Interaction of Straight Dislocations With Impurity Atom: Elastic Interaction Between Edge Dislocations and Solute Atoms
Interaction of Straight Dislocations With Impurity Atom: Elastic Interaction Between Edge Dislocations and Solute Atoms
Interaction of Straight Dislocations With Impurity Atom: Elastic Interaction Between Edge Dislocations and Solute Atoms
Impurity Atom
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than that of a similar atom in an undistorted portion of the
lattice This difference in energy is the interaction energy U]. Ul
is invariably negative in sign for impurities that are attracted
towards the dislocation. This happens because the edge
dislocation has both tensile and compressive hydrostatic
components of stress field which lower the potential energy
of the impurity depending on the sign of the volume
dilatation that the impurity produces in the surrounding
crystal. The self-energy of the edge dislocation is also lowered
in the process. The impurity will then move to a lowest energy
site in the immediate proximity to the edge dislocation This
may happen as the dislocation glides through the crystal
containing these defects or by annealing at a high enough
temperature enabling the defects to diffuse to such favorable
locations When such a large number of these impurity atoms
redistribute themselves around the edge dislocation. there is
an overall reduction in the free energy of the crystal and a
higher stress is needed to dislodge the dislocation from such
an atmosphere of foreign atoms. This kind of interaction not
only is of the elastic type. but can also be chemical interaction
(formation of second-phase particles and electrical interaction
(Columbic type)The screw dislocations. however. do not
interact can interact if the distortions are asymmetric in
anisotropic materials. screw dislocations have zero interaction
-with spherical foreign atoms
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Interaction between Screw Dislocations and
Solute Atoms Tetragonal Distortions:
In the previous analysis. only the hydrostatic stress
component was considered which yielded a direct first-order
interaction with a solute which caused a spherically
symmetric distortion in a matrix. Interaction could then occur
with any dislocation which has some edge character. A screw
dislocation which does not possess a hydrostatic component
of stress does not exhibit this interaction to the first order-
But the former interaction is only a special case of a more
general size effect interaction Solute atoms in crystals occupy
interstitial sites in the matrix which are far from spherical and
thereby produce asymmetric distortions. Such centers of
distortion can interact with both hydrostatic and shear stress
fields and hence can interact with both edge and screw
dislocations. A case of special interest IS C in bcc -fcc
causing tetragonal distortion around it. which can then
interact with the shear stress field of a screw dislocation and
thus a second-order interaction
Yield point:
Carbon and nitrogen, even in concentrations as low as 0.005
wt%, in iron lead to a sharp transition between elastic and
plastic deformation in a tensile test performed on ferritic iron
(Fig. 2.4a). Decarburization of the iron results in the
elimination of this sharp transition or yield point, which
implies that the solute atoms are in some way responsible for
this striking behavior. Frequently the load drops dramatically
at the upper yield point (A) to another value referred to as the
lower yield point (B). Under some experimental conditions,
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yield drops of about 30% of the upper yield stress can be
obtained. Following the lower yield point, there is frequently
a horizontal section of the stress-strain curve BC) during which
the plastic deformation propagates at a front which can move
uniformly along the specimen. This front is referred to as a
Luders band( Fig. 2.4b), and the horizontal portion, BC, of the
stress-strain curve as the Luders extension. The development
of Luders bands can be much less uniform and, e.g., in
pressings where the stress is far from uniaxial, complex arrays
of bands can be observed. These are often referred to as
stretcher strains, but. they are still basically Luders bands.
When the whole specimen has yielded, general work
hardening commences and the stress-strain curve begins to
rise in the normal way. If, however, this deformation allowed
to rest either at room temperature or for a shorter time at
100-150C
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Fig. 2.4 (a) Schematic diagram of yield phenomena as shown in a tensile test. (b) Luders bands in deformed steel
specimens (Hall.Yield Point Phenomena in Metals and Alloys (c) Luders bands in a notched steel specimen
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u
Figure 4.36 Stress-strain curve of -Fc containing some carbon in solid solution y and yl are the
Upper and lower yield points. Respectively
.
2-When the precipitates are small, weak, closely spaced and
coherent with the matrix; dislocations may penetrate the
precipitates and shear them
.
