Poisson ratio prediction

The mechanical properties of polymers are controlled by the elastic parameters the three moduli and the Poisson ratio these four parameters are theoretically interrelated. If two of them are known, the other two can be calculated. The moduli are also related to the different sound velocities. Since the latter are again correlated with additive molar functions (the molar elastic wave velocity functions, to be treated in Chap. 14), the elastic part of the mechanical properties can be estimated or predicted by means of the additive group contribution technique. 

The speeds of longitudinal and transverse (shear) sonic waves can be estimated, c.q. predicted via two additive molar functions. From these sound velocities the four most important elastic parameters (the three elastic moduli and the Poisson ratio) can be estimated. 

When an FEA model is run, several elements must be present. These include the CAD data, material properties, loads acting on the part, and the boundary conditions used. Table 4.3 shows the typical input to conduct an analysis through one of the software programs. The minimum input for structural analysis is the modulus of elasticity, Poisson ratio, and density. For thermal predictions, the minimum inputs are coefficient of thermal expansion, specific heat, and thermal conductivity. For modal analysis, the minimum inputs are modulus of elasticity, Poisson ratio, and density. 

The application of pure hydrostatic stress leads to the prediction of the bulk modulus, K, for the rubber-toughened epoxy. For an isotropic material, the value of the bulk modulus is related to the value of Youngs modulus, E, and the Poisson ratio, v, by... 

Prediction of the Poisson Ratio. Earlier work performed using the cylindrical-material model found good agreement between predicted and... 

Figure 5. Comparison of experimental and predicted values of the Poisson ratio of epoxy resin filled with glass beads. Experimental values are shown with error bars, and predicted values were obtained from the two material models.

The critical values are generally obtained from a standard tensile test. Once the critical values are obtained the application of any (or all) of these criteria in conjunction with a dependable stress analysis is straightforward. Here we demonstrate the method by a simple example. Let us assume that it is desired to determine the torque required to cause failure of a 25 mm in diameter shaft constructed of an homogeneous isotropic plastic with a failure stress in tension, o, of 7 x 10 N/m. Assume further tlmt the modulus of elasticity, E, for this plastic is given by 3 x 10° N/m, and that is has a Poisson ratio of 0.3. We will explore the prediction of the three criteria just discussed. 

Lastly, it has been predicted that auxetic properties (i.e. negative Poisson ratio orthogonal contraction upon axial compression, and vice versa), which are closely related to NTE, may also arise in MOFs, but this highly sought after behaviour is yet to be reported. 

In this way, the Poisson ratio can be calculated from any two moduli. However, the identity required by Equation (11-5) is not always experimentally found for polymers, since such measurements are often carried out under different loading periods, and, consequently, the viscoelastic contributions become marked. On the basis of theoretical limiting Poisson ratios, however, it can be predicted that the shear modulus must always lie between 1 /3 and 1 / 2 of the E modulus. 

More recently, different approaches (cellular models, percolation analogy, fractal structure, blobs and links) have been proposed, to account for the porous volume and calculate the evolution of the meehanical properties as a function of the structural characteristics. Such models seem attraetive to deseribe the meehanieal properties of gels for several reasons. In eontrast with the empirieal relationships, they try to relate the physieal properties to a description of the mean strueture, or to the aggregation process they also predict that the Poisson ratio is with the fraetion of the solid phase, which is an experimental result demonstrated for aerogels and PDA. Another interesting feature of those models is that they prediet a power law evolution of the meehanieal properties as a funetion of the... 

Mills and Zhu [73] used the same microstructural model assuming 60% of the polymer in the cell faces and compression in the (001) direction. Cell edges were bent and compressed axially, while cell faces acted as membranes. The predicted Young s moduli were slightly low (Fig. 10) because compressive face stresses were ignored, but the Poisson ratio was correctly predicted. The predicted value of collapse stress for polyethylene foams was close to the experimental value. 

Figure 8. Relative bulk modulus of porous alumina (red) and zirconia (blue) modified exponential relation for spherical pores and matrix Poisson ratio 0.2 (green solid curve), predictions for alumina with intrinsic bulk modulus 2.139 (red solid curve) and for zirconia with intrinsic bulk modulus 2.724 (blue solid curve), dashed and dotted curves are relative bulk moduli calculated via elasticity standard relations using the differential and Mori-Tanaka prediction, respectively.

The Halpin-Tsai model was also compared with the ellipse model predictions. The modulus for montmoriiionite (178 GPa), the Poisson ratio for the nylon 6 (0.35), the Poisson ratio for the montmoriiionite (0.20), and the modulus for nylon 6 (2.75 GPa) that were employed in the Eornes-Paul evaluations were utilized in the calculations [5]. The volume fractions for the dispersed phase were similar to the experimental values determined in the Eornes-Paul work. When the major axis a of the dispersed-phase ellipse is aligned with the direction of the applied stress and the ratio of ui/ui is greater than 1 (1 is the ratio for a disk morphology), YJYp is greater for the ellipse when compared to the disk morphology. The application of the ellipse morphology in place of the disk morphology allows for predicted values for Y jYp to be closer to the experimental results provided in the work of Eornes and Paul. 

In the smdy of mechanical properties of particulate filled polymers, numerous models were developed to predict the effect of the particles on tensile or shear modulus. Most of these were derived from rheological models such as Einstein s, Eilers and Mooney s equations. A strong relationship exists between rheology and mechanical properties measurements and such correlations were studied by Gahleitner et al [66], as well as by Pukansky and Tudos [67]. There seems to have a direct relation between viscosity and shear modulus [59]. However, compensation has to be taken for matrix s Poisson ratio which is lower than 0.5 as shown by Nielsen and Landel [59]. Nevertheless, these equations on modulus predictions can be broadly classified under two groups. 

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