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A model for sliding mode crack closure part I: Theory for pure mode II loading
~
Pergamon
Engineering Fracture Mechanics Vol. 52. No. 4, pp. 599-611, 1995
0013-7944(95)00044-5
ElsevierScienceLtd Printed in Great Britain. 0013-7944/95 $9.50+ 0.00
A MODEL FOR SLIDING MODE CRACK CLOSURE P A R T I: T H E O R Y F O R P U R E M O D E II L O A D I N G J. TONG,~" J. R. YATES and M. W. BROWN SIRIUS, Department of Mechanical and Process Engineering, University of Sheffield, P. O. Box 600, Sheffield S! 4DU, U.K. Abstract--A physical model has been developed to quantify the sliding mode crack closure (SMCC) experienced in cyclicmode I1 loading conditions. Idealizedcrystallographic surface geometry and Coulomb friction behaviour were assumed in the model. The results reveal that faceted crack surfaces will generate a local wedging mode I component while frictional attenuation tends to reduce the effectivemode II stress intensity factor range. It is found that asperity angle and friction coefficientare two fundamental factors influencing near-threshold fatigue crack growth behaviour during the entire fatigue cycle. The local stress-strain field is significantlyaltered due to the wedging mode I component and frictional attenuation of the nominal mode II component. It is concluded that ideal mode II loading may never be achieved in practice for polycrystallinematerials due to the inevitableexistenceof local mixed mode loading conditions.
NOMENCLATURE a crack length b friction free crack length F, N friction and normal forces p, q maximum normal and tangential stresses in a linear distribution Q, P local opening and shearing forces r, ~b local coordinates R load ratio Pu/Pm~x K stress intensity factor u, v mode II and mode 1 displacements u.0 initial nominal mode II displacement u,r actual tangential displacement upon convergence U closure ratio friction angle A range of displacement or K friction coefficient 0 facet angle a, z normal and tangential stresses Subscripts
eft II lit Iw
effective value of displacement or K nominal mode II component displacement or K due to crack flank tangential stresses displacement or K due to wedging.
1. I N T R O D U C T I O N NEAR-THRESHOLD fatigue crack p r o p a g a t i o n ( N T F C P ) b e h a v i o u r in m o d e I has been extensively investigated in m a n y structural materials a n d the characteristics of N T F C P b e h a v i o u r have been rationalized by using the concept o f crack closure [1-4]. A variety o f different m e c h a n i s m s can be involved in the d e v e l o p m e n t o f crack closure, a m o n g them r o u g h n e s s - i n d u c e d closure a n d oxide-induced closure have been proposed [5-7] to a c c o u n t for crack closure in the near-threshold region where plane strain c o n d i t i o n s are generally expected. I n m a n y metals a n d alloys near-threshold crack a d v a n c e takes place primarily a l o n g a single slip system, resulting in crystallographic or generally faceted fracture m o r p h o l o g i e s a n d local mixed m o d e l o a d i n g c o n d i t i o n s [2]. Oxide debris, formed as a result o f relative a b r a s i o n between faceted © Crown Copyright (1995). ~To whom all correspondence should be addressed at Dept of Materials Scienceand Technology, University of Surrey, Guildford GU2 5XH, Surrey, U.K. Eru 52/~B
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crack surfaces, can build up to 0.2 pm [7] in the threshold region, comparable to the scale of crack tip displacements. The occurrence of crystallographic plane separation or oxide built up on the fracture surface or both can lead to the creation of asperities on the fatigue fracture surfaces, which may further promote oxide-induced crack closure. The first analysis to quantify the effect of fracture surface roughness on mode I fatigue crack closure was presented by Purushothaman and Tien [8] who suggested that the closure stress intensity Kc~ could be estimated by comparing the change in fracture surface asperity height to the crack opening displacement. Suresh and Ritchie [9] proposed a simple geometric model in which the contribution of local mode II displacement was considered. A physically based model was proposed by Beevers et al. [10], who used a "local" stress intensity factor induced by symmetric concentrated point forces in deriving Kc~.The effect of oxide-induced closure was also included in the last analysis as the oxide debris may be simply treated as asperities. Displacement was used directly or indirectly to estimate Kd in almost all of the above analyses, a methodology consistent with Elber's original concept and the experimental method to obtain the load vs crack face opening displacement curve. However, if a crack is subjected to mode II loading, the situation becomes more complex as the effect of fracture surface roughness on NTFCP behaviour can be very different from the response under mode I loading, depending on the R-ratio, loading