SCIENTIFIC - PHYSICS

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• Original Classification: C
• Document Page Count: 8
• Document Creation Date: December 22, 2016
• Document Release Date: July 15, 2011
• Sequence Number: 1008
• Publication Date: July 1, 1949
• Content Type: REPORT

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C'_+.SSIF ?\TION - SAT
CENTRAL I Y
IMFORMATION FROM
FOREIGN DOCUMENTS OR RADIO BROADCASTS
COUNTRY USM
SUBJECT Scientific - Physics
HOW
PUBLISHED Monthly periodical
WHERE
PUBLISHED Moscow
DATE
PUBLISHED Jun 1946
LANGUAGE
DATE OF
INFORMATION 1948
REPORT
CD NO.
DATE DIST. I ?,u;l 1949
NO. OF PAGES 8
SUPPLEMENT TO
REPORT NO.
THIS IS UNEVALUATED INFORMATION
Zhurnal 1 ri7ees.; -I Inoi Teoraticheako Fiziki, Vol XVIII,
No , 1,
V. Zrd:mov and L. Tilchonova
Siberian Physiec -Tech Inst
Tbmak State U
25 Nov 194T
ffiguree referred to are appended]
.The stability of a monatomic cubic face-centered crystal lattice
during unilateral compression a:yi extension is discussed. It is
shown that the collapse of a lattice during compression and extension
has an essentially distinct and dbfinite character and. that the lattice
reefs+s compression considerable ese than it resists extension.
1. It is veil known that residual inelastic stra:as in macroscopic solids
are a secondary result of the collapse of microscopic crystal. lattices in
certain minute regions, which appear initially as regions of overstrain.
Stepanov, for example, shoved experimentally that the phenomenon of the collapse
in solids coreicte of tVo phases. (l) t-.e appearance of centers of disintegration
or collapse "nuclei" and (2) the development of these minute nuclei into a
macroscopic formation.
In those regio a serving as collapse nuclei or centers )f disintegration
in solids, stress and strain evidently can attain large values. It can be
assumed that the mechanical stability of these regions and the moron
inelastic deformations attain the same values as those calculated from the
theory of crystal lattices Insofar as the behavior of thece regions of
overstrain is determined by the development of plastic strains and fractures,
it to essential to know the conditions governing stability in ideal crystal
lattices for various conditions of stress. These conditions should tatormine
the mazie?m stress at which thv lattice can st?,ll be elastically deforaed
(defining the stabl ityy of the lattice), and its maximum elastic deformations.
CLASSIFICATION
DATE MW I
ARIrY NR FBI
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DI'S71I1l m
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They should slag ohc?,:n hsv >r, idea` c.ya: i lattice collapses when
stress exceeds the mnximun permltted for olastic stress values. Of
course, a theory based on the concept of 'n ideal crystal lattice can-
not fully explain the dynamos of collapse in a lattice, but an idea
of the initial phase of disintegration in s collapsing lattice can be
completely developed.
Because of the rather large elastic strains in the regions
of o.Jistrain, deformations naturally cannot be described by the
theory of elasticity as based on Hooke's law. Nonlinear terms can
be introduced into Hooke's law to describe such strains, as was done
by Mixnaghan in the case of a continuous medium and by Fuerth for
cubic crystal lattices. These authors supplemented Hooke's law
with second-order terms relative to components of homogenous strains.
For these problems, spch a ge"e tlization yields five constants
characterizing the elastic properties of material., instead of the
two in Hooke' s law. , However, elastic strains in an ideal lattice
can be so large, e.g., amounting to one third of the initial volume,
that calculations of only the first-order and second-order tome, and
the discarding of third-order terms and above will become baseless.
The, calculation of still h1'_hfr-order terms will lead to nsv: constants
of elaasticity in such large numbers that their physical significance
and the description of elastic properties of solids will be vague.
It seems more expedient to us, in cases of great stresses and
strains, to use:iooke's law, i.e., "linear" relationships between
stress components and strain components, taking into account the
dependence of ordinary coefficients of elasticity (according to
Fuerth'e first-order coefficients of elasticity) upon strain nr
stress.
From this point of view, the coefficients o, elasticity
lose their significance as characteristics of the materitu. itself
and become functions of stress; but with their help we can draw
an ordinary "classical" picture of the elastic properties of a
solid in a strained state without'requiring new coefficients of
elasticity (second-order coefficients and higher).
