Kinematics of a continuously dislocated medium

KINEMATICSOF A CONTINUOUSLY DISLOCATED MEDIUM (KINEMATIKA KONTINUAL'NO - DISLOKATSIONNOI SREDY) PMM Vol.

30,No.5,1966, pp. 950-955 V.A.

BABKIN

(Moscow) (Received

November

29, 1965)

The theory of deformation of a medium with a continuous distribution of dislocations is studied. Such a medium will be called continuously dislocated, in contradistinction to a discretely dislocated medium in which the number of dislocations is finite, albeit very large. 1. Def~matiot~. [‘(i

The motion of a point in a solid medium with Lagrangian

= 1, 2, 3) is examined

Cartesian

coordinate

in three-dimensional

system

Euclidean

zi (i = 1, 2, 3). All kinematic

space, properties

coordinates

in which we introduce will be referred

a

to the

&axes. The present

work deals

a small relative

is the total differential ters xs

(tie

with the following

displacement

of the points

with respect

to [’

and certain,

hypothesis:

At a fixed time t,

coordinates

t’

as yet undefined,

and 4’ + d [’

physical

parame-

r ) (S = 1, -em, S ). du =

where n= rc’([i,

xsfcit

Hereinafter,

2)~ t)

summation

By definition,

au ag,

over paired indices

a relative

relative

dci + $-

dx’

3 i is the displacement

displacement dul =

is an elastic

fundamental

with Lagrangian

displacement,

while

of a point in Cartesian

coordinates.

is understood.

with xs = const

au

w

(1.1)

(s = 1, 2, . . . . S)

d$

(1.2)

x8 =comt a relative

displacement

(1.3) is a parametric parameters

(inelastic)

relative

displacement,

because

it is related

to a change

in the

xs.

The physical reason for introducing the parameters x ’ is, that their introduction for the division of small relative displacements into two parts, (1.2) and (1.3). Thus,

1129

allows the

V. A. Babkin

kinematic description becomes more detailed: The the general displacement of the point t’, but also described

by the parameters

at every

accompany

the displacement.

xs which depend on the coordinates must he closely related to the relative

The parameters introduction,

their

which

xs,

x’, t ) indicates not only processes in the medium,

vector u (5 the physic81

[‘ and t, by the very reason displacement, characterizing

point and at every moment. They may he scalars, vectors be found by experiment and from physical considerationa.

generally

reason for introducing x ’ is, that (1.1) in a continuously dislocated medium.

The mathematical relative displacements Let

us rewrite

(1.1)

or tensors,

permits

for it

but they

must

the description

of

in the form

au

ax* dfi =

2-g. Since

da is the total

differential

with

respect

to [‘,

D2u

da is also

and since

the total

differential

we have

I%

04irtgj=----Y-

above),

(1.4)

(1.5)

WD<

with

respect

to 6’

and xs

(see

the hypotheses

we have (1.6)

But it can

be easily

verified

that

(1.7) Hence,

the integral

over

any

closed

contour

C given

by (1.8)

yields, implies

by definition [l that the medium

aod 21, a B urgers under consideration da is given

If the relative displacement contains no dislocations. The

Burgers

vector

may

vector which, in view of (l.?), has continuously distributed

also

be given

by (1.2)

and (1.31,

then,

is non-zero. dislocations.

clearly

This

the medium

by the integral

(1.9)

In the above

formulas,

which takes pect to [‘, is the partial derivative partial derivative. The

components

the symbol

Du/Dt’

into account with respect

of the final

strain

2Pij = pAi,i where fixed

3Ai

form the basis

coordinates.

in moving

denotes

the dependence to [’ considering

tensor :oij

Lagrangian

sre -’

the total partial derivative with resd~,/d~~ of u on x s (5 ‘1. The quantity xs constant. This is the elastic

given 3A,3^j-

coordinates

by

[3] 3”,30j while

(1.10) :Yi

form the basis

in

Kinematics

Utilizing

known methods

and Formula (1.4),

we readily

of a continuously

of expressing obtain,

dislocated

1131

medium

the strain tensor in terms of displacements

in the Cartesian

coordinate

au”

au,

system

au,

3x8

ax8

ax'

