Hot Spots from Dislocation Pile-up Avalanches
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Hot Spots from Dislocation Pile-Up Avalanches William R. Grisé Department of Industrial and Engineering Technology, Morehead State University, Morehead, KY 40351, U.S.A. ABSTRACT The model of localized adiabatic heating associated with release of a dislocation pileup avalanche is described and re-evaluated. The model supplies a fundamental explanation of shear banding behavior in metal and non-metal systems. Now, a dislocation dynamics description is provided for more realistic assessment of the hot spot heating, for both straight dislocation pile-ups and circular loop pile-ups. Such localized heating effect was overestimated in the earlier work, in part, to show the dramatic enhancement of the work rate, and corresponding temperature build-up, potentially occurring in the initial pile-up release, say, at achievement of the critical dislocation mechanics-based stress intensity for cleavage. Proposed applications are to potentially brittle metal, ionic, and energetic material systems. INTRODUCTION The dislocation pile-up model is shown in Fig. 1. Two important aspects of the avalanche-assisted enhancement of the local material plastic work rate are derived from the critical condition: n (τa – τo) = τc* for which n is the number of (free) pile-up dislocations, τa is the applied shear component of stress, τo is the lattice friction stress resisting individual dislocation movement, and τc* is the critical component of shear stress. First, substitution of the linear dependence of n on effective stress and slip diameter gives a microstructural stress intensity, ks, evaluated at the (highest) crack nucleation limit as πGb1/2/4α, where G is the shear modulus, b the dislocation Burgers vector and α = 2(1-ν)/(2-ν), with ν being Poisson’s ratio [2]. Thus, n has its largest value at this τc*. Secondly, at sudden pile-up release, the first now free dislocation is driven by the effective stress, (n – 1) (τa - τo), and the one behind by (n - 2) (τa - τ0), and so on [2]. The combined result is an appreciably enhanced work rate with greatest potential temperature rise.
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Figure 1. Stages of dislocation pile-up release [1]. Formation of the pile-up is an isothermal process, followed by catastrophic collapse that releases stored strain energy adiabatically. DISLOCATION AVALANCHES The temperature rises for such dislocation avalanches were over-estimated by the relations 1 ⎡ ⎤ ⎛ 2v k l s ⎢ ⎥ ⋅ ln⎜ 2 K ⎞⎟ ΔT ≤ ⎢ 16 π K ⎥ ⎜ c * v b ⎟ ⎠ ⎢⎣ ⎥⎦ ⎝ or 1 ⎡ ksl 2 ⎢ ΔT ≥ ⎢ 16 π ⎣⎢
(2)
1 ⎤ ⎛ 2 ⎞ v 2 ⎥ ⋅⎜ ⎟ ⎥ ⎜ c * bK ⎟ ⎠ ⎦⎥ ⎝
dependent on whether [2K/c*vb] > 1.0, or < 1.0, respectively [1]. The material constants for metals and ionic solids fit the first condition and those for molecular energetic materials fit the second condition. Substitution of a thermally-activated dislocation velocity for v ⎡ G0 − ∫ b A dτ th ⎤ v = v0 exp ⎢− (3) ⎥ k T ⎣⎢ ⎦⎥
(
)
led [3], then, with A = Wo/bτth and τth proportional to an exponential dependence on the drop-weight height for 50% probability of initiation, H50, to prediction of
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