High Temperature Strain of Metals and Alloys Part 5 pdf

The typical Burgers vectors, slip planes and unit dislocation vectors have been selected for examination of sub-boundaries.. From our point of view, in metals with little stacking fault

Fig 4.11 The jog formation in a dislocation emitted by a

low-angle sub-boundary (a) The initial position,  b iare the

Burgers vectors;  ξ iare the unit dislocation vectors (b) The

same as (a) after changing the signs of the vectors  ξ3and

ξ4.P1andP2 are the slip planes;  V1 and  V2are the velocity

vectors (c) Dislocations with jogs after emission of the

dislocation  ξ1

The typical Burgers vectors, slip planes and unit dislocation vectors have been selected for examination of sub-boundaries The results are presented

in Table 4.2 The angles< b1ξ1and< b2ξ2are not equal to90◦ This means

that the dislocations of both systems contain screw components

Tab 4.2 The crystallography of low-angle sub-boundaries

Lattice Slip < ξ1ξ2 b1 ξ1 < b1ξ1 b2 ξ2 < b2ξ2

plane

f.c.c. {111} 90◦
a

2 [110] √2

2 [¯110] √2

2 [¯110] 0◦

60◦
a

2 [110] √

2

2 [10¯1] √2

2 [10¯1] 0◦

60◦
a

2 [011] √

2

2 [¯1¯10] 120◦
a

2 [¯110] √2

2 [0¯11] 120◦

60◦
a

2 [110] √2

2 [¯110] √2

2 [011] 60◦

b.c.c. {110} 90◦
a

2 [111] √2

2 [110] 35.3 ◦
a

2 [1¯11] √2

2 [1¯10] 35.3 ◦ 73.2 ◦
a

2 [111] √2

2 [¯1¯10] 144.7 ◦
a

2 [1¯11] √6

6 [1¯21] 19.5 ◦

4.7 Significance of the Stacking Faults Energy 57

4.7

Significance of the Stacking Faults Energy

The processes of high-temperature strain are dependent upon the nature of a metal, especially, upon peculiarities of dislocations in its crystal lattice Metals have different values of the stacking fault energy which results in a different ability to change the slip plain, i.e to climb into parallel slip plains This difference leads to various types of macroscopic behavior at high temperature

In Ref [24] four crept metals with face-centered crystal lattices: aluminum, nickel, copper and silver were investigated The subgrain misorientations were measured with the X-ray rocking method at discrete time moments Tests were carried out at the tensile rate of 0.5MPah−1 The total dislocation density was calculated from the misorientation angles

All four metals reveal linear dependences of the misorientation angle on strain at room temperature In Fig 4.12 the data of tests at0.45 Tmare shown The linear dependence remains only for silver At T = 0.68 Tm all depen-dences η(ε) have a certain curvature (Fig 4.13) The curves are ordered in

the order of the stacking fault energies: Al, Ni, Cu, Ag, 290, 150, 70, and 25mJ m−2, respectively.

Pishchak [24] considers the dependence of deformationε on stress σ At

room temperature there is a linear dependence,ε ∼ σ At high temperatures

he assumes the empirical equationε = Aσmto be the most appropriate The exponent of the power function,m, turns out not to be a constant value but to

increase with temperature fromm = 1 to m = 2 The temperature of the m

change is equal to 0.30, 0.35, 0.40 and 0.60Tmfor Al, Ni, Cu, Ag, respectively

Fig 4.12 The average subgrain misorientation versus strain

in four metals with face-centered crystal lattice Temperature

is equal to0.45 Tm B, aluminum; C, nickel; D, copper; E,

silver The results were calculated from the data of Ref [24]

Fig 4.13 The same as in Fig 4.12, but at temperature

0.68 Tm

From our point of view, in metals with little stacking fault energy, the climb of the dislocation edge components is hindered and dislocations cannot change their slip plane A higher temperature is needed in order for regular sub-boundaries to be formed

4.8

Stability of the Dislocation Sub-boundaries

As has been noted above, the sub-boundaries are both the sources of and the obstacles for deforming dislocations

Let us consider the effect of external stress and temperature on the sub-boundary dislocation emission By dislocation emission we mean a thermally activated release of a dislocation from an immobile boundary and its sub-sequent transformation into a mobile deforming dislocation Our aim is to de-termine a threshold stress, above which the sub-boundaries are unstable and can be destroyed without the thermal activation We shall analyze the effect

of applied stress and temperature on the sub-boundary dislocation emission Consider the boundary built by two perpendicular systems of equidistant parallel screw dislocations (Fig 4.14) Assume first that there is no dislocation

2 in a slip plane P1 The components of stress affecting a sub-boundary dislocation 1 in the slip plane (y = 0) are given by [18]

σ yz=



(µb sinh 2πX)(1 − cos 2πZ) 2λ(cosh 2πX − 1)(cosh 2πX − cos 2πZ)



−



µb 2πλX

 (4.22)

4.8 Stability of Dislocation Sub-boundaries 59

Fig 4.14 A sub-boundary formed in theyOz-plane by two

systems of screw dislocations.λ is the distance between

adjacent dislocations,D is the subgrain size, P1 (xOz) is

the slip plane; a boundary dislocation under consideration

is denoted by 1, another dislocation in the slip plane outside

the boundary is denoted by 2

σ xz =2πλX µb ; σ xy= 2λ(cosh 2πX − cos 2πZ) µb sin 2πZ (4.23) whereµ is the shear modulus, X = x/λ, Y = y/λ, Z = z/λ.

