THE GEOMETRY OF SLIP PROCESSES AT A PROPAGATING FATIGUE CRACK--II*?

P. NEVMANN;jj

A previously published model of fatigue crack propagation is relined and the resulting slip geometry at the crack tip is calculated. The essential features of this refmed model are: The slip on both sides of the crack is assumed to occur on slip bands instead of the discrete slip planes used in the old model. Siip is represented by finite matrices, shape changes of the crack tip as well as the anisotropy of slip are fully taken into account. It is shown, that the geometric mean of the average strains on both sides of the crack cannot be chosen independently but is a unique function of material parameters cl, ce (characterizing the inhomogeneity of slip on both slip systems involved) and the angle o? between the two slip planes. Whenever such strai-which must typically be of the order of one--csn be produced by the applied stress, ductile crack propagation is possible (ductile fracture criterion). The crack tip angle and the ratio of crack advance per cycle and crack opening displacement are also given in terms of cl, ct and cx. Finally it is demonstrated that only the surface production by slip on intersecting slip planes cannot be reversed by slip reversal in the compression phase.

LA GI?OXltZeTRIE DE PROCESSUS DE GLISSEMENT 4 II%= CRAQVFLVRE DE FATIGVE ES PROPAGATIOX-II

Vn modble precedemment publili de propagation de craquelure de fatigue est ragme et on calcule la geometric resultante de glissement a 1s. c&e de la crequelure. Ces carecteristiques essentielles de oe mod&e raf%n~ sont les suivantes: le glissement /de chaque cot6 de la craquelure est suppose avoir lieu sup des bandes de glissement au lieu de plans de glissement comme sur l’ancien modele. Le gliiment est rep&sent+5 par des matrices Snies, les cbangements de forme de la c&e de craquelure sinsi que l’anisotropie du glissement sont pris en consideration. 11 est montre que la moyerme geometrique des con- traintes moyennes des deux c&es de la craquelure ne peut pas Btre independemment choisie mais qu’elle est une fonction unique des parametres du m&&au C,C., (qui caracterisent l’inhomogedite du glissement sur les deux systemes de glissement impliques) et l’angle c( entre les deux plans de glissement. Lorsque de telles contraintes--qui doivent typiquement &r-e de l’ordre de l’unit&--peuvent dtre produites par la contrainte appliqued, une craquelure ductile peut se propeger (critere de fracture ductile). L’angle de la Crete de craquelure, le taux d’avancement de la craquelure par cycle et le deplacement de l’ouverture de craquelure sont egalement don& en fonction de C,, Ct et c(. Enfln, il est demontre que seule la pro- duction de surface par glissement 8u.r des plans de ghssement qui s’interseetent ne peut etre inversee par inveraement de glissement dans la phase de compression.

DIR GEOMETRIE VON GLEITPROZESSEN AX EINEJI FGRTSCHREITENDES ERXUDUNGSRIR

Ein friiher veroffentlichtes Modell der Ausbreitung van Ermtidungsrissen wird verfeinert und die Gleitgeometrie an der Rlspitze wird berechnet. Die wesentlichen Merkmale dieses verfeinerten Xodells sind: Die Annahme, da4 die Gleitung auf beiden Seiten des Risses auf Gleitbandern und nicht, wie im alten Xodell beschrieben, auf .&&n&en Gleitebenen stattf!indet. Die Gleitung wird durch endliche Matrizen dargestellt. Sowohl Gestaltsanderungen an der Risspitze als such die Anisotropie der Gleitung werden voll beriicksiohtigt. Es wird gezeigt, da13 das geometrische Mittel aus den mittleren Abgleituugen auf beiden Seiten des Risses nicht unabhangig gew&hlt werden kann, sondern da6 es eine eindeutige Fur&ion der Yaterialparameter cl, ca (die die Inhomogenit& der Gleitung in beiden beteihgten Gleitsystemen charakterisieren) nnd des Wink& ct zwischen den Gleitebenen ist. Immer wenn soiche Abgleitungen-die von der Gr6Benordnung eins sein miissen-durch die M3ere Spanmmg eneugt werden konnen, wird duktile RiDausbreitung moglich (Kriterium fiir Duktilbruch). Der Winkel en der Riflspitse und der Quotient aus dem RiDfortsohritt pro Zyklus und dem COD werden als Funktion van cf. cI und cc angegeben. SchlieRlich wird gezeigt, da0 nur die durch Gleitung 8Uf sich schneidenden Gleitebenen erseugte Oberflache dumb Gleit~kehr in der Komp~~io~phase nicht annihiliert werden kann.

INTRODUCTION

In most of the existing models of fatigue crack p~pagation(l,~) emphasis was placed on the connection between the local values of stress and plastic strain at the crack tip and the applied stress. Little attention w&s devoted, however, to the problem, how these values of stress and plastic strain lead to an extension of the crack. Since the work of Laird and Smith(a) it is generally accepted that plastic deformation at the crack tip is responsible for the crack advance. Because of the lack of a detailed model, however,

l Received February 6, 1974. t Submitted as “Habilitationsschrift” at the University of

Gottingen, West Germany.

Vtsz? onne Xational Laboratory, Argonne, Illinois 60439,

. . . 8 Present address: MPI f. Eisenforschung, 4 Diisseldorf,

Germany.

ad 7toc ‘Lfracture criteria” are commonly used. Some author@“) used a cumulative damage criterion, i.e. the crack is assumed to advance if the total amount of strain in front of the crack exceeds a

critical value. Other authors(Ge) simply assume that the crack extension is equal to the crack opening displacement produced by the plastic deformation.

