yifc P*** Hm

RtFERENC:

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

IC/79/72

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

DISLOCATIONS, THE ELASTIC ENERGY MOMENTUM TENSOR

AND CRACK PROPAGATION

Lung Chi-wei

1979 MIRAMARE-TRIESTE

IC/T9/T2

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Orgaaiiation

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

DISLOCATIONS, THE ELASTIC ENERGY MOMENTUM TENSOR AND CRACK PROPAGATION •

Lung Chi-vei

International Centre for Theoretical Physics, Trieste, Italy,

and

Institute of Metal Research, Academia Sinica, ShengyangiPeople's Republic of China,

ABSTRACT

Based upon dislocation theory, some stress intensity factors can be

calculated for practical cases. The results obtained by this method have been

found to agree fairly well vith the results obtained by the conventional fracture

mechanics. The elastic energy momentum tensor has been used to calculate the

force acting on the crack tip. A discussion on the kinetics of migration of

impurities to the crack tip was given. It seems that the crack tip sometimes

may be considered as a singularity in an elastic field and the fundamental lav

of classical field theory Is applicable on the problems in fracture of materials.

MIRAMAHE - TRIESTE

July 1979

* To be submitted for publication.

I. INTRODUCTION

Dislocation theory has teen successfully used to explain the plastic

deformation processes of solids , but here it is applied to a deeper under-

standing of the brittle fracture of materials. It seems that this problem

interests many people since the progress in the understanding of brittle

fracture.

There are tvo quite distinct roles of dislocation theory in fracture

mechanics and the fracture process. First, the presence of the crystal

dislocations themselves in crystalline material enables us to understand how a

fracture may be Initiated and how all the plastic relaxation phenomena

associated vith the presence of cracks and microscopic inhomogeneities take

place. Second, we must also examine the use of the dislocation concept

in macroscopic fracture mechanics. This paper reports the author's vork on

the latter problem, the future progress of vhich will depend largely on the

interactions between solid state physics and fracture mechanics.

A macrocrack may be considered as arrays of dislocations. Based on this

concept and the theory of continuous distribution of dislocations we may get

a relationship betveen the stress field of dislocations and that In the neigh-

bourhood of a crack tip.Furthermore, a maerocrack may also be considered as

a big dislocation in a continuum medium. They correspond to each other

according to their modes of deformation-, for example, the mode I deformation

in fracture corresponds to the climbing edge dislocations, the mode II

deformation corresponds to the gliding of an edge dislocation, and mode III

deformation corresponds to the gliding of screw dislocations. Based on this

concept and the continuum theory of elastic singularity, we may analyse the

crack opening force and the kinetics of migration of impurities to the crack

tip.

II. ARRAYS OF DISLOCATIONS AMD STRESS FIELD IB THE NEIGHBOURHOOD OF ACRACK TIP

There are many similarities between the mechanical behaviour of a

macrocrack end dislocations piled up against a locked dislocation. Both have

stress concentration effects. The Griffith formula for brittle fracture of

Bolide may be derived from the fundamental properties of dislocations piled up2)

against a locked dislocation. But the physical concept of a crack

dislocation is not the same as an ordinary dislocation • The extra half

atomic plane does not really exist, and crack dislocations are defined by

discontinuities occurring naturally across unwelded cuts when a body is

stressed. Perhaps the simplest way to regard a crack dislocation Is as a

-2-

type of imperfect dislocation whose associated sheet of "bad crystal is a

missing plane of atoms. The existence of the applied force is necessary for the

existence of crack dislocations. It should be noted that crack dislocation

densities cannot be any distribution other than that satisfying the condition

of the stress-free surface on crack planes.

1. The fundamental properties of dislocation arrays

If the dislocation density is «5(x') and the stress applied at the point

X on the crack plane is o (x) (Fig.l); the dislocation density distributionyy

function can be determined by the condition that the crack planes should be

free surfaces; then

(1)

A • 2nfi_vi > G i s t n e shear modulus, b is the Burgers vector. SolvingEq.(l), the dislocation density distribution function 2>(x') can te determined.

