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1. Mechanical Behaviour of Materials
Chapter Three

2. Grain-boundary sliding
To prevent the formation of internal voids or cracks during
polycrystalline diffusional creep, additional mass-transfer
must occur at the grain boundaries. This results in grain-
boundary sliding and the diffusional creep rate must be
balanced exactly by the grain-boundary sliding rate if
internal voids are not to form.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

3. Grain boundary sliding causes
incompatibilities at both ends of the planes,
A and B, on which sliding occurs. This must
be relieved by another mechanism for
sliding to continue.

4. Diffusional flow and grain-boundary sliding, therefore,
can be considered sequential processes in which
 Mass is first transported by N-H and/or Coble
creep and a grain shape change and separation
are effected.
 This is followed by "crack healing" via grain-
boundary sliding.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

5. Since the grain-boundary sliding and diffusional flow
processes occur sequentially, the net creep rate is the
lesser of the separate creep rates.
In general, the accommodating grain-boundary sliding
creep rate is, well, accommodating. That is, it takes place
rapidly relative to the diffusional flow creep.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

6.  Special boundaries across which atoms match up
more-or-less as they do across a coherent boundary
exist in a two-phase material; the sliding rates of
these boundaries are rather low.
 Grain-boundary ledges (steps on the order of several
atomic diameters in height) occur frequently and these
structural features impede grain-boundary sliding.
However, there are exceptions. For example,
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

7. Except when grain-boundary sliding is restricted
by second-phase particles, the diffusional flow
processes causing sliding are the same as those
causing Coble or N-H creep. The result is that
even when the grain-boundary sliding creep rate
is inherently less than the diffusional flow rate,
can still be used to describe the creep rate.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

8. When accommodation does not occur, grain-
boundary voids form. This is associated with the
initiation of creep fracture .
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

9. At high temperatures, grains in metals and ceramics can
move against each other. This process is called grain
boundary sliding.
Boundary Sliding
The strain rate of grain boundary sliding cannot be
estimated as simply as for the other processes. It is
d
D
A GB
n
GBS
GBS

 

Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

10.  AGBS is a material parameter,
 δ is the thickness of the grain boundary,
  the externally applied stress,
 DGB the diffusion coefficient of grain boundary
diffusion,
 d the diameter of the grain.
 n the creep exponent of grain boundary sliding
usually takes values between 2 and 3
d
D
A GB
n
GBS
GBS

 

Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

11. In metals, grain boundary sliding usually contributes
only slightly to the overall deformation, but it is
nevertheless important for two reasons:
First, in diffusion creep, grain boundary sliding ensures
the compatibility of the grains during the deformation.
Second, at points where three grain boundaries meet
(triple points), movement of the grain boundaries by
sliding can cause a large concentration in local stresses
and thus induce damage by rupture of the grain
boundaries.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

12. In ceramics, the strength at high temperatures is
often limited by grain boundary sliding.
The reason for this is the presence of a glassy phase at
the grain boundaries.
These amorphous regions have a much lower softening
temperature than the grains themselves. This
‘lubricating film’ eases sliding of the grains, without
dislocation movement inside the grains being
necessary.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
One important goal in manufacturing ceramic high-
temperature materials is thus to reduce the amount of
glassy phase as much as possible.

13. Creep Mechanisms Involving
Dislocation and Diffusional Flow
The linear dependence of creep rate on stress for
diffusional creep is not observed under conditions of
moderate applied stress. Instead the value of the
stress exponent n ranges from about 3 to 8 (with n =
4.5 being observed as often).
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

14. Under these conditions, creep involves dislocation, as
well as diffusional flow. The process goes by several
names;
 dislocation creep and
 power law creep (PLC)
The term power law creep arises because the creep
rate varies with stress to a power greater than unity.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

15.  The second mechanism dislocation "climb-
glide" creep.
A number of mechanisms have been proposed for
PLC. We discuss two of these.
 The first, "solute drag" creep, appears to be well-
understood and its applicability to certain metallic
alloy systems is well-established.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

16. Solute atoms can interact with dislocations by
several mechanisms:
a) elastic interaction,
b) modulus interaction
Those are long range interaction; i.e. they are
relatively insensitive to temperature and
continue to act to about 0.6 Tm.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
Solute drag creep

17. Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

18. Elastic interaction between solute atoms and dislocation
arises from the mutual interaction of elastic stress
fields which surround misfitting solute atoms and core
edge dislocation.
The relative size factor is
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
Where: a is the lattice parameter and
c is the atomic concentration of the solute.

