MULTIPLICITY OF A PLANE SIX-ATOM MODEL

M. Ya. Leitane UDC 539.3

The force-distance curve has been calculated for a plane six-atom model under tensile loading. The shapes of the model in its stable and unstable equilibrium positions have been determined.

The mechanical twinning of crystals is one of the possible types of residual- plastic change of shape. Mechanical twinning is known to occur in single cry- stals deformed under certain conditions. The presence or absence of twinning is largely determined by the temperature and by the orientation and magnitude of the deformation. Two crystals simultaneously present in a given region of a solid are in the position of twins if their lattices are linked by a symmetry operation about some crystallographically important axis or plane. In the mech- anical twinning process, under the influence of the stresses individual volumes of the crystal or the entire single crystal goes over into a position symmetric with respect to the starting part of the crystal. The twinned part of the crys- tal has a plane, axis,or center of s~anmetry with respect to the starting part of the crystal. The twinning effect is reversible; the application of stresses of the opposite sign leads to the disappearance of the twins [1-3].

Apart from twinning, plastic deformations in single crystals may take the form of slippage, kinks, dislocations, etc.

Nowadays the interaction between particles in crystals should be considered from the standpoint of quantum mechanics, but for molecular crystals, such as many polymer crystals, the equations of quantum mechanics are so complicated that it is necessary to turn to the older theory of central forces, which is capable of giving a qualitative picture. In [4] the following notation is used for the energy of interacting particles:

~ ( r ) = n-n~,- -~ -7-, ' - ~ - r '

where m, n, ~0, and r0 are coefficients characterizing the material.

The expression for the interaction between two particles as a function of the distance r takes the form

~t~_(rA_ ,~,o,.. ( to"' f ( r ) - ~," - - i Y - E ~ .... ' - - - o r f ( r ) . . . . . . . . . . 1 - - - ,

/ -~ T 1 l *m ~- ! l -n - - n l

Two interacting particles occupy a stable equilibrium position in space at a distance r 0 apart.

The mechanical twinning effect is illustrated by the four-atom model con- sidered in [I]. The system consists of four interacting particles. In the stable position the four particles form a rhomb (Fig. I); this position of the atoms is possible owing to the presence of forces of repulsion and attraction acting between the particles. If a tensile force is applied to particles i and 3, then, at a certain value of the length of the diagonal, the four-atom model

(1)

( 2 )

Riga. 1974.

Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Translated from Mekhanika Polimerov, No. 4, pp. 584-588, July-August, Original article submitted January 18, 1974.

©1976 Plenum l~¢btishing Corporation, 22 7 West 17th Street, New York, iv'. Y. ] 0011. No part o f this pu blicatio~z may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or othe~vise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15, 00.

4 9 7

2

/ , / / / X

¢

Fig. i

A

Fig. 2

Fig. I. Twinning of four-atom model.

Fig. 2. Characteristic curve showing the diagonal of the four-atom model as a function of the applied tensile force.

has a second equilibrium position - twinning occurs.

The characteristic curve representing the dependence of the length of the diagonal of the four-atom model on the applied tensile force is shown in Fig. 2. On this curve it is possible to identify three equilibrium positions: I, III denote the first and second stable quilibrium positions; II denotes the unstable equilibrium position. The coordinates of the characteristic points on the twinning curve may vary with the parameters %, r 0, m, and n and the degree of compression of the system and are different for different materials; however, the curve retains its wavelike character.

Thus, the system consisting of four atoms reflects the elastic, plastic and brittle properties of the solid and can be used even for explaining the pro- cesses of thermal expansion and melting of solids.

In order to obtain a clearer understanding of the plastic deformations in a single crystal it is proposed to consider an atomic model consisting of fixed interacting particles located in the same plane. The coordinate system is so arranged that particles 3 and 6 are located on the x axis (Fig. 3). The dis- tances between particles are given by

t '+ i=~(x j - -x i )2- - (y j - -y i )2 ; i = l . . . . . 5; ] = 2 . . . . . 6 (i=/=]).

In accordance with the Born formula (2)

A 1 f~J = ~--.,,W+--F . , , - . ,

.~ j r i j ; i = l . . . . . 5; ] = 2 . . . . . 6 ( i¢]) .

