Elastic behavior prediction

The JKR model predicts that the contact radius varies with the reciprocal of the cube root of the Young s modulus. As previously discussed, the 2/3 and — 1/3 power-law dependencies of the zero-load contact radius on particle radius and Young s modulus are characteristics of adhesion theories that assume elastic behavior. 

Displacement Strains The concepts of strain imposed by restraint of thermal expansion or contraction and by external movement described for metallic piping apply in principle to nonmetals. Nevertheless, the assumption that stresses throughout the piping system can be predicted from these strains because of fully elastic behavior of the piping materials is not generally valid for nonmetals. 

An analysis of the transfer function of this system can be made using the matrix method described by Okano et al. (1987). However, the stiffness of the rubber pieces is highly nonlinear. Okano et al. (1987) found that the measured transfer function does not fit theoretical predictions based on a constant stiffness. A nonlinear elastic behavior must be taken into account. Another problem with the metal-stack system is that the resonance frequency is around... 

The structural organization of the components of a cheese, especially the protein network, affect the cheese texture in particular the stress at fracture, the modulus, and work at fracture could be predicted very well from the size of the protein aggregates (Wium et al., 2003). Cheeses having a regular and close protein matrix with small and uniform (in size and shape) fat globules show a more elastic behavior than cheeses with open structure and numerous and irregular cavities (Buffa et al., 2001). 

In addition to strength and WOF of FMs, the elastic behavior of these architectures should be considered. Simple brick models were proposed to accurately predict elastic properties of FMs [1, 24], Figure 1.8 shows the elastic modulus versus orientation for uniaxially aligned Si3N4/BN FMs with experimentally measured values, indicating that there is very good agreement between experiment and prediction. This prediction can be used for FMs with multiaxial architectures. 

For random crosslinking rf may be assumed to be equal to r, the corresponding mean square end-to-end distance for unconnected chains of the same molecular length. Because A is inversely proportional to (Eq. (1.2)), the only molecular parameter that remains in Eq. (1.8) is the number N of elastically effective chains per unit volume. Thus, the elastic behavior of a molecular network under moderate deformations is predicted to depend only on the number of molecular chains and not on their flexibility, provided that they are long enough to obey Gaussian statistics. 

The corresponding relation is shown in Figure 1.8. It illustrates a general feature of the elastic behavior of mbbery solids although the constituent chains obey a linear force-extension relationship (Fq. (1.1)), the network does not. This feature arises from the geometry of deformation of randomly oriented chains. Indeed, the degree of nonlinearity depends on the type of deformation imposed. In simple shear, the relationship is predicted to be a linear one with a... 

The molecular theory of rubberlike elasticity predicts that the first coefficient, Ci, is proportional to the number N of molecular strands that make up the three-dimensional network. The second coefficient, C, appears to reflect physical restraints on molecular strands like those represented in the tube model (Graessley, 2004) and is in principle amenable to calculation. The third parameter,, is not really independent. When the strands are long and flexible, it will be given approximately by 3X, where Xm is the maximum stretch ratio of an average strand. But is inversely proportional to N for strands that are randomly arranged in the unstretched state (Treloar, 1975). Jm is therefore expected to be inversely proportional to Ci. Thus the entire range of elastic behavior arises from only two fundamental molecular parameters. 

Moving to a larger scale, let us now look at the influence of microstructure on elastic behavior. As indicated in the previous section, the elastic constants are a fundamental property of single crystals through the geometry and stiffness of the atomic bonds. Thus, one may expect elastic behavior to be controlled simply by the choice of material. By using composite materials, however, one can control the final set of elastic properties with some precision, i.e., by mixing phases with different elastic constants. Clearly, it is useful to be able to predict the elastic constants of a composite from those of its constituents. This has been accomplished for many types of composite microstructures. For this section, however, the emphasis will be on (elastically) isotropic composites, i.e., composites contain-... 

Studies of the viscous-elastic behavior of OMC with various water-glass silica moduli and MGF-9 contents have revealed the considerable influence of these factors on composite viscous-elastic properties. Evidently, study of the viscous-elastic properties of OMC is an effective method for prediction of physical, mechanical, and operating characteristics of the composites investigated. 

Predictive model for the morphology variation during simple shear flow under steady-state uniform shear field was developed. The model considers the balance between the rate of breakup and the rate of drop coalescence. The theory makes it possible to ctunpule the drop aspect ratio (p = 01/02), a parameter that was directly measured for PS/PMMA =1 9 blends. Theoretically and exptaimtaitally, p vs. shear stress shows a sharp peak at the stresses cturesponding to a transition from the Newtonian plateau to the power-law flow, i.e., to the onset of the elastic behavior (see Fig. 9.9) Lyngaae-J0rgensen et al. 1993, 1999... 

One may conclude, therefore that the microfibrils in the structure must have a sufficiently low diameter in order to induce hard elastic behavior. There are two justifications for this conclusions. Firstly, we have observed that hard elastic polymers contain a large surface tension component in their retractive stress, an observation supported by Miles et. al. [6] and postulated by others [7-9]. The surface component of the stress in Gore-Tex as shown by our experimental evidence and predicted by Equation 1, is simply too small to induce a suitable retractive force. 

Unfortunately, Hooke s Law does not accurately enough reflect the stress-strain behavior of plastics parts and is a poor guide to good successful design. Assuming that plastics obey Hookean based deformation relationships is a practical guarantee of failure of the part. What will be developed in this chapter is a similar type of basic relationship that describes the behavior of plastics when subjected to load that can be used to modify the deformation equations and predict the performance of a plastics part. UnUke the materials that have been used which exhibit essentially elastic behavior, plastics require that even the simplest analysis take into account the effects of... 

The equation for ideal elastic behavior can be expressed as equations 11 or 12, depending on the importance of topological chain entanglements. These can then be compared in predicting of equilibrium shear modulus, G, in the fully cured elastomers. 

Correlation coefficients in composite micromechanics to predict ply elastic behavior. 

The first attempts in the 1960s and 70s proposed to predict elastic properties by using simple analytical models derived from the analysis of fiber-filled media. Takayanagi [210] described the elastic behavior of miidirectional oriented processed fibers. Halpin and Kardos [211] proposed to use the Halpin-Tsai model [212]. Phillips and Patel [213] also applied this model to PE with an adjustable paiameter linked to the crystallite shape ratio. However, these models require the assmnptiou that lamellae be regarded as fibers. Moreover, they are known to fit only the experimental data at low volume fr actions of filler and this is not the case for semicrystalline materials, for which the crystallinity can often reach 60 to 70%. 

More recently, Bedoui et al [221,222] showed that a classical inclu-sion/matrix model (Figure 1.17) associated with a differential scheme provides satisfactory results for predicting the elastic behavior of isotropic semicrystalline polymers. 

For many applications, however, the small-strain behavior is at least as important as the large deformations. As the prediction of elastic behavior of heterogeneous materials is the most known and well-established, it is worth to know if the cmrent micromechaiiical models are able to predict the behavior of semicrystalline polymers. Firstly, the elastic properties of the composite components should be determined. 

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