combinations, loading history and pre-crack conditions. Even when loading history and pre-crack conditions are set aside, crack closure is strongly influenced by load ratio and the mixed mode combination. Under mode II loading conditions the crack flanks will always be in contact (Fig. 1). Frictional attenuation and wedging over opposing asperities would be the leading factors in determining the real crack driving force. The term "sliding mode crack closure" (SMCC) is used here, after Tschegg [11], to specify the effect of reduction of nominal stress intensity factor because of friction, abrasion and similar dissipative processes in mixed mode loading cases, analogous to fatigue crack closure in mode I. Although experimental evidence [12, 13] seems to support the above arguments, to quantify the closure effect under mode II loading appears to be formidable, due to various uncertainties. Experimental difficulties arise in obtaining the relevant parameters, e.g. friction coefficient (assuming Coulomb friction), the extent of wear debris or even the real displacements along the crack flanks. In a study of mode II fatigue crack growth in a structural steel, Smith and Smith [12] identified three sources of crack growth attenuation: compressive residual stress ahead of the fatigue pre-crack, interlocking of fracture surface asperities and gross plastic deformation of the interlocking asperities. The unlocking response was simulated by assuming that crack flank frictional stresses obeyed a constant interfacial shear stress friction law. However, no analytical solution was proposed from the investigation to describe SMCC quantitatively. In the current study a physical model has been developed to quantify the SMCC experienced in mode II loading conditions. NTFCP behaviour is modelled by considering frictional attenuation and facet wedging as the major influential factors, The model has been evaluated for a variety of friction coefficients and inclination angles under both loading and unloading conditions. The extension of the model to mixed mode I and II cases, and the application of the model to experimental results on a structural steel BS4360 50D will be presented and discussed later in Part II of this paper.
2. P R O P O S E D M O D E L 2.1. Characteristics and assumptions
The proposed model aims at quantifying the onset condition of branch cracks under mixed mode I and II loading. However, the analysis will begin with a nominal mode II loading case without losing generality. The characteristics of the model are largely based on the current experimental conditions and assumptions are made accordingly. In experiments, the pre-crack conditions were carefully controlled and detailed elsewhere [14]. Briefly, the pre-cracks were produced using decreasing stress intensity tests at a high R-ratio (R Kmin/Kmax), terminated just above the mode I threshold so the closure induced by plasticity during the pre-cracking process would be minimized. =
A model for sliding mode crack closure. Part I
601
T
t~ .
"C
t~
ol Mode
II
Mode
I
Fig. 1. Schematic illustration of crack closure during mode II and mode I loading.
At maximum load, pre-cracks fully unlock up to the crack tips and crack surfaces slide along the whole crack length under limiting friction. However, shear displacements at both the crack mouth and tip are small compared to facet length such that slips will be confined on one facet. The SMCC effect is most active within the crystallographic area[13]. The crack tip driving force is reduced to a varying extent by frictional attenuation and wear debris, depending on the R-ratio. Among the proposed sources for frictional resistance [12], only the compressive normal stresses due to the crack surfaces wedging open over asperities are considered. The compressive residual stresses due to plasticity-induced crack closure are ignored, in view of the pre-cracking method adopted. The frictional resistance due to gross deformation of interlocking asperities is also overlooked, as it is generally regarded as a transient process of frictional attenuation. An idealized crystallographic surface geometry is assumed. The pre-crack is treated as an elastic line crack, where an idealized crystallographic surface geometry is assumed. The notch and pre-crack as a whole are idealized as a partially loaded edge crack in a semi-infinite plate (Fig. 2). 2.2. The model Assuming Coulomb friction with a friction coefficient p, an idealized zig-zag angle 0 for the crack path taken to the direction of the external shear stress. A previous study [12] based on direct crack surface displacement measurement shows that compressive normal stresses due to wedging arise as the pre-crack tip is approached [Fig. 2(a)], a fact confirmed by the present SEM analysis as rougher fracture topographies are observed right behind the pre-crack tips than at positions near the pre-crack mouths. For simplicity, a linear distribution is assumed for normal stresses tr(x) and tangential stresses r(x) [Fig. 2(b)].