The distance r 0 is taken equal to unity. Each particle interacts with all the other particles of the system. For example, particle I is acted upon by a net force whose vector under extension is equal to the vector p which in the equili-

brium position is equal to zero: f~=p; n = 6; i = I. j=2

In solving specific problems the vectors of the particle interaction forces are written in coordinate form:

h~~=lhJl xJ-x----! ; f,~ '~= If~Jl u.~-y, - - ; i = l . . . . . 5; ] = 2 . . . . . 6 (iva]).

For each particle we obtain two equations, for example, for the first p a r t i c l e

~ n

• .x -- , l-~)/ A ~,,, --p, A fov=q ( i= l ;n=6) , j=2 j=2

4 9 8

!

\ y I y

2

3 ~

J 4

y / t r

_ _ 3

6

F i g . 3

-7

0

y F

!

4

Fig. 4

Fig. 3. Equilibrium position of the six-atom model: !, ili, V) stable, II, IV) unstable equilibrium positions.

Fig. 4. Force-distance curve for the six-atom model: I, !II, V) stable, II, IV) unstable equilibrium positions.

where p and q are the x and y components of the vector p.

In developed form the equations for the first particle are written as follows:

ro ~-r" ) Y 2 - g l +rl3_,~ ~ I 1 - + r12 " I'~ -" 1 1 ( 1 2 . - ni t rl 2 g13 n - m r13

(to,,-,,,) ] + rl6 - ' '~ - I 1 Y 6 - g l = P ;

/" 16;'1 - m t F i r

[ ( . . . . . ( . . . . . A r le -m- - I I . . . . . . . ..}_rl 3 ,,, i 1 r~z"- '* ' t - - - ~ t 3 - - 4-

t. t2~l - m FI 2

4- r14 - ' ~ - 1 1 r14 ' ' - m t - - - ~ i 4 - q - r l 5 - m - ~ l r15 n - m r!5 - Jr

ron--rn I X 6 - - X I

Similar equations are obtained for the other particles.

The problem now consists in calculating a system with 12 nonlinear equa- tions. In general form

@:,(x~ . . . . . x ~ ) = p ; + y ( y ~ . . . . . y 6 ) = q .

In order to compute this nonlinear system of equations we use the program given in [5]; the system is solved by progressive minimization (zero minimum) of the following target function:

q)~ p~: (+~- ~- :~i~. t

The minimum was found for a given accuracy of 10 -4 . B~SM-3M computer.

The problem was solved, on a

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If the coordinate system of the six-atom model is selected as shown in Fig. 3-I, and the force is applied in the direction of the x axis to points 3 and 6 only, the problem is simplified since for all the points q = 0, and the force applied to points 3 and 6 is expressed in terms of p/A.

At n = 3, m = 6, the most probable values of the coefficients for polymer crystals, the calculations showed that the relation between the distance r36 and the force p/A is represented by the curve shown in Fig. 4.

For five values of r36 the force p/A = 0. The configuration of the six- atom model for these cases is shown in Fig. 3. In stable equilibrium positions I, III, and V the particles form parallelograms, whereas in the unstable equili- brium positions they form one or two squares, which characterize the instability of the system. As compared with the four-atom model, the six-atom model has three rather than two stable equilibrium positions. The results obtained show that the six-atom model not only has the twinning property but has three equili- brium positions. This further confirms the existence of plastic deformations in single crystals; it may be assumed that more complicated atomic models may have several equilibrium positions, i.e., may be subject not only to twinning but to higher orders of replication.

SUMMARY

i. The force-distance curve for the six-atom model has a total of five equilibrium points, three stable and two unstable; this means that more compli- cated atomic models may better express the mechanism of residual strain forma- tion.

2. In a six-atom model consisting of two unit cells, under deformation each cell may twin independently of the other and it is therefore possible to speak of the microtwinning of the model.

LITERATURE CITED

I. A. K. Malmeister, Elasticity and Anelasticity of Concrete [in Russian], Riga (1957).

2. A. K. Malmeister, in: Problems of Dynamics and Dynamic Strength, Vol. 1 [in Russian], Riga (1953), p. 23.

3. M. V. Klassen-Neklyudova, Mechanical Twinning of Crystals [in Russian], Moscow (1960).

4. M. Born and Heppert-Meyer, Theory of Solids [Russian translation], Moscow (1938).

5. A. F. Kregers, State Fund of Algorithms and Programs [in Russian], Inv. No. P000337.

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