Reuss model

We prove an existence of solutions for the Prandtl-Reuss model of elastoplastic plates with cracks. The proof is based on a special combination of a parabolic regularization and the penalty method. With the appropriate a priori estimates, uniform with respect to the regularization and penalty parameters, a passage to the limit along the parameters is fulfilled. Both the smooth and nonsmooth domains are considered in the present section. The results obtained provide a fulfilment of all original boundary conditions. 

Mechanical tests indicate that these blends do not behave like conventional blends and suggest that the polystyrene phase is continuous in the substrate. The moduli of the blends as a function of blend composition is plotted in Figure 10.6. The Voigt and Reuss models are provided for comparison (Nielsen, 1978) These are the theoretical upper and lower bounds, respectively, on composite modulus behavior our data follows the Voigt model, suggesting that both the polystyrene and polyethylene phases are continuous. In most conventional composites of polystyrene and HDPE, the moduli fall below the Voigt prediction indicating that the phases are discontinuous and dispersed (Barentsen and Heikens, 1973 Wycisk et al., 1990). 

The other limiting case (Reuss model) of a two component material is a sandwich structure of alternating layers of high and low modulus materials loaded perpendicular to the layer plane In this case the stress is uniformly distributed within the sample. The resulting modulus is given by ... 

We can see from Equations (90) or (91) that in the frame of the Voigt model of the crystallite interactions the relative peak shifts do not depend on the Miller indices, which frequently is contradicted by experiment. The Reuss model gives such a dependence. 

Reuss Model. In the Reuss hypothesis the intergranular stress in the sample system is zero and then ... 

Kroner Model. A model for crystallite interaction that is better than the Voigt or the Reuss models was proposed by Kroner. According to Kroner every crystallite is an inclusion in a continuous and homogenous matrix that has the elastic properties of the polycrystal. For the isotropic polycrystal the strain in the inclusion is the following ... 

X3 = 5,54 = 55 = 56 = 0, one obtains (sh) = (3Ai + 82/2)3. There is no dependence of (sh) on the Miller indices for the Voigt and the Kroner model. For the Reuss model the dependence is similar to those from Equation (115) and Table 12.8 but with only one refinable parameter for all Laue groups, the macrostress 5, and very probably the refined value of 5 will be wrong. 

This approximation is called the Voigt model, and the value of the elastic modulus is often known as the Voigt bound. The expression is identical to that for a continuous aligned fibre composite under a longitudinal load, and gives the elastic modulus when the load is applied parallel to the sheets. Similarly, if the stress is applied perpendicular to the layers, and an iso-stress condition applies (the Reuss model), the elastic modulus is ... 

FIGURE41.3 Stiffness vs. volume fraction for Voigt and Reuss models, as well as for dilute isotropic suspensions of platelets, fibers, and spherical particles embedded in a matrix. Phase moduli are 200 and 3 GPa. 

Calculate using the Voight and Reuss models the bounds for Young s modulus of Mg0-Al203 composites as a function of volume fraction. 

As shown in Figures 14.5 and 14.6, the magnitude and draw ratio dependence of the moduli 11 13 < 44 < 66 2nd are well predicted by the Reuss model. However, or is close to the Reuss bound... 

Figure 14.21 Draw ratio dependence of the (a) axial and (b) transverse Young s moduli of PC + VBlO. The theoretical predictions based on 3 and j values derived from the Voigt and Reuss model are shown as solid and dashed curves, respectively. The dot-dashed curves are theoretical predictions obtained by taking the observed moduli of Vectra B as 3 and . (Adapted from [10] by permission of the Society of Plastic Engineers.)...

The rule of mixtures is useful in roughly estimating upper and lower bounds of mechanical properties of an oriented fibrous composite, where the matrix is istropic and the fiber orthotropic, with coordinate 1 the principal fiber dfrection and coordinate 2 transverse to it. For the upper bound, the Voight model is used (Fig. 12.3), where it is assumed that the strain is the same in the fiber and matrix. For the lower bound, the Reuss model is used, where the stress is assumed to be the same. This gives the following equations for composite moduli ... 

The rule of mixtures equations have several drawbacks. The isostrain assumption in the Voight model implies strain compatibility between the phases, which is very unlikely because of different Poisson s contractions of the phases. The isostress assumption in the Reuss model is also unrealistic since the libers cannot be treated as a sheet. Despite this, these equations are often adequate to predict experimental results in unidirectional composites. A basic limitation of the rule of mixtures occurs when the matrix material yields, and the stress becomes constant in the matrix while continuing to increase in the fiber. 

Fig. 17.7 Fittings on the evolution of the modulus of LCFo i-based biocomposites versus filler volume fraction content (Takayanagi, Voigt and Reuss models). Reproduced with permission (Averous and Le Digabel 2006). Copyright of Elsevier...

The mechanics of materials approach is the simplest and least useful. It assumes that each phase is subjected either to the same strain (the Voigt model Equation 9.4) or the same stress (the Reuss model Equation 9.5). This yields the relationships ... 

Figure 9.4 Comparison of Voigt and Reuss models with experimental data for a urethane methacrylate polymer (no polyol soft-block) filled with silica sand...

Figure 2.14 Nonlinear rheological models, (a) Saint-Venant model (b) PrandtI-Reuss model (c) Bingham model.

Nonlinear models of rheological behavior can be approximated by step functions, whereby the existence of a finite yield stress G plays a dominant role. Three typical nonlinear models include the Saint-Venant model of ideal plastic behavior, the Prandtl-Reuss model of an elastoplastic material, and the Bingham model of viscoelastic behavior. The first model can be mechanically approximated by a sliding block, the second by a Maxwell element and a sliding block in series, and the third by a dash pot damping element and a sliding block in parallel (Figure 2.14). 

Below the yield stress 0 the rate of shear deformation (strain) (d /dt) is zero in all three nonlinear models. This is also true of the shear strain /for the Saint-Venant and Bingham models, whereas in the Prandtl-Reuss model /increases slowly with the shear stress t from zero to the yield stress 0. At this point, the value of /increases limitless as a step function, as it does in the two other models (Figure 2.15). 

Vitallium (CoCrMo alloy), 38-6-38-10 Viviani, E, 77-7 Voigt and Reuss models, 41-2-41-3 Volta, A., 29-3 Volterra-Wiener approach... 

Both Voigt and Reuss models provide initial estimates of the upper and lower bounds of elasticity of multiphase composites with the only consideration of the inclusion volmne fraction but irrespective of inclusion shape/geometry, orientation and spatial arrangement. 

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