The experimental evidence presented in the paper preceding this publication’nn (cited as “I” in the fo~o~g) indicates that a simple shape of the crack tip can be obtained in single crystals or when the crack advance per cycle is small compared to the grain size. Under such conditions the crack tip is F-shaped and has a well defined vertex, The crack grows wider and longer in such a way that the crack

tip angle stays constant. Compressive strains close ACTA XETALLCRGICA, VOL. 22, SEPTEXRER 1954 1167

Ilea ACTA. XETALLVRGIC.A, VOL. ‘2, i9i-t

the crack. The closure starts at the tip and proceeds backwards in such a Kay that the remaining open part of the crack tip &ill has the same constant crack tip angle. Consistent and well reproducible results were obtained under these conditions (see I). Therefore in this paper we shall consider from a theoretical point of view the slip processes associated with such plastic growth of cracks.(l’)

THE MODEL

Independent of all detailed assumptions about the stress-strain distributions around a crack there can be no doubt, that the tendency for plastic de- formation to occur will be strongest at the vertex of the crack tip. This and the strong anisotropy of plastic deformation in metals leads to the postulate, that at any moment slip should occur on slip planes only, which pass through the immediate neighbour- hood of the current crack tip. Under the circum- stances described in I this postulate is fuElled extremely well.

An immediate consequence of this postulate can be drawn for f.c.c. metals: There the maximal number of slip planes which have a common line (the crack front) is two. Thus in f.c.c. metals there are only two slip planes which have to carry the dominant part of t.he plastic deformation at the crack tip. This was verified experimentally even under extremely adversive conditions (see Fig. 6(a), i of I).

If we assume for a moment that the slip lines pass exactly through the current vertex, the reader can easily verify, that there are only the following two possibilities : If there is only one slip system active, a moving vertex can be produced only by a monotonously increasing displacement along one single slip plane. It is obvious, that crack growth over long distances cannot be produced in this manner. If there are two active slip planes which are not. parallel and both pass through the crack tip, then the sequence shon?l in Fig. 1 of I is the only possibility to move the crack tip with &nite displace- ments on the activated slip planes.

In spite of the extreme idealizations of this “coarse slip model “GW) it describes all essential qualitative features of ductile fatigue crack growth {see I). For a quantitative explanation of measurable crack shapes it definitely is, however, not flexible enough. In order to improve the model with a minimum increase in complexity rre only drop the assumption that any active slip plane must pass exactZy through the vertex of the crack tip by allowing activation of slip lines which pass the crack tip vertex at a

-I

FIG. 1. Change of crack tip ABO’ due to successive activation of parallel slip planes 1, 2, 3, . . . which pars by the current crack tip .at a distance ,of one slip line spacing (snccesslvear~~~t!~,~ ,of the right crack side

3 ). ,-I ,...

maximum distance of one slip line spacing. Figure 1 shows the typical slip line pattern at a crack tip which become possible now: Let X30’ be the crack tip with the vertex at B and let 0 be a slip line passing exactly through B. Assume then that slip has stopped on 0 due to the back-stresses of dis- locations piled-up on this slip plane. If we assume furthermore that the crack front is parallel to the intersecting line of two different slip planes me have only two possibilities to proceed :

(1) A slip plane of the second kind is actirated. Since a not yet work-hardened slip plane of this kind passes through B, it is most natural to assume, that the newly activated slip plane will pas exactly through B. This is the old case of Fig. 1 out of I and does not reveal anything new.

(2) Another slip plane 1, which is parallel to 0 and which does not pass through B is actirated. This moves the right-hand side of the crack from 0’ to 1’. (There seems to be less driving force for the activation of a slip plane which is parallel to 0 but on the left hand side of 0.)

At the new rertex C of the crack tip the process 1 or 2 will happen again. In general ,n~ parallel slip planes will be activated followed br the successive activation of ,n slip planes of the other type, which again are followed by m slip planes of the former t,ype and so on. In Fig. 1 activation of slip planes 1-T leads successively to the positions l’-7’ of the

right hand side of the crack tip. Then for the first time a slip plane of the other system (8 in Fig. 1) is activated producing the &al shape of the crack tip, which is marked in Fig. 1 by a thick line. If

SEI;NldSS: SLIP AT PROPAGATISG FATIGUE CRACK-II 1169

we proceed in this manner (.m slip planes of the same type, 12 of the other type, m, of the former type and so on) we arrive at the connation of Fig. I, where we replaced the serrated crack borders (e.g. B D in Fig. 1) by straight lines. In some areas the slip lines of different type do necessarily intersect. These regions of multiple slip are situated within the black areas in Fig. 2. hl, p, are the thickness and the periodicity of t,he slip bands on the left hand side of the crack (‘Up system 1”) measured parallel to the slip direction of the other system. h,,p, have t,he cor- responding meaning. From the comparison of Fig. 2 and Fig. 1 out of I the main differences between them are obvious : The sharp slip lines are replaced by slip bands of llnite width &&a’ = 1,2) and along the crack surface there is a layer of multiply slipped area, which is everywhere thinner than the maximal value of h, and A,. It should be emphasized, that every slip band forms like a Liiders band with its propagating front at the vertex of the crack tip. This makes it possible that at any instant of time the maximal distance of the currently active slip plane from the current vertex position is much smaller than the slip band widths hi.

