Lecture #15/
Dr.Haydar Al-Ethari
Solid Solution Hardening:
• Solid Solution- – When two or more elements are combined such that single phase
microstructures are maintained.
• A series of elastic, electrical, and chemical interactions occur between the stress
fields of the solute atoms and the dislocations that occupy the lattice.
• These interactions ultimately produce an alloy that is (usually) stronger than the pure
metal.
• Solute atoms (‫ )ذرات اﻟﻤﺬاب‬of roughly similar sizes (i.e., within ±15% of the radii of
the parent atoms) can occupy points in the crystal lattice of solvent atoms. We call
this a substitutional solid solution (‫)اﻟﻤﺤﻠﻮل اﻟﺠﺎﻣﺪ اﻻﺳﺘﺒﺪاﻟﻲ‬.
• When solute atoms are considerably smaller than the solvent atoms (‫)ذرات اﻟﻤﺬﯾﺐ‬, i.e.,
up to 57% of the radii of the solvent atoms, they occupy interstitial sites in the solvent
lattice. We call this type of solid solution an interstitial solid solution( ‫اﻟﻤﺤﻠﻮل اﻟﺠﺎﻣﺪ‬
‫)اﻟﺒﯿﻨﻲ‬.
• There are special rules that determine whether or not a substitutional solid
solution will form. They are called Hume-Rothery rules.
1- Atomic size factor (i.e., “size effect”)
If the atomic diameter of the solute atoms differs from that of the solvent atom by
more than ±15%, the extent of solid solution is small (i.e., usually less than 1%).
When atomic size differences are less than 15%, the extent of solid solution is high
(i.e., the formation of a solid solution is favorable).
2- Electrochemical effect
Metals that are far apart on the electromotive series tend to form intermetallic
compounds (i.e., the more electropositive the one component and the more
electronegative the other component, the greater the tendency for the two elements to
form a compound).
3- Relative valence effect
The solubility of a metal of higher valence in a solvent of lower valence is more
extensive than the reverse situation.
4) Crystal structure : Similar crystal structure of metals of both atom types are
preferred.
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Principles:
• Inhomogeneities in a crystal lattice will produce a stress field within a lattice.
• Dislocations will interact with those stress fields.
– The type of interaction will determine the degree of hardening or softening.
• In general, solute atoms increase the strengths of crystals. However, under certain
conditions they can decrease the strength.
• Dislocations interact with solutes that lie on, above, and below slip planes. The most
intense interactions will occur in close proximity to the slip plane (i.e., near the
dislocation core).
Solid solution strengthening is dominated by elastic interaction and modulus
interaction. These interactions are relatively insensitive to temperature and are strong
up to ~0.6 – 0.7Tmp.
Elastic interaction:
Solute atoms “stretch” (i.e., dilate) the lattice producing different types of stress fields
surrounding the solutes.
Small substitutional
solutes segregate to
compressive region
Large substitutional
solutes segregate to
tensile region
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There is an interaction between the stress fields around solute atoms and the
stress fields around dislocations. This interaction is based on reducing the strain
energy associated with dislocations and solute atoms. Both attractive and repulsive
forces between solutes and dislocations will inhibit the motion of dislocations, thus
increasing strength.
- Substitutional solutes generally stretch the lattice uniformly producing hydrostatic
(“spherical”) stress fields around the solutes, so they can interact only with defects
that have a hydrostatic component in their stress fields, as happens to be the case with
an edge dislocation. Screw dislocations, by contrast, have a stress field of a pure shear
character; that is, the hydrostatic component of a screw dislocation is zero. Therefore
there is no interaction between screw dislocations and substitutional atoms.
- For the case of an interstitial atom (carbon or nitrogen in α-iron), the minimumenergy position at an edge dislocation is the dilated region near the core, they not only
produce a dilational misfit (in volume), but also induce a tetragonal distortion.
The important effect of the tetragonal distortion is that the interstitial atoms such as C
and N in iron will interact and form atmospheres at both edge and screw dislocations
and will lead to a more effective impediment to the movement of dislocations than in
the case of substitutional atoms.
Stress fields around solutes:
• FCC lattice: – Substitutional solute: dilatational (hydrostatic) strain.