J. T O N G et al.
602
j"
I_
P
"
-4
(a) Real pre-crack condition
|
O
'
•
't
"
t jj
!
L
x
i
b
(b) Idealized pre-crack condition Fig. 2. Real (a) and idealized (b) pre-crack condition (mode II), showing local stressesexerted on the upper fracture surface due to crack face contact.
Partially loaded linearly distributed normal and shear loads due to wedging and rubbing can be obtained by integrating the theoretical solution [15] for concentrated normal and shear forces on an edge crack.
(f°o(x)g(x/t,Odx K~,
= -n
~" z(x)g(x/a)dx
x/~
%//-~ __ X 2
Here a ,(x,O) =
- tr(x) =
z~,.(x,O) =
--z(X)
•
=
x-b a - b p'
x-b a--b
q,
"
(1)
A model for sliding mode crack closure. Part I
603
g(x/a) = g ( c ) = 1 + (1 - c2)(0.2945 - 0.3912c-' + 0.7685c 4 - 0.9942c 6 + 0.5094c 8)
(2)
and
which yields
(3)
K.,J
where K,w is the stress intensity factor due to wedging, K., is the attenuation in the applied stress intensity caused by the tangential stress acting along the crack flanks. Displacements at the end o f the crystallographic length (distance a - b f r o m the pre-crack tip) were derived by integrating the displacements due to the concentrated forces [16] to give: H
o'(x)dx (4)
2u = ~
"H
where
:E,.6,1_0.,84(:)°1.
(5)
F r o m eqs (4) and (5) an integrated form is obtained:
{
4 h(b/a){~} a '
2vw'~=
(6)
2u, J
where Vw and u, are displacements due to wedging and frictional attenuation, respectively, E' = E/(1 - v2), E is Y o u n g ' s m o d u l u s and v is Poisson's ratio. T h e integrated results forf(b/a) and h(b/a) are shown in Fig. 3.
1.5
1.0 ,Jc
0.5
0
.2
.4
.6
.8
1.0
b/a
Fig. 3. Geometry factors for stress intensity and displacement for a partially loaded edge crack in a semi-infinite plate.
604
J. TONG et al. LOADING
N
Q Q = Ptan (O+c¢)
(a)
UNLOADING
Q = Ptan(O-ct)
(b) Fig. 4. Normal and friction force decomposition. (a) Loading, (b) unloading.
Under limiting friction, the friction angle, ~, the normal force, N, the friction force, F, and the friction coefficient, ~, are related as p = tan ~ =
F
~.
The normal and friction forces produce a resultant force, which in turn can be decomposed into vertical P and horizontal Q components (Fig. 4). As the direction of the friction force will be reversed in going from loading to unloading, the cases of loading and unloading are treated separately in Fig. 4(a,b). For the loading case, Q = P tan(~ + 0),
(7)
Q = P tan(0-a).
(8)
and for the unloading case,
The displacement field for the current specimen was computed using the finite element program ABAQUS [17] under nominal mode II loading conditions. The local coordinate system at the crack tip is shown in Fig. 5. Considering two nodes, i and j, on opposing crack surfaces at a distance a - b from the crack tip, the shear displacement due to nominal shear stress at the end of the crystallographic length [x = - (a - b)] can be obtained as u i - uj = 2u.0. From
(9)
a simple geometrical consideration, the normal displacement induced by wedging
would be 2vw0 = 2u,0 tan 0.
(10)
A m o d e l f o r sliding m o d e c r a c k c l o s u r e . P a r t I
.I !
x
o
a-b
605
j
Fig. 5. Local coordinate system at the pre-crack tip.
The sliding displacement u.0 would be an upper bound value for the loading case as no frictional attenuation has yet been taken into account. The normal stress due to wedging may be determined from eq. (6) as ~zE'
- 2Vwo. po- 4h(b/a)a
(11)
For loading, the shear displacement attenuation due to friction would be 2U,o-
4h(b/a)a E'n q0 =
tan(~ +
O)tan(O)2u,o,
(12)
and by superposition the instantaneous effective mode II elastic displacement would be 2urn = 2uli0 - 2ut0.
