Periodic planar dislocation networks and phase boundaries

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Periodic planar dislocation networks and phase boundaries

C. Rey, G. Saada

To cite this version:

C. Rey, G. Saada. Periodic planar dislocation networks and phase boundaries. Journal de Physique,

1977, 38 (6), pp.721-725. �10.1051/jphys:01977003806072100�. �jpa-00208631�

PERIODIC PLANAR DISLOCATION NETWORKS AND PHASE BOUNDARIES

C. REY and G. SAADA

Laboratoire des

propriétés Mécaniques

^et

Thermodynamiques

des Matériaux Université

Paris-Nord,

^avenue

Jean-Baptiste-Clément,

⁹³⁴³⁰

Villetaneuse,

^France

(Reçu

^{le 10}

janvier 1977, accepté

^{le 3 mars}

1977)

Résumé. ²⁰¹⁴ La méthode décrite par Rey ^{et Saada} (1976) pour calculer les propriétés élastiques ^des

réseaux de dislocations plans ^et périodiques ^est appliquée entre autres cas au réseau nid d’abeilles.

Un accent particulier ^{est mis sur} le calcul de l’énergie élastique. ^D’autres

applications

^sont suggérées.

Abstract. ²⁰¹⁴ The method outlined by Rey ^{and Saada} (1976) ^to calculate the elastic properties ^of

periodic planar

dislocation networks is

applied

^{to cases such as}

honeycomb

^networks. Emphasis ^is put ^{on the} calculation of the elastic energy.

Applications

^are suggested.

Classification

Physics ^Abstracts

7.221 1 ^- 9.124

1. Introduction. ^- In a recent paper

[1] the

^authors

have

given

^a discussion of the elastic

properties

^of

planar periodic

dislocation networks.

Assuming

^the

network is in the

plane

^xlox2

(see Fig.

^{la and}

b)

^one

may

^define

^following

^Kroner

^[2]

^a dislocation

density tensor

in the form

a(xl, X2) 6(X3)-

^{Since a is}

periodic

in the

plane xlox2 it

^{can be}

developed

^{as a Fourier}

sum :

Here g refers to the

reciprocal

^{vectors of the}

periodic

dislocation network.

The

incompatibility q

^{is then}

easily

calculated as :

A and B are

periodic

^{in the}

plane

^{XIOX2 with the same}

periods. They

^{can be}

developed

in Fourier _sums, the

corresponding

coefficients

being ^A(g)

^and

^B(g).

^Com-

plete general expressions

^{for A and B have been}

given in [1].

. The stress functions _x is then calculated in all ^cases of interest and from it the self _energy can be obtained.

In this _paper we wish to

apply

these results to the

case of more

complicated

networks such as those described in

figures

^la and lb. Section 2 of this _paper will be devoted to the derivation of such formulae and their

application

^{to a}

simple

^{case. It will be shown}

that the method outlined in

[1]

^{leads to}

simple

^calcula-

tions which are more

general

and easier to

perform

FIG. la. ^- Dislocation network made of two sets [(I) ^and (2)]

of parallel dislocations : : Dislocations ; - - - : ^Unit cell ;

----+ : Translation vectors of each set.

FIG. lb. ^- Honeycomb ^{network : :} Dislocations ; - - - : ^: Periodic lattice ; ^{- :} Translation vectors of the periodic

lattice.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003806072100

722

than those

already published [3, 4].

^{In section 3 the}

results will be

generalized

^{to the case of}

general phase boundary

^where

long-range

^{stresses exist.}

We conclude in section 4.

All calculations are made in the

approximation

^of

isotropic elasticity.

2. Self _{energy of}

planar

dislocation networks formed from two

independent

^{sets of} dislocations with no

long-

range ^{stresses. -} 2 .1 GEOMETRY. - We shall consider

as in

[1] (see Fig. la)

^{two sets} of dislocations such that each set

(p)

^is characterized

by :

t(p)

Unit vector

along

the dislocation line.

À (p)

Vector

perpendicular

^{to À (p).}

characterizing

^the

period of pth

^set.

b(p)

The common

Burgers

^vector of the dislocations.

g(p)

^The

reciprocal

^{lattice vector.}

All these

symbols

have been defined in

[1].

^We

need to add to them :

Pp, Qp

and the unit cell are defined in

figure

^{1 a.}

Having independent

^{sets means}

[5]

that the follow-

ing equations (2a)

^and

(2b)

^{have to} be verified simul-

taneously :

It is convenient to make use of the

angle (P

^between

t(l)

and

t(2).

As shown

by

^Frank

[4]

this network _may transform into a

honeycomb

^network

(Fig. 1b)

^{with a unit cell}

made of three segments I(P) such that the well-known node condition is verified.

Then as in

[ 1]

^{we define}

a(1), a(2) ; I(1), I(2),

^{I(3) the}

dislocation segments

having

^their

origin

^{at the node N}

The node relation is

The

corresponding

dislocation tensors,

incompatibi- lity

^{tensors and the}

corresponding

^{A and B tensors}

have been defined and calculated in

[1].