Before ‘rve discuss detailed geometric features of these slip processes Ke list the advantages and disadvantages of such a treatment if compared to more conventional elasto-plastic considerations :

(1) The strains at the crack tip are of t,he order of unity (see the following paragraphs). Therefore they cannot be linearized with reasonable accuracy. In our treatment they will be represented without any approximations by finite matrices and strange but

Fm. 2. Slip bands on both sides of a crack tip which propagates according to the mechanism shown in Fig. 1. Black areas are deformed on another slip system than the adjacent slip bands, or they are multiply deformed. Parameters used for the drawing (cf. appendix): c1 = c2 = 4, fg cc = 15f8, c = 5, (y/, = 90’).

8

important properties like their non-commutivity are fully taken into account.

(2) Changes in the shape of the crack tip due to the plastic deformation are considerable because of the magnitude of the strains. This is also rigorously taken int.0 account. In fact this change of the crack shape is nothing but the crack propagation itself and should not be neglected in any theory, which tries to describe ductile criick advance.

(3) On a microscopic scale plasticity in all metals is highly anisotropic (only distinct crystallographic slip planes are allowed), which is also fully incor- porated in our calculations.

(4) An important disadvantage of our method lies in the fact, that we do not calculate stress-strain distributions, but postulate, that active slip planes should always pass through the immediate neighbour- hood of the current crack tip. Experiments can be devised (see I) such that this postuIate is fulfilled extremely well.

Summarizing we may say, that we have tried to treat correctly all local features at the very crack tip \I-hich are important for the mechanism of the crack propagation. On the other hand the dependence of these events on remote boundary conditions, like the applied stress, was neglected. Therefore the results of our calculations like the equations (AX and A36) have to be interpreted as boundary conditions at the crack tip for elasto-plastic continuum mechanics calculations which connect stresses and strains at the crack tip with the macroscopic loading conditions.

DISCUSSION OF THE RESULTS

In order to facilitate the presentation of the essential results we leave out all deductions in the following discussion. The deductions are given in the Appendix in the appropriate sequence which some- times deviates from the sequence used in the sub- sequent discussion, Equations out of the Appendix, which are repeated without modification in the discussion are listed in both places under the same number.

The definition of all vectors which are necessary for the calculations is given in Pig. 5. It is important to perform the calculations in a coordinate system which is appropriate to the problem. We have chosen an inclined coordinate s+ystem with the basis vectors ie (see Fig. 5) paralle1 to the slip direc- tions. In general re is not perpendicular to ,e. There- fore we have to distinguish between contravariant and covariant vector and matrix components, but t.his disadvantage is overcompensated by the

11;0 ACTA SIETALLURGICA, VOL. 32, 1974

extreme simp1icit.y of the resulting equations (see (A%A%)).

For the crack geometry and the average strains on both sides of a crack tip like that of Fig. 2 the internal structure of the slip bands is irrelevant. We therefore assume for the following (as in Fig. 2), that these bands are strips of homogenous shear strain of the magnitude ei(i = 1,1) and of height hi (measured in the inclined coordinate system as

indicated in Fig. 2). Further we assume that ail

slip bands on each side of the crack are identical, i.e. have the same local strains and heights. A look at Fig. 2 shows that the periodicity of the slip bands on one side of the crack is determined by the total displacement in the slip bands on the other side. Thus the pi are determined by the Q, h,. Further- more the hi can be easily eliminated by the proper selection of the 1ength.s of the ie (as x, the angle between the slip planes, is eliminated by the choice of the directions of the je). Thus in the appropriate coordinate system we have o&y two independent variables.

In the calculations we have chosen as independent variables

The ci specify the inhomogeneity of slip on the two slip systems. We shall therefore call ci the “coarseness of slip on the slip system d”. The inhomogeneity of slip is usually a material property and at least in f,c.c. metals, where there are only crystallo- graphically equivalent slip systems available, we will usually have e, = c2. It should be stated explicit.ly, that the definition (AZO) is different from t,he definition used inou for the coarseness of slip. ?Ye adopted this new notation because of considerable simplification of central equations (see (A21 to A%)). The relation between the c1 and the old definition used in(“) is given in (AZOa).

ci + co indicates infmitely sharp slip bands as in Fig. 1 out of I, whereas ci = 2 requires h, = pi. In the latter case the slip bands just touch (see Fig. 4). Even smaller values of the ci Kould require overlapping slip bands, which is unrealistic because of lvork-hardening. Therefore we may assume, that the ci are always larger than two.

The crack tip angle

In general we have to start out with an arbitrary crack tip tith some angle y at the vertex of the crack tip (e.g. BUE in Fig. 5) and then we have to perform the shearing processes sholvn in Fig. 5. As mentioned

earlier, we assume that the parameters si, h,(i = 1, 2) which characterize one pair of slip bands are the same for all successive pairs. In spite of that, the crack tip angle will be different after every additional pair of slip bands (see Fig, 5). It is shown, hou-ever, in the Appendix that these crack tip angles converge very rapidly with the number of successive slip pairs ~b towards one stationary crack tip angle yj. The quality of convergence is in the worst case as good as that of the sequence 1 f 4/16” (see A-G?). Because of this fast convergence we shali confine ourselves to discuss only the stationary case, where the crack tip angle is always equal to yJ. If the initial value of y is chosen to be ys, we start out already in the stationary case and the crack tip angle stays constant (see Fig. ‘7).