2. A condition governing stability in crystal lattices is
positive free energy, considered :is a quadratic function of the
components of unilateral stn-t-, of the lattice, preliminarily
deformed by externe.l siresses. These are represented by positive
coefficients of elasticity in the determinant and its chief minors.
Because the det-,)xminant is of the sixth degree, there will be a
total of six conditioao governing stablity, expressed by the
coefficient of elasticity. P. specifi, physical significs. ce can be
attached to each of the conditions governing stability; therefc.f
during the collapse of this or that condition the lattice will
collapse in a specific manner. They nature of the W lapse of the
lattice is evidently determined by that condition governing 3tablitiy
which is the first to collapse.
3. In the present work we shall consider the stability of a
monatomic c"bic crystal lattice under monoaxtal stress -- unilateral
compression or extension along one of the edges of an elementary nucleus
We shall set the coordinate axes along the edges of an
elementary lattice cube and set the stress along the Z-axis so that
Xx= Y,, = 0. zz-f ; YZ~ZX~~(y~ a. (1)
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CO AL.
Then the elementary nucleus takes on rhe form of a rectangular
parallelepipedon and the la"..rice aoswaee to;,tagonal symmetry; the
lattice will be characterized by A;ix r;oefficients: C11, C33, C44,
Chti, C12, C23. With this we ;btain the following corditions of
stability;
I) `ii -C~arte'D; aJ c!+" ',a~ 0; 33(C,,+ e, z.) C,,--20,4>0.
4)c,~>O 5) ca>'? 41 c,,>0 7 rw?O? (2)
We assume that the energy of interaction of two particles is
expressed in the form
a-g,Y ~t 1 ry t Y ) (3)
where e is the energy of dissociation of the d -=tomic molecule; to
is the' equilibrium distance in the molecule and d' is the distance
between interacting particles.
For numerical tale""latious we assume m,n ? 6,2.2. In view of
the problem's symmetry, t., parameters of state can be selected as the
length a of the edge perpendicular to stress and the ratio ec of the
length of the edge, parallel to stress, to the length a,.
In their work Zhdenov and Konueov briefly discussed a method
of calculating the elasticity of a crystal lattice as a function of the
staves upon it, .nd set up general formulas for the equation of state
and coefficients of elasticity. The general formulas were quite
unwieldy; therefore we shall state here only the s npler formulas for
the case of 'temperatures near absolute zero.
The equations of state then will be:
Sae+2` S7r+2 ~a~~
~(Sf31 S(S) y~l1
~,{2 7r+2
c, 7.
The consideration of the first ease should result in. an essentially
different relation of p rd), from that given In Figure 1. During
coa*re. lion of a face-centered lattice a stable volume-centered. lattice
should occur; that Is, the tangent of the curve pfel) at the 'point o(
should be positive and hence the curve should have the form as shown
in Figure 3. Deformation due to compression according to the curve
in Figure 3, should lead not to simple collapse in tae cubii face-
centered crystal lattice, but to its polymorphous transformation
into a volume-centered lattice.
Thus the problem of central forces can distinguish two types of
face-centered cubic 1-ttices; lattices of one type collapse during
unilateral compression and lattices of the other type undergo poly-
morphous transformation.
Peng and Power etudied the stability of a monatomic face-
centered cubic lattice (m, n, = 6, 12) relative to extension (and
compression) along the major diagonal of a cube. They established
that during extension, the face-centered lattice passes into a
free simple lattice and that with further extension the free simple
lattice passes into a volume-centered one. Considering the lattice
energy as a function of the parameter of extension .Y , they fur?.her
found that face-centered and volume-centered lattices are stable
because they correspond to miniasom energy and that a simple lattice
is unstable because it, corresponds to ^.aximim- elnergy. As a matter
of fact, the energy surface at the point of the volume-centered
lattice forms a saddle -- a minlimm relative to the parameter of
Pang aid Power and a maximum relative to other parameters and in
rerticular to our parameter cC ; therefore, such a lattice for the
law of forces given above is unstable.