A DThe total stress o + a can be determined at any point of the solid

material and

<£*(rj = i>J *8y (x'J <^'(x -x.'.y,(2 )

o?,(r) Is the stress at point r due to the dislocation, the core of which is

at x1 and the Burgers vector of which is [010].

DSince is much larger than In the neighbourhood of the

crack t i p , we need to consider only a .

Displacing the origin of co-ordinates to the crack t i p , le t s = x - a,

and be taken as small. After certain calculation, the expressions for a (s)

and %(e) may be obtained

(3)

>« = - *

-3-

In comparing formula (3) with that in conventional fracture mechanics, we

see that they are similar in mathematical forms.

Therefore

<r(s)

(5)

(6)

We do not know whether they are similar or equivalent, because we do not

know whether a is equal to unity or not. Although the general solution of

IT from Eq.(l) Is the same as that of KC in symposium ASTM, 3TP 381 In the

central symmetrical crack case, we think, it is necessary to compare the

calculated numerical results from Eqs.(3). and (h) vith that from coventlonal

fracture mechanics. According to the definition of the streps intensity

factor.

(7)

from

(8)

Solving^integral Eq.(l), 2(x) is obtained and then IT may be

obtained from Eq..(8). Expressions C O and (8) were described by Bilfcy and

Eshelby ; but no paper reporting practical numerical results calculated

with this method has been seen up to now. Perhaps there are some

practical problems to be solved. For example> it is not so easy to solve the

integral equation for J5(x') because of the variety of distribution of a (x).

Furthermore, some problems should be solved for edge cracks or bend specimens

though it Is easier for central symmetrical cracks (the total sum of the

number of dislocations is zero).

Tn this paper two methods are use± for obtaining practical

results: 1) solving the integral equation with the Chebyshev polynomials,

aS(x) is easily obtained; and using the semi-empirical method associated with

-k-

the semi-theoretical method, any arbitrary distribution of stress may be

expressed in polynomials; 2) the results for edge cracks calculated by

conventional fracture mechanics vas used in our calculation. Then, anJ> k)

analytical formula of IT was obtained and vas applied to some practical

cases.

2. Solving the integral equation vith Chebyshev polynomials

T (x) and U (x) are a pair of normalized orthogonal polynomialsn n 2 ~i /? 2i/9

between (-1, +l) , the weight functions of which are(l-x ) and (l-x ) ' .respectively.

(9)

The-relation between T (x) and U (x) is

(10)

3. The stress intensity factor of a single edge crack under arbitrary

distribution of applied stresses in an infinitely large plate

If

= £

(ID

n - * . has the form of (9)- Comparing (9) with (l), we obtain

(12)

-5-

Using the relation of (10), we may obtain J5(TI) and then the expression for

IT. The stress intensity factor is multiplied by 1.12 for the edge crack

case (we did not suppose £9(0) » 0)

«* = at F(«L) JKCL (13)

ilk)

Comparing (lit) with that in the handbook edited by Sih 5' (Pig.2),

(15)

then (ill) changes i ts form to

(16)

in the handbook 5)

(17)

(16) and (17) are similar.

After certain steps of calculation, formula (lit) was applied to a

rotary plate (steam turbine wheel) with an edge crack at the central hole. The

result obtained by this method was compared with that of the finite element

method. The differences in percentage are about £-15!J, The calculation

based on the dislocation theory is simpler (Fig.3).

Chebyshev polynomials may "be applied to more general cases. Any

function a(x) may be considered as a "vector" in the Hilbert space. The

orthogonal fundamental function sets (such as U (x) or T (x)) are the base

vectors in Hilbert space. If the orthogonality and weight functions were

known, the components of every base vector (the coefficient of certain U (x)n

terms in the expansion of the function) can be calculated by the

-6-

expansion of c(x). So that. In principle, any stress distribution function

o(x) can be expressed in terms of U n(x).