In the Appendix it is derived that yS is given by

ys = x + 1so

1 + 2 are tg ctg x - sin ( J 5 (C< - 1 )C&

; id,2 EI, (Q-1) )

k= 1,2;k#i, (X35)

where ci are the coarsenesses of slip, z is the angle between the two sets of slip planes and q are the average strains on either side of the crack tip. It should be emphasized that this equation is identical with the corresponding equation (1) ino”’ because of (A20a). For infinitely coarse slip (ci -+ co) (A35) reduces to

as we expect from Fig. 1 of I. For the symmetrical case c1 = c2 = c (A35)

reduces to

ys = a + 180 + 2 arctg(otgz-J$&); i-1,2

k=l,Z;k#i. (2)

y&l/c) is pIotted in Fig. 3 for < = F2 and for different values of CC as a parameter (which can be obtained from a = y,(l/c = 0), see (1)). For finite coarseness yS is always larger than a. The a-values of 109,4i0 and 10,53* are the only ones possible in f.c.c. metals (cos a = +)). For a = 109,47’ and c = 2 (quasi- homogeneous slip) ys can become up to 30’ larger than a.

The expression given for yd is useful in specifying the shape of the crack tip which in turn is important for the calculation of the stress and strain distributions.

SE’CiXASS: SLIP AT PROP_AGhTISG FATIGGE CRACK-II 1171

FIG. 3. Stationary crack tip angle y, aa a function of the inverse coarseness of slip l/c for e, = c2 = c, 8; = S; with a as a parameter. a can be obtained from

a = y, (l/e = 0).

The ductile fracture criterion

It was mentioned earlier that the total displace- ment due to one slip band on one side of the crack determines the periodicity of the slip bands on the other side and vice versa. This ~te~ependence of the average strains r< on either side of the crack is expressed in equation (A27)

rlE; = (CiC, - 1)2

C2Ca(C1 - l)(c* - 1) sin2 CL . (1127)

This equation is again identical with the correspond- ing equation (4) in(ii) due to (AZOa). The reader can easily verify by differentiation, that the right- hand side of (A27) is a monotonously decreasing function of the cj, which therefore assumes its extreme values at the extreme values of its arguments. This yields

1 3 -< Jz<-. sin u 2sincr

(3)

Since sin Q will always be of the order of 1 (in f.c.c. metals it is sin a = 0.94) we have the important result: crack advance by double slip requires average strains of the order of unity on either side of the crack. Variations of the order of 50 per cent are possible and are dependent on the coarseness of slip as described by (A27).

These large strains make long range unidirectional ductile crack propagation impossible in most materials, because the stresses required are so large that brittle fracture takes place instead. In fatigue, holyever,

only very narrow strips of material have to be deformed to this extent. (Their thicknesses must be about equal to the crack adyance per cycle). It will be discussed in detail in a later paragraph how slip reversal during the crack closure continually remoTes the residual stresses produced in the pre- ceding crack opening phase. Therefore the ductile crack advance mechanism is of much greater im- portance for fatigue than for unidirectional fracture.

For the symmetrical case c1 = cz = c (837) reduces to

1+l

Jg+--$. (4

For c = co, a = 90’ and < ri = < = 1. This special already by McClintook.(i~)

= < we therefore have result %-as discussed

It is important to not,e, that (AS) is the only condition, which must be fulfilled to get crack propagation by slip. There is no deeohesion process in the sense of discontinuous separation of atoms which would make it necessary to fulfill any other fracture criterion. On the other hand new surface is produced, because the crack is getting longer But this additional surface is produced in a radically different way as it is in brittle fracture. The elementary process of surface production can be easily envisioned in Fig. 5: Activation of slip band 1 stretches the surface between points C and F to the much larger area between C and 1’ (A27) is just the condition, which must be fulfilled in order to continue with this ductile kind of surface pro- duction without limits. In this sense we may call (A27) the ductile fracture criterion. It must be fulfilled at any vertex, Thich mores in a ductile manner with the help of two systems under conserva- tion of the crack tip angle.

The ductile fracture criterion is a strain criterion, since it specifies a crit’ical value of the geometric mean of the average strains on either side of the crack tip in terms of material constants like ci and CL It can therefore be interpreted as the boundary condition at the very crack tip for the strain distribu- tion, and can be used in this manner in elasto- plastic calculations of stress-strain distributions around the crack.

It should be noted that the strains entering the criterion for crack propagation are not accumulated strains but the local averaged strains occurring during one cycle at the crack tip. In this respect the criterion deviates from others used for fatigue crack advance.(a.S) On the other hand it is a IveIl

1172 ACT-1 JIETALLURGICA, VOL. 22, 1974

established esperimental fact that the fatigue limit correlates with the VTS, i.e. the 50~ stress at the largest possible unidirectional strain. This seems to be quite reasonable in the light of the above criterion, since it requires strains of the order of unity.

Crack opening cli.splawment

The well defined V-shape of the crack tip raises the question, whether there is a unique relation between the crack opening displacement (COD) and the crack advance produced by one pair of slip bands (UX in Fig. 5). Separate expressions are derived in the Appendis for both quantities as follows

cs = h,(c, - 1) w 5 Cl(C2 - 1) + cos 3: ‘f sin’x Gc*(c1 - 1) )

and

(X30)

X ,4/~--JI::_cos~~+sin2.. (-131)

For the symmetrical case c1 = cy and < = < these equations reduce to

Fix = 2h,(c - 1) COY ; ;

thus

a @3 ;,

UX i -=-. COD ~ I 1

(5) l-k-

C

Since z is usually about 90°, ctg a/2 is of the order

of 1 (in f.c.c. metals ctg u/2 = 6 or l/A) and since c > 2, we obtain for ci = c2 and g = g: The crack advance per cycle is of the order of the crack opening displacement. The exact and general relations are given by (A30 and A31).