(13)
This reduced displacement implies a reduction of the estimated wedging displacement to Vwl,with corresponding values for p~, qj and ut., such that 2U112 =
2UI|0 - -
2u,1.
(14)
As the actual normal and shear stresses p and q will depend on the final effective mode II displacement to give a compatible set of displacements, the above procedures are repeated so that 2Urn+ I =
2U.o - 2uti
=
2UllO- tan(~ +
O)tanO2u.,.
(15)
The formula converges to a constant value after n iterations. On convergence, 2UI|,,+ I = 2 U n ,
(16)
2u.0 1 + tan(~ + 0)tan0 '
(17)
and
2u,,, =
so the actual tangential displacement 2u~f would be tan(~ + 0)tan0 2u,. = 2u.0 - 2u.,, = 1 + tan(~ + 0)tan0 2u,0.
(18)
The attenuation in the applied stress intensity caused by the tangential stresses acting along the crack flanks is: Kilt =
F(b/a)E'£2u,.,
(19)
where the geometry factor F(b/a) =f(b/a)/2h(b/a). The mode I stress intensity factor due to wedging would be:
K.w = F(b/a)E'tan 0 N/-~--n 2(u,.0 u
u,r).
(20)
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Ull
~~ri
~:ion locking dwell
reversed loading
~ ~ ; ~ reversed unloading
KII~
g-
Fig. 6. Schematic of variations of nominal and effective mode 11 displacements and stress intensities during a fully reversed fatigue cycle.
The effective mode II stress intensity factor can then be resolved as K~I,, = K i t - K . ,
(21)
where K. is the applied stress intensity factor at the crack tip and may be calculated from the displacement field: 8~rG 1 KH -- (2nr~)i/2 (2k + 2)
1 1
( u , - u,j) 2 ' .
(22)
3
(2k + 3, sin(~ 4~,)+ sm(~ ~b,) where k = 3 - 4v for plane strain, G is the modulus of rigidity, r, q~refer to the local coordinate system (Fig. 5) at the crack tip, u,i, u,j are the displacements in the vicinity of the crack tip, symmetrically situated with respect to the y-axis. Note here that K . , Kl,, u, and Vware strongly interdependent via p, 0 and b due to contact pressure and friction on opposing fracture surfaces. These effects are incorporated in the terms p and q, which govern the mode I and II responses, respectively. For the unloading case, the frictional force will reverse its direction. Under the limiting friction condition, the displacement induced by the tangential stress acting along the crack flanks would be 2Utf =
tan(0
-
~)tan
1 +- t--~(-O • ~
0
0 2u.0,
(23)
and the corresponding stress intensities can be expressed in the same forms as in eqs (19)-(21). The more general case would be to consider a fully reversed loading cycle, including reversed loading and unloading phases, as well as loading and unloading phases. Figure 6 shows a schematic of variations of nominal and effective mode II displacements and stress intensities during an entire fatigue cycle at R = - 1. During loading the pre-crack is unlocked to the crack tip and the mechanical behaviour from a to b may be depicted by eqs (18)-(22). Load reversal at maximum load locks the pre-crack. Unlocking would not take place until the reduction in the applied load is such that the limiting friction condition is achieved and the behaviour from c to d may be described by eqs (19)-(23). Further reversed loading causes the opposite facets to make contact and the
A model for sliding mode crack closure. Part I
607
behaviour would repeat itself and the mechanical responses from d to e and from f to g may be modelled again by eqs (18)-(22) and eqs (19)-(23) accordingly. Generalization of these results to cover both loading and unloading mechanical behaviour under limiting friction gives: tan(0 + ~ ) t a n 0 1 + tan(0 + ~ ) t a n 0 2u.0,
(24)
tan 0 K~w = F(b/a)E' / :- 1 + tan(0 + ~ ) t a n 0 2u,0, va
(25)
KII,
F(b/a)E, f-ff
4a
where ~ = 1 for loading, ~ = - 1 for unloading and Klleff =
KII - - K . , .