We shall now investigate the conditions of stability (2).
a. 3)Ci-'-e,,>44cu>4Q)4 Al_ these conditions have
a tendency iovards disruption ituring extension. This tendency is
lacking or weakly expressed during ;onpression. Figure 4 above
the left parts`.of conditions 3), 5), 6), for TowO. The change in
piassure was taken in the range determined by the equations of state,
C I I I W I AL
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cc
The expression C11 C1~ eo?als the coefficient of shear
in the plane of the "face of a rhombic dodechahedron"; Cll - C12 during
extension decreases to zero at p/p,.= $.j1 am A thereupon becomes
negative. Consequently, during axtensicn the lattice decreases its
resistance to elaetio shear in the plane cf the face of the rhombic
dodecahedron and at p/po $.5 loses stability relative to these shears.
The stress at which C11 - C12 becomes zero, we shall conditionally call
to "third" stability of the lattice.
The coefficient C33 characterizes the resistance of the
lattice to extension iarallel to the basic stress. By increasing the
extension stress the coefficient decreases, but remains positive for
all values of pressure, and becomes zero at p=p nmx. Thus at T = 0
the "fifth" stability agrecoa with the absolute stability of the lattice.
The coefficient C44 is characterized by resistance to shear in the plane
of the face of the cabs; perpenciicular.to the applied stress p. As can
be seen in Figure 4, the lattice does not lose stability relative to
such shear, although its resistance can be greatly weakened.
b. 7) C66 >O . ^-i.e condition of stability has a tendency to
disruption during compress:..: of the lattice. The left part of
this condition for T+s+ 0? is represented in Figure 4.
Coefficient C66 characterizes resistance to shear in the
plane of the face of the cube, parallel to the basic stress p. The
graph clearly shows that the lattice remains stable relative to
these shears, although the resistance to such shears during the
prerence of "basic" compression is weakened.
c. 1) (G?+C,s)Cs3 2,19J?0. This condition has a strong
tendency toward disruption in case of great compressive and extensional
stresws.
d. 2) C,,?C, . >0,4)G?>O. The left parts of these conditions
of stability, for all values of pressure, remain quite large. Therefore
their investigation is not essontial..
Thus investigations of conditions of stability yield the following
conclusions:
Exteneio:.- -- During extension the lattice collapses because of
she4rs in the llane of the face of a rhombic dodecahedron; the oorrer
eponding "third" atr.uility appears as the lowest of all the stabilities
of the lattice.
Compression -- In the case of compression the accuracy of
calculation (accuracy of calc..l.ating see) proved to be inadequate. There-
fcreuno definite conclusions can be mode concerning the character of
lattice collapse during compression. Because the first condition is
disrupted during rompression in the limits of abio.lute atabtlity,
collapse can be asetmted to take place by the appearance of a complex
eoexdinatlon of the particles. This can be contthaod by the ultimate
compression -- the appearance of P. free volu.ae-centered lattice.
We did not investigate the important problem of the simulteneoue
influence of monobasic stress and temperature upon elastic characteristics
and stability of the lattice. This problem will be investigated
separately.
CAL
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We wish to thank Proioeao Ya. I. Fi?, kel' for bia valuable
comments during preliminary discussio. of this work.
[5 1 V. Zbd.aacv and L. Tikhonova, Theses and Annotations TV
44
BIBI.IOCBAPHT
A. V. Stepanov, Zh''1'F 17, 601, 1947-
F, D. )4urnc han, An. Jour. Math 59, 235, 1947.
B. Puerth, Proc. Boy. Soc. 180, 285, 1942.
47 4 .] V. Zhisnov and V. Honuso?, Trans. Siberian FTI 24, 30, 1947.
,
Conference of Young Students of Tamale, 19
[6' J M. Born and B. luerth, Proc. Cambr. Phil. Soc. 38, 454, 1942.
[7 _ V.. Born, Proc.Cambr. Phil. Soc. 37, 160, 1941.
[8 ?] R. D. liera, Proc. Carabr. Phil. S- 36, 466, 1940.
[9 ] B. W: Pang and S. C. Power, Proc. Cambr. Phil. Soc. 38, 67, 1942.
DA peaded figures foliow_7
CO ML
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P,
-7/10 io
q
Figure X. The relation between the
Parameters of equilibriui of the
lattice. Volume-centered lattice is
stable.
Figure 2. Formation of a
volume-centered lattice
from a face-centered one.
Figure 3. The relation betveen
the parmetere of equilibriua of
the lattice. the volume-centered
lattice is stable.
CINMAI
Figure h. Coefficient of
elasticity, as a function of
stress (pZI0 .
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