Crack closure and irreversibility of surface production

The reversal of strain at the crack tip is of vital importance for crack propagation since it removes the residual stresses produced during crack opening without removing the just formed crack surface. We shall discuss both points separately: As the crack

opening proceeds, more and more dislocations of the same sign are pushed along the activated slip planes into the crystal and get stuck (assuming that we have contained plastic 50~ at the crack tip only). Since the strains required are of the order of unity, this produces large back-stresses, which become the larger the further the crack advances during one crack opening phase. This explains the strong dependence of the crack propagation rate (crack advance per cycle) on the applied stress G for contained plastic 50~ (proportional to cr”, n = 2 to Wj)). During crack closure all dislocations run back again and the residual st,resses are removed. The new part of the crack is, however, not removed but. just closed, so that in the next crack opening phase crack propagation can continue at the old position but lvith considerably reduced residual stresses (see I, Figs. 1 and 12).

During the compressive phase the new part of the crack closes and does not disappear although all strains are reversed. This irreversibility of surface production is characteristic of slip on inter- secting slip planes: The reader can easily verify, that perfect slip reversal is necessary in order to annihilate the produced surface. That means that all dislocations have to run back and in addition to that those on intersecting slip planes must run back in exactly the opposite sequence to that in which they came. This last condition becomes important only for double slip. There it is extremely unlikely to be fulfilled. Therefore the surface pro- duced by slip on intersecting slip planes will not be annihilated just by reversal of slip on the average.

The above argument holds independently of the coarseness of slip. But for small coarseness another effect becomes important: due to intersecting slip bands of finite width there are regions of multiple slip which undergo extremely large strains leading to large shape changes. Figure 4 illustrates this situation for cr = c2 = 3, x = 903, Fr = g. The inital crack tip (with quite a large yj = 143,13” due to (2)) is lying at ABCDE. Since both ci are assumed to be equal to two, the slip bands touch. The slip directions are horizontal and vertical (a = 90’). After the activation of three pairs of slip bands the crack tip is at A’B’C’D’E’ (A, *4’; B, B’ and so on represent the same points before and after the deformation). The crack advance is tirn the upper left to the lower right corner. The two parts of the figure thus represent the crack tip before and after the deformation. They are drawn in such a relative position, that identical points in the undeformed lower right corner do coincide. Hatched areas below

SECX_%SS: SLIP IT PROPIG_ATISG F_%TIGCE CRhCh-II 1173

FIG. 4. -Areas of multiple slip at a propagating crack tip. Crack tip which initially is at ABCDE, is moved to A’B’C’D’E’ by the activation of three touching slip bands (hatched). The bIack and white areas to the left of A’B’C’D’E’ will be multiply deformed into the black and white areas on the right of A’B’C’D’E’. Crack propagation from the upper left to the lower right. Parameters used: c1 = cs = 2, 6 = q, c( = 90”,

(‘/* = 143.13”).

A’B’C’D’E represent areas of single slip, i.e. the

touching slip bands, whereas the black and white

areas below A’B’C’D’E’ are areas of multiple slip. The very same areas before deformation are indicated

in the same manner and are situated below- ABODE, Thus these areas will become areas of single or multiple slip respect,ively. The picture shows dearly the extreme deformations and the mixture of different

parts of material lying underneath the crack tip.

It is very unlikely that the reverse deformation will happen in such a way to unravel this mechanical

mixture again and to restore the old interrelations

including the surfaces.

The reader can easily verify that alternate back

slip of the slip bands is important for c, > 2 (e.g.

Fig. 3), since simultaneous reversal would not even

close the crack properly. It is shown in the Appendix, however, (see A37 and A38) that the crack is closed exactly if the reversed strains are spread homo-

geneously also between slip bands, keeping the average strains G constant. Such completely homo-

geneous slip reversal would not unravel the mechanical mixture at all. Xost likely a combination of these two extreme cases will be found in reality.

SUMMARY

The coarse slip model of fatigue(1*,13) is refined by allotig activation of slip planes which pass by the vertex of the current crack tip at a maximal distance of one slip line spacing (the coarse slip model allows

only activation of slip planes which emanate emctly from the vertex). This refined model is characterized

by the following features : (1) The plastic deformations at the crack tip are

represented by finite matrices without neglections. (2) The shape change of the crack tip due to these

deformations is fully taken into account. This,

in fact, is the crack propagation itself.

(3) The anisotropy of plasticity in metals is

incorporated in the model. (4) The coarseneases of slip cl, c.) on the two

activated slip planes and t’he angle x between these

two slip planes are the only material properties entering the model. The ratio of the average strains

on both sides of the crack El/< is the only free

parameter. (5) The very nature of alternating slip at the

crack tip requires that the average strains on either

side of the crack cannot be chosen independently. They must fulfill one condition, which may be cailed

ductilefracture criterion. It states, that the geometric

mean of these average strains on either side of the

crack is a unique function of the material properties ci and ~1. Typically these strains must be of the order

of unity. (6) Expressions are derived which gire in terms

of the ci, x, q/e; the crack tip angle and the ratio

of crack advance per cycle and crack opening

displacement. (7) Because of the nature of slip on intersecting

slip planes the surface production during the crack opening phase is highly irreversible, i.e. during the

compression phase the crack is closed without

annihilation of crack surface.

