Bicerano [11]. The orientation angle is defined as the angle between the symmetry axis of the platelets and the direction of deformation, so that it is 90° if the platelets are aligned perfectly along the direction of deformation while it is 0° if the platelets are aligned completely perpendicular to the direction of deformation. The curves are labeled by the platelet aspect ratio Af. The platelet volume fraction was =0.025, the platelets were assumed to have a Young s modulus of 100 times that of the matrix, and a Poisson s ratio of 0.4 was assumed for both the matrix polymer and the platelets in these calculations.

The applications of Hooke s law [equations (2-14) and (2-18)] discussed above have assumed that the volume of the material is invariant with strain during a tensile deformation. However, because the pressure is not zero, this may not be the case, and the strains in each direction must be known to account for this. By measuring the actual transverse (yyy) and longitudinal (y ) strains, one can define the ratio of these two strains as a material property. This is called Poisson s ratio ju, and is defined as ... 

---