Creep diffusional

The curvature has the value 2/R at z = 2np/3 and approximately — 1/p at 2 = 0 (neglecting terms of order p/R). The average curvature gradient —3/(27rp2) can be inserted into Eq. 16.33 for an approximation to the total accumulation at the neck (per neck circumference), 

The corresponding neck surface area is approximately p (per neck circumference), and therefore the neck growth rate is approximately 

Putting Eq. 16.32 into Eq. 16.35 and integrating yields the neck growth law, 

Equation 16.36 predicts that x(t) oc f1/5 and that the neck growth rate will therefore fall off rapidly with time. The time to produce a neck size that is a given particle-size fraction is a strong function of initial particle size—it increases as R4. Equation 16.36 agrees with the results of a numerical treatment by Nichols and Mullins [11].9 

Mass diffusion between grain boundaries in a polycrystal can be driven by an applied shear stress. The result of the mass transfer is a high-temperature permanent (plastic) deformation called diffusional creep. If the mass flux between grain boundaries occurs via the crystalline matrix (as in Section 16.1.3), the process is called Nabarro-Herring creep. If the mass flux is along the grain boundaries themselves via triple and quadjunctions (as in Sections 16.1.1 and 16.1.2), the process is called Coble creep. 

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Ultimately, a knowledge of kinetics is valuable because it leads to prediction of the rates of materials processes of practical importance. Analyses of the kinetics of such processes are included here as an alternative to a purely theoretical approach. Some examples of these processes with well-developed kinetic models are the rates of diffusion of a chemical species through a material, conduction of heat during casting, grain growth, vapor deposition, sintering of powders, solidification, and diffusional creep. 

Equations 3.71 and 3.72 can be further developed in terms of the self-diffusivity using the atomistic models for diffusion described in Chapters 7 and 8. The resulting formulation allows for simple kinetic models of processes such as dislocation climb, surface smoothing, and diffusional creep that include the operation of vacancy sources and sinks (see Eqs. 13.3, 14.48, and 16.31). 

Because only relatively small variations in cy occur in typical specimens undergoing sintering and diffusional creep (Chapter 16), we prefer to carry out the analyses of surface smoothing, sintering, and diffusional creep in terms of atom diffusion and the diffusion potential using Eq. 3.72. In this approach, the boundary conditions on a can be expressed quite simply.15... 

In a process termed diffusional creep, the applied stress establishes different diffusion potentials at various sources and sinks for atoms in the material. Diffusion currents between these sources and sinks are then generated which transport atoms between them in a manner that changes the specimen shape in response to the applied stress. 

Equation 13.3 was first obtained by Herring and is useful in modeling the kinetics of diffusional creep [5] and sintering [6] in pure metals. 

MORPHOLOGICAL EVOLUTION DUE TO CAPILLARY AND APPLIED FORCES DIFFUSIONAL CREEP AND SINTERING... 

CHAPTER 16. MORPHOLOGICAL EVOLUTION DIFFUSIONAL CREEP, AND SINTERING... 

To show that boundary sliding must participate in the diffusional creep to maintain compatibility, suppose that all of the SA, SB, and Sc sliding displacements are zero. Equations 16.44 require that the LA, LB, and Lc must also vanish. Therefore, nonzero Sl s (sliding) are required to produce nonzero grain-center normal displacements. 

Figure 16.5 Deformation mechanism map for Ag polycrystal a = applied stress, p = shear modulus, grain size = 32 pm, and strain rate = IGF8 s 1. The diffusional creep field is divided into two subfields the Coble creep field and the Nabarro-Herring creep field. From Ashby [20].

Finally, mechanisms besides diffusional transport of mass between internal interfaces can contribute to diffusional creep. For instance, single crystals containing dislocations exhibit limited creep if the dislocations act as sources and sinks, depending on their orientation with respect to an applied stress (see Exercise 16.3). 

R. Raj and M.F. Ashby. On grain boundary sliding and diffusional creep. Metall. Trans., 2 1113-1127, 1971. 

B. Burton. Diffusional Creep of Poly crystalline Materials. Diffusion and Defect Monograph Series, No. 5. Trans Tech Publications, Bay Village, OH, 1977. 

E. Arzt, M.F. Ashby, and R.A. Verrall. Interface controlled diffusional creep. Acta Metall., 31(12) 1977-1989, 1983. 

Consider the diffusional creep of the idealized two-dimensional polycrystal illustrated in Fig. 16.4 and discussed in Section 16.2. Each boundary will be subjected to a normal stress, an, and a shear stress, as, as illustrated in Fig. 16.11. Suppose that all boundaries shear relatively slowly at a rate corresponding to... 

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