When the dislocation 1 deviates from the boundary the shear stress com-ponentσ yzacts on it (For a screw dislocation the stress component is parallel

to the dislocation line.) The value of this component depends upon the co-ordinates The results of the calculations of the shear stress are shown in Fig 4.15

Fig 4.15 The stress componentσ yzin units ofµb/2λ as a

function of distance On the left:z/λ = const, on the right:

x/λ = const.

The curves have singularities atx = 0 Within the sub-boundary (in the

initial position) the stress components are therefore equal to zero Thus, the dislocation inside the boundary is affected by the forceF (0) = 0 The force

reaches its maximum value near the node at a distance equal to the dislocation core radiusr0 It is reasonable to assume thatF (r) is a linear function within

the range 0 < r < r0 FurtherF (r) = −bσ yz if r0 ≤ r < r1, wherer1

is a distance at which the interaction force between the dislocation and the boundary is close to zero

The calculated dependences of force and energy on the distance from the deviated dislocation 1 to sub-boundary are shown in Fig 4.16 One can see that the maximum returning force is achieved at a distance of the order of the dislocation core This force acts in the opposite direction

Fig 4.16 The force at which the sub-boundary acts on the

emitted dislocation 1, and the activation energy versus the

distance.r0 = 2b is assumed.

Assume that the applied external stress isσ.

The energy to be consumed by the emission is expressed as

U = −

 r0

0

F (r)dr −

 r1

The stress field of the sub-boundary tends to return the dislocation 1 to the sub-boundary Thus, the dislocation is pinned with pinning point density1/λ

and is emitted by means of thermal activation According to the theory of the rates of reactions [25] the dislocation can be regarded as a linear crystal with

D/b degrees of freedom The number of thermal activations per unit of time

4.8 Stability of Dislocation Sub-boundaries 61

can be represented by an expression of the form

Γ = νeffexp

∆U kT



(4.25)

where νeff is a pre-exponential factor;∆U is the activation energy and kT

has its usual meaning From Eqs (4.22), (4.23) and (4.24) we obtain for one degree of freedom (z = 0)

U = µb 2π3lnαe α r1

whereα = b/r0 Taking into account the work of the external stress we obtain

The activation energy is essentially less if there aren ≥ 2 slipping

dislo-cations in the same slip plane One can show that in this case the factor n

appears before the second term on the right-hand side of Eq (4.27)

In Fig 4.17 the calculated curves of the influence of temperature and stress

on theΓ value are shown for two metals The probability of dislocation

emis-sion from the sub-boundary is strongly affected by the temperature and the number of dislocations in the slip plane

Fig 4.17 The number of thermal activations per unit time

as a function of stress and temperature Solid lines, one

dislocation in slip plane, dashed lines, two dislocations in

slip plane (a) Nickel, (b) vanadium.r0 = 2b and r1= λ is

assumed

The condition of the stability of the boundary during strain is

1

Γ > τcreep.

HereΓ −1 is the time interval before the emission begins andτcreepis the time interval during which the creep deformation occurs; e.g for a creep time

of105s thenΓ < 10 −5s The results in Fig 4.17 show the temperature and

stress intervals where the sub-boundaries are observed

From Eqs (4.26) and (4.27) we obtain the condition of the inactivated emis-sion of dislocations from the sub-boundaries:

Assumingλ = 50nm, n = 2, α = 0.5 we obtain σ ≥ 2 × 10 −3 µ for nickel.

When the external stress is higher than this value then the sub-boundaries are unstable and are destroyed

4.9

Scope of Application of the Theory

A well-read reader may ask: what is the distinction between this theory and the model published by Barrett and Nix [11]?

This excellent article was the first to examine deeply the motion of jogged dislocations as a process which controls the strain rate However, the authors conceived the jogs as being a result of thermal activation The equations pro-posed by them take into account only a thermodynamic equilibrium number

of jogs in the dislocations In their opinion, the screw components therefore contain equidistant alternating jogs of opposite signs They wrote: “The av-erage spacing between jogs, λ, has never been measured directly”, so they

assumed a parameterλ which could not be measured The quantitative

eval-uation of the strain rate was out of the question at that time, of course

As a matter of fact, the adjacent jogs of opposite signs slip along the dis-location line easily and would simply annihilate each other The equilibrium values ofλ can affect neither the dislocation velocity nor the creep rate.