– Interstitial solute: dilatational (hydrostatic) strain.
• BCC lattice: – Substitutional solute: dilatational (hydrostatic) strain
– Interstitial solute: distortional (shear) strain. This component is
asymmetric!
• Interactions expected between solutes and dislocations in different lattices are:
• FCC Lattice: – Screw dislocations – little or no interactions with solutes
– Edge dislocations – strong interactions with both types of solutes
• BCC Lattice: – Edge dislocations – strong interactions with both types of solutes
– Screw dislocations – strong interactions possible with interstitial
solutes; no interactions with substitutional solutes
• HCP lattice: – Which solutes will cause the most potent hardening? (H.W)
Modulus interaction:
• Since foreign atoms generally have different shear moduli than the parent atoms,
they impose additional stress fields on the lattice of the surrounding matrix.
• When solutes with smaller shear moduli than the solvent (i.e., Gsolute < Gsolvent), the
energy of the stress fields around dislocations will be reduced (i.e., elastic strain
energy is reduced) which causes an attraction (‫ )ﺟ ﺬب‬between the solutes and the
dislocations.
• Both edge and screw dislocations are subject to this interaction. Generally, however,
their contributions (‫ )ﻣﺴﺎھﻤﺔ‬are less important than the size effect.
Mechanical Effects Associated with Solid Solutions:
Many important mechanical effects are associated with the phenomenon of solid
solution.
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• In the case of steels, solute-dislocation interaction leads to a migration of interstitial
solute atoms to a dislocation such that the total elastic strain could be minimized thus
minimizing the elastic strain energy of the system (i.e., solutes will move to locations
that minimize the distortion of the lattice caused by the presence of defects).
• This segregation of solutes results in the formation of solute atmospheres called the
Cottrell atmosphere. This atmospheres has the effect of locking-in the dislocation,
making it necessary to apply more force to free the dislocation from the atmosphere.
• According to this, the dislocations in annealed steels are locked-in by the interstitial
solute atoms (carbon). When stress is applied to such a steel in a tensile test, it must
exceed a certain critical value to unlock the dislocations. Dislocation motion will be
restricted (‫ )ﻣﻘﯿ ﺪة‬until the applied stress becomes large enough to “break the
dislocations free” from the solute atmospheres. At this point, dislocation velocity
increases significantly resulting in a significant reduction in stress.
• The stress necessary to move the dislocations
is less than the stress required to unlock them
- hence the phenomenon of a sharp yield drop
and the appearance of an upper and lower yield
point in the tensile stress-strain curve.
•Note: The yield point phenomenon has also been
Observed in other metals such as Fe, Ti, Mo, Cd,
Zn, Al alloys
• In that region where σ remains relatively
constant with increasing ε, we often see Lüders
bands form.
• Temperature is an important variable in the migration of solute atoms to a
dislocation. If the temperature is too low, the solute may not be able to diffuse in a
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reasonable length of time. At very high temperatures (>0.5Tm) the mobility of foreign
atoms will be much higher than that of dislocations, with the result that they will not
restrict dislocation motion.
• The explanation for the formation of Lüders bands is intimately related to the cause
of the appearance of the well-defined yield point. The unlocking of dislocations that
occurs at the upper yield point is, initially, a localized phenomenon. The unlocked
dislocations move at a very high speed, because the stress required to unlock them is
much higher than the stress required to move them, until they are stopped at grain
boundaries. The stress concentration due to the dislocations that accumulate at grain
boundaries unlocks the dislocations in the neighboring grains.
• Lüders bands mar the surface finishes of materials leading to surface roughness and
unsightly surfaces. This phenomenon is particularly undesirable in metalworking
operations.
• An aspect of great technological importance is the formation of Lüders bands during
the stamping of low-carbon steels, with the consequent irregularities in the final
thickness of the sheet. This problem is tackled, in practice, in two ways:
1. By changing the composition of the alloy to eliminate the yield point. The addition
of aluminum, vanadium, titanium, niobium, or boron to steel leads to the formation of
carbides and nitrides as precipitates, which serve to remove the interstitial atoms from
the solid solution.
