Stress diffusion models

An account of the mechanism for creep in solids placed under a compressive hydrostatic stress which involves atom-vacancy diffusion only is considered in Nabarro and Herring s (1950) volume diffusion model. The counter-movement of atoms and vacancies tends to relieve the effects of applied pressure, causing extension normal to the applied stress, and shrinkage in the direction of the applied stress, as might be anticipated from Le Chatelier s principle. The opposite movement occurs in the case of a tensile stress. The analysis yields the relationship... 

However, if a Reynolds-stress model is used to describe the turbulence, a modified gradient-diffusion model can be employed ... 

In transported PDF methods (Pope 2000), the closure model for A, V, ip) will be a known function26 ofV. Thus, (U,Aj) will be closed and will depend on the moments of U and their spatial derivatives.27 Moreover, Reynolds-stress models derived from the PDF transport equation are guaranteed to be realizable (Pope 1994b), and the corresponding consistent scalar flux model can easily be found. We shall return to this subject after looking at typical conditional acceleration and conditional diffusion models. 

The literature contains reviews of air quality modeling that stress special purposes. Some concentrate on meteorologic aspects, and others combine this with air chemistry. Proceedings of several conferences are another information resource. Recent surveys have been addressed specifically to photochemical modeling problems. It may be concluded that, although they are relatively complex, the photochemical-diffusion models perform as well as, if not better than, available inert-species models. 

Traditional turbulence-diffusion models (based on the boundary layer adjoining a solid wall) imply that n = 2/3, but a value n = 1/2 is appropriate for a boundary layer adjoining a free surface (Jahne and Haussecker 1998). The appropriate value of n depends on the wind stress and the surfactant loading of the surface. Soloviev and Schluessel (1994) have described a procedure for estimating gas transfer velocities from measurements of heat transport, assuming that transport is adequately described by the classical (Danckwerts) surface renewal model. The key relationship can be written in the form ... 

To incorporate the dispersive stress effect, the diffusion model must first be written as a form of force balance. Including the hindered settling correction, the appropriate equation 27 would be... 

Various types of coupled non-linear Fickian diffusion processes were numerically simulated using the free-volume approach given by equation [12.8], as well as non-Fickian transport. The non-Fickian transport was modeled as a stress-induced mass flux that typically occurs in the presence of non-uniform stress fields normally present in complex structures. The coupled diffusion and viscoelasticity boundary value problems were solved numerically using the finite element code NOVA-3D. Details of the non-hnear and non-Fickian diffusion model have been described elsewhere [14]. A benchmark verification of the linear Fickian diffusion model defined by equations [12.3]-[12.5] under a complex hygrothermal loading is presented in Section 12.6. 

We first consider the ].( ) term for arbitrary bead-spnng models in which all beads have the same mass m and the same fnction coefficient C then we investigate the i (2) term for the Rouse chain model. Next we give a derivation of a stress-diffusion relation for the simplified model of Hookean dumbbells (that is, a Rouse chain with N = 2), which makes use of the j ,(2) term. Then, we show how the use of the jj(3) term leads to a different result. These discussions and Appendix B are helpful in understanding the nature of the series expansions and some of the problems associated with them, because they are not expansions in a physical parameter. 

A Stress-Diffusion Relation for General Bead-Spring Models... 

Incident-related releases of air pollutants often occur within seconds or minutes. In order to be able to estimate these stresses, the model of an exhaust air plume, which is the basis for the basic equation (7-1), is replaced by a model which, in addition to a turbulent diffusion at right angles to the wind direction, also includes a turbulent diffusion in the direction of the wind. In this manner, short-term emissions or emissions... 

The previous drying experiment ona gel rod is used to assess the model. Moisture and temperature evolution are fairly well reproduced by the simulation and maximal tensile stresses are obtained that are similar to those computed with the diffusion model However, the numerical Uquid pressure attains unrealistically low values, which is a result of the non-penetration assumption for gas. Here, the missing link of the macroscopic model to microstructural properties, that is, pore size, becomes obvious. 

In the absence of the last two terms, we obtain w = [— stress-diffusion couphng. Finally, we remark that our dynamic model ensures the nonnegative definiteness of the heat production rate [10]. Then we can generally prove that the system tends to a homogeneous equilibrium state with W — I = V = w = 0 as t ->QO in the absence of macroscopic flow field. 

Intercalation-induced stresses have been modeled extensively in the Hterature. A one-dimensional model was proposed to estimate stress generation in the lithium insertion process in the spherical particles of a carbon anode [24] and an LiMn204 cathode [23]. In this model, displacement inside a particle is related to species flux by lattice velocity, and total concentration of species is related to the trace of the stress tensor by compressibihty. Species conservation equations and elasticity equations are also included. A two-dimensional porous electrode model was also proposed to predict electrochemicaUy induced stresses [30]. Following the model approach of diffusion-induced stress in metal oxidation and semiconductor doping [31-33], a model based on thermal stress analogy was proposed to simulate intercalation-induced stresses inside three-dimensional eUipsoidal particles [1]. This model was later extended to include the electrochemical kinetics at electrode particle surfaces [2]. This thermal stress analogy model was later adapted to include the effect of surface stress [34]. 

Figure 16 The solvent-squeeze model for the shear-enhanced concentration fluctuations driven by stress-diffusion coupling. Based on Saito, S. Matsuzaka, K. Hashimto, T. Macromolecules 32,4879. Copyright (1999) American Chemical Society.

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

The interdiffusion of polymer chains occurs by two basic processes. When the joint is first made chain loops between entanglements cross the interface but this motion is restricted by the entanglements and independent of molecular weight. Whole chains also start to cross the interface by reptation, but this is a rather slower process and requires that the diffusion of the chain across the interface is led by a chain end. The initial rate of this process is thus strongly influenced by the distribution of the chain ends close to the interface. Although these diffusion processes are fairly well understood, it is clear from the discussion above on immiscible polymers that the relationships between the failure stress of the interface and the interface structure are less understood. The most common assumptions used have been that the interface can bear a stress that is either proportional to the length of chain that has reptated across the interface or proportional to some measure of the density of cross interface entanglements or loops. Each of these criteria can be used with the micro-mechanical models but it is unclear which, if either, assumption is correct. 

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