High Temperature Strain of Metals and Alloys Part 6 doc

Hence, the strain rate is strongly affected by the sub-boundary dislocations spacing.. 5.7 The sub-boundary dislocations spacing versus time for niobium tested at 1370K.. 5.4 Density of

Fig 5.2 Dependence of strain and structural parameters

on time for nickel The computer simulation uses the set of

12 ordinary differential equations Two curves in each of six graphs correspond to two intersecting systems of parallel slip planes.T = 1073K, σ1= 17.5MPa, σ2= 16.5MPa.

time

t = 0

deformation

γ1= γ2= 0 dislocation density

ρ1= ρ2= 2 × 108m−2

5.3 Results of Simulation 73

dislocation spacing in sub-boundaries

λ1= λ2= 50 nm subgrain size

D1= D2= 3 µm coefficients of the dislocation multiplication and emission, respectively,

δ = 2 × 104m−1 ; δ s = 4 × 104m−1

The test time is 5h =1.8 × 104s.

Comparison of the obtained results (Fig 5.2) with the experimental data shows remarkable overall agreement

Further analysis of data from the model leads to some interesting con-clusions: There are some differences in how the processes in both plane sets proceed One can see an increase in strain in Fig 5.2 The steady-state stage of creep occurs earlier under the lower stress The strain value of 2% is observed

in1.5 × 104s after the load has been applied

For comparison with the model data the experimental results are presented

in Fig 5.3 and Fig 5.4 The inter-dislocation spacings in Fig 5.4 were deter-mined from X-ray measurement data, as described in Chapter 2 There is an obvious fit of model and experimental data which is evidence that the physical model is adequate

Fig 5.3 Strain versus time for nickel tested at 1073K To be

compared with the first graph in Fig 5.2 Two specimens

B,σ1= 10MPa; C, σ2= 14MPa

Fig 5.4 The sub-boundary dislocation spacing versus time

for nickel tested at 1073K To be compared with the third

graph in Fig 5.2 Experimental data for the same two

specimens as in Fig 5.3

The density of dislocations increases during the high-temperature strain from2 × 108m−2to(4.0–4.5) × 1011m−2 The dislocation density increases

very quickly after loading, within 75–100 s At the steady-state stage the values

ρ1andρ2are almost constant, hence the rates of the mobile dislocation gen-eration and of the dislocation annihilation (immobilization) are equal The sum of the positive terms on the right-hand side of Eq (5.3) is equal to the sum of the absolute values of the negative terms

The velocity,V , of the mobile dislocations decreases gradually At the

steady-state it is of the order of 20 nm s−1, i.e 80 interatomic spacings per second

V is less in the planes where the applied stress is less In contrast, the climb

velocityQ increases with time Q is two orders less than V

The spacing between jogs in mobile dislocations1),λ, decreases from 50 to

35 nm in exactly the same way as in reality, see Fig 5.4 The decrease inλ = l

correlate with the decrease in theV value.

The change in the subgrain sizeD (the last graph) differs somewhat from

the observed one The calculated values drop too quickly

The proposed model enables one to examine the influence of structural parameters on strain and on the strain rate It is convenient to study the evolution of the investigated values

The strain decreases when the initial valueλ is decreased or the initial

value D is increased by means of suitable treatment For example, if the

initial average subgrain sizeD is 12µm instead of 3µm then the strain drops

from 0.02 to 0.005 As regardsλ this value is in the exponents in Eqs (4.8),

1) In Fig 5.2 and 5.5 the sub-boundary dislocation spacing is denoted as l.

5.3 Results of Simulation 75

(4.10), (5.4) Hence, the strain rate is strongly affected by the sub-boundary dislocations spacing

A seeming paradoxical result is of interest When the initial value of the mobile dislocation density,ρ, increases sharply then the annihilation of

dis-locations progresses As a result the strain of the specimen decreases In con-trast, when the initial valueρ is decreased, e.g from 2 × 108to5 × 107m−2,

the strain increases from 0.02 to 0.10

It might seem that the coefficients of the dislocation multiplications are chosen somewhat arbitrarily Undoubtedly, the real valuesδ and δ sare un-known We are forced to consider them as fitting coefficients However, it

Fig 5.5 Dependence of strain and structural parameters on

time for niobium The computer simulation uses the set of

12 ordinary differential equations Two curves in each of the

six graphs correspond to two intersecting systems of parallel

slip planes.T = 1370K, σ = 17.0MPa, σ = 16.5MPa.

Fig 5.6 Strain versus time for niobium tested at 1370K,

σ = 44.1MPa.