19.  Substitutional solutes only impede the motion of
edge dislocation.
 Interstitial solutes have both dilation and shear
components, and can interact with both edge and
screw dislocations.
The strengthening due to elastic interaction is directly
proportional to the misfit of the solute.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

20. Modulus interaction occurs if the presence of solute
atom locally alters the modulus of the lattice. If the
solute has a smaller shear modulus than the matrix
the energy of the strain field of the dislocation will be
reduced and there will be an attractive between solute
and matrix.
The modulus interaction is similar to the elastic
interaction but, because a change in shear modulus is
accompanied by a local change in bulk modulus, both
edge and screw dislocations will be subjected to this
interaction.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

21.  At higher temperatures, solute atoms are mobile. And if
the dislocation velocity is not too high (i.e., if the creep
rate is not too high), the solute atoms move along with
edge dislocations, acting as a "drag" on their motion.
Drag depends on several factors:
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
An equation predicts how the velocity of the
moving dislocation relates to these factors as
well as to the applied stress, ;
o
2
b
solute
c
D
~



Dsolut is the solute diffusivity,
co its concentration, and
b , the misfit parameter

22.  The size misfit parameter between solute atoms and
edge dislocations leads to restriction of dislocation
motion. At low temperatures, the solute atoms are
immobile; thus, the effect they have is to increase
the flow stress required for dislocations to move by
them.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
 Edge dislocations are subject to both size and modulus
interactions, whereas
 Screw dislocations experience only the modulus
interaction.
The physics of solute drag creep are straight forward
and relate to models of solid-solution strengthening.

23.  One is the solute atom diffusivity. Provided the solute
atoms are able to keep up with the moving dislocation,
high diffusivities lead to a lesser drag and vice versa.
 Second, greater size misfit parameters lead to a greater
binding energy between the dislocation and the solute
atoms and result in a greater drag.
 Third, the greater the solute atom concentration the
greater the drag effect.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

24. The creep rate is then linked to
dislocation velocity and mobile
dislocation density by an equation
v
b

 

Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

25. The dislocation density, , increases with  and
analysis indicates that ~2. Thus, the solute drag
creep rate, , can be expressed as
o
2
b
3
solute
SD
c
D
~




Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
The equation contains
another dimensionless
term (/G)2.
Such a term is absent in diffusional creep, and it is
this additional stress-dependent term that makes
solute drag creep more stress sensitive.

26. Schematic stress-strain curve of
a material manifesting serrated
flow.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

27. element Atomic radius
(nm)
G(Pa)
Ni 0.125 86
Al 0.143 23
Fe 0.124 81
Cu 0.128 45
W 0.137 159
Nb 0.143 45
1) The elements listed are being
considered as substitutional
solid- solution strengtheners
for nickel. Calculate the values
of s for them, and list the
elements in order of decreasing
ability to solid-solution
strengthen nickel.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
2) Calculate values of the parameter (s)3/2/700 for the
above elements. Are these solid solution atoms "hard"
or "soft" obstacles? Explain.
Example:

28. Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
Solution:
We first calculate (estimate) the separate terms that go into s
and then determine their sum. There are "assumptions“ made .
 First, we assume that the lattice parameter increases linearly
with composition and that it scales with the "average" atomic
radius (considering both solute and solvent) of the solution.
This assumption is used in calculating b .
 Also assume (used in calculating G ) that the shear modulus
increases linearly with composition.
 Finally, we have taken  = 3 in calculating s .
s = '
G -  b 

29. element b G G’ s s
3/2/700
Ni ….. ….. ….. …..
Al 0.144 -0.733 -0.536 0.968 0.0014
Fe -0.008 -0.058 -0.056 0.032 0.0000084
Cu 0.024 -0.477 -0.385 0.457 0.0004
W 0.096 0.849 0.596 0.308 0.0002
Nb 0.144 -0.477 -0.385 0.817 0.0011
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
Listed in order of decreasing ability to strengthen (that is,
first element listed is the most potent hardener) we have:
Al, Nb, W, Cu, and Fe. Note that the value of the parameter
s
3/2/700 is low even for the most potent hardening
elements. Thus, these solid-solution strengthening atoms are
"weak" obstacles.

30. ii. Dislocation climb-glide creep
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
In precipitation strengthening alloys an
obstacle holds up a moving dislocation on
its slip plane. The applied stress is less
than that needed to overcome the obstacle
via dislocation glide alone.
The dislocation can climb by diffusional
processes to a parallel slip plane and
continue its glid.

31. Schematic representation of
the climb of a dislocation
that has encountered an
obstacle in its slip plane.
Climb
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

32. The climb process permits the dislocation to then glide
on the new plane until it encounters another obstacle
and the process repeats itself. In a sense, the climb
process is the high-temperature equivalent of cross-slip
by which screw dislocations may circumvent slip-plane
obstacles at low temperatures. Since dislocation motion
involves both dislocation glide and climb, this type of
creep is referred to as climb-glide creep.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
With climb, the creep rate is not dependent on the grain
size, but the rate of climb does depend very strongly on the
stress,

33. climb velocity < glide velocity;
Since glide and climb are sequential processes, the
climb-glide creep rate is determined by the lesser of
the glide and climb rates. In most circumstances,
the creep rate is determined by the climb rate.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
At lower temperatures, creep is not entirely climb-
controlled and higher exponents are observed.