(26)
3. M O D E L B E H A V I O U R
From the above analysis, it is apparent that the friction angle and angle of inclination of facets on the crack profile are two fundamental factors which significantly influence the behaviour of the model. The variation of these factors can lead to substantial changes of the local tangential force and normal wedging force, which therefore change the crack tip stress-strain field. The influence of friction angle ~ and facet angle 0 on SMCC have been investigated numerically, and the results were characterized by the parameters xK, t and xK~w, where
Z-
F(b/a)
E'2u,,o "
In Fig. 7 the attenuation of the mode II stress intensity factor due to friction is presented for loading and unloading cases, respectively. The dependence of the wedging force on the angle of friction for loading and unloading cases is shown in Fig. 8. An examination of the curves in Figs 7 and 8 reveals that the effects of both friction and asperity angle are significant. The local tangential force will increase with the friction and asperity angles on loading. Upon unloading the tangential force will again see an increase for the cases of ~ > 0, no change for ~ = 0 and a decrease for ~ < 0, as predicted from eqs (19) and (23) and illustrated in Fig. 7(b). On the other hand, the normal wedging force will decrease with the friction angle on loading, and increase upon unloading, leading to a reduced wedging stress intensity factor range with friction angle. In the pulsating loading cases studied currently, however, the tangential and wedging forces during the loading course are much more sensitive to the frictional attenuation compared to the unloading process, due to the differences in initial external displacements 2u,0. This may be appreciated from Figs 9 and 10, where the predicted variations of the crack flank displacements 2U.ogK.~ and 2u.,,zK,~tan 0 are plotted as a function of the friction angle for the angle of inclination of 20 ~, 2u.o = 1 F~m and R = 0.1. For zero inclination angle all local stress intensity factors are zero. This corresponds to the case of a perfectly flat crack face. The combination of large friction and asperity angles will tend to lock the crack flanks and fatigue crack propagation becomes impossible. 4. DISCUSSION The results presented here suggest that the contributions of crystallographic features of the crack faces and the frictional attenuation to the alternating stress-strain field in the vicinity of the crack tip are substantial. The faceted crack surface tends to generate a local wedging mode I displacement, while frictional attenuation tends to reduce the mode II displacement, reducing the mode II stress intensity factor and limiting the local mode I stress intensity factor at the same time. The local tangential and wedging forces are shown to be functions of the external force and the crack morphology leads to a change of loading combinations at the crack tip. As wedging seems to
608
J. TONG
et al.
be unavoidable in real cases, it is not unreasonable to suspect that it is possible, after all, to achieve pure m o d e II case in practice. T o various degrees, a mixed m o d e loading condition will always exist near the crack tip, whatever the external loading might be. The branch crack direction seems to be less sensitive [18] to the change of the local stress field, mainly because the angles measured were actually global average values. As soon as the branch crack has initiated, the crack tip condition will tend to be d o m i n a n t l y controlled by the external m o d e I opening c o m p o n e n t and the local forces between facets will become negligible. 0.8
(a)
0.6
0= 35
==
,¢
0.4
0.2
0
10
20
30
40
50
40
50
Friction angle a (degrees)
(b)
0.4
~~--35 0.2
0
-0.2 0
10
20
30
Friction angle a (degrees)
Fig. 7. (a) Variation of frictional attenuation of shear stress intensity factor, zK., with friction angle ct (loading). (b) Variation of frictional attenuation of shear stress intensity factor, ;(K,, with friction angle ct (unloading).
A model for sliding mode crack closure. Part I
(a)
609
0.5 O= 35
0.4
O=20
0.3
o=15 0.2 a=10
e=5
0.1
e L
0
10
20
30
40
50
Friction angle a (degrees)
(b)
0.8
0.6
e=3
~
0.4
0 =20 J
0=15
------
0=10
0.2
e=5
0
.
0
.
.
.
'
10
.
.
.
.
'
20
.
.
.
.
.
.
.
30
'
'
40
.
.
.
.
50
FdclJon angle = (degrees) Fig. 8. (a) Variation of the induced opening mode stress intensity factor zK~ , with friction angle :t (loading). (b) Variation of the induced opening mode stress intensity factor, zK~, with friction angle ~t (unloading).
