Phases elasticity

Here t is the resulting shear stress, 6 is the phase shift often represented as tan(d), and (O is the frequency. The term 6 is often referred to as the loss angle. The in-phase elastic portion of the stress is To(cosd)sin(wt), and the out-of-phase viscous portion of the stress is To(sind)cos(complex modulus and viscosity, which can be used to extend the range of the data using the cone and plate rheometer [6] ... 

Mondragon et al. [250] used unmodified and modified natural mbber latex (uNRL and mNRL) to prepare thermoplastic starch/natural rubber/montmorillonite type clay (TPS/NR/Na+-MMT) nanocomposites by twin-screw extrusion. Transmission electron microscopy showed that clay nanoparticles were preferentially intercalated into the mbber phase. Elastic modulus and tensile strength of TPS/NR blends were dramatically improved as a result of mbber modification. Properties of blends were almost unaffected by the dispersion of the clay except for the TPS/ mNR blend loading 2 % MMT. This was attributed to the exfoliation of the MMT. 

Here, the complex modulus G is expressed as a combination of a real element G (representing the in-phase elastic response, the so-called storage modulus), and an imaginary element G" (representing the dissipated energy, the so-called loss modulus). The ratio of these two moduli is termed the loss factor (tan 6) ... 

In mathematical models of healthy human joints, cartilage is often represented as a single-phase, elastic material with homogeneous and isotropic properties. This approximation is valid, provided that only the short-term response of the tissue is of interest when cartilage is loaded for 1 to 5 seconds, its response is more or less elastic (Hayes and Bodine, 1978 Hori and Mockros, 1976 Mak, 1986). In the long term, however, say more than 1 minute, the response of the tissue is dominated by the nonlinear, viscoelastic properties of creep and stress relaxation (Hayes and Mockros, 1971 Mow et al., 1984). 

Mondragon et al ° reported that unmodified and modified NR latex were used to prepare thermoplastic starch/NR/MMT nanoeomposites by twin-screw extrusion. After drying, the nanoeomposites were injection moulded to produce test specimens. SEM of fractured samples revealed that chemical modification of NR latex enhanced the interfacial adhesion between NR and thermoplastic starch (TPS), and improved their dispersion. X-ray diffraction (XRD) showed that the nanoeomposites exhibited partially intercalated/exfoKated structures. Surprisingly, transmission electron microscopy (TEM) showed that clay nanoparticles were preferentially intercalated into the rubber phase. Elastic modulus and tensile strength of TPS/NR blends were dramatically improved from 1.5 to 43 MPa and from 0.03 to 1.5 MPa, respectively, as a result of rubber modification. 

The steady-state deformation of isolated droplets decreases with increasing dispersed phase elasticity for the same imposed capillary number. A linear relationship between critical capillary number for droplet breakup (Kn-i,) and dispersed-phase Weissenberg number (Wi[Pg.934]

Beginning with the paper by Jackson [20], disturbance stabilization in a fluidized bed is usually associated with the action of specific normal stresses inherent to the dispersed phase. These stresses impede volume deformations of the dispersed phase. Despite this fact having been understood for a long time, comprehensive development of a stability theory is hindered by the almost total absence of reliable information concerning the dependence of dispersed phase stresses (or of the corresponding bulk moduli of dispersed phase elasticity) on the suspension concentration and on the physical parameters. This lack of information partly invalidates all theoretical inferences bearing upon hydrodynamic stability in suspension flow. 

Klusemann B and Svendsen B (2010), Homogenization methods for multi-phase elastic composites Comparisons and benchmarks , Technische Mechanik,yi A),... 

The second important factor is the calculation of draw ratio 2, for composites. This problem is due to heterogeneity of composite structure, consisting of two phases elasticity modulus of one of them is essentially (in some orders of value) more, than the second. Differently speaking, during composite tension process only one phase is extended, namely, rubber, owing to nominal value X not already does describe adequately composite deformation process. This effect is well known and for its quantitative accounting for there are two models. The first of them uses the equation [7] ... 

Since stress and strain are out of phase, elastic energy is dissipated during each cycle. Calculate the energy dissipated per volume in each cycle Hint Use the relation... 

