13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- In Eq. (2), G is shear modulus, b is dislocation Burgers vector, and 1 ≥ k ≥ (1- ) for screw or edge dislocation character respectively with being Poisson’s ratio. Substitution for n in Eq. (1) and taking τ0 = τys - τi, leads to the Hall-Petch equation for yield stress, τys: (3) In Eq. (3), i is a “friction” stress and k'y is now known as the microstructural shear stress intensity. Taking the pile-up length, L, to be equal to the average polycrystal grain diameter, ℓ, then Eq. (3) can be written in terms of tensile stresses: (4) One may view yielding as a process in which the local shear stress at the tip of a slip band builds up with each additional dislocation and eventually reaches a level at the boundary or nearby in the adjacent grain to trigger transmission of plastic deformation through the boundary. The microstructural stress intensity parameter, ky, clearly controls this process and as such is, in principle, both temperature and strain rate dependent in addition to having sensitivity for other aspects of the materials microstructure. The build-up to reach the critical stress is most rapid as the first few dislocations pile up against a boundary. The effect of adding one dislocation to the pile up, n = 1, in Cottrell’s Eq. (1) gives: (5) Thus, the fractional decrease in stress necessary to reach the threshold for spreading plasticity decreases with each additional dislocation added to the pile-up. In essence, this describes the reduction in Hall-Petch stress provided by increasing the slip length (by increasing the grain size), so as to facilitate overcoming an obstacle to plastic flow. The model result is illustrated in Fig. 1. Figure 1. Normalized yield stress vs. reciprocal root of normalized grain size in metals. The core radius is ro [5]. The integers represent the number of dislocations in the pile up. Figure 1 also illustrates the lowered flow stress with an increased number of dislocations. It also illustrates the reason for “scatter” in the mechanical properties of ultra fine grained materials and nanocrystalline materials. Since slip bands cannot have fractional numbers of dislocations, a range
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