IC/79/72 yifc P*** Hm RtFERENC:

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

DISLOCATIONS, THE ELASTIC ENERGY MOMENTUM TENSOR

AND CRACK PROPAGATION

Lung Chi-wei

INTERNATIONAL ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1979 MIRAMARE-TRIESTE

I. INTRODUCTION

IC/T9/T2 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 International Atomic Energy Agency interests many people since the progress in the understanding of brittle and fracture. United Nations Educational Scientific and Cultural Orgaaiiation There are tvo quite distinct roles of dislocation theory in fracture

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 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 DISLOCATIONS, THE ELASTIC ENERGY MOMENTUM TENSOR AND CRACK PROPAGATION • interactions between solid state physics and fracture mechanics.

A macrocrack may be considered as arrays of dislocations. Based on this Lung Chi-vei concept and the theory of continuous distribution of dislocations we may get International Centre for Theoretical Physics, Trieste, Italy, a relationship betveen the stress field of dislocations and that In the neigh- and bourhood of a crack tip.Furthermore, a maerocrack may also be considered as Institute of Metal Research, Academia Sinica, Shengyangi People's Republic of China, 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 ABSTRACT deformation corresponds to the gliding of an edge dislocation, and mode III Based upon dislocation theory, some stress intensity factors can be deformation corresponds to the gliding of screw dislocations. Based on this calculated for practical cases. The results obtained by this method have been concept and the continuum theory of elastic singularity, we may analyse the found to agree fairly well vith the results obtained by the conventional fracture crack opening force and the kinetics of migration of impurities to the crack mechanics. The elastic energy momentum tensor has been used to calculate the tip. 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 II. ARRAYS OF DISLOCATIONS AMD STRESS FIELD IB THE NEIGHBOURHOOD OF A CRACK TIP of classical field theory Is applicable on the problems in fracture of materials.

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 MIRAMAHE - TRIESTE Bolide may be derived from the fundamental properties of dislocations piled up July 1979 2) 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

* To be submitted for publication. 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 In comparing formula (3) with that in conventional fracture mechanics, we missing plane of atoms. The existence of the applied force is necessary for the see that they are similar in mathematical forms. 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. (5)

1. The fundamental properties of dislocation arrays Therefore

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

G is tne fracture mechanics. According to the definition of the streps intensity A • 2nfi_vi > shear modulus, b is the Burgers vector. Solving Eq.(l), the dislocation density distribution function 2>(x') can te determined. factor. The total stress oA + aD can be determined at any point of the solid material and (7) <£*(rj = i>J *8 (x'J <^'(x -x.'.y, y (2) from

o?,(r) Is the stress at point r due to the dislocation, the core of which is (8) at x1 and the Burgers vector of which is [010]. D Since is much larger than In the neighbourhood of the crack tip, we need to consider only a . Solving^integral Eq.(l), 2(x) is obtained and then IT may be obtained from Eq..(8). Expressions CO and (8) were described by Bilfcy and Displacing the origin of co-ordinates to the crack tip, let s = x - a, Eshelby ; but no paper reporting practical numerical results calculated and be taken as small. After certain calculation, the expressions for a (s) with this method has been seen up to now. Perhaps there are some and %(e) may be obtained 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 (3) 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

-3- -k- the semi-theoretical method, any arbitrary distribution of stress may be Using the relation of (10), we may obtain J5(TI) and then the expression for expressed in polynomials; 2) the results for edge cracks calculated by IT. The stress intensity factor is multiplied by 1.12 for the edge crack conventional fracture mechanics vas used in our calculation. Then, an case (we did not suppose £9(0) » 0) J> k) analytical formula of IT was obtained and vas applied to some practical cases. «* = a F(«L) JKCL t (13)

2. Solving the integral equation vith Chebyshev polynomials T (x) and U (x) are a pair of normalized orthogonal polynomials ilk) n n 2 ~i /? 2i/9 between (-1, +l), the weight functions of which are(l-x ) and (l-x ) ' . respectively. Comparing (lit) with that in the handbook edited by Sih 5' (Pig.2),

(15)

(9) then (ill) changes its form to

The-relation between T (x) and U (x) is (16)

(10) in the handbook 5)

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

distribution of applied stresses in an infinitely large plate (16) and (17) are similar.