The errors in the above calculation were not too large,though some

approximations were introduced during calculations. From the above, we may-

say that the fractural behaviour of the dislocation model gives approximately

the same results as that calculated by other methods of fracture mechanics. .

Over and above, the author has used this method for calculating stress

intensity factors for bending specimens

III. ELASTIC ENERGY MDMBJTUM TEMSOR AND FORCE ACTIBG ON THE CRACK TIP

A macrocrack may he considered as a big dislocation in a continuum

medium. In general, point defects, dislocations, crack tips, etc. and all

other defects may be considered as singularities in an elastic field. Their

mechanical behaviour obeys the law of the general elastic field theory.

Their differences reflect their specificities. Eshelby has discussed the

interaction between the applied stress and the defects .

1. Elastic energy momentum tensor

The Lagrangian density for the free elastic field

(IB)

with an external force density f. is not taken into account in the Lagrangian.

The equation of motion is

2L = o(19)

The methods of field theory enable us to derive an energy momentum tensor,

where n,X = 1,2,3,li, X, = t, u, = 0 and their components are

(20)

- 7 -

IJc ^. i

ana P, J* 1J 1 ,*and the relation

3Wwas used.

The physical significance of T,, i s the I component of momentumthe J" 8)

flowing through^unit area which i s perpendicular to the Jt axis in unit time ,

They are tensors. For the s ta t ic case in the absence of a body force, only

P,^ and W of T . come into existence.

Eshelby used an elast ic singularity motion model to derive the

following expression:

(22)

where F is the i-component of the forces acting on a l l the sources of

internal stresses as well as e las t ic inhomogeneities in region I (Fig.lt). These

forces are caused by sources of internal stresses and elast ic inhomogeneities

in region II and by the image effects associated with boundary conditions.

2. Force acting on the crack tip

Suppose there is only one crack tip in region I, F^ may be taken as

the force acting on the crack tip. The formula (22) is the form

in three dimensions.' If we use the form in two dimensions, it is similar

to the J integral in fracture mechanics.

L e t 1 = 1 ,

F,= oT- :.(.,-,,) dsf

(23)

ds. dx , Td. l

<T. n and ds. denotes the line element.IJ j j

l)' F = G for mode I deformation,

2) mode II and combined mode deformation9)

-The crack extends along the ^ direction under pure mode I deformation,

i.e. the direction of the crack extension coincides with F ^ If node II

deformation Is mixed with mode I, the direction of F l Q would not coincide

with the :L direction. Suppose it has an angle a with x 1 (Tig.5),

TC

= P co.s s C n

with

(25)

for maximum F, ,

(26)

because3a

< 0, so that a denotes the direction of maximum

(27)

Suppose Ft0(K,K2) > FJt0o , the crack extends; this is the

criterion of fracture under combined mode deformation. That is to say, F

is the effective crack extension force of an elastic field acting on the

crack along the i direction, FgQ Is the maximum value of F^ and F £ O c

is the critical value of Pjj0, which is a material constant.

-9 -

The author's results obtained by this method agreed with Hellen et al.

though the steps of calculation and numerical results were a l i t t l e different.

In fact, Ckn in formula (27) is small, and the errors of calculation may be1/2smaller, and i ts effects on «T (its dimension is [F,Q] )would be much

smaller.

In comparison with the methods of maximum, strain energy density

factor S,G = —, (K + Kt) and his stress projection method, Wang

found that their differences are quite small and all are good from the

engineering point of view. He pointed out that the fracture stress of the

singularity motion model, G value and his stress projection method are much

simpler (Fig.6).

On the problem of fracture angle, different authors have different12)

opinions . The author of this paper thinks that o^ calculated from

formula (26) denotes the direction of Fj (a vector) but it does not denote

the direction of the fractural plane {which should be determined by some

tensors, such as O" ). This reflects the contradiction between the variety

of modes of applied stress (e.g. mode I, mode II, mode III or the. complex

mode etc.) and the single model of actual fracture mechanism (only mode I ) .