Fig 5.7 The sub-boundary dislocations spacing versus time

for niobium tested at 1370K B and C are two crystallites of

the same specimen To be compared with the third graph in

Fig 5.5

turned out that changes in these values, even if by orders of magnitude, have only a small effect on the results of the calculations For example, varyingδ

from2 × 104to3.2 × 105m−1does not affect the deformation curve or the

dislocation density The increase inδsfrom4 × 104to1.6 × 105leads to an increase inρ at the steady-state stage up to 7 × 1011m−2.

In Fig 5.5 the model data are presented for niobium The typical experi-mental curves for niobium tested at 1370K are shown in Figs 5.6–5.8 These curves show the parametersε, λ = l, and D, respectively, during steady-state

strain One can see that the physical model fits the experimental data well

5.4 Density of Dislocations during Stationary Creep 77

Fig 5.8 The average subgrain size versus time for niobium

tested at 1370K B and C are two crystallites of the same

specimen

5.4

Density of Dislocations during Stationary Creep

At the steady-state deformation the rates of the dislocation generation and annihilation seem to be equal and the parameters λ and D are constant.

Thus, we should solve the system of equations to calculate the density of deforming dislocations during the constant strain rate (see Section 4.3):

whereρ is the real root of the following equation:

δρV + δsρsV − 0.5D2V ρ 2.5 − ρ V D = 0 (5.14)

These equations were solved numerically (the Newton method was used) The dislocation densities were computed for different values of the structural parametersD and λ The results are presented in Fig 5.9.

The obtained results seem to be quite reasonable The density of deforming dislocations is of the order of1011m−2 This density is strongly affected by the

subgrain size as well as by the distance between sub-boundary dislocations The larger the subgrains the smaller the density of dislocations that contribute

to the deformation process It is obvious that sub-boundaries of relatively large misorientation are sources for moving dislocations

Fig 5.9 The computed density of deforming dislocationsN

versus the sub-boundary dislocation spacing B, the subgrain

sizeD = 60 µm; C, D = 30 µm; D, D = 10 µm.

The experimentally measured data (Table 3.5) are of the same order,

1011m−2 For example the measured density is(1.3–9.5) × 1011m−2in nickel, (1.6–5.3) × 1011m−2in niobium and so on.

From Figs 5.9 and 5.10 it can be seen that the total dislocation density,

ρ, is one order greater than the deforming dislocation density, N Since the

dislocation density is affected by structural parameters the minimal strain rate depends on their values, too The different values ofD and λ can be

obtained by a preliminary treatment of the metal

Fig 5.10 The computed total dislocation densityρ versus

the sub-boundary dislocation spacing B, the subgrain size

D = 60 µm; C, D = 30 µm; D, D = 10 µm.

5.4 Density of Dislocations during Stationary Creep 79 Tab 5.1 Structural parameters in nickel after preliminary

deformation and annealing

In order to test the influence of the D and λ parameters, we deformed

specimens of nickel by 3% and 7% at room temperature and then annealed them at 873K As a result we obtained the various average subgrain sizes and misorientations (Table 5.1)

The specimens revealed after treatment an improved creep resistance (Fig 5.11) The difference in the creep rate for specimens with 7% deforma-tion is one order less than for specimens without preliminary deformadeforma-tion However, at relatively high stress the values of the steady-state strain rates become equal

The sub-boundary dislocation distance decreases if the stress increases (Fig 5.12)

Fig 5.11 The steady-state strain rate of the preliminary

de-formed nickel specimens versus applied stress Temperature

873K B, without deformation; C, deformation 3%;

D, deformation 7%

Fig 5.12 The distance between sub-boundary dislocations

in the preliminary deformed nickel specimens versus applied

stress Temperature 873K B, without deformation; C,

deformation 3%; D, deformation 7%

5.5

Summary

A system of differential equations has been proposed to simulate the processes

of high-temperature deformation in metals Two intersecting crystalline sys-tems of parallel slip planes are considered The dislocation slip velocity in each system is controlled by vacancy-producing jogs and depends on the dis-tances between the sub-boundary dislocations in the parallel planes of another system

The evolution of six values in each system was studied: the shear strain; the total dislocation density; the slip velocity of the dislocations; the climb veloc-ity of the dislocations to the parallel slip planes; the mean spacing between parallel dislocations in sub-boundaries; the mean subgrain size

The formulas that describe the changes in each parameter as a function of time and of the other parameters have been derived; a system of 12 ordinary differential equations was obtained The Runge-Kutta methods were used for integration of the system

The quantitative model results show a satisfactory fit with experiments The processes in each plane set happen somewhat differently

The density of mobile dislocations increases during the high-temperature strain from2 × 108m−2to(4.0–4.5) × 1011m−2 Experimental data have the

same order of1011m−2 The total dislocation density,ρ, is one order greater

than the deforming dislocation density,N.