2. By prestraining the sheet to a strain greater than the yield point strain such that the
strains during the stamping operations occur in the strain-hardening region.
Strain Aging:
Strain aging is a type of behavior, usually
associated with yield point phenomenon, in which
the metal increase in strength while losing ductility
after being heated at relatively low temperature or
cold-working.
• Reloading at X and straining to Y does not produce
yield point.
• After this point if the specimen is reloading after
ageing (at room temp. or ageing temp) the yield point will reappear at a higher value.
• This reappearance of the yield point is due to the diffusion of C and N atoms to
anchor the dislocations
• N has more strain ageing effect in iron than C due to a higher solubility and
diffusion coefficient.
Blue Brittleness:
Carbon steels heated in the temperature range of 230 and 3700 ◦C show a notable
reduction in elongation. This phenomenon is due to the interaction of dislocations in
motion with the solute atoms (carbon or nitrogen). When the temperature and the
strain rate are such that the speed of the interstitial atoms is more than that of the
dislocations, the dislocations are continually captured by the interstitials. This results
in a very high strain-hardening rate and strength with a reduction in elongation. With
increasing strain rates, the effect occurs at higher temperatures, as diffusivity
increases with temperature. Called blue brittleness, this effect refers to the coloration
that the steel acquires due to the oxide layer formed in the given temperature range.
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Strengthening from fine particles:
Because of the finely dispersed second-phase particles, these alloys are much more
resistant to recrystallization and grain growth than single phase alloys. Because there
is very little solubility of the second phase constituent in the matrix, the particles
resist growth or overaging to a much greater extent than the second-phase particles in
a precipitation-hardening system.
In dispersion-hardening, we incorporate hard, insoluble second phases in a soft
metallic matrix. Here, it is important to distinguish dispersion-strengthened metals
from particle-reinforced metallic composites. The volume fraction of dispersoids in
dispersion strengthened metals is generally low, 3-4% maximum. The idea is to use
these small, but hard, particles as obstacles to dislocation motion in the metal and
thus strengthen the metal or alloy without affecting its stiffness. In the case of metallic
particulate composites, the objective is to make use of the high stiffness of particles
such as alumina to produce a composite that is stiffer than the metal alone.
Improvements in strength, especially at high temperatures, also result, but at the
expense of ductility and toughness.
Examples of dispersion-strengthened systems include Al2O3 in Al or Cu, ThO2 in Ni,
and more.
Two important alloy systems that exploit precipitation-hardening, or age-hardening
are aluminum alloys and nickel-based super alloys. Fig.1 shows examples of
precipitates in some systems.
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• The degree of strengthening provided by particle dispersions depends on:
– Particle size
– Particle volume fraction
– Particle shape
– Nature of the interface between the particle and the matrix
– Structure of the particle
Precipitation-, or age-, hardening.
The supersaturated solid solution is obtained by sudden cooling from a sufficiently
high temperature at which the alloy has a single phase. The heat treatment that causes
precipitation of the solute is called aging. The process may be applied to a number of
alloy systems. Although the specific behavior varies with the alloy, the alloy must,
at least:
1. Form a monophase solid solution at high temperatures.
2. Reject a finely dispersed precipitate during aging
The precipitation treatment consists of the following steps:
1. Solubilization. This involves heating the alloy to the monophase region and
maintaining it there for a sufficiently long time to dissolve any soluble precipitates.
2. Quenching. This involves cooling the single-phase alloy very rapidly to room
temperature or lower so that the formation of stable precipitates is avoided. Thus, one
obtains a supersaturated solid solution.
3. Aging. This treatment consists of leaving the supersaturated solid solution at room
temperature or at a slightly higher temperature. It results in the appearance of finescale precipitates
In the initial stages of the aging treatment, zones that are coherent with the matrix
appear. These zones are nothing but clusters (‫ )ﻣﻨ ﺎطﻖ ﺗﺠﻤ ﻊ‬of solute atoms on certain
crystallographic planes of the matrix (in aluminum-copper alloy the zones are a
clustering of copper atoms). The zones are transition structuring and are referred to as
Guinier-Preston zones, or GP zones. The GP zones are very small and have a very
small lattice mismatch with the matrix. Thus, they are coherent (‫ )ﻣﺘﻄﺎﺑﻘ ﺔ‬with the
matrix; that is, the lattice planes cross the interface in a continuous manner (see
fig.A).