This also reflects the fact that fracture processes are closely connected

with the properties of materials. In fact, hardly anyone saw a sliding

mechanism during fracture. To answer the question why this contradiction

exists is not a pure fracture mechanical problem. It would be a problem of

solid state physics or metal physics. All are doubtful whether to take the

argument of vector 1 " J^ - iJ- as a fracture angle as in (10) and (12)

or to take the direction of F,, as that of crack extension as in (9). oL

and fracture angle are not the same thing but it would seem that there exists

a relation between them. In Ref.£>, the author suggested a relation

= arc ig ( + f(26)

The calculated results of 6 Q agree qualitatively with the experiments. Further

work is needed.

3. Elastic energy momentum tensor and experimental measurement of the

J integral

l) In the integral expression of the force F, , the condition

Cljki,m *" ° i B re(luired» t h e P a t h ot t h e integral may envelop inhomogeneous

-10-

regions as well as sources of Internal stresses, but must not run across them

In any case, the condition C.,, • ^ 0 would make the measurement of the J

integral lose physical significance.

2) If there are many sources of internal stress and inhomogeneities

7)

I-within > , C. J k i > m * 0. F. gives the combined force on them; there is no way

T)of separating the two contributions

It seems that no attention has been paid to the two points In the

experimental measurement of the J integral. Perhaps these effects would be

cancelled out automatically if the formula for the J integral calculation in

Ref.15 were used.

under the plain strain condition,

v is the Poisson ratio, then

For mode I deformation,

V

(31)

(32)

IV. KINETICS OF MIGRATION OF IMPURITIES TO CRACK TIP AND HYDROGEN DELAYEDFRACTURE

It is well known that delayed fracture of ultra-high strength steel is

closely connected vith the diffusion process of hydrogen atoms. Under a

certain amount of load, hydrogen atoms in steel will diffuse to the stress

region near the crack t ip . It will be bri t t le as the concentration of the hydrogen

atom reaches a certain critical value. Every step of fracture may only

make a leap of a finite distance, the crack propagation will be arrested as

the crack tip approaches the region of low concentration of hydrogen atoms.

Then, the hydrogen atoms diffuse to the new crack tip again, the process

is repeated again and again, the crack propagates discontinuously.

The crack tip may be considered as big dislocations in a continuum

medium as above, ve may use the method of kinetics of migration of

impurities to the crack tip to analyse this process, *ut the singularity of

the crack tip has its own characteristics.

1. Interaction energy between point defects and stress field near a crack

tip

Let P be the pressure at a point near a crack t ip , and V the

expansion due to a point defect. The interaction energy

E = pv

and

jb = - | ( «1 +• ff? t oj

-11-

(29)

(30)

(33)

Then

(34)

2. The number of point defects arrived_at the crack tip during time t

and the time interval between two .steps of crack extension

i fIn line with the general formula of Bullough

E = " V

the number of point defects arrived at the lattice defect is

(35)

(36)

In our case n • —, A = —i.—» —±-y and the angle-dependent factor may be* J (2TT)S

neglected,as Bullough has pointed out, since i t has l i t t l e effect on this

analysis,

-12-

(37)

If H(t) reaches a critical value, the crack extends; then t_ (the time

interval between steps of crack extension) is

(36)

andJA

3. The relation between the macroscopic process of delayed fracture and the

microscopic diffusion process of hydrogen atoms

Zhurkov and others have measured the time of fracture under varied

stress of many kinds of solid materials, such as metals, alloys, glasses,

high polymers. They found a general empirical formula

= %ezf>((U-/<-)//lT) (39)