2.2 CALCULATION OF THE SELF ENERGY IN THE CASE OF A HONEYCOMB NETWORK. -

Following

^{the method}

outlined in

[1]

^and

using

^identical

symbols,

^{we consider}

two identical

planar periodic honeycomb

dislocation networks

exhibiting

^no

long

range ^stresses

(i.e.

^such

that B(0) is

0)

translated with respect ^{to each other}

perpendicular

^{to the}

plane

^of the network. Let _xlox2 be the

plane

^of the network and r the distance between

both

networks ;

their interaction _{energy E} is defined

by

^Kroner

[2]

where q

and x ^are

respectively

^the

incompatibility

tensor and the stress function.

They

have been cal- culated in

[1].

^Since

they

^are

periodic

in XlOX2 with the same

periods

^one may define an interaction _energy per unit surface

EI :

The self energy per unit surface

Es

is obtained

by

^the

I usual

procedure

It must be

emphasized

that the self _energy defined in

(5b)

^{is not}

unique,

^but

depends

^on where the mathe- matical cuts are made. Therefore the dislocation core

energy ^must be added to the results we obtain. We shall come back to this

point

^{in our} conclusion.

Using

the results of

[1]

^one obtains after a

lengthy

but

simple

calculation

In these

formulae g

^{is the}

length

^{of the}

reciprocal

vector g, p is the shear modulus and v the Poisson ratio.

To

study EI

when r varies we notice the

following

09

^{is a} linear combination of terms such as

while

Y,

^{is a linear} combination of terms such as

therefore

Y’g

^and

«PI

exhibit the same variation in _g as

As shown in

[1]

Two cases must be considered : If _{g is} on a

straight

line D (p) such that

sin

(ngl)

^and

g(ap -1)

^{tend to zero in the same way}

and I a;. ^is independent of g.

If _g is not on D(P)

then I rxfj ^I

decreases with _g as

!g!".

From these considerations it is clear that the sum

(6a)

converges for all values of ^r except ^r

equal

^to

zero.

For

large

^values

of r,

^the

exponential

^{term allows}

the sum to be restricted to the next nearest

neighbours.

For small values of _r, i.e. r much smaller than the size of the

network,

^{the sum has to be calculated}

numerically.

^However the behaviour for _very small values of r is

easily

^obtained

by restricting

^{the summa-}

tion to the three

straight

^lines

^{D (p).}

^The calculation is then

formally

^{similar to} that made for sets of infinite dislocation lines.

Therefore,

^{for small r :}

It must be

emphasized

^that

restricting

the calculation of the sum

(6a)

^{to the set of}

points belonging

^{to the}

three

straight

^lines

Dp

^indeed

^gives

^{the correct beha-}

viour of

EI (or Eg)

^{for small r.} However the

procedure gives only

^a

rough approximation

^{of the exact nume-}

rical values. To obtain these values a numerical

computation

is needed. As will be shown in section 2.4 the numerical calculation is _very

simple.

2.3 TWO SETS OF PARALLEL INFINITE DISLOCATIONS.

- In the case described in

figure

^la

equations (5a)

and

(5b)

hold. The

calculation ^{of EI (or EJ}

^{is very}

simply expressed

^{in terms of the tensors}

^C(p)

^and

^D(p)

defined as :

Making

^{use of the same method one obtains :}

When r -+ 0

From which we obtain :

The result is _very similar to that obtained in the _pre- vious section.

2.4 APPLICATION TO THE CASE OF A PURE TWIST BOUNDARY IN THE

(111)

^{PLANE OF F.C.C. METALS. -}

2.4.1

Honeycomb

^{network. - The} geometry ^{is well} known and the minimum _energy

configuration

^corres-

ponds

^{to the}

symmetrical position

^{with screw disloca-}

tions of

equal length

^{I. In that} case, both Tr A and Tr B

are 0 and use of

(5)

^and of formulae

(6) gives

the result :

nl and _n2 are

integer

^numbers.

f (x)

^{has to} be calculated

numerically.

^We

give

ⁱⁿ

figure

^{2 the}

plot

^of

f (x)

which for small values of x

724

FIG. 2. ^- Plot of f(x) ^versus Log ^x given by ^formula (14a).

(x ^0.1)

^{is a}

logarithmic

function of x. For small

values

^{of x :}

Since the twist

angle

^{0 is :}

We may

write

for small values of x :

2.4.2 Network made

of

^two

independent

^sets

of parallel

dislocations. ^- The characteristics of the two networks are summarized in table I.

Using

^for-

mulae

(11), (12), (13)

^{the self} energy is calculated as :

2.4.3 Remarks. ^- As indicated in the introduction the method for the calculation of the self _energy of a

honeycomb

^{network is more}

general

and easier to

perform

^{than that}

published

^elsewhere

[3, 4].

^{Here the}

numerical calculation is restricted to the calculation of a sum which can be

performed

^{with a small com-}

puter. The calculation for the case of two sets of infinite dislocations is

simple

^{too and more}

general

than that

published by

^{Read and}

Shockley (1950)

who obtained formula

(17a).