According to our experimental results the sources of jogs of the same sign

in mobile dislocations are the immobile sub-boundary dislocations and we be-lieve that the substructure formation plays a key role during high-temperature strain, being the process that affects the strain rate

The present theory is understood to be valid within certain limitations When the temperature is relatively low, the dislocation climb is depressed

4.9 Scope of the Theory 63

and hence regular sub-boundaries cannot be formed The lower limit to give

a sufficient climb rate is about 0.40 or 0.45Tm The low-temperature defor-mation is controlled by other processes, e.g the overcoming of the Peierls stress in the crystal lattice

The stable sub-boundaries are of major significance in the process of high-temperature strain for pure metals and solid solutions The upper stress limit of the sub-boundary stability depends upon the metal properties and temperature The lower the shear modulusµ and the higher the

tempera-ture, the lower the limit An estimation, for instance, shows that in nickel

at 0.6Tmsub-boundaries are destroyed by a stress of2 × 10 −4 µ in 30h The

analysis shows that when the applied external stress is higher than about

2 × 10 −3 µ inactivated emission of dislocations from sub-boundaries occurs

and the sub-boundaries break up The upper limit of temperature is (0.70– 0.75)Tm Diffusion creep takes place (the mechanism of Herring-Nabarro) at higher temperatures and relatively lower stresses It is necessary to empha-size that an adequate understanding of dislocation processes in these ranges

of temperature and stress is of great practical importance Most heat-resistant metals, steels and alloys operate at temperatures between 0.40 and 0.75Tm The area of temperature and stress where the proposed mechanism of high-temperature deformation takes place, is shown in normalized coordinates in Fig 4.18 Construction diagrams (maps) of this type were proposed by Ashby, e.g in Ref [26]

Fig 4.18 The deformation map of nickel The shaded area

represents the interval of temperature and stress where the

physical mechanism under consideration takes place

The numbers on the curves denote strain rates in s−1

Fig 4.19 The same as in Fig 4.18 but for iron

It is known that in iron allotropic transformation occurs at 0.65Tm(Fig 4.19) The mutual arrangement of the deformation areas is in other respects similar

to the previous one, however, there is a quantitative difference The strain rate

of iron, which has a body-centered crystal lattice is considerably greater For example, at 0.5Tmunder a stress of6 × 10 −4 µ strain rates for Ni and Fe are

equal to10−7and10−3s−1, respectively.

4.10

Summary

The dislocation density increases at the beginning of plastic strain In the pri-mary stage of deformation a part of the generated dislocations form discrete distributions Dislocations penetrate low-angle sub-boundaries The interac-tion of dislocainterac-tions having the same sign is facilitated by high-temperature and applied stress These conditions make it easier for edge components of dislocations to climb The immediate cause of the formation of dislocation walls is the interaction between dislocations of the same sign resulting in a decrease in the internal energy of the system

At the end of the substructure formation the dislocation arrangements are ordered Then a steady-state stage of strain begins During this stage a dislocation emission from sub-boundaries takes place

In metals with small stacking fault energy the climb of the dislocation edge components is hindered The ordered dislocation sub-boundaries require a higher temperature in order to form

The low-angle sub-boundaries are built up of parallel equidistant dislo-cations that contain screw components The sub-boundary dislodislo-cations are

4.10 Summary 65

sources, emitters for mobile dislocations, which contribute to the specimen strain Emission of mobile dislocations from sub-boundaries leads to the for-mation of the equidistant one-signed jogs The distance between jogs at mo-bile dislocations is close to the distance between the immomo-bile sub-boundary dislocations

The jogged dislocations can slip when there is a steady diffusion flux of generated vacancies from jogs The emitted dislocations are replaced in sub-boundaries with new dislocations, which move under the effect of applied stress Having entered a sub-boundary a new dislocation is absorbed by it The relay-like motion of the vacancy-emitted jogged dislocations from one sub-boundary to another one is the distinguishing feature of the high-temperature strain of single-phase metals and solid solutions

The velocity of dislocations depends exponentially on the applied stress The exponent contains the sum of the activation energies of vacancy generation and vacancy migration

Processes of dislocation multiplication, annihilation, sub-boundary emis-sion and immobilization occur in metals during the high-temperature strain The balance equation, which characterizes the change in the mobile disloca-tion density, has been derived

Three groups of physical parameters are needed for estimation of the steady-state strain rate ˙ε:

• External parameters: temperatureT and stress σ.

• Diffusion parameters: the energy of the vacancy generation Ev and the energy of the vacancy diffusionUv

• Structural parameters: the average subgrain size ¯D and the mean distance

between dislocations in sub-boundaries ¯λ.

The computed values of ˙ε fit the experimental data satisfactorily at certain

temperature and stress conditions

The rate of the stationary creep correlates with the amplitude of atomic vibrations at high temperatures

The developed theory is valid within certain limits of the temperature and applied stress When the temperature is relatively low, the dislocation climb

is depressed and hence the regular sub-boundaries cannot be formed The lower limit for sufficient climb rate is about 0.40Tm or 0.45Tm, the upper limit is 0.70Tmor 0.75Tm

The upper stress limit of the sub-boundary stability depends upon the metal properties and temperature The lower the shear modulusµ and the higher

the temperature, the lower the limit An inactivated emission of dislocations from sub-boundaries occurs when the applied external stress is higher than about2 × 10 −3 µ, the sub-boundaries then break up.