Appendix 2 The Characteristic Length L |
For a physical picture of the length L, consider two compressive stresses s1 and s2, and a small element of material that is part of, and embedded within, a larger extent. If s1 is imposed on the element north-south and s2 is imposed east-west, a north-south shortening strain rate will be present, --- (s1-s2)/4.(viscosity) in plane strain or (s1-s2)/3.(viscosity) with cylindrical symmetry . Now consider a different situation where one site within the material is compressed hydrostatically by s1 and a site not far away is compressed hydrostatically by s2; in this set-up, there is radial shortening at the high-stress s1 site because of self-diffusion of material away to the low-stress s2 site. The rate of radial shortening depends on the separation-distance of the two sites, and there is some separation distance such that the radial shortening rate by self-diffusion equals the viscous shortening rate in the first situation. This particular separation-distance is the length L (or, in some formulations, a small multiple of it such as L/2).
In most practical situations, L is less than a micrometer. In fact, there is an inherent awkwardness: the manner in which L is defined above assumes that the material is ideally continuous, whereas both creep and self-diffusion depend on the material having microstructure, such as atoms and dislocation loops; and L is so short that, on the scale of L, one sees the microstructure, --- one cannot reasonably treat the material as a continuum. A resolution is as follows: we admit that every material is atomic; this includes admitting that "homogeneous" plane strain involves atoms moving around, dislocation loops expanding or shrinking and so on; then in homogeneous plane strain, there is an average distance an atom moves in contributing to the strain process. Where the dominant mechanism is dislocation-climb, for example, the average distance would be of the order of magnitude of the length of a dislocation or the separation of one dislocation from the next. A second view is that L is an estimate of this average distance. Fortunately, the two views of L are, I think, wholly compatible. If one works wholly at the macro-scale using material slabs as in Figure 6, one can measure viscosity N and diffusivity K, respectively in Pa-sec and m2-Pa-1-sec-1. Then (4NK)1/2 is a distance, --- determined macroscopically but informing us about the microstructure; I think it tells us the average distance a participating atom moves when the material undergoes homogeneous deformation, or is a good indicator of that distance. (Of course, (NK)1/2 --- without the 4 --- is also a distance. As a purely technical point, to define L as (4NK)1/2 leads to neater equations, but currently estimates of K are so uncertain that the factor of 4 has no practical significance.) A third view of L or L2 is gained if we use the idea of a material’s mobility m, with m = 1/N. Then L2 = 4K/m or m = 4K/L2. For an atomic material with self-diffusion coefficient K, the shorter the distance L that an atom has to travel in contributing to change of shape, the greater the mobility m with which the material will deform. |