Lecture 14 Dislocations and Friends

Lecture 14 Dislocations and Friends - Dislocations...

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Interaction of dislocations with… Case 1: Point defects We’ll start with a vacancy, then generalize. Recapping: u r = C/r 2 ; C needs to be determined some other fashion. u θ = u φ = 0 e rr = du r /dr = -2C/r 3 e θθ = e φφ = u r /r = C/r 3 = 0 σ rr = -2 μ C/r 3 = -2 σ θθ = -2 σ φφ Note that = 0, i.e. there is no volume change for a vacancy. This seems like a contradiction since a vacancy “logically” has less volume. We have to be a little careful about this. Looking again at a vacancy, consider that we create it by cutting out material inside some radius R to create an atomic size void. Assuming that the atoms want to come closer together (reverse signs for the other way) we model this as equivalent to some negative pressure –P. Similar to the precipitate problem, the void will contract until σ rr matches this pressure, i.e. σ rr –P = 0 or P = -2 μ C/R 3 …. Which we can solve for C if P is known. By doing this we have reduced the radius from R to R(1+u r ), i.e. the outer wall has moved in by R*u r . Suppose next that the vacancy sits in a region with some hydrostatic pressure P H , where P H is negative for compression, positive for expansion. I will assume that the vacancy is small enough that we don’t need to be concerned about variations of the hydrostatic pressure across it, i.e. it is essentially a point. There will be an extra work term for moving the outer wall in by R*u r of W = - R*u r *P H (Sign sanity check; u r is negative, contraction with our origin in the center of the vacancy so an expansion has a positive work.) We can lump R*u
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Lecture 14 Dislocations and Friends - Dislocations...

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