High Temperature Strain of Metals and Alloys Part 4 ppsx

The conclusion is drawn that the sub-boundary dislocations generate mobile dislocations with vacancy-producing jogs in their screw com-ponents... In Section 4.6 and in Supplement 2 one c

3.8 Summary 41

type for the cubic body-centered crystal lattice The fifth distinctive structural feature is the presence of jogs in mobile dislocations Moreover, the distances between jogs are very close to the distances between immobile dislocations

in small-angle sub-boundaries The following conclusion can be drawn: the mobile dislocations arise from the sub-boundary dislocations It is as if the former bear a “stamp”, an imprint of the latter

The generation of vacancies during the process under consideration is the last feature Loops and helicoids are formed when vacancies collapse It

is logical conclude that sources of vacancies are activated during the high-temperature deformation process It should be noted that some features described above, especially, the formation of small-angle boundaries were observed in many studies However, we should to take into account all the structural peculiarities for an adequate understanding of the phenomenon under consideration

Now we can proceed to describe the physical mechanism of high-temperature deformation of pure metals and single-phase alloys Our aim is to relate the microstructural observations to the measured strain rates

3.8

Summary

Typical structural features are observed in pure metals and solid solutions, which are loaded at high temperatures These features are caused by certain physical mechanisms of deformation

The average subgrain sizes,D and the average subgrain misorientations,

η, have been systematically directly measured during high-temperature strain

of the metals and alloys under investigation The values ofD are of the order

of 0.7–2.0µm, the values of η are of the order of 2.9–6.0 mrad.

The substructure is formed inside crystallites during the primary stage of creep The value ofD decreases and η increases during the primary stage.

The origin of the steady-state stage coincides with the end of substructure formation The steady-state creep occurs at almost constant mean values of both parameters It is the process of substructure formation in the primary stage that causes the decrease in the strain rate and the beginning of the steady-state stage

The values ofD and η are strongly dependent upon stress The greater

the applied stress the greater the misorientation angles and the smaller the subgrains

Investigations of single-phase two-component alloys do not reveal any qual-itative differences in the structure evolution from the processes occurring in

pure metals Larger values ofη are observed in solid solutions as compared

with metals

Most of the dislocations in specimens are associated in sub-boundaries The parallel sub-boundary dislocations are situated at equal distances from each other It follows from results of the Burgers vector determinations and from the repeating structural patterns that the parallel sub-boundary dislocations are of the same sign Two intersecting dislocation systems are often found inside sub-boundaries

The sub-boundaries that have been formed are the sources of slipping dislocations At the same time the sub-boundaries act as obstacles to the movement of deforming dislocations

Kinks and bends are observed in dislocations inside subgrains They are caused by jogs in the mobile screw components A great number of vacancy loops and helicoids are present in the structure The slipping dislocations with equidistant one-sign jogs generate vacancies

It has been found that the average distances between sub-boundary dislo-cations,λ, and the mean spacings between jogs in mobile dislocations, z0are close in value The conclusion is drawn that the sub-boundary dislocations generate mobile dislocations with vacancy-producing jogs in their screw com-ponents

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

43

4

Physical Mechanism and Structural Model of Strain at High

Temperatures

4.1

Physical Model and Theory

It follows from the obtained data that complex processes occur in the crystal lattice at high-temperature strain These processes are inherent to pure metals and to solid solutions at certain temperatures and under the applied stresses The quantitative theory, which we are going to develop, will be based on the experimental results presented in Chapter 3

The dislocation density increases at the beginning of the plastic strain

In the primary stage of deformation some of the generated dislocations form discrete distributions They enter into low-angle sub-boundaries The interac-tion of dislocainterac-tions having the same sign is facilitated by the high-temperature conditions and applied stress These conditions make it easy for dislocations

to move and for the edge components of dislocations to climb The edge dislocations can change their slip planes

Why do the dislocations form ordered sub-boundaries spontaneously? The immediate cause of the formation of the dislocation walls is the interaction between dislocations of the same sign that results in a decrease in the internal energy of the system The elastic energy of dislocations that are associated in subgrains,Es, is less, than the energy of dislocations distributed chaotically

in the whole volume of the material,Ev One can compare these energies The values ofEvandEsare expressed as [18]

Ev≈

ρµb2

4π

 ln

L b



(4.1)

Es≈

ρµb2

8π

 ln

F

ρb2



(4.2)

where ρ is the dislocation density, µ is the shear modulus, b is the Burgers

vector,L is the size of a crystal, and F is the fraction of the crystal volume

that is occupied by sub-boundaries

Assuming reasonable values ofρ = 1012m −2,L = 0.3cm, F = 0.05, one

can obtain

Ev

Es = 2.4

It follows from this ratio that an appreciable decrease in internal energy takes place due to the formation of sub-boundaries

