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We will first investigate the
interaction between two straight and
parallel dislocations of the same
kind. |
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If we start with screw dislocations, we have to distinguish the following
cases: |
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In analogy, we next must consider the
interaction of edge dislocations, of edge and screw dislocations and finally of
mixed dislocations. |
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The case of mixed dislocations - the general case
- will again be obtained by considering the interaction of the screw- and edge
parts separately and then adding the results. |
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With the formulas for the stress and
strain fields of
edge and
screw dislocations
one can calculate the resolved shear stress caused by one dislocation on the glide plane of the other one and get everything from there. |
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But for just obtaining some basic
rules, we can do better than that. We can classify some basic cases without calculating anything by just
exploiting one obvious rule: |
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The superposition of the stress (or strain)
fields of two dislocations that are moved toward each other can result in two
basic cases: |
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1. The combined stress field is now
larger than those of a single dislocation.
The energy of the configuration than increases and the dislocations will
repulse each other. That will happen if
regions of compressive (or tensile) stress from one dislocation overlaps with
regions of compressive (or tensile) stress from the other dislocation. |
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2. If the combined stress field is
lower than that of the single dislocation,
they will attract each other. That will
happen if regions of compressive stress from one dislocation overlaps with
regions of tensile stress from the other dislocation |
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This leads to some simple cases (look
at the stress / strain pictures if you
don't see it directly) |
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1. Arbitrarily curved dislocations with identical b on the same glide plane will always repel each other. |
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2. Arbitrary dislocations with opposite b vectors on the
same glide plane will attract and annihilate each other |
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Edge dislocations with identical or opposite Burgers vector b on
neighboring glide planes may attract or repulse each other, depending on the precise
geometry. The blue double arrows in the picture below thus may signify
repulsion or attraction. |
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The general formula for the forces
between edge dislocations in the geometry shown above is |
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Fx |
= |
Gb2
2π(1 ν) |
· |
x · (x2 y2)
(x2 + y2)2 |
Fy |
= |
Gb2
2π(1 ν) |
· |
y · (3x2 + y2)
(x2 + y2)2 |
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For y = 0, i.e. the same glide plane, we have a
1/x or, more generally a 1/r dependence of the
force on the distance r between the dislocations. |
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For y < 0 or y > 0 we find zones of repulsion
and attraction. At some specific positions the force is zero - this would be
the equilibrium configurations; it is shown below. |
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The formula for Fy is just given for the sake of
completeness. Since the dislocations can not move in y-direction,
it is of little relevance so far. |
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The illustration in the link gives a
quantitative picture of the
forces acting on
one dislocation on its glide plane as a function of the distance to another
dislocation. |
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© H. Föll