^Formulae

(16)

^and

(17)

allow for the calculation of energy AE

gained by transforming

the network of

figure

^{la into the}

honey-

comb network. It is

easily

verified that AE is

positive (i.e.

^the

honeycomb

^{network is}

stable) provided

⁰

is smaller than a critical value

Be.

In the case calculated

above, assuming

^{v ~}

1/3

and r -

ble

^{one obtains :}

It must be made clear that the result

(18) depends

on the

assumption

^{made for the core} size and that the

core energy should be taken into account to obtain

an accurate value of

Oc.

3. Effect of

long

range ^{stresses. -} One of the

advantages

of the method outlined in

[1]

is that it allows the calculation of _energy even when there exist

long

range ^stresses. The condition for the exis- tence of such stresses is that the tensor

B(0)

defined in

[1]

be different from zero. In that case it has been shown that the

long

range ^stress is uniform in each half _space limited

by

^the

boundary.

Therefore the

only meaningful

situations corres-

pond

^{to an}

assembly

^of

parallel

boundaries such that the uniform stress is zero outside the slab limited

by

the boundaries.

For such a slab of width ^r the elastic _energy can be written :

K

depends

^{on the}

boundary

and has been calculated in this _paper.

g(r)

^{is a function}

decreasing

^{with r.}

B(O)

corresponds

^{to one}

boundary.

TABLE I

From

(19a)

^and

(19b)

^{it is seen} that there must exist

a critical

value rc

for which the _energy is a minimum :

Such a situation occurs for

example

when disloca- tions

split

outside the

boundary

^{as shown in}

[1]

^{in a}

simple

^{case. More}

complicated

situations _may occur

which can be calculated

by using

^the

technique

^deve-

loped

^{in this} paper. It ^must be

emphasized

^however

that the term Lr may

give

^{rise to} three-dimensional instabilities such as those observed in stainless steel

[8].

Lamellar

compounds

^{can be}

approximately

^treated

within the frame of this method and an

approximate

width of the lamellae can be obtained

through

equa- tions

(19).

A detailed calculation would involve

taking

^into

account the

long-range

interaction between lamellae and differences between elastic constants.

However formulae

(19) provide

^a

limiting

^condition

for the

stability

^{of such lamellar}

compounds.

^For

small rc

^{the function}

g(r)

^{is well}

approximated

^as

-

Log r therefore rc

^is

simply given

^{as :}

Where K and L are

easily

calculated from formulae

(16), (17)

^and

(19).

^A condition for the

stability

of lamellar

compounds

^{is then}

that r be larger

^{than b.}

4. Conclusion. ^- The results obtained in

[1] ]

^and

in this _paper allow

simple

calculations of the elastic

quantities

^{related to}

periodic planar

distributions of dislocations.

As shown

previously [5]

^{a correct} formulation of the elastic

problem

^{enables one to} describe the _geo- metry ^of

grain

boundaries and to show that Boll- mann’s

theory

^is

only partially

^correct

[9].

^Moreover

it is well known

[10]

that the transformation relation-

ship

^between

periodic

lattices is not

unique.

^{To each}

transformation

corresponds

^{at least one} dislocation distribution. To choose between those distributions

one needs an energy calculation which can be made

using

the results of this _paper.

The use of this kind of calculation is however

severely

limited for the

following

^{reasons :}

1)

^Core

effects,

local relaxation and

point

^defects

are not included in the calculation.

They

^are

obviously important

^for

large-angle

boundaries

[5].

2)

Three-dimensional instabilities such as those described

by Bollmann,

Michaut and Sainfort

[8]

cannot be taken into account.

3)

Most of the effects of elastic

anisotropy

^are

difficult to include in such calculations except ^{for the}

case of small

angle

boundaries

[11].

References

[1] REY, C. and SAADA, G., Phil. Mag. ³³ (1976) ^825.

[2] KRÖNER, E., Kontinuumstheorie der Versetzungen ^und Eigen-

spannungen (Berlin : Springer Verlag) ^1958.

[3] HOKANSON, ^{J. L. and} WINCHELL, ^P. G., ^J. Appl. Phys. ³⁹ (1968) 3311.

[4] YOFFE, ^E. H., Phil. Mag. ⁵ (1960) ^161.

[5] SAADA, G., 4th International Conference ^{on the} strength of metals and alloys,

edited

by E.N.S.M.I.M.I.N.P.L. (Nancy, France) ^1976.

[6] FRANK, ^F. C., Report ^{on the} Conference ^on Defects ⁱⁿ Crystalline

Solids (London : Physical Society) ^1954.

[7] READ, ^{W. T. and} SHOCKLEY, W., Phys. ^{Rev. 78} (1950) ^275.

[8] BOLLMANN, W., MICHAUT, B., SAINFORT, G., Phys. Stat. Sol.

A 13 (1972) ^637.

[9] BOLLMANN, W., Crystal Defects ^and Crystalline Interfaces (Berlin, Heidelberg, New York : Springer Verlag) ^1970.

[10] BULLOUGH, R. and BILBY, ^B. A., ^Proc. Phys. ^{Soc. B 69} (1956) 1276.

[11] SAADA, G., ^Phil. Mag. ³⁴ (1976) ^639.