High Temperature Strain of Metals and Alloys Part 7 pot

6.13 Interaction of deforming dislocations with particles in the EI867 alloy.×60000... 6.3 Interaction of Dislocations and Particles 93particles cannot be obstacles for deforming disloca

engines and also in various applications up to 1023K (750◦C) such as turbine

blades, wheels and afterburner parts The specimens were solution treated for 1h at 1273K, air cooled and aged for 16h at 973K

Figure 6.6 presents the results of X-ray investigations on this superalloy The loading of specimens results in an increase in the misorientation angle,

η, and a decrease in the average subgrain size, D The mean values of the

parameters under investigation are almost unchanged at the steady-state stage and are equal to 3mrad and 0.6µm, respectively Consequently, fragmentation

of theγ crystallites is also intrinsic to superalloys This is due to the formation

Fig 6.6 Dependence of the elongation, average subgrain

size and subgrain misorientation angle on time for the

EI437B superalloy T = 973K ◦, •: σ = 570MPa;
:

σ = 700MPa.

6.3 Interaction of Dislocations and Particles 89

Fig 6.7 Dislocation sub-boundaries in the matrix of the

tested EI437 superalloy The steady-state stage of creep

T = 973K; σ = 450MPa ×150000.

of dislocation sub-boundaries in theγ matrix Decrease in D and increase in

η is observed when the applied stress increases.

Transmission electron microscopy is difficult to apply to the alloy because it contains many small coherentγ
particles The contrast at the matrix–particle

boundary is known to have a deformation origin and hence the borders of the particles seem to be fuzzy The average particle size in EI437B superalloy was found to be 14nm after the initial heat treatment The dimensions of the particles increase to 22nm after creep tests and the borders of the particles become more distinct One can see the dislocation sub-boundaries in Fig 6.7 The dimensions of the subgrains are about 0.3–0.5µm This is close to the values estimated with the X-ray method

6.3

Interaction of Dislocations and Particles of the Hardening Phase

Typical pairs of deforming dislocations are seen in Fig 6.8 The dislocation lines, which slip under the effect of the applied stress, are parallel and inter-sect the particles Transmission electron microscopy evidence supports the cutting ofγ
particles by slipping dislocations The dislocations cut the

coher-ent particles of theγ
phase without changing the slip plane which is mainly

of the type {111} However, during the tertiary stage of creep the particles coarsen and their coherent bond with the matrix is broken Orowan bowing

Fig 6.8 Electron microphotographs of the EI437 superalloy

during the steady-state stage of creep.T = 973K; σ =

450MPa ×150000 (a), ×200000 (b).

occurs as the rate-controlling strain mechanism It is the Orowan mechanism that dominates in tertiary creep deformation

In Fig 6.9 one can see that dislocations cut small particles and bow the big ones The dislocation loops around particles remain when the dislocation lines have passed The bowing of particles takes place till cavitation occurs and the specimen ruptures

EI 867 is a superalloy strengthened by chromium, aluminum, molybde-num, tungsten and cobalt The standard heat treatment consists of solution treatment at 1493K for 2h, quenching in air and ageing at 1223K for 8h This heat treatment produces cuboidalγ
particles, which are on average 130 nm

in size along the cube edge The edges of the cubes are oriented along the

< 100 > direction (Fig.6.10).

The electron micrographs taken during the steady-state stage of creep are presented in Figs 6.11–6.13 Parallel deforming dislocations are seen They move inside the ordered zones one after the other It is at once apparent from Fig 6.11 that the particles are obstacles for the moving dislocations

A plane sequence of dislocations is pressed to the edge of the γ
particle.

The spacing between successive dislocations decreases as the distance to the particle is reduced, as if the dislocations “are waiting” to enter the particle After entering the particle the dislocations continue to move The dislocation loops that expand from the interface of the phases are seen in Fig 6.13

6.3 Interaction of Dislocations and Particles 91

Fig 6.9 Electron micrographs of the tested EI437 superalloy

during the tertiary stage of creep.T = 973K; σ = 450MPa.

×200000.

Fig 6.10 Electron micrographs of the EI867 superalloy in

the initial state Particles ofγ
phase (a) Replica,×20000;

(b) thin film,×100000.

At the stage of the tertiary, accelerating creep the shape of the particles becomes irregular A rafting process of theγ
structure occurs because of

development of diffusion coarsening Now the incoherent irregular rafted

Fig 6.11 Electron micrographs of the EI867 superalloy during the steady-state stage of creep T = 1173K;

σ = 215MPa Interaction of deforming dislocations with γ

particles.×130000.

Fig 6.12 Interaction of deforming dislocations withγ

precipitates in the EI867 superalloy.×90000 (a);

×40000 (b).

Fig 6.13 Interaction of deforming dislocations with particles

in the EI867 alloy.×60000.

6.3 Interaction of Dislocations and Particles 93

particles cannot be obstacles for deforming dislocations As a result disloca-tion networks are formed (Fig 6.14) The networks fill the volume between the particles and spread inside the particles At the same time the strain rate

of the specimen increases

Fig 6.14 Dislocation networks during the tertiary stage of

creep in the EI867 superalloy.×90 000.