T is the fracture time, U-. is the cohesive energy of atoms or molecules, v

is a parameter depending on structure, k is the Boltzmann constant and T

is a certain constant vhlch is the fracture time as U = V-,

In formula (39), when a approaches zero, we have

HZ '* "Ar) (1*0)

and,on the other hand, when the variation of T can be neglected as compared

with exp(U/kT) in a certain temperature range, formula (38) approaches

(1*1)

-13-

then (1*0) and (111) are similar. It means that the results calculated from the

analysis of tha microscopic-process under certain conditions are similar to

the empirical relation of tbe macroscopic fracture process. Therefore,

that 1B, the activation energy of fracture is nearly equal to the activationik)

energy of diffusion of hydrogen atoms. Johnson and Willner obtained the

activation energy of fracture (9,000 cal/g-atom)to he equal to the activation

energy of diffusion of hydrogen atoms, from measuring rr (*tl ) at

varied temperatures.

li. Microscopic mechanism of a delayed fracture

One thinks of using (39) as a general method to determine the

microscopic mechanism as formula (1*2) in the course of nature. But, the author

of this paper thinks, this should be done with care.

1) It is known from (39) that the activation energy in appearance

(U--YO) approaches UQ only if 0" approaches zero.

2) After comparing (38) with C<0), the microscopic mechanism

approximately represents the macroscopic phenomenon only if it is under

certain conditions.

As a consequence of the above, it would seem that tha experimental

values of tL-'VO'are a function of c and U- night be obtained as a"

approaches zero. We would rather compare UQ with that of a microscopic

process. In fact, only the experimental activation energy of fracture due to

low stresses corresponds to the activation energy of a diffusion process

of hydrogen atoms . Perhaps a. simple exponential form such as {k0) would

be doubtful under low loads. ThiB calls for further experimental work.

V. FUHDAMEMTAL PROPERTIES OF AN ELASTIC FIELD

All the point defects, dislocations and crack tips obey the fundamental

law of the elastic field theory, although they have different lattice

structures and properties of their own.

In Sec.I, Eq.(l) is fundamental. The most fundamental physical

quantity ii the force acting on dislocations and it may be derived from

fundamental properties of an elastic field

-ll*-

T) The interaction ener^ between

point defects and crack tips may also be calculated from fundamental

properties of the elastic field.

In Sec.II, it has been shown that the elastic energy momentum tensor

is a fundamental physical quantity of the J integral in fracture mechanics.

In consequence, to consider the crack tip as a concrete special

singularity in an elastic field and to apply the fundamental lav of the

field theory to crack extension is an important problem in fracture of

materials.

ACKNOWLEDGMEHTS

The author would like to thank Professor Abdus Salam, the

International Atomic Energy Agency and UNESCO for hospitality at the

International Centre for Theoretical Physics, Trieste.

-15-

REFERENCES

1) R. Bullough, in Theory of Imperfect Crystalline Solids (IAEA, Vienna

1971), pp.101-218.

2) J.P. Hirth and J- Lothe, Theory o.fLDislocations (McGraw-Hill. 1968),

p. 691*.

3) B.A. Bilby and J.D. Eshelby, in Fracture, Ed. H. Liebowitz (Academic

Press, Mew York 1968), Vol.1, pp.100-178.

h) Lung Chi Wei, Acta Metallurgica Sinica (in Chinese), 1^, No.2, 118

(1976). t**'

5) G.C. Sih, Handbook of Stress Intensity Factors, Lehigh University,

Pennsylvania, 1973, pp.l-2U.

6) Lung Chi Wei, see Second Peking Symposium on Fracture Mechanics" Peking

1977 (in Chinese}.

7) J.D, Eshelby, in Solid State Physics, Eda. F. Seitz and D. Turnbull(Academic Press, Hew York 1956), Vol.3, pp.79-11*1*.

8) L.D.Landau and B.M.LlfBhits,Field Theory (in Chinese) (Peking 1961),

pp.90—9I*.

9) Lung Chi Wei, Acta Metallurgica Sinica (in Chinese) 12., tfo.l, 68 (1976).