*Process (1) encourages uniformly distributed slip because it
is harder to force a second dislocation past a strong obstacle
that already is surrounded by a dislocation loop
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Different Types of Interactions between
Dislocations and Impurity Atoms:
Addition of solute atoms generally raises the yield strength
and the level of the stress -strain curve of many metals. alloys
and ceramic materials. This is due to the interactions that
arise between a dislocation and an impunity during
dislocation glide. Principal types of interactions between
dislocations and impurity (solute) atoms can be considered
under the following classification:
7- Snoek ordering
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(4.130). Both these interactions are equally strong and the
latter can
Electrostatic Interaction:
Unlike the elastic interaction. the electrostatic Coulombic
interaction is different in character and magnitude in different
types of crystals (Nabarro. 1967). In ionic crystals. a straight
edge dislocation line contains excess charges of alternating
positive and negatiye signs. Then an electrical interaction
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arises because such a dislocation can attract clectrostatically
charged impurities of both signs (vacancies or interstitials
carrying charge if). However. this attraction can occur only
over very short distances. because the alternating charges
along the dislocation line annul at large distances In metals.
the tensile and the compressive hydrostatic stress field
associated with an edge dislocation produce a redistribution
of the free conduction electrons around the dislocation
creating an electric dipole which can interact with differently
charged impurities at short distances. Since the upper half of
the glide plane is compressed (material more dense) and the
lower half is under tension (expanded lattice). as well as the
Fermi energy has to remain the same everywhere. electrons
leave the denser region and enter the expanded region.
Thereby lowering their kinetic energy and acquiring a positive
electrostatic potential energy a. The charge redistribution
along the edge dislocation line will then resemble a line
dipole. The dipole can interact with a solute atom of a
different valiancy. Because of the screening by the conduction
electrons. this interaction gets reduced to negligible
proportions compared to the elastic size effect and the other
inhomogeneity effects such as modulus interaction. In ionic
crystals. however. where there are no screening effects. this
interaction is significant particularly b at charged jogs and can
surpass the elastic interaction.
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from that of the matrix- For example. a substitution impurity
which is softer than the solvent will have a loner modulus than
that of the matrix Since the self-energy of a dislocation is
proportional to the modulus. a solute softer than the matrix
atom will then attract a moving dislocation. This is not always
an attractive interaction. but both edge and screw dislocations
interact with the defect. if the solute atom is hard compared to
the solvent atom. there m'll be repulsion between the
dislocation and the solute atom Consider the geometry in
Figure 4.49. where an edge dislocation is shown to interact
with a misfit solute atom with the elastic constant different
from that of the matrix
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Short-range order interaction Fisher:
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SOLUTE SEGREGATION AT GRAIN BOUNDARIES:
Let A be the solvent atom. B the solute atom. XbA. XbB. X A and
XB their mole fraction at the grain boundary and in the bulk.
respectively. Equation (7.2) is then expressed as:
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A number of models and theories of grain boundary
segregation have been proposed. The simplest of these.
McLeans model is basically an application of the Langmuir-
type surface adsorption to grain boundary adsorption and
assumes mono-layer segregation of single adsorbate without
interference between solvent and solute atoms (no site-to-site
interaction). The assumption of no interference means that
for any site. the probability of solute segregation assumption
of the regular solution model McLean suggested the major
driving force to be the elastic strain energy around solute
atoms in the lattice that results from lattice distortion The
lattice distortion energy, W(E1). generated by a solute atom in
the bulk can be calculated by using elastic continuum theory
as:
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surface energy difference and mixing enthalpy. According to
these authors, the heat of segre gation of a solute atom h is
expressed as:
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Since the surface energy is. in general, 2-3 times the grain
boundary energy. the effect of the interfacial energy
difference should be higher for the surface than for the grain
boundary. Solute segregation increases as the value of mixing
enthalpy increases so that there is an increased tendency of
having a miscibility gap in the phase diagram. The three
contributions to segregation enthalpy in liq. (7.6) vary with
the system concerned. However, the variation of the elastic
strain energy contribution is usually the largest in metal(For
grain boundary segregation in ionic compounds)
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of solute atoms a similar dragging effect results. Calm, and
Liicke and Sttiwe have theoretically analysed the solute drag
effect of a grain boundaryTo calculate this drag force the
solute distribution around the moving boundary must be
found
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distribution around the grain boundary is symmetric for a
stationary boundary and asymmetric for a moving boundary
The solution also shows that the concentration of the solute
segregated at the boundary decreases as the boundary
velocity increases. A temperature increase also reduces the
segregation concentration The drag force exerted by the
segregated solutes against the boundary movement, Pg, is
expressed as
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Metallurgical and Materials Engineering
Department Faculty of Petroleum and Mining
Engineering Suez University
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Metallurgical and Materials Engineering
Department Faculty of Petroleum and
Mining Engineering Suez University
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Metallurgical and Materials Engineering
Department Faculty of Petroleum and
Mining Engineering Suez University
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