If 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 (ID 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 n-*. has the form of (9)- Comparing (9) with (l), we obtain 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 (12) terms in the expansion of the function) can be calculated by the -6- -5- expansion of c(x). So that. In principle, any stress distribution function . i

o(x) can be expressed in terms of Un(x). IJc ^ 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. . 3W Over and above, the author has used this method for calculating stress and the relation was used. ana P, J* 1J 1,* intensity factors for bending specimens The physical significance of T,, is the I component of momentum the J" 8) flowing through^unit area which is perpendicular to the Jt axis in unit time ,

III. ELASTIC ENERGY MDMBJTUM TEMSOR AND FORCE ACTIBG ON THE CRACK TIP They are tensors. For the static case in the absence of a body force, only P,^ and W of T . come into existence. A macrocrack may he considered as a big dislocation in a continuum Eshelby used an elastic singularity motion model to derive the medium. In general, point defects, dislocations, crack tips, etc. and all following expression: other defects may be considered as singularities in an elastic field. Their (22) 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 . where F is the i-component of the forces acting on all the sources of internal stresses as well as elastic inhomogeneities in region I (Fig.lt). These 1. Elastic energy momentum tensor forces are caused by sources of internal stresses and elastic inhomogeneities in region II and by the image effects associated with boundary conditions. The Lagrangian density for the free elastic field

(IB) 2. Force acting on the crack tip Suppose there is only one crack tip in region I, F^ may be taken as

with an external force density f. is not taken into account in the Lagrangian. the force acting on the crack tip. The formula (22) is the form The equation of motion is in three dimensions.' If we use the form in two dimensions, it is similar to the J integral in fracture mechanics.

2L = o Let 1=1, (19) F,= oT- :.(.,-,,) dsf The methods of field theory enable us to derive an energy momentum tensor,

(20) (23)

ds. dx , T

-7- The author's results obtained by this method agreed with Hellen et al. l)' F = G for mode I deformation, 9) though the steps of calculation and numerical results were a little different. 2) mode II and combined mode deformation In fact, Ckn in formula (27) is small, and the errors of calculation may be -The crack extends along the ^ direction under pure mode I deformation, 1/2 smaller, and its effects on «T (its dimension is [F,Q] )would be much i.e. the direction of the crack extension coincides with F^ If node II smaller. deformation Is mixed with mode I, the direction of FlQ would not coincide In comparison with the methods of maximum, strain energy density with the :L direction. Suppose it has an angle a with x1 (Tig.5), 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 TC simpler (Fig.6). On the problem of fracture angle, different authors have different 12) = P co.s s C n 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 with 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

(25) 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) for maximum F, , 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

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

ig ( + = arc f (26) because < 0, so that a denotes the direction of maximum 3a

The calculated results of 6Q agree qualitatively with the experiments. Further (27) work is needed.

Suppose Ft0(K,K2) > FJt0o , the crack extends; this is the 3. Elastic energy momentum tensor and experimental measurement of the criterion of fracture under combined mode deformation. That is to say, F J integral is the effective crack extension force of an elastic field acting on the l) In the integral expression of the force F, , the condition crack along the i direction, FgQ Is the maximum value of F^ and F£Oc Cljki,m *" ° iB re(luired» the Path ot the integral may envelop inhomogeneous is the critical value of Pjj0, which is a material constant.

-9- -10- 7) regions as well as sources of Internal stresses, but must not run across them under the plain strain condition, In any case, the condition C.,, • ^ 0 would make the measurement of the J integral lose physical significance. (31)

2) If there are many sources of internal stress and inhomogeneities v is the Poisson ratio, then within > , C.Jki>m * 0. F. gives the combined force on them; there is no way I- T) V of separating the two contributions (32)

It seems that no attention has been paid to the two points In the For mode I deformation, 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.