[ 33

x’s

---&--$1-~+

auj

au,

a$

ax8

69

au,

au,

au” ai

+qg-gax”-g-asj,xra,i-7,t7 For infinitesimal

aUa ax d a$

ax ax ax’ a~

(1.11)

displacements,

(1.12) The

strains

in (1.12)

may be divided

into elastic

scejij and inelastic,

or parametric

,(PLj p,.. = I+-!. + e(P) 23

13

(1.13)

{j

where ~Wj=;(~+~), Similarly,

= L (-5” 2 ax”

e’qj

the tensor of infinitesimal

rotations

ad

+

$E?$

may be represented

(1.14)

as the sum of two

components

However, 9ij3i3’

neither

the tensor of finite strain

may be split into elastic

eij3f3j

and parametric

parts,

nor the tensor of finite rotation as may be seen,

for example,

from

(1.11). For the medium under investigation u = The velocity

of the point

r -

r. = u (e, x8 (k*, t), t)

au v=E

-I-af

I

p, $=const

The components

(1.16)

4“ is given by

of the strain rate tensor

atI9

zi=const

V(d + +)

(1.17)

=

are given by [ 31

(1.18) Taking

into account

(1.141,

(1.17)

rate tensor from the velocity in case S does not depend explicitly d”i/dx

and (1.18).

when +/&i on t:

we obtain the components does not depend

explicitly

of the strain on xs

and

2. Geometric treatment of strain theory. Let us examine the strains of a continuously dislocated medium, and introduce a moving Lagrangian coordinate system with the basis

1132

Y.A. Babkin

(2.1) If the medium under study is considered

as some space

whose

metric

is

g”tj=3”i3*jr”g”rj”;2&*ij where g*ij = 3O*3OjZiD d 3°iform turns out that the above In fact,

space

it is easily

Rfkl

a basis

-

&2

computation

ari,m

arklm z=.

state

in Euclidean

space,

then it

is Euciideau.

shown by direct m

(2.2)

in the initial

T

that

r,,mrklP-

+

rkpmr 4p G

(2.3)

0

(2.4) Here Rikl m are the components torsion

tensor

The coefficients

of the curvature

are the coefficients

and r*.k

of bktnection

D3^, -= w This transference space, i.e.

result

is entirely

of each point

g(e)

ni j

natura1,

since

the entire

strain

a part of Euclidean

(2.5)

of the body is the result space

into another

dislocated medium under study may be considered g(B)ij3f3j is defined by z

3(e), 3@).

to the tensor

2

of elastic

strains

as motion of the medium will result

Direct

computation

as well

(2.7)

strains,

the metric

coordinates

tensor

by the formulas (2.8)

in the dependence

as through the parameters

in Cartesian

in

(2.6)

Ixs=const

- goij = 2E@)Jj 13

Inasmuch

a space

= ar at’

of

Euclidean

gW ij $e)~$eli = g(e),j 343i

3’

g@) n.,

dinates [‘, explicitly tion are given by

of the

coordinates,

space.

3”‘. I (i = 1, 2, 3) will be called the elastic basis. From (l.lO), (2.6) and (1.141, we find that, for small is related

in Cartesian

r-ijk3^,= riik3,

3W.

&r)tj3i3j

written

by

in the body from one Euclidean

the body remains

But the continuously which the metric tensor

tensor,

of connection

are defined

S.. k are the components v

yields

x ‘,

of

3(“),

the coefficients

on the coorof connec-

Kinematics of a continuousty dislocated

1133

medium

Formulas (2.10) and (2.11) show that s continuously dislocated medium may, from a geometric point of view, be considered as a three-dimensional space with affine connection t cej3 with torsion

tensor

~(e~~~~3~3~3~.

If on the other hand we define a moving reference pondence with a deformable medium by

system

and space

metric

in corres-

(2.12)

g(P)“” ‘ , = 3(~).3(~? * 23

Then we obtain

formulas

similar

~~(P}*j~3~

&$“) n ij 3W

3’

to (2.10)

= (Fiji

_

-

3i 3j

3,=

Z aV L)z?* &r’

Dg’P’

_ $?$.