This section contains an outline of the theory and results of experimental studies of the elastic properties of nematics. First, a short introduction of the standard theories is given and the characteristic quantities, used to describe nematic phase elasticity are introduced. After an overview of the standard methods of measuring elastic constants, a summary of the experimental results is given. In particular, we list a collection of papers dealing with the extensively explored cyanobiphenyls and the standard substance 4-methyloxy-4 -butylbenzylidene-aniline (MBBA). The next part is devoted to the less-common surface-like elastic constants, and this is followed by a sketch of the theoretical approaches to the microscopic interpretation of elastic constants and the Landau-de Gennes expansion. The section is concluded by a brief discussion of elastic theory for biaxial nematic phases. 

The variety of data on this homologuous series allows for a comparison of the reliability of different methods. Moreover, the octyl homologue (8-CB) [165-180] exhibits a nematic phase followed by a SmA phase and is therefore particularly interesting for the study of the critical divergence of the Kjj and K22 elastic coefficients on approaching the transition from the nematic to the SmA phase. Elastic coefficient measurements show that short-range pretransitional smectic-Uke order is found more or less in all n-CB homologues [98]. In Fig. 2a-c the elastic constants of the cyanobiphenyl series, as determined by Karat and Madhusu-... 

Lerdwijitjarud W, Sirivat A, Larson RG. Influence of dispersed-phase elasticity on steady-state deformation and breakup of droplets in simple shearing flow of immiscible polymer blends. J Rheol 2004 48(4) 843-862. 

For many processes the vapor pressure can be neglected and also the last two terms must only be considered for small particles (surface influence) or nucleation inside of a solid phase (elastic strain). They can be neglected for any bulk growth processes from the liquid or vapor phase. These assumptions are the basis for the presentation of the technical important T — x phase diagrams. Equation (1.10) is reduced for the case of a two-component system A-B to... 

In order to try to quantify the above elastic effect, we now consider the usual situation, encountered above and throughout the previous chapter, in which the net force on a fluidized particle (or element of the particle-phase) is regarded to be a function of, among other things, particle concentration a. This seems a more appropriate variable, when considering particle-phase elasticity, than void fraction e, with which it is, of course, readily interchangeable ... 

The analysis presented in Chapter 8 was solely in terms of the conservation equations for the particle phase of a fluidized suspension. However, the full one-dimensional description is in terms of the coupled mass and momentum conservation equations for both the particle and fluid phases eqns (8.21)-(8.24). These equations correspond to those derived in Chapter 7, except for the inclusion of the particle-phase elasticity term on the extreme right of eqn (8.22). 

The reason for this paradoxical state of affairs is that particle-phase elasticity in the particle bed model formulation is a purely fluid-dynamic... 

The one-dimensional particle bed model has been formulated in terms of the primary fluid-particle interaction forces, which alone may be considered to support a fluidized particle under steady-state equilibrium conditions, together with particle-phase elasticity, which provides a force proportional to void fraction gradient (or particle concentration gradient) and so comes into play under non-equilibrium conditions. Only axial components of these interactions have been considered so far. Generalizing these considerations to encompass lateral force components is a straightforward matter, but, as we now see, calls for some modification in the constitutive relation for drag in order to unify the axial and lateral constitutive expressions. The following derivations are expressed in terms of volumetric particle concentration a rather than void fraction e a= - . 

In the following section, we make use of this equilibrium relation to provide an estimate for particle-phase elasticity. 

This completes the assembly of the primary force interactions for the two-dimensional formulation. In order to arrive at the two-dimensional counterpart of the particle bed model, it only remains to consider the effect of particle-phase elasticity. 

The total force acting on a single particle is the sum of the net primary force and the force resulting from particle-phase elasticity. The axial component Fp is thus obtained from eqns (16.12), (16.16) and (16.17) ... 

In analogy to the gas phase, elastic scattering of atoms and molecules from surfaces can provide information about the gas-surface potential. Diffraction intensities and bound-state resonances can be analyzed to provide Fourier coefficients of the interaction potential. Classical mechanical scattering patterns exhibiting "rainbows" and "shadows" may provide additional information about the topography of the potential energy surface. Thus, after a lot of effort, box 1 of Fig. 1 shows promise of providing considerable useful data about gas-surface interaction potentials. 

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