Well-formed sub-boundaries were observed in the experiments described

in Chapter 3 In fact the dislocation subgrains are two-dimensional defects They are sources, emitters of mobile dislocations, which contribute to strain

In Section 4.6 and in Supplement 2 one can find evidence that emission

of mobile dislocations from sub-boundaries leads to the formation of jogs in them

It is the dislocation sub-boundary that generate jogs in mobile dislocations The screw components of emitted dislocations “keep their origin in their memory” They contain equidistant one-signed jogs, although this is only in the physical model, actually, the distances between jogs,z0, are distributed values

A jog is a segment of dislocation, which does not lie in the slip plane The jog cannot move without generation of point defects, i.e vacancies The jogged dislocation can slip if there is a steady diffusion of vacancies from

it The nonconservative slipping of jogged dislocations is dependent on the material redistribution The shorter the distance between jogs, the lower the dislocation velocity Hence, it is the diffusion process that controls the velocity

of the slip of deforming dislocations

Note that we consider vacancy-producing jogs only The interstitial-produ-cing jogs are practically immobile because the energy of formation of the interstitial atoms is several orders greater than that of the vacancies That is why we have observed so many vacancy loops and helicoids in the structure

of tested metals (Section 3.6) It appears that a subsequent coalescence of vacancies leads to cavity formation and rupture This seems to be the essence

of the tertiary stage of creep

The distance between dislocations in sub-boundaries decreases during the primary stage of deformation and the role of sub-boundaries as obstacles for slipping dislocations increases

At the end of the substructure formation the dislocation arrangement is ordered and a steady-state stage begins During this stage a dislocation emis-sion from sub-boundaries takes place The emitted dislocations are replaced

in sub-boundaries with new dislocations, which move under the effect of ap-plied stress Having entered a sub-boundary a new dislocation is absorbed by

4.2 Velocity of Dislocations 45

it Jogs of the same sign appear along the screw components of the emitted dislocations The distance between jogs at mobile dislocations is equal to the distance between the immobile sub-boundary dislocations, see Eq (3.2)

Figure 4.1 demonstrates the described model of steady-state strain (creep)

It can be seen that atoms are required in order to complete the extra-planes during the motion of the jogs Vacancies are generated in the vicinity of jogs since atoms are consumed in completing the extra-planes Consequently, the relay-like motion of the vacancy-emitted jogged dislocations from one sub-boundary to another is the distinguishing feature of the high-temperature strain

Fig 4.1 The physical model of the steady-state strain at high

temperature The sub-boundaries are built of two systems:

the pure screw and the60◦-dislocation systems Emission

of mobile dislocations from sub-boundaries is shown The

jogs at the mobile screw components have the same sign

and generate vacancies

4.2

Velocity of Dislocations

Consider a screw dislocation that is situated along the Oz axis of the coordinate

system A dislocation with jogs moves in the direction Ox (see Fig 4.2) The

material parameters are different in the volume and inside the dislocation

“tube” Let us denote the coefficients of the diffusion of the vacancies byDv

andDd, respectively

In Figs.4.2 and 4.3 the energy of vacancy generation is denoted byE The

energy of vacancy diffusion is denoted byU Subscripts j, d, and v refer to the

jog, dislocation and volume, respectively For example,Ujdis the energy of

Fig 4.2 Scheme of energetic barriers for the motion of

the screw dislocation with vacancy-producing jogs The

dislocation is slipping along Ox from left to right (see text

for further explanation)

the vacancy displacement from the jog in the dislocation “tube”;Udvis the energy of the vacancy displacement from the dislocation in the crystal volume and so on

The following elementary events determine the process of the dislocation slip: the generation of vacancies near jogs (energy of activationEd); the diffu-sion of vacancies along the dislocations (energy of activationUd); transition

of vacancies to the volume (Udv); diffusion of vacancies in the volume (Uv) The external applied stress performs work,dA, and facilitates the generation

Fig 4.3 Scheme of energetic barriers for the motion of the

screw dislocation with vacancy-absorbing jogs

4.2 Velocity of Dislocations 47

of vacancies:

The vacancy concentration in the vicinity of the jogs,cjp, increases under the effect of stress:

cjp= c0exp

dA + ε0

kT



= 1

b3

− Ev

kT

 exp

dA + ε0

kT

 (4.4)

wherec0is the equilibrium vacancy concentration in the volume at a given temperature,ε0 is the energy of bonding a dislocation and a vacancy,ε0 =

Udv− Uvd.