2

02b:

>

s (P)iikg blksz

~(P)i~k~(~)~~- ~(~)~~~~(~)~~ +

is again found to be a metric

space

with affine

connection

~(p)~~~3~3~3~. By comparing

(2.10)

with (2.14),

we find

‘s(e). .k = 13 Thus, dingly,

the continuously point of view,

from a physical

or inelastic considered located

dislocated

as a space

simultaneously,

It is easily

connection

as either elastic strain

tensor.

L ce13 or L@),,

tensors

a Euclidean of the Lte),

from the

and correspon-

with an incompatible

If the elastic

are compatible,

point of view,

shown that the curvature

and with

(2.16)

medium under study may be considered,

then the total strains

medium is, from a geometric

L(P),

S(P). .k 23

with affine

point of view,

also with an incompatible

(2.15)

r(~)..k~(P)k~ 13

torsion

geometric

(2.14)

flrg@).. 23

and the space tensor

(2.13)

and (2.11)

l+$k)

NP)ts 1 -1 2-g-+--L---(

3t~)i == g(~),j

strain tensor

and inelastic

strains

and the continuously

are dis-

space. and L@)3

spaces

equal zero*

All considerations of this section apply to infinitesimal as well as finite strains. Bowever, for finite strains, the formulas (2.8) do not determine the components of elastic deformation, for clearly, in the case of finite strains, such a tensor cannot, in general, be separated from the strain tensor; instead, they determine the components of what may be called a quasi-elastic tensor. Similarly,

in the case

of finite strains,

p-

the formulas

. . - g”*j ^- 2E@)Aij

(2.17)

x3

determine

the components

The torsion tinuous definition

tensor

dislocations,

of a quasi-parametric ~(e)*~3~3j3~

the tensor

of the dislocation

of the density

density

tensor.

of the L(e)s tensor

[I]

space

is, in terms of the theory of con-

of dislocations

atj3i3P

In fact,

from the

,

(2.18) where S is the surface

supported

on C. By Stokes’ Ct*k3k=e

ilrn

D -7 Dx’

theorem, au ax

(2.18)

yields (2.19)

1134

V.A. Babkin

Whence,

(2.19)

and (2.10)

yield

(2.20) Here

ei?3i3~3,

and eir,3%&3m

to f+ 1) for even permutations similar

to (2.20)

of the indices,

were previously

The formulas

expressing

are given in Cartesian

are antisymmetric

obtained

and to (-

in [ 51

the components

coordinates

this section,

it should

we can introduce

. tensor

in terms of displacements

-& einm-&srn eIij

be noted that a continuously

point of view,

a space

metric

as a space

(2.211 dislocated

with affine

medium

connection

and torsion

rj = gfl,...*cr)^

$l'...'Q)i =

equal

Formulas

by

may be considered, from the geometric different from ,!, cej3 and ,!, @IS. In fact,

of third order,

1) for odd permutations.

of the torsion

SteiijP=: .Scp) jik = In concluding

unit tensors

3% . . . . ‘It,30

ax8 ar as X~=conet T

f...l9

(2.22.l

3

(s = 1, . . ., (S -

q))

(2.23)

m=1,...,p 2sfl,..., ri),,k=

@,....ri).ff_. 13

v

The coefficients

of connection

are given

J+-P);~~

(2.20

by

&p'...'")* =

r(l,...9)*ijk3(l'...,q)k

(2.25)

DQ In passing

to another

according to transformation coefficients of connection. The torsion tensor

by (2.20),

incompatibility, taneously. It is readily L(P) 3, equals

tensor since

coordinate

system,

the coefficients

rft****q)A~~k

transform

formulas of the Christoffel symbols, and therefore they are The proof of the above is similar to that given in [3] .

S@’ “.q)i$F&3~3~ the dislocation

is no longer related

density

tensor,

when all, and not only certain

ones,

seen that the number of spaces

to the dislocation

by definition, of the parameters

with affine connection

xs

3. Equations [6].

t(1***‘@3,

simulincluding

(2.26)

coefficient.

of Equilibrium.