The proposed model principally serves to quantitatively illustrate the marked importance of crystallographic features and friction force in promoting crack closure in shear mode loading for the near-threshold regime. The physical insight provided by the model is essential to understand the underlying mechanisms of branch crack threshold behaviour under mode II loading. However, some considerations should be given with respect to the limitations of the model. Firstly, the proposed model is two-dimensional and no account was taken of crack front shape; secondly, the operative mechanism may be a fretting process which is far more complex than that depicted by the model assuming the simple friction law; the actual profiles o f p and q can be very different from the linear
610
J. TONG et al. 0.5 Inclination angle 0 = 20 °
0.3 E: ¢
==
v, O4
0.1
unloading
-0.1 0
10
20
30
40
50
FdclJon angle a (degrees)
Fig. 9. Variation of 2uHozK.,with friction angle ct. simplification; and most of all, although crystallographic features 0 may be estimated from the knowledge of microstructure, the friction coefficient # may have to remain as a fitting parameter, which certainly limits the model's prediction capability. Nevertheless, the proposed model, as a preliminary attempt towards quantitatively modelling the SMCC behaviour, does provide a physical insight into the mechanisms of crack closure. It is believed that the proposed model may be extended to cover mixed mode I and II loading cases, where pre-crack face interference is expected under predominantly mode II loading conditions.
Inclination angle O = 20 °
0.15
E=
~
loading
0.10
O4
0.05
unloading
0
10
20
30
40
50
Friction angle ct (degrees)
Fig. 10. Variation of the induced opening displacement. 2u.,,zK,,, tan O. with friction angle a.
A model for sliding mode crack closure. Part I
611
5. CONCLUSIONS
The general conclusions are summarized as follows: (1) A physical model has been proposed to quantify the sliding mode crack closure experienced in the mode II loading condition. The results show that the asperity angle and friction coefficient are two crucial factors influencing near-threshold fatigue crack propagation during the entire fatigue cycle. (2) The model reveals that the faceted crack surface tends to generate a local wedging mode I displacement, while frictional attenuation tends to decrease the mode II displacements, consequently reducing both external mode II and local mode I stress intensity factors at the same time. (3) In view of the real crack profiles, an ideal mode II loading may never be achieved in practice due to the inevitable local mixed mode loading conditions. Acknowledgement--The financial support of the Science and Engineering Research Council of the U.K. is gratefully acknowledged.
REFERENCES [I] S. Suresh and R. O. Ritchie, A perspective on the role of crack closure. Fatigue Crack Growth ThreshoM Concepts, 227-261 (1984). [2] K. Minakawa and A. J. McEvily, On crack closure in the near-threshold region. Scripta Metall. 15, 633-636 (1981). [3] P.K. Liaw, Overview of crack closure at near-threshold fatigue crack closure. Mechanics of Fatigue Crack Closure, A STM STP 982, 62-92 (1988). [4] J. Schijve, Fatigue closure: observations and technical significance. Mechanics c?f Fatigue Crack Closure, A S T M STP 982, 5-34. [5] R. O. Ritchie and S. Suresh, Some considerations on fatigue crack closure at near-threshold stress intensities due to fracture surface morphology. Metall. Trans. 13A, 937-940 (1982). [6] G. T. Gray, III, J. C. Williams and A. W. Thompson, Roughness-induced crack closure: an explanation for micro structurally sensitive fatigue crack growth. Metall. Trans. 14A, 421~,33 (1983). [7] S. Suresh, G. F. Zamiski and R. O. Ritchie, Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behaviour. Metall. Trans. 12A, 1435-1443 (1981). [8] S. Purushothaman and J. K.Tien, Strength of metals and alloys. Proc. ICSMA5 Conf. Vol. 2, pp. 1267-1276 (1979). [9] S. Suresh and R. O. Ritchie, A geometric model for fatigue crack closure induced by fracture surface roughness. Metall. Trans. 13A, 1627-1631 (1982). [10] C.J. Beevers, K. Bell, R. L. Carlson and E. A. Starke, A model for fatigue crack closure. Engng Fracture Mech. 2, 93-100 (1984). [11] E. K. Tschegg, Sliding mode closure and mode III fatigue crack growth in mild steel. Acta Metall. 31, 1323-