Electron micrographs of the EP199 superalloy (Table 2.1) are shown in Fig 6.15 Parallel dislocations can be observed The dislocations move one after the other and intersect particles of theγ
phase ((a), (b)) The dislocation

sets are formed at the tertiary stage of creep ((c), (d))

Electron microstructural examination of the crept test specimens of super-alloys has indicated that a rate-controlling process is the precipitate cutting,

or shearing During high-temperature exposure the precipitates coarsen, and the rate-controlling mechanism becomes dislocation bowing

It follows from the obtained data that deforming dislocations are slowed

by the coherent particles and then cut them Hence, the thermally activated overcoming of particles is the process that controls the constant strain rate However, under the effect of applied stress and high temperature the raft-ing of particles occurs and the deformraft-ing dislocations can bow between the obstacles This results in accelerating tertiary creep and rupture

Fig 6.15 Electron micrographs of the EP199 superalloy.

T = 1173K; σ = 110MPa (a), (b) At the end of the

steady-state stage of creep;×65000 (c), (d) The stage of

the tertiary creep;×48000.

6.4 Creep Rate Length of Dislocation Segments 95

6.4

Dependence of Creep Rate on Stress The Average Length of the Activated Dislocation Segments

The experimental dependences of the minimum creep rate, ˙ε, on the applied

stress for five superalloys are presented in Fig 6.16, where ln ˙ε is plotted

againstσ Results for three alloys are shown in Fig 6.17 A linear dependence

is observed for all superalloys Hence, the minimum creep rate is dependent exponentially on stress

The activation volume v of an elementary deformation event can be

cal-culated from these data according to Eq (1.5) Further, we may compute the average length of a dislocation segment ¯l that must be activated in order that

the dislocation can move ahead:

¯l= v

In Table 6.2 the values ofl and the average particle dimensions 2¯r are listed.

The lengths of the activated dislocation segments are one order less than the average particle sizes Values of the ratio of ¯l/2¯r lie within the range 0.07 to

0.14, more precisely0.12 ± 0.04.

Fig 6.16 Logarithm of strain rate

versus stress for superalloys: B,

Ni+18Cr+2.6Al T = 1023K Data from

Ref [35] C,Ni + 19Cr + 0.8Al + 2.1Ti.

T = 1023K Data from Ref [35] D,

Ni + 9Cr + 4.5Al + 5W + 14Co (EI867).

T = 1173K Data of the present

au-thor E,Ni + 19Cr + 0.8Al + 2.5Ti.

T = 973K Data from Ref [35] F, Ni+20Cr+2.2Al+2.0Ti+3.3W+5Fe.

T = 1023K Data from Ref [32].

Fig 6.17 Logarithm of strain rate versus

stress for superalloys: B,Ni + 9Cr +

5.0Al + 2.0Ti + 1Nb + 12W + 10Co.

T = 1144K Data from Ref [36] C,

Ni + 21Cr + 0.8Al + 2.5Ti (EI437B).

T = 973K Data of the present author F,

superalloy C263.T = 973K Data from

Ref [37]

Tab 6.2 The length of activated dislocation segments

Alloying elements in Ni-based alloy T , K l, nm 2¯r, nm Particle shape Ref.

20Cr + 2.2Al + 2.0Ti + 3.3W + 5Fe 1023 8.9 100 spher 32

6.5

Mechanism of Strain and the Creep Rate Equation

The applied stress is insufficient to let a dislocation cut a particle under normal creep conditions The ordered structure of the γ
phase requires that two

dislocations in theγ phase must combine in order to enter the γ
phase as

a superdislocation The associated anti-phase energy inγ
posseses a large

barrier to the dislocation entry

A mechanism involving diffusion-controlled movement of dislocations in the orderedγ
phase seems to be the most probable one There is good reason

to believe that, in these conditions, the slip of the deforming dislocations is

6.5 Mechanism of Strain and the Creep Rate Equation 97

controlled by diffusion processes, indeed, with ordering, the activation energy

of diffusion increases as well as the creep strength

A mechanism of diffusion-controlled dislocation displacement through the orderedγ
phase is presented in Fig 6.18 An arrangement of atoms in a

su-perdislocation is shown The susu-perdislocation is dissociated into two partial dislocations that are separated by the band of the anti-phase boundary A va-cancy approaches the first partial dislocation as a result of thermal activation The atomic row shears under the effect of the applied stress, and a relaxation

in the vacancy area occurs, thus, a double bend is formed in the dislocation line and the adjacent rows displace This is equivalent to the expansion of both