With th e aid of geometric

We repeat identities

and R’e’~j~~3~3~~‘~3~ it is possible theory of dontinuous

complete

change

c~+c~+...+c[i=25 where C, k is the binomial

density

characterizes

dislocations

to obtain [21,

here almost

interrelating

verbatim

the complete

gfe)$&~,

set of static

of Kunin

,$fs)ijk3*3%k

equations

for the

f41 and [61

#e)ij 3i3j= 0, CurI,o(e)ij3i3j = aij3i3j, Di\r Here ru(e)i. sre th e components I

the discussion

the tensors

of the elastic

distortion

#ePi= ~(diiZ7~ w(e)rmj3.1 J tensor

Kinematics of a continuously dislocated

,(e)i,

CC_

=

(3.2)

ad

3

The operations

1135

medium

of Curl and Div on a tensor

ai%+3j

are defined

by

Rakm

Curl aij3i3j=

cilm Dz’

3,3,

,,$ Div cii3i3j= The known tensor

of the density

3

(3.3)

3j

of dislocations

atj3tgj

should

be subjected

to the

condition Divai’3i3j Clearly, interrelating

however,

we may obtain

similar

= 0 results

(3.4) with the aid of geometric

identities

the tensors g(nl,j

Curl w(P)ij a,$

=

a’j3i3j,

-

R(P)

S(o)ijk3t3j3k,

3”$,

Div p(oltj

. l . ijk Bia3Bkal

p(pjij =

3, 3j = 0,

h(rr’tjlrn ,@lm

(3,5)

Here ,(p)i

are the components If both systems is permissible

of the parametric of equations

since

.3

aui

ax’

a$

azf

distortion

tensor.

are combined,

the equations

are linear,

Curl Wij3,3j

combining

Divpij3i3j =O, .. PI3 = h(eliilm m(e) Im + The first equation

in (3.7)

is satisfied

of equilibrium

equations,

which

Jj = W’“‘ij + uwj pij =

(3.7) are the equations

corresponding

we obtain

0,

z

(3.6)

#e)ti

@)iilm

identically;

and equations

+

#P)fj

(3.7)

,(p)I,

the second

of state,

and third equations

in

respectively.

If the medium is in a state of static equilibrium, then the parameters of the medium must satisfy all three systems of equations simultaneously, from which it follows that in a medium with dislocations defined

by (3.2),

there must exist

but also the parametric

not only the elastic

stress

tensor

The system of equations (3.7) is not complete. dynamic system will be obtained in future work.

stress

p(P)ij3i3j,

A complete

In conclusion, the author expresses thanks to L.D. interest in this work and for their valuable advice.

tensor

defined

p(P)ij3,3j,

by (3.5).

set of equations

Sedov and V.V.

Lokhin

for a for their

BIBLIOGRAPHY 1.

Eshelby J., Kontinual’naia vo inostr. liter., M., 1963.

2. Landau L.D. and Lifshits M., 1965.

teoriia

dislokatsii

E.M., Teoriia

(Theory

uprugosti

of continuous

(Theory

dislocations).

of elasticity).

Izd-vo

IzdNauka,

V.A.

1136

Babkin

3.

Sedov L.D., Vvedenie v mekhaniku solids). Fizmatgiz, M., 1962.

4.

Kosevich A.M., Dinamicheskaia lJFN, Vol. 84, No. 4, 1964.

5.

Kondo K., Memoirs of unifying study of the basic problems in engineering geometry, I. II. Tokyo, Cakujutsu Bunken Fukyn-Kai, 1955, 19.58.

6.

Kunin

I.A.,

Teoriia

dislokatsii

sploshnoi

teoria

(Theory

sredi

dislokatsii

(Introduction

(Dynamic

to the mechanics

theory

of dislocations). Dopolnenie Tenzomyi analiz

(Supplement to the monograph) by Skhouten, J.A., analysis for physicists). Izd-vo Nauka. M., 1965.

Translated

of

of dislocations).

by means

k monografii dlia fizikov

by H.H.

of

(Tensor