10) T.K. Hellen et a l . , Int . J . Fracture 11,, 605 (1975).

11) Wang Zen-Tong, Research paper of Chekians University on Fracture

Mechanics, 1976 (in Chinese).

12) Research paper of Inst i tute of Mechanics, Acadesnia Sinica, 1976

(in Chinese).

13) See, Int . J . Fracture 11, No.5 (1975).

ill) H.H. Johnson and M.A. Willner, Appl. Materials Research k_, 3*< (1965).

15) See, Acta Metallurgica Sinica 12., 162 (1976).

16) Lung Chi Wei, A report of Institute of Metal Research, Academia

Sinica, 1977 (in Chinese).

-16-

w -4

FiK.l The crack dislocation model.

-17-

a(X)v

A single edge crack under arbitrary distribution of appliedstresses in a seml-infinitely large plate.

-18-

I

1.2

0.8

Icc

U. 0.4

dislocation

Finite element

0.2 0.4a

0.6 0.8

R2—

Fig.3 The stress intensity factors of a rotary plate with an edge

crack at the central hole.

IOTP MEPSINTS AND TNTiifillU, REPORTS

W

— Fi theory(v=0.3)

Sc theory(v =0.333)

Ic/79/2IHT.KSP.*

Ic/79/3

IC/79/4IC/79/5INT.REP.*IC/79/6IHT.REP,*

Ic/79/7INT.REP."IC/79/8

1ST .HBP,*

IC/79/lOINT.REP.*

IC/79/llINT.HKP."

Ic/79/12IFr.REF.*

IC/T9/l3INT.EiEP.*

Ic/79/ l4INT .HEP.*

IC/79/l5INT.REP.*

MUBARAK AHMAD: Pew-body 3ystem anA [.article resonances.

K.O. AKDiSHIS ani 4. 3MAILACIC; Clruiuical solutions for ferroionicmodels.

H.3. BAAJfLIHI and ABDU3 SA.LAM: Non-n.etrical aupergravity for a spin-2nonat.

V. ELIAS and S. RAJPOOTt Confinement, perturbation theory 4nrtSterman-Veinberg j o t s ,

MUBAEAK AiiMAD and M. 3HAPI JAlI.Us Heavy a«ark and ma^netio aoaent.

K. KAWARABAYASHI and T. SUZUKI: Dynamical quark mass and the U(l)problem,

K. KAMAHABArA.SHI , 5. KITAKMO and T. lNAMts flavour and spin s t ruc-ture of l inear 'baryons.

0, VIK133ICH: A priori bounds for relat ions of tiro-point boundaryvalue problems using different ia l inequal i t ies .

0. VID0S3ICH: On the existence of positive solutions for non-lineare l l i p t i c equations

0. VIDOSSICHt Comparison, existence, imiquanana and successiveapproximations for tvo-point boundary valna problems,

G. VIDOSSICHj Two remarkB on the s t ab i l i t y of ordinary different ia lequations.

G, VID03SICHj Comparison, eiiRtenne, unirjuaneag and auooessivsapproximations for ordinary and e l l i p t i c boundary value problems.

0. VIDOSSICH: Two fixed-point, theorems related to eigenvalues with_the solution of Kazdsn-Warner's problem on e l l i p t i c equations.

Oi VIDOSSICHt Existence anduniqoentsss resul ts for boundary valueproblems froffi the comparison of eirenvilues.

20

I c / 7 9 / 1 9 A. NOBILE and M. i ' . TOGIj Inho:;io>-enbi >.•,• ':l'i'bctr, i n t h e e x o h a n p e h o l eI N T . R E P , * f o r a q u a s i - o n e - d i m u n s i o r j a l o h n i n .

IC/

Fii r.6 The relationship betveen (c Va) and B .

IC/79/22 "&.¥.. !"il.: Chiiru. I'ra.'inen tat. i^n I'un'jh .11 T O I . neutrino l i s t s .I N T . R E P , "

- 2 1 -

* I n t e r n a l R e p o r t s : L i m i t e d i l i a t r i bu t i o n .