In the case of vacancy-absorbing jogs the sequence of events is the inverse, i.e., the generation of vacancies in the volume(Ev), diffusion of vacancies

in the volume (Uv), transition of vacancies in the dislocation “tube” Uvd, diffusion of vacancies in the “tube”Ud, joining the vacancy in the jogUdj The applied external stress facilitates joining of vacancies:

In both cases the work of applied stress is given by

dA = σ zz b3+ σ yz z0

whereσ zzandσ yzare the components of the stress tensor;a is the height of

the jogs As super-jogs are unstable we may assume thata ≈ b and b  z0 Hence

Now we will consider the velocity of the dislocations

The velocity of the screw components with vacancy-producing jogs is given

by the following expression [19]

Vp=πνr0z0

bF (α) exp

− Ev+ Uv+ ε0

kT

  exp

σ yz b2z0+ ε0

kT



− 1

 (4.8)

where ν is the Debye frequency, r0 is the radius of the dislocation “tube”,

α = Vpr0/2Dv,F (α) is a weak function.

The vacancy-absorbing jogs have velocity

Va= πνr0z0

bF (α) exp

− Ev+ Uv+ ε0

kT

 

1 − exp

− σ yz b2z0+ ε0

kT



(4.9)

In practice we always haveσ yz b2z0 kT Therefore Vp Va The velocity

of the dislocations is exponentially dependent on stress One can see that the exponent (4.8) contains the sum Ev + Uv This implies that the effective energy of the jogged dislocation motion is close to the activation energy of diffusion The activation volume in Eq (4.8) is equal tob2z0.

The computed values ofVpvary in the range10−11to10−2cm s−1 As is

shown in Fig 4.4 the velocity of dislocations with vacancy-absorbing jogs,Va,

is less by many orders These dislocation components are immobile and do not control the strain rate

In Fig 4.5 the velocity of jogged dislocations inα-iron is shown The

dis-tance between jogs strongly influences the velocity of the dislocations

Fig 4.4 Velocity of screw dislocations in nickel The distance

between jogs is 36nm 1, 2, 3: jogs generate vacancies;

1 , 2  , 3 : jogs absorb vacancies 1 and1: 673; 2 and2: 873;

3 and3: 1073K

4.3 Dislocation Density 49

Fig 4.5 The effect of temperature, stress and distance

between jogs on the velocity of screw dislocations inα-iron.

1: 773K,z0 = 35nm; 2: 813K, z0 = 57nm; 2: 813K,

z0 = 75nm; 3: 973K, z0= 52nm; 3: 973Kz0= 75 nm

4.3

Dislocation Density

The slip strain rate is given by [20]

where N is the density of deforming dislocations and V is their average

velocity The total mobile dislocation density is assumed to be equal to the sum

whereNais the density of annihilating dislocations UnlikeN the density Na

does not contribute to the macroscopic strain

Processes of the dislocation multiplication, annihilation, sub-boundary emission and immobilization, occur in metals during the high temperature strain The balance equation, which characterizes the change in the mobile dislocation density, can be written as

˙ρ = ˙ρm

d + ˙ρm

s + ˙ρa

d+ ˙ρa

where ˙ρm

d is the rate of the density increase due to the dislocation multipli-cation, ˙ρm

s is the rate of the density change on account of the sub-boundary emission, ˙ρa

dis the annihilation rate, and ˙ρa

sis the rate of the immobilization of dislocation by sub-boundaries (Subscript d refers to dislocations, subscript s

to sub-boundaries, superscript m to multiplication and emission, superscript

a to annihilation and immobilization) Therefore, two terms of Eq (4.12) are determined by the interactions of dislocations and the other two are related

to the effect of sub-boundaries

Consider each term of Eq (4.12) separately The number of newly generated dislocation loops is directly proportional to the dislocation density and to the dislocation velocity Hence the rate of multiplication of the mobile slip dislocations is given by

˙ρm

whereδ is a coefficient of multiplication of the mobile dislocations The

co-efficient has the unit of inverse length

Similarly, the rate of emission of the mobile dislocations out of sub-boundaries

is directly proportional to the sub-boundary dislocation density and to the dis-location velocity:

˙ρm

whereρs= η/bD = 1/λD is the density of dislocations in sub-boundaries;

δsis a coefficient of dislocation emission

Let us assume there are n+ positive dislocations and n − negative ones inside a subgrain The annihilation rate of dislocations of opposite signs is given by

˙na

d= −2 n+n − V

wherel = ρ −0.5is the mean distance between dislocations inside subgrains.

Sinceρ = n/D2,ρ+= ρ − = ρ/2, one can obtain

˙ρa

Immobilization of the mobile dislocations occurs when they are captured

by sub-boundaries The rate of immobilization is assumed to be directly pro-portional to the dislocation density and velocity and inversely propro-portional to the subgrain size:

˙ρa

Only deforming dislocations of density N come out of sub-boundaries;

annihilating dislocations of densityNaare eliminated inside subgrains