ACKNOWLEDGEMENTS

The author would like to express his thanks to

‘c’. F. Kocks, R. 0. Scattergood and D. Bacon for

many stimulating discussions. The author is par- ticularly grateful to S. L. Peterson and P. G. Shewmon

for their continuous support and to P. Haasen for helpful comments and for reading the manuscript. This work was performed under the ampices of the

U.S. Atomic Energy Commission.

REFERENCES

1. R. 31. PELLOES, Pmt. Air Force Co??f. on Futigue am? Fracture of Aircraft Stmcturee and Mate&&, Xiami Beach, p. 4Q9 (1969).

2. J. C. GROSSKRECTZ, Rhys. SiatusSolidi (b)47, 359 (1971). 3. C. LAIRD and G. C. SJIITE, Phil. Xag. 8, 1945 (1963). 4. F. A. JXCCLISTOCK, in: iFracture of Solida, p. 05, edited

by D. C. DR~CIZZR and J. C. Grmmx. Interscience (1963). 5. J. WEERTXAS, Int. J. Fract. Illech. 2, 460 (1966). 6. R. W. LARDX‘ER. Canad. J. Phys. 46, 22 (1965). 7. B. Toxxr~s, PitiE. -Wag. 18, 1041 (196s).

117-l ACT-1 JIET-%LLCRGIC,I, VOL. 24, 1974

S. L. S. JICC.~RTSET and B. GUE, Proc. Roy. Sac. Land. A323,337 (19i3).

9. C. A~l-rs~ssos and D. L. CLEXEXTS, dcto met. 21,55 (19i3). 10. P. SEC>uSS, dcta meb. 22, 1155 (19i4). 11. P. S~~xxiix, III. Int. Co@. ora Fracture, Xunich, 1233

(1973). 1”. P. ?u'Enu?;S, Z..MetaZlk. 58, 780 (196;). 13. P. SEUXAXT, dcta met. 17, 1’719 (1969). 14. F. -1. J~CCLISTOCS. Fracture, Vol. 3, p. 136, edited by

Et. LIEBOWITZ, Academic Press (1971).

APPENDIX

Fi,g.xre 5 shows all important vectors and the consecut,ire positions of the crack t.ip (BUE, AVU, FXY) for the formation of one pair of slip bands on the two slip systems iwolved. Since all successive slip pairs are identical we have to calculate all interesting quantit,ies for one slip pair only.

For a quantitative description of the situation shown in Fig. 5 we will use a vector and matrix notation as follows: A vector is denoted by a bold face letter, e.g. S, a matrix by a bold face capital letter which is underlined once, e.g. A. Contravariant components are represented by capi% letters with a superscript, covariant components with a subscript, and mixed components of a matrix with both. Any vector may also be written in the

form ii , 0

or (S, S2), or in the form W, which

denotes the vector pointing from point U to point V. Summation is assumed over any index appearing as a superscript as well as a subscript within one product. Subscripts which are not numbers of components but eharaeterize yectors are written 6efore the main letter, e.g. ie.

FIG. 5. Crack tip before (Blj’E) and after (WXY) the activation of one pair of slip bands. In the figure all

symbols are defined, ahich are used in the appendix.

All vector and matrix components will be given-if not otherwise noted--&h respect to the two base vectors re and +, which are defined as follo\vs: re is parallel to the slip planes 1 and has a Iength which is equal to the thickness h, of slip band 2 measured along the slip planes 1. ?e is defined correspondingly.

With the covariant metric tensor gik of this coordinate system we have for arbitrary rectors X. Y

X*Y = g&_LXiYb (*Al) Since

X * Y = Pie f Ykke, we have

gis = ie . ke = h12 i

h2 n h cos ct h - cos 5c 1

with 1

(AZ)

jL+ (-13) 1

and the angle between the slip planes r.. We define now vectors .S and .T (I& = 0, 1, . . .)

as follows : $3 (,T) is parallel to the Ieft (right) side of the crack tip after ?z slip pairs. All these vectors point away from the current crack -tip (see Fig. 5). From Fig. 5 it is obvious, that $5’2 and $7” must be negative, because other&e the slip bands 1 and 2 lvould produce unreasonable deforma. tions of the crack tip. Therefore we can define the length of ,S and ,T by setting

,,Ss = ,Tf = -I.. WI This choice of the lengths ensures that the S reach from one border of slip band 1 to the other (,T correspondingly with slip band 2).

The local slip in slip band 1 can be represented by the matrix

a, > 0. (as)

a, is connected with t.he plastic shear strain e1 in slip band 1 by

a&2 El=-.

h, sin a bw

4 transforms & into 4 - ,S = W, rhich must be antiparallel to ,T, since the new crack tip is now at V (see Fig. 5). Kate that ,T, the T-vector after one pair of slip events, is indeed parallel to VU since VU is parallel to XT.). We have

,T z --“,_A*($

= -*$ -p’)( “‘i)

= --VI f 8 -I- a,

--I Yl > 0 ) ; (A7a)

?;ET;MASS : SLIP AT PROPAGATISG FATIGUE CRACK-11 1173

and froin (A7a) and (M)

Since VA is parallel to U’B and ,T is determined by ,,S alone, ,,T is irrelevant for the shape of the new crack tip. Therefore, the S-vector i* su$Xent for the c~~~te~~~io~ of a crack tip, as far as its flute de~eZo~~~t is co?scemed. (If we would consider the slip pairs in the other sequence: first slip on system 2 and then on system 1, then the T-vectors were

which states that the shape of the crack tip will stay unchanged after any number of slip pairs if and only if ,S is an eigenvector 3 of the product of the two slip matrices B, * A. We shall consider t,his very interesting case first in detail and prove then, that it is always the limiting case after a large number of repetitions of elementary slip pairs.