THESE I'liEPHIWTS AHE AVAILABLE FTiOH THE nil*; < : . ; r : •.;;.; T 'T1C£ , 1CTI', ! ' . O . R1X

5 8 6 , 1 -34100 TRIESTE, 1TW.T. IT l.i :JOT Wi.. ...V; Y ':•. :,';.; iTE ",(, TIL-; AiJTiJ0i:3.

IC/79/24 HIAZUDDIN and PAYTTAZUDDIN: Gluon corrections to non-laptonic hyperondecaya.

IC/79/25 H.P. MITAL and II, HAilAINi Dieloctronic rtcorcbination in sodium iao -INT.REP.* electronic sequence,

IC/79/26 MUBARAK ilHM.M) and M. 3HAFI JALl.IIi lal ' i 'and and Tsetl in technique andINT.HEP,* heavy quarks.

IC/79/27 Workshop on d r i f t waves in hip;h temperature plasmas - 1-5 SeptemberINT.SEP.* 197S (Reports and sumniarias ) .

IC/79/28 J .G. ESTEVE and A.P. PACHECO: Renoraslization group approach to theINT.REP.* phase diapram of two-dimensional Hfiiesenbarp snin aystacis.

Ic/79/29 E. MAHDAVE-HE2AVEH) Production of li.-ht leutons in a rb i t ra ry beamsINT.HEP," via many-vector bo3on aiohan?.'e.

IC/79/30 M, PABHIW3LL0 and H.r. TOSI: Analytic solution ofthe mean sphericalINT.REP.* approximation for a multicomponent plasma.

IC/79/31 H, AKQAYi The leading order behaviour at' tho two-photon sca t te r ingINT.REP." amplitudes in QCD.

IC/79/32 M.S. CHAIGIE and H.P. JOtTESi On the intorfaoe between ammll-p^ non-perturbat ive and lar#e-an#le rierturbative -hysics in QCB and tKeparton model.

IC/79/31 J . LORENC, J . PB2T3TAWA and A.P. CMCVIJiii.: A comrnant on the chainINT.REP,* subduction c r i t e r i o n .

IC/79/34 H.A. NAHAZIE and D. 3TO EVi Suparsyiiiaiuti-io •:iuantisation of l inear izedsuper gravi ty .

IC/79/36 J . TAREKIt Hamarks on a confo!T..til-invitri in' thuory of . r a v i t y ,INT.REF.'

Ic/79/37 L. T&TEIj Additive qiiirV i.,t>del v.'ith :;iINT,HEP.•

IC/79/39 A.O. BARUTs Hadronic ini,ltir.letB ir. IINT,REP." part iclesi An already-unifi eu thi-irj .

stable

IC/79/41 A. TAGXIACOZZTJ and E. TO.^Ti'I: ;. 1.1'.••'.INT.REP.* charge and spin dens i ty »av«E.

Ic /79/44 filAZliraill and FArYAZIlMJ!.: a.arj: 11 u-••:IHT.RSP.* paeudoaealar ciesons pa i r .

i f :•; . n-orV.i t co.plinr on

:--,;i-.:: ii.j t.o .'lolustone

IC/7£'/46 M.S. CRA1GIE an.i "Jd i s t a n c e s in .in

IC/79/47 A.O. BAKU'!1!TBT.REP." cociv-osite ay

IG/79/4S ' : ' . " . fiifiiif'i f 1

i;: .;,-.L A!-1-1 ; ' n t i r . w ^ I ' : ' - J I . : ; ;•

t t i e o r i e ^ w i t h : : i c : . ' u r . o t :

u - n o i r . r - o n u n t w a v v i : • • . • • ! ' , 1 -jr.

ica]ar ;j'-.irtons at shorttrokun colour Kyd:iEetry.

jsrfiribe r e l a t iv i s t i c

i1 1 «r m h i S