The coefficient of multiplication of mobile dislocations is found to be of the order of2 × 104m−1

5.5 Summary 81

The velocity of the mobile dislocations, V , decreases gradually At the

steady-state stage it is of the order of 20 nm s−1 The value ofV is less in the

plane set where less applied stress operates In contrast, the climb velocity,

Q, increases with time The value of Q is two orders less than that of V

The jog spacing in mobile dislocations decreases when the strain increases,

as in reality A decrease inλ correlates with a decrease in the V value The

strain rate of the specimen is strongly affected by the sub-boundary dislocation spacing

A preliminary decreased value ofλ and increased value of D lead to the

strain rate decreasing when the applied stress is relatively low

High Temperature Strain of Metals and Alloys, Valim Levitin (Author)

Copyright c
2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim

ISBN: 3-527-313389-9

6.1

γ
Phase in Superalloys

The high-temperature strength requirements of materials have increased with new developments in engine design The continual need for better fuel ef-ficiency has resulted in faster-spinning, hotter-running gas turbine engines One of the most important requirements is resistance to high-temperature deformation This has created a need for alloys that can withstand higher stresses and temperatures for the hot zones of modern gas turbines The de-velopment has led, during past decades, to a steady increase in the turbine entry temperatures (5K per year averaged over the past 20 years) and this trend

is expected to continue Other crucial material properties are crack resistance, stiffness, resistance to oxidation and an acceptable density

Such alloys – superalloys – have been developed The largest applications of superalloys are in aircraft and industrial gas turbines, rocket engines, space vehicles, submarines, nuclear reactors and landing apparatus

The structure of the majority of nickel-based superalloys consists of a matrix i.e of theγ phase and particles of the hardening γ
phase Theγ phase is a solid

solution with a face-centered crystal lattice and randomly distributed different species of atoms By contrast, theγ
phase has an ordered crystalline lattice of

typeL12(Fig 6.1) In pureNi3Al phase atoms of aluminum are placed at the vertices of the cubic cell and form the sublattice A Atoms of nickel are located

at the centers of the faces and form the sublattice B In fact the phase is not strictly stoichiometric, there may exist an excess of vacancies in one of the sublattices, which leads to deviations from stoichiometry Sublattices A and B

of theγ
phase can dissolve a considerable amount of other elements Many

of the industrial nickel-based superalloys contain, in addition to chromium, aluminum, and titanium, also molybdenum, tungsten, niobium, tantalum and cobalt These elements are dissolved in theγ
phase.

84 6 High-temperature Deformation of Superalloys

Fig 6.1 Crystal structure of theγ
phase The face-centered

cubic cell contains 3 atoms of B-type (6 atoms/2 adjacent

cells) and 1 atom of A-type (8 atoms/8 cells) The chemical

formula isB3A

The crystal lattice parameter of theγ
phase is close to the parameter of

the solid solution, so the misfit between two lattices is relatively small The misfit,δ, between precipitates and matrix is defined as

δ = (a γ
− a γ )/

a γ
+ a γ

2



(6.1)

The value ofδ is negative for current commercial superalloys The

magni-tude and sign of the misfit also influence the development of microstructure under the operating conditions of stress and high temperature

Furthermore, because their lattice parameters are similar, theγ
phase is

co-herent with theγ phase with a simple cube–cube relationship ([001] γ

[001] γ) Dislocations in theγ phase nevertheless find it difficult to enter γ
, because

of stresses on the boundary and partly because theγ
is the atomically ordered

phase The particles reduce the velocity of the deforming dislocations and thus act as obstacles The order interferes with the dislocation motion and hence strengthens the alloy The small misfit between theγ and γ
lattices is

impor-tant When combined with the cube–cube orientation relationship, it ensures

a lowγ/γ
interfacial energy The ordinary mechanism of precipitate

coars-ening is driven entirely by the minimization of the total interfacial energy

A coherent or semi-coherent interface therefore makes the microstructure stable, a property which is useful for high-temperature applications

The solubility of theγ
phase is dependent on temperature This

depen-dence has been studied in [31] using a high resolution triple crystal diffrac-tometer and high energy synchrotron radiation (150keV,λ = 0.08) The

re-Fig 6.2 Dependence of the solubility of theγ
phase on

temperature

sults are shown in Fig 6.2 Solution of theγ
phase in the matrix, i.e in theγ

solid solution, is observed above 1173K (900◦C) The usual heat treatment of

superalloys consists of heating above the temperature of theγ
-solution and

subsequent hardening

Theγ
phase has remarkable properties, in particular, an anomalous

de-pendence of strength on temperature Theγ
phase first hardens, up to about

1073K, and then softens This peculiarity is reflected in a similar depen-dence of the yield strength upon temperature in superalloys This is shown

in Fig 6.3 The yield stress increases as the temperature increases from 573

to 1073K

Fig 6.3 Dependence of the yield strength of a superalloy

on temperature Experimental data (symbols) and predicted

quantities (dashed lines, see below) Reprinted from [30]

with permission from Maney Publishing