Elasticity model elastic properties

It is likely that most biomaterials possess non-linear elastic properties. However, in the absence of detailed measurements of the relevant properties it is not necessary to resort to complicated non-linear theories of viscoelasticity. A simple dashpot-and-spring Maxwell model of viscoelasticity will provide a good basis to consider the main features of the behaviour of the soft-solid walls of most biomaterials in the flow field of a typical bioprocess equipment. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

To address this issue, several researchers have developed models of chiral self-assembly. In this section, we review these models. We begin with models based on nonchiral mechanisms and argue that they all have limitations in identifying a mechanism to select a particular tubule radius. We then discuss models based on the elastic properties of chiral membranes and argue that they provide a plausible approach to understanding the formation of tubules and helical ribbons. Most of this discussion was previously presented in our recent theoretical review article.139... 

In this section, we review models for the self-assembly of tubules and helical ribbons that are not based on chiral elastic properties. Our overall assessment is that these models have not been successful in explaining the characteristic length scale for the curvature of tubules and helical ribbons. 

The earliest approach to explain tubule formation was developed by de Gen-nes.168 He pointed out that, in a bilayer membrane of chiral molecules in the Lp/ phase, symmetry allows the material to have a net electric dipole moment in the bilayer plane, like a chiral smectic-C liquid crystal.169 In other words, the material is ferroelectric, with a spontaneous electrostatic polarization P per unit area in the bilayer plane, perpendicular to the axis of molecular tilt. (Note that this argument depends on the chirality of the molecules, but it does not depend on the chiral elastic properties of the membrane. For that reason, we discuss it in this section, rather than with the chiral elastic models in the following sections.)... 

More recently, Smith et al. have developed another model based on spontaneous curvature.163 Their analysis is motivated by a remarkable experimental study of the elastic properties of individual helical ribbons formed in model biles. As mentioned in Section 5.2, they measure the change in pitch angle and radius for helical ribbons stretched between a rigid rod and a movable cantilever. They find that the results are inconsistent with the following set of three assumptions (a) The helix is in equilibrium, so that the number of helical turns between the contacts is free to relax, (b) The tilt direction is uniform, as will be discussed below in Section 6.3. (c) The free energy is given by the chiral model of Eq. (5). For that reason, they eliminate assumption (c) and consider an alternative model in which the curvature is favored not by a chiral asymmetry but by an asymmetry between the two sides of the bilayer membrane, that is, by a spontaneous curvature of the bilayer. With this assumption, they are able to explain the measurements of elastic properties. 

The simplest model of tubule formation based on chiral elastic properties was developed by Helfrich and Prost.180 They considered the elastic free energy of... 

ICC Termination Act of 1995, 25 331, 326 Ice. See also Water entries elastic properties, 5 614t hydrogen-bonded structure of, 26 15 properties of, 26 17t Ice wines, 26 315 Iceberg model, 23 95 Ice formation, in food processing, 72 82 Iceland, bioengineering research program, 7 702... 

The relaxation spectrum H is independent of the experimental time t and is a fundamental description of the system. The exponential function depends upon both the experimental time and the relaxation time. Such a function in the context of this integral is called the kernel. In order to describe different experiments in terms of a relaxation spectrum H or retardation spectrum L it is the kernel that changes. The integral can be formed in time or frequency depending upon the experiment being modelled. The inclusion of elastic properties at all frequencies and times can be achieved by including an additional process in the relaxation... 

For crystals with molecule-like constituents, like the BO, " and BO4 " groups in some borates, semi-quantitative models of the molecular component as a gas-phase entity have been proposed (Oi et al. 1989). This is conceptually similar to the approximation made for species in solution, although in practice most studies of crystals consider additional frequencies that reflect inter-molecular vibrations. The spectroscopic data on these vibrations (which typically have lower frequencies than the intra-molecular vibrations) are often available, at least approximately, from infrared and Raman spectroscopy and elastic properties. This type of hybrid molecule-in-crystal model has been applied to many minerals in theoretical studies of carbon and oxygen isotope fractionation, the most noteworthy being studies of calcite (Bottinga 1968 Chacko et al. 1991) and sihcates (Kieffer 1982). Because specfroscopic dafa are always incomplete (especially for subsfances substifufed wifh rare isolopes), some amounl of vibralional modeling is necessary. 

Hsu and Berzins used effective medium theories to model transport and elastic properties of these ionomers, with a view toward their composite nature, and compared this approach to that of percolation theory. ... 

For coarse-grained models of linear biopolymers—such as DNA or chromatin— two types of interactions play a role. The connectivity of the chain implies stretching, bending, and torsional potentials, which exist only between directly adjacent subunits and are harmonic for small deviations from equilibrium. As mentioned above, these potentials can be directly derived from the experimentally known persistence length or by directly measuring bulk elastic properties of the chain. 

Beyond pure geometry, the two-angle model is also useful to predict some of the physical properties of the 30-nm fiber, for instance, its response to elastic stress [17]. In an independent study on the two-angle model by Ben-Haim et al. [76] this question has been the major focus, and as demonstrated by Schiessel [72], the elastic properties of the two-angle model as a function of 6 and are analytically solved completely by combining the results from both papers. 

H.M. Princen Rheology of Foams and Highly Concentrated Emulsions I. Elastic Properties and Yield Stress of a Cyhndrical Model System. J. Colloid Interface Sci. 91, 160 (1983). 

Mapping of the elastic modulus of the glassy and rubbery blocks and clay regions was probed by employing Hertzian and Johnson-Kendall-Roberts (JKR) models from both approaching and retracting parts of the force-distance curves. In order to determine the elastic properties of SEBS nanocomposites in its different constituting zones, the corrected force-distance curve was fitted to the Hertz model ... 

No model is really exact unless the geometry of the phases is known as well as the elastic properties of the phase in its form as present (which may differ from the values in a bulk phase). 

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