The nature of precipitate/matrix interface produced during the aging treatment can be
coherent, semicoherent, or incoherent.Coherency signifies that there exists a one-toone correspondence between the precipitate lattice and that of the matrix. (See Figure
A and B). A semicoherent precipitate signifies that there is only a partial
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correspondence between the two sets of lattice planes. The lattice mismatch is
accommodated by the introduction of dislocations at the noncorrespondence sites, as
shown in Figure C. An incoherent interface, shown in Figure D implies that there is
no correspondence between the two lattices. Such an interface is also present in
dispersion-hardened systems.
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Precipitates are generally very small in the early stages of precipitation. They coarsen
with time at temperature.
1. Small particles are generally coherent with the matrix
2. Intermediate particles are often partially coherent with the matrix
3. Large particles are generally incoherent with the matrix
a) Coherent interface
– Two crystals/phases match perfectly at the interface plane so that the two lattices are
continuous across the interface. Strengthening arises from lattice parameter mismatch
between phases.
b) Semi-coherent interface
– Matching is imperfect between two crystals/phases. Mismatch strains are reduced
by formation of dislocations at inter-phase boundaries.
c) Incoherent interface
– There is no matching between phases at the interface plane. Atomic planes do not
line up. No coherency whatsoever. Thus no coherency strain.
Effect of Interface Character:
• Precipitates can have strain fields associated with them.
• Strain fields around particles, when present, creates a larger effective particle
volume.
• Dislocations will interact with the strain fields surrounding coherent particles in the
same way that they do with the strain fields around solute atoms (i.e., just like solid
solution hardening).
Loss of coherency
• When precipitate particles grow, coherency can be lost. WHY?
– The energy of the strained interface between the particle and matrix becomes
greater than the energy for an incoherent interface. This occurs when the particle size
exceeds a critical value.
• When you lose coherency, you lose the coherency strain and associated hardening.
This same thing occurs in dispersion hardening materials where there is no coarsening
and no coherency.
• What happens to strength?
• It increases anyway! WHY?
Dislocation- Precipitate interaction:
The amount of strengthening is determined by the manner in which the dislocations
and particles interact.
For deformation to proceed dislocations must:
1. Cut through particles or
2. Extrude between (loop around or bow between) particles.
Depending on both the nature of the precipitate and the crystallographic relationship
between the precipitate and the matrix, we can have two limiting cases:
1. The precipitate particles are impenetrable to the dislocations.
Particles can bypass incoherent precipitates by looping around them. This also
applies to non-deformable particles.
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After the dislocation bows between the particles it rejoins other bowed segments and
passes on, leaving a loop around each particle. These loops in turn repel subsequent
dislocations, effectively decreasing the distance between the particles The result is
that a higher stress is required for them to pass.
For a given volume fraction, Vf, the distance, d, decreases when particles are finer.
An estimate of the effect of particle size can be made by assuming that the particles
intersecting the slip plane are arranged in a square pattern and that r is the radius of
the particle as it is cut on the slip plane. Then the area fraction equals the volume
fraction,
Vf = π r 2/d 2. Solving for d = r (π/Vf)1/2, and substituting into the above equation
τ = 2Gb(Vf/π)1/2/r.
Since the increase of tensile yield strength, ∆σ, caused by the precipitate particles is
proportional to the shear stress, τ,
∆σ = αGbVf 1/2/r
where α is a constant. This predicts that the strength increases with increasing volume
fraction and decreasing particle size.
2. The precipitate particles are penetrable to the dislocations.
The second possibility for dislocation motion in a particle-hardened matrix is that
dislocations cut through the particles, even though they are harder than the matrix.
In this case the overall strength is given by the rule of mixtures,
τ = τmat(1 − Vf) + τpartVf
where τ mat and τ part are the shear strengths of the matrix and particles, respectively.
Because Vf is usually small,
τ ≈ τmat + τpartVf.
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