First we determine the eigenvalues of

necessary for the characterization of the crack tip.} by solving

~3rres~ondingly the slip in slip band 2 can be --a1 represented by

Ox I--r: I -332 1 + a1132 - L

B =j+ ’ ’

( > ; a,>o. VW = (1 - A)2 + (1 - &a, - (z1a2

-a2 1 or

KO~V we have (1 - /?)2 y= )_ _ (2 - 1)’

a,& ala2 E.2 = -

h, sin CC (89)

ala2

for 1 - i., which yields

and ?.I,, = 1 f U#*

(+J:+$j *

It is easy to verify that

1, ’ as = 1 .

Because of (ALI) we have

1 o<,&= 44

1 a2 + -

; I~ = _-l . ( >

(AlOb)

@as’ f “1

We can now give a necessary and su~~ient eon- dition for ,S, which ensures, that we will indeed be able to find positive v, and p, for all subsequent slip pairs: vi > 0 requires due to (A7b)

@S’ > --a,. (IW

This is also sufficient: because (AlOb) and (Aloe) it is p1 > 0. Furthermore me have

$1 = /41 > 0 > -a,

\.* -

Now We can easily of a,, a2, 1,:

For a1 * a2 > 0 there are two real eigenvalues, which are both positive. Using the plus sign in (A13) gives the largest eigenvalue which we call i.; the smafler one we call 1. We shall show later on that any ,S (with &P > -al) converges towards the eigenvector Avith the largest eigenvalue. Therefore n;e xill contine ourselves to consider 1 and the corresponding eigen- vector S. The eigenequation for the first component is (see All)

!s - a,S” = ZP.

Since S2 = -1, the eigenvectors tith the correct lengths are

s= (&t-l), &= ~~:-I)* (916)

calculate other rectors in terms

which is exactly (Aloe), but now for $, so that we can repeat the arguments proving v2 > 0, pn > 0, uv 2S1 > -a, and so on.

The stationary mae

From (AlOa) and (A’i) we have

s 0 ’ ( AlOd)

1176 ACTA METALLURGICA, VOL. 22, 1971

the latter because of (Al2). Then u-e get from (X7a), (A16) and (A4)

(A17)

Further

so that we have

ux=uv+vx= (

d - 1 1

j. - 1 -- ,- - 1 . (Al9)

a!2 a1 1

UX is the cracli advance due to one elementary slip pair because the old craclr tip is BUE and the new one is WXY. Since everything repeats after that, the contravariant components of the crack advance vector UX are the periodicities p,, p, of the slip band patterns measured along the other slip direction (see Fig. 2). The thicknesses of the slip bands measured along the slip directions are unity in our coordinate system due t.o the choice of the lengths of le, se. Thus the (UX)i are the periodicities in units of the thicknesses of the slip bands h,, h, (see Fig. 2) and thus characterize the coarseness of slip on the two slip systems. Therefore we define

i, k = 1, ‘3; k # i (-420)

as the coarseness of slip on system i. If the slip bands are very thin compared t,o their distance, the ci approach infinity; when the individual slip bands just touch (hi = pi) me get ci = ‘3, and for heavily overlapping slip bands the ci approach 1. Sote that this definition used throughout this paper is different from that adopted inIl’), where we defined other parameters (here called cprd) to characterize the coarseness of slip. The relation between cpld and ci is given by the following equations

h. 1 cold = 1 _ -1; c’!id = 1 _ - . L Pi ’ ci - 1

(A20a)

Equations (Al2 and A20) represent three conditions for the five variables a,, i., ci. Therefore rre can

express every quantity in t.erms of two cf them, e.g. cr, c2: Inserting (A20 into 912) ae get

. 1 A = c1c2 ; 2 = - . (-411)

Cl%

Using this in (A20) yields

1 ai = ck - -;i,k=l,?;k+:. (-422)

‘i

Thus Tee have from (Al& Al?O); (A17, APO), (in the stationary case is T = rT)

w = (C?, -l)?

vx = C-1, Cl),

ux = (C? - 1, Cl - 1). (824)

The cracli opening displacement due to one slip pair can be calculated as follows: The right-hand neighbourhood of U (see Fig. 5) stays undeformed and is translated into the right-hand neighbourhood of Y. The same is true for the left-hand neighbour. hood of F and 1’ respectively. Thus the crack opening displacement vector COD is

COD = YV - UF = YX + XV - S

=UV+IT+XV-S

( 1 1 = c*---, --c 1'

Cl cz J

Our ha1 result as expressed in the equations (A21-A%) is of remarlrable simplicity. This is due to t,he choice of the coordinate Fstem and of the independent variables ci. With the help of the met,ric tensor (A’>) of this coordinate bytern we are able nor to derive in a straightforward and unified nay the much more complicated expressions for macroscopic variables like distances and angles.

(a) The coar.se slip rupture criterion

Of special interest are the average strains on either side of the crack beyond the region of multi-slip in Fig. 2 6 = h&pi. With t,he help of (A20, 96, A9 and ,422) a-e have

G_ hiei_ 2 __& ai

Pi ci - 1 hi (Ci - 1) sin Y.