Fig 6.18 The atomic mechanism of the

dislocation diffusion displacement inγ

phase Arrangement of atoms in two

par-allel slip planes of [111] is shown (a) Ideal

crystal lattice Twelve rows are shown

Along the face diagonal[10¯1] atoms of

aluminum and nickel are altered Atoms

of Ni that are denoted as 1 and of Al

that are denoted as 3 are located in the

first slip plane Atoms of Ni 2 and of Al

4 are located in the second parallel slip plane (b) Partial dislocations and the anti-phase-boundary (APhB) The Burgers vector (arrows) is[10¯1] At the row 11 a vacancy 5 (
) is formed (c) The shear

of the atomic row The vacancy
has migrated to the next atom A double bend has been formed at the moving superdislocation

boundary energy also plays an important role This value has been determined

to be approximately 180mJ m−2for industrial single crystal superalloys For

example, the energy of the anti-phase boundary ofNi3Al increases from 180

to 250 mJ m−2 when substituting 6 at.% Ti for 6 at.% Al Thus, thermal

activation is necessary in order for the segmentl to advance The work which

is performed is equal to the increment of the thermodynamic potential of the system dislocation-obstacle

The average slip velocity,V , of the dislocation segment ¯lis given by

whereΓ is the frequency of attempts of the dislocation to overcome the

po-tential barrier,A is the average area swept out by the segment released during

the thermal activation, ¯l is the average length of segment, which is activated

inside the particle The displacement of the segment is equal to the Burgers vector length|b|.

On the basis of the theory of the activated reactions rate [25] the value ofΓ

can be represented by an expression of the form

Γ =

n

j=1 ν j

n−1

j=1 ν
j

exp

− ∆Φ kT



(6.5)

In Eq (6.5)ν jare the normal frequencies of oscillations of the segment in a crystal lattice in the initial state, the total number of these oscillations isn (all

the atoms that take part in overcoming the potential barrier are considered) The values ofν

jare(n − 1) frequencies of oscillations in the activated state

at the peak of the potential barrier The increment of the thermodynamic potential,∆Φ, (of the Gibbs free energy) is given by

where∆S is the increment of entropy, ∆U the increment of internal energy

as the segment overcomes the barrier, τ the shear stress, v the activation

volume

6.5 Mechanism of Strain and the Creep Rate Equation 99

Applied stress affecting the crystal does some work τv in order for the

segment to move forward It is possible to express∆S for vibrations with a

small amplitude as

∆S = k

n−1

j=1

lnν ν j

j

(6.7)

The first frequency ν1 = νb/2¯l is an efficient frequency of attempts to

overcome the barrier, ν is of the order of the Debye frequency Combining

Eqs (6.4)–(6.7) we find that the dislocation segment velocity can be expressed as

V = νbA 2¯l2 exp

− ∆U kT

 exp

τb2¯l kT



(6.8)

The value of∆U in Eq (6.8) is close to the activation energy of generation

and migration of vacancies in the ordered phase The sum of these values is known to be the energy activation of diffusion in theγ
phase The value of

areaA is assumed to be expressed as

Substituting Eq (6.9) in Eq (6.8) we obtain

V = νb2 2l exp

− ∆U kT

 exp

τb2l kT



(6.10)

whereτ is the shear stress in the slip plane.

We have obtained the theoretical equation for the velocity of movement of dislocations through particles Recall that it describes the diffusion-controlled mechanism of the cutting of particles by dislocations

It is important to connect the velocity of dislocations with the average parti-cle size We can use the correlation which has been obtained experimentally:

l

Substituting ratio (6.11) into Eq (6.10) we find

V = 0.24 · 2r νb2 exp

− ∆U kT

 exp

0.12τb2· 2r kT



(6.12)

The last equation is a semi-empirical one, because we have used the results

of measurements of the particle sizes and the data of the strain rate tests

First, we can conclude from Eq (6.13) that the dependence of the strain rate upon the average particle size2r is a function with a minimum There

is a particle size, under which the deforming dislocations move at the least velocity Indeed, if the value of2r is small the pre-exponential factor is large

and the exponent factor is close to unity In this case, if the particle size

2r increases, the exponent increases more rapidly than the pre-exponential

factor Hence there is an optimal size of particles2r0, which is dependent on

T and σ Taking the derivative and solving ∂V/∂(2r) = 0 we find that for

EI437B superalloy at 973K andσ = 400MPa the value of 2r0= 12.2 nm The

result fits the measured value 14 nm satisfactorily

It is of importance to estimate the energy∆U.

Taking the logarithm of Eq (6.13) we obtain

∆U = −kT ln ˙ε + 0.12τb2· 2¯r + kT ln

νb3

0.24 · 2¯r



+ kT ln[f(c)N] (6.14)

It is essential to know the last term on the right-hand side of Eq (6.14) in order to calculate a mean activation energy∆U.

Rae et al [38] have measured the dislocation density in the crept CMSX-4 superalloy The density of dislocations was estimated to be2 × 1012m−2by

measuring the total length of dislocations in an area One may reasonably assume that the product of the concentration and the dislocation density will

be in the range1010–1012m−2, most likely1012m−2.

Applying Eq (6.14) for tests of the same superalloy under two stressesσ1 andσ2and denoting the sums of the first three terms asA1andA2one can write

Thus, the logarithm of the ratio of the dislocation densities is given by

lnN2

N1 =A1− A2