Shrinkage 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... 

BostrOm, B., Svan0, G., Horsrud, P., Askevold, A., 1998. The shrinkage rate of KCTexposed smectitic North Sea shale simulated by a diffusion model. SPE 47254, Proc. Eurock 98, Trondheim, 8-10 July. 

Mahabadi and O Driscoll also studied the effect of dissolved polymer upon the termination rate coefficient [82, 83]. They derived a theoretical relationship for the dependence of kt upon conversion [82], based on a previously derived segmental diffusion model [100] that allowed for a concentration dependent linear expansion coefficient and polymer-solvent interactions. Also Mahabadi and O Driscoll pointed out that the rate of segmental diffusion would increase if the segment density gradient was increased as a result of coil shrinkage. In a simplified form, the dependence of kt upon polymer concentration, C, could be represented by ... 

Fig. 1. Shrinkage isotherms for Alcoa A-14 plotted according to the grain boundary diffusion model.

Fig. 5. Shrinkage isotherms for pure and titania-doped Linde Cl.O alumina plotted according to the volume diffusion model. T = 1300°C. Data from Bagley (12). 

An expression of the same linear form will be obtained for the growth of a needle if the tip is modeled as a hemisphere. Further results bearing on the diffusion- or interface-limited growth (and shrinkage) of particles have been reviewed by Sutton and Balluffi [6]. 

Solution The full numerical model needs to include shrinkage since the material is 50 percent water initially and the thickness will decrease from 100 to 46.5 lm during drying. Assuming the layer is viscous enough to resist convection in the liquid, diffusion is the dominant liquid-phase transport mechanism. 

Abstract A united mathematical model for the rheological and transport properties of saturated clays is proposed. The foundation of the model is the unification of filtration s consolidation theory and the theory of the stability of lyophobic colloids, which is based on the conception of disjoining pressure as a surplus in relation to hydraulic pressure. This pressure is caused by surface capacities and exists in water films between clay particles. In this work it is shown that the problem of the shrinkage of a clay layer can be reduced to the well known problem. We obtained the approximate solution for pressing the water out of a clay layer. The solution that we obtained requires introduction of a concept for the limit shear stress for clays. We investigated the model, and explained some characteristic features of transfer processes in clays (the existence of anomalous high pressures in clays, the flocculation at diffusion in clays, etc.). It is shown that solutions which we received are in harmony with results of experiments. 

Further development of this model incorporating diffusion, bulk flow, transmembrane flux, and matrix shrinkage [42-44] showed that the cell membrane is the main barrier to mass transfer only for single cells or thin slices of tissue. When the thickness of the sample increases, the extracellular space may become the limiting factor [45]. 

Johnson DL, Cutler IB (1963) Diffusion sintering. 1. Initial stae sintering models and their application to shrinkage of powder compacts. J Am Ceram Soc 46 541-545... 

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