IV. KINETICS OF MIGRATION OF IMPURITIES TO CRACK TIP AND HYDROGEN DELAYED FRACTURE

It is well known that delayed fracture of ultra-high strength steel is closely connected vith the diffusion process of hydrogen atoms. Under a (33) certain amount of load, hydrogen atoms in steel will diffuse to the stress Then region near the crack tip. It will be brittle 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 (34) 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. 2. The crack tip may be considered as big dislocations in a continuum The number of point defects arrived_at the crack tip during time t and the time interval between two .steps of crack extension 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 In line with the general formula of Bullough if the crack tip has its own characteristics. E = " V 1. Interaction energy between point defects and stress field near a crack (35) tip the number of point defects arrived at the lattice defect is Let P be the pressure at a point near a crack tip, and V the expansion due to a point defect. The interaction energy (36) E = pv (29) ± and In our case n • —, A = —i.—» — -y and the angle-dependent factor may be * J (2TT)S jb = - | ( «1 +• ff? t oj (30) neglected,as Bullough has pointed out, since it has little effect on this analysis, -12- -11- 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 (37) the empirical relation of tbe macroscopic fracture process. Therefore,

If H(t) reaches a critical value, the crack extends; then t_ (the time that 1B, the activation energy of fracture is nearly equal to the activation interval between steps of crack extension) is ik) 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 (36) energy of diffusion of hydrogen atoms, from measuring rr (*tl ) at varied temperatures. and JA 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 3. The relation between the macroscopic process of delayed fracture and the (U--YO) approaches U only if 0" approaches zero. microscopic diffusion process of hydrogen atoms Q

Zhurkov and others have measured the time of fracture under varied 2) After comparing (38) with C<0), the microscopic mechanism stress of many kinds of solid materials, such as metals, alloys, glasses, approximately represents the macroscopic phenomenon only if it is under high polymers. They found a general empirical formula certain conditions. As a consequence of the above, it would seem that tha experimental

are a = %ezf>((U-/<-)//lT) (39) values of tL-'VO' 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 T is the fracture time, U-. is the cohesive energy of atoms or molecules, v low stresses corresponds to the activation energy of a diffusion process is a parameter depending on structure, k is the Boltzmann constant and T of hydrogen atoms . Perhaps a. simple exponential form such as {k0) would is a certain constant vhlch is the fracture time as U = V-, be doubtful under low loads. ThiB calls for further experimental work. In formula (39), when a approaches zero, we have

HZ '* "Ar) (1*0) V. FUHDAMEMTAL PROPERTIES OF AN ELASTIC FIELD

All the point defects, dislocations and crack tips obey the fundamental and,on the other hand, when the variation of T can be neglected as compared law of the elastic field theory, although they have different lattice with exp(U/kT) in a certain temperature range, formula (38) approaches 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 (1*1) fundamental properties of an elastic field T) The interaction ener^ between

-13- -ll*- point defects and crack tips may also be calculated from fundamental REFERENCES properties of the elastic field. 1) R. Bullough, in Theory of Imperfect Crystalline Solids (IAEA, Vienna In Sec.II, it has been shown that the elastic energy momentum tensor 1971), pp.101-218. is a fundamental physical quantity of the J integral in fracture mechanics. 2) J.P. Hirth and J- Lothe, Theory o.fLDislocations (McGraw-Hill. 1968), In consequence, to consider the crack tip as a concrete special p. 691*. 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 3) B.A. Bilby and J.D. Eshelby, in Fracture, Ed. H. Liebowitz (Academic materials. 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, ACKNOWLEDGMEHTS Pennsylvania, 1973, pp.l-2U.

The author would like to thank Professor Abdus Salam, the 6) Lung Chi Wei, see Second Peking Symposium on Fracture Mechanics" Peking International Atomic Energy Agency and UNESCO for hospitality at the 1977 (in Chinese}. International Centre for Theoretical Physics, Trieste. 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 al., 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 Institute 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- -15- w - 4 a(X) v

A single edge crack under arbitrary distribution of applied FiK.l The crack dislocation model. stresses in a seml-infinitely large plate.

-17- -18- 1.2

dislocation 0.8 Finite element I

I cc U. 0.4

0.2 0.4 0.6 0.8 a R2—

Fig.3 The stress intensity factors of a rotary plate with an edge crack at the central hole. IOTP MEPSINTS AND TNTiifillU, REPORTS

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