34. High-temperature deformation mechanism all
depend on
atom or ion diffusion
but differ in their sensitivity to other variables
such as
G, d, and 
A particular strengthening mechanism may
strengthen a material only with regard to a
particular deformation mechanism but not
another.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

35. An increase in alloy grain size will suppress
Nabarro-Herring and Coble creep along with
grain boundary sliding, but will not
substantially change the dislocation creep
process.
For example
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

36. As a result, the rate controlling creep
deformation process would shift from one
mechanism to another.
Consequently, marked improvement in alloy
performance requires simultaneous
suppression of several deformation
mechanisms.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

37. Creep mechanism
The various creep mechanisms discussed so far differ
in their:
 temperature dependence because the activation
energy of the mechanisms is different.
 stress-dependence. The creep exponent takes
values between 1 in diffusion creep and 3 in
dislocation creep, with even higher values occurring
in reality. Thus, depending on the external
conditions, different creep mechanisms dominate
the behaviour.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

38. Steady-state creep rate vs.
stress for UO2 polycrystals
with a grain size of 10 µm.
At low stress levels,
diffusional flow dominates;
at higher stress levels
dislocation creep does.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

39. Name of Mechanism m q Description
Diffusional flow (N-H
creep) 1 2
Vacancy diffusion through the
crystal lattice
Diffusional flow (Coble
creep)
1 3
Vacancy diffusion along grain
boundaries
Grain boundary
sliding
2 2 or 3
Sliding accommodated by
vacancy diffusion through the
crystal lattice (q = 2) or along
grain boundaries (q = 3)
Dislocation creep
(Power law creep) 3 to 8 0
Dislocation motion, with climb
over microstructural obstacles
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

40. Alloy Structure T (C) m
Qc
(kJ mol)
Single phase
Ni3Al(10Fe) P 871-1177 3.2 327
Ni3Al(11Fe) P 680-930 2.6 355
Ni3Al(Zr, B) P 860-965 4.4 406
Ni3Al(Zr, B) P 760-860 2.9 339-346
Ni3Al(8Cr,Zr, B) P 760-860 3.3 391-400
Multiphase (precipitation strengthened)
Ni-20.2Al-
8.2Cr-2.44Fe
/'-Cr 777-877 4.1 301
Oxide-dispersion strengthened Ni3Al
Ni3Al(5Cr, B) 2 vol.% Y2O3 1000-1200 7.2, 7.8 650, 697
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials
The activation energy for diffusion of Ni in Ni3Al varies
from 273 to 301 kJ mol-1

41. Deformation mechanism maps
So-called deformation mechanism maps allow to
read off the dominant mechanism under different
conditions. A schematic deformation mechanism
map. In the diagram, the temperature and the
external stress, normalised by the relevant material
parameters (melting temperature and shear
modulus), are used as axes so that the dominant
deformation mechanism can be read off.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

42. A schematic deformation
mechanism map. The axes of the
map are homologous
temperature (T/Tm) and stress
(normalized by the shear
modulus). The stress
temperature combination
determines the primary
deformation mode. At the
boundary lines, deformation is
due equally to two mechanisms
and, at the intersection of the
lines, to three mechanisms.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

43. A deformation mechanism map for the alloy MarM-200 with a grain size of 100 µm. The
intended stress-temperature use for a turbine blade of this material is indicated. The
dominant creep mechanism is Coble creep and the operational strain rates are in the
range of 10-8 to 10-10 S-I. (b) The deformation mechanism map for this same alloy with d =
1 cm. Coble creep still dominates creep, but the strain rates are much less.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

44. Deformation mechanism maps for several metals (AI and W), all with a grain
size of 32 µm.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

45. Deformation mechanism maps for two nonmetals (G and MgO), all with a grain
size of 32 µm. Nonmetals are somewhat more resistant to dislocation glide than
metals. The covalent solid, Ge, is quite resistant to both dislocation and diffusional
creep.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

46. Deformation mechanism
map of silver as function
of the grain size
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

47. For creep in Ni of grain size d
= 200 μm, determine the
boundary between Coble and
Nabarro–Herring creep on a
deformation mechanism map
using the data in the Figure.
The activation energy for
lattice diffusion and grain
boundary diffusion in Ni is QL
= 286 kJ/mole and Qgb = 115
kJ/mole. Tm of Ni =1453 C. Deformation mechanism map for pure
nickel with a grain size of d = 32 μm.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials

48. Solution: The boundary is where the two mechanisms contribute equally
to the strain rate. Writing Equations:
as ˙εN−H = AL(σ/d2)exp(−QL/RT ) and εC = Agb(σ/d3)exp(−Qgb/RT ),
and equating
AL(σ/d2)exp(−QL/RT)=Agb(σ/d3) exp(−Qgb/RT ),
AL/Agb = d exp[(−QL + Qgb)/(RT )] = d exp(−171,000/RT ).
Substituting
from Figure, d=32 μ and T=1280+273=1553 K AL/Agb =(32 μm)
exp[−171,000/(8.314 × 1553) = 5.667 × 10−5 μm. Now solving for T,
T =(−171,000)/{R ln[(5.667 × 10−5/200]} = 1364  K = 1090 C.
The boundary for 200 μm is at T = 1364 K = 1090 C.
Note that the boundary is vertical because both the Nabarro–Herring and
Coble creep rates are proportional to stress.
Mechanical Behaviour
of Materials
High Temperature Deformation of
Crystalline Materials