Effective medium approach Elasticity

It has been argued that the metals do not act as a collection of atoms, but rather as a uniform entity. Thus various properties of the metal, such as elastic constants, are not accurately described by pair potentials [146]. If that is so, a model which assumes a pairwise atom-atom interaction will perhaps have to be replaced with one which incorporates the influence of electron density from a distributed electron gas rather than from localized ones. Such theories can furthermore be based on the Hohenberg Kohn theorem [211], namely that the energy is a functional of the electron density. Thus several approximate theories are based on the electron gas concepts, for example, the embedded atom and diatom methods, or the effective medium approach [210]. 

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. ... 

More recently Morse produced a complete microscopic tube theory for stiff polymers that successfully interpolates between the rigid-rod and flexible chain limits. This theory explains many features of semiflexible polymer rheology, including the two mechanisms for plateau moduli described above (which depend on a comparison of timescales), with the tube diameter being the sole fitting parameter as in the Doi-Edwards theory. More recently, Morse successfully computed a tube diameter from two different approaches (self-consistent binary collision and continuum effective medium) that give similar results, e.g. modulus G p and respectively). An elastic network approximation... 

The self-consistent (SC) solutions are an approximate approach, in which a particle of one phase of given shape (e.g., spheres, needles or discs) is surrounded by the composite material. This is illustrated in Fig. 3.12, in which a spherical particle is surrounded by an effective medium that represents the elastic properties of the composite. For the spherically shaped particles, one can again use Eq. (3.14) but now H=4pJ i is used for the bulk-modulus calculation and... 

The SC solutions appear to run into difficulties when there is a large elastic mismatch between the constituent phases, for example, at high concentrations of a rigid phase in a compliant matrix or of a porous phase in a stiff matrix. The latter situation will be discussed in Section 3.6. One approach to this problem is known as the Generalized Self-Consistent Approach, the concept behind which is illustrated in Fig. 3.14. Instead of a single inclusion in an effective medium, a composite sphere is introduced into the medium. As in the composite sphere assemblage discussed in the last section, the relative size of the spheres reflects the volume fraction, i.e., V = a bf as before. Interestingly, this approach leads to the HS bounds for the bulk modulus. The solution for the shear modulus is complex but can be written in a closed form. 

Figure 3.14 The generalized self-consistent approach in which a composite particle is embedded in an effective medium that possesses the elastic properties of the composite. The relative sphere size in the composite particle represents the volume fraction of the phases.

The water content is the state variable of PEMs. Water uptake from a vapor or liquid water reservoir results in a characteristic vapor sorption isotherm. This isotherm can be described theoretically under a premise that the mechanism of water uptake is sufficiently understood. The main assumption is a distinction between surface water and bulk water. The former is chemisorbed at pore walls and it strongly interacts with sulfonate anions. Weakly bound bulk-like water equilibrates with the nanoporous PEM through the interplay of capillary, osmotic, and elastic forces, as discussed in the section Water Sorption and Swelling of PEMs in Chapter 2. Given the amounts and random distribution of water, effective transport properties of the PEM can be calculated. Applicable approaches in theory and simulation are rooted in the theory of random heterogeneous media. They involve, for instance, effective medium theory, percolation theory, or random network simulations. 

Zhao (1994) presented a model of coupled coal deformation and methane migration based on a consolidation theory of elastic medium with Darcy fluid flow and the Terzaghi effective stress law, and its numerical solution technique and applications to practical problems. Works using similar approaches were also reported in Liang et al. (1995,1996), Sun and Xian (1999), Ding et al. 

The results of Clayfield and Lumb relate entirely to the loss of configurational entropy of the polymer chains on close approach of the particles, due either to the presence of the impenetrable surface of the opposite particle or the polymer chains that are attached to that particle. In the early papers, the effect of the solvent on the conformation of the macromolecules was ignored but an attempt was made to include the role of solvency in some of the later publications. Notwithstanding this, essentially what Clayfield and Lumb calculated was the elastic contribution to Ae repulsive free energy of interaction between sterically stabilized particles. As such, their results are manifestly unable to explain the observed flocculation of sterically stabilized particles that is induced by decreasing the solvency of the dispersion medium. Even if only for this reason, the assertion by Osmond et al. (1975) that the Clayfield and Lumb theory was the best available at that time is clearly untenable. 

Fracture mechanics for brittle materials is basically derived from the theory presented by Griffith (1920). Initially it was proposed for an elliptical crack in an infinite elastic and homogeneous medium subjected to distant tensile forces and the conditions for crack propagation were formulated for that case. Later, the brittle fracture mechanics were developed for various situations in real structural elements, with non-negligible plastic deformations and with several complications necessary to account for heterogeneity of materials, time effects, etc. In that approach the crack s appearance and propagation is considered as a basic effect of loading and as phenomena directly related to final failure. 

A study was made of gas decompression failures in elastomeric seals using a fracture mechanics approach with considerations of gas permeation. An equation is proposed for the tearing energy associated with crack growth from internal gas bubbles in a finite thickness elastic media. The effects of gas pressure, temperature, rate of decompression and mechanical strain were studied for a range of elastomers used in oil and gas sealing applications. A theoretical treatment is presented based on a fracture mechanics criterion for fracture from an internal disc shaped flaw in a thick elastic medium. Permeation theory provides a quantification of the amount of gas available internally to initiate failures. 21 refs. 

A set of analytical expressions has been developed from the perspective of LBA for the elasticity, extensibility, and mechanical strength of low-dimensional systems in terms of bond order, bond length, bond strength, and their response to the coordination environment, temperature, and stress field. The effect of a broken bond on the identities of the remaining bonds between the undercoordinated atoms dominates the mechanical performance and thermal stability of the mesoscopic systems. The presented approaches connect the macroscopic properties to the atomistic factors by developing the functional dependence of the measurable quantities on the bonding identities and the response of the bonding identities to external stimulus, which complement the classical theories of continuum medium mechanics and statistic thermodynamics that have demonstrated the limitation to mesoscopic systems. The developed approaches also provide predictive... 

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