1

h Ck --

= &Ci

‘i

- 1) sin Q ; i, k = 1,1; kf i. (A26)

SEUJIASS: SLIP AT PROPXGATISG FATIGUE CRACK-II 1177

Multiplication and division of these equations yield

-- &lEZ =

(C& - l)? c2c1(c1 - l)(c? - 1) sin’ x

(Xi)

and

H ‘=h’ - > c*(c, - 1) or h = 5 Cl(C1 - 1)

q Cl(C1 - 1) J q cz(cy - 1) * (A%)

(b) Crack adcance and crack opening displacement

Because of (Al and A?) the length of an arbitrary vector V is

IV1 = h, IPj J(\ $ + cos x)1+- sin?x. (29)

Therefore the crack advance per slip pair is due to (it”4 and X8)

4 ’ 2

luxl = h,(c, - 1) qcl(ct - 1)

’ &(cr - 1) +cosy. + sin’x

(-130)

and the corresponding crack opening displacement

x 4 _&&l - 1) E.2 cr(cz - 1)

+ coa r )z + sin’ x.

(-131)

(c) The crack tip angle

The angle y between the two arms of the crack tip-v is obviously given by (see Fig. 5)

y = 360” - O(S, 2e) - Q(T, le) + ‘x. (-132)

Since

~0s 0 (% 2ef = ,s, . ,ze,

it is easy to calculate from (Al, A?, -423 and 929) that

Cl

cos 4 (S, se) = ctg CC - h sin r

Cl ctg cc - - h sin z

fl

A well known trigonometric formular yields from that

ctg 0: (S, se) = ctgx - -. h sin x

(133)

Correspondingly it can be shoxn that

hc ctg ==$I (T: le) = ctg x - 2.

sin 2 (-134)

Replacing h with the help of (ASS) and inserting (A33 and A34 into A31) yields for the stationary case

q2 ( 1 arc tg ctg x - -

sin x J F (Ci - l)c,c, -1. q (c,-1) 1 ;

(cl) Crack do&g

k = 1, 2; k f i. (-133)

Beyond the region of mechanical mixing (see Fig.2) the slip bands define average strains G which are given by (A%). The transformations n-hich belong

to these average strains we call &, i and they arc

We shall show in this paragraph that the strains

&-‘, B-l, if applied to both sides of the opened craci: as homogeneous deformations, mill exactly close the crack along the line of crack advance UX. Kc

need to prove only 71-l .XW=XUandB-l.XY= XU: Using Fig. 5, (AZ, AH, A23, A94) yields

i

1 C? - -

Cl = l-

Cl - 1 Ii i 1-k

0 1 -cl $1

1 - C? =

i 1 =XU (-137) -cl + 1

and correspondingly

g-l.XY=E-‘-(VU-T)

0

1 -cz + 1

= i

1

Cl - 2 c_ 1 cf? - 1 )I l-1

c2 1 -cg +- 1

= i ) = xu. (A3C)

-cl + 1

11;s ACTA XETSLLCRGICA, VOL. 22, 1974

It is worthlvhile noting that the closed crack are taken with respect to the eigensystem and not

fulfills the condition (Bloc) and thus is suitable with respect to our usual coordinate system.)

for crack propagat,ion again: In order to show the Son we need an estimate of ,,S2al&‘1a. The trans-

validity of (AlOc) for an ,,S, which is parallel to XU formation matrix from the usual coordinate system

we have to show because of (A% and X22) to the eigensystem is given by-

ca - 1 -- Cl - 1

>- &-I ( > Cl

Because of cl > 2 this is equivalent to

-+a - l)G1 > --CICf(C1 - 1) -j- Cl - 1

Or

0 > -c&1 - 2) - 1

1 ’ A-1 --La1 D= - a,(i. -L i 1) -(i - 1) -a1 i ’

(A&l)

since it can be easily verified with the help of (A15)

that

D*S=(l,O) and D*i=(O,l).

Therefore we have

which is obviously correct.

The convergewe of an arbitrary crack tip towards the stationary crack tip

$le = l al(>. + 1)

((2 - l),S’ + ia,); ($’ = -1).

If we start out from an arbitrary crack tip, rrhich is

characterized by ,S, the resulting crack tip after G repetitions of the elementary slip pair fz, & will be

according to (AlOd) characterized by

Since i. > 1 this is a linearly increasing function of Jr, which therefore accepts its minimal value at the

minimal allowed value of ,,S1 which is due to (Aloe)

-a,. This y-ields

$ = .X,( B * b)“*S; 2, > 0, .sr = -1. (A39)

We will prove in this paragraph that ,S converges

towards the eigenvector S of B * &, which belongs

to the largest eigenvalue 2, if ,S fulfills the condition

(AlOc).

@ss > & > 0. -

Further we get from (A41 and Al5)

From (A21) rre know, that for ai > 0 JJ * 4 has two

different eigenvalues h and j(A = l/l = clcz > 4 for

not overlapping slip bands). Therefore -there are

always two non-collinear eigenvectors S, S. In this

eigensystem of B - A t,he matrix representing p * & -- is diagonal with the eigenvalues 3, and 112 as diagonal

elements. Therefore (A39) can be rewritten in the

eigensystem as

$P -(i. - 1)&P P n, -Z @se (2. - 1)&P f ;.a, ’

which is a monotonously decreasing function of

,,Sl (the derivative does not become zero). Since

we thus have

-a, < OS1 < co

Therefore ,S is parallel to (1, b J tit h

(The sunerscrint e indicates that the components

lbnl < Al < $ I.’

(e.g. b3 < A) . (A42)

Thus $ is converging towards S more rapidly than

the series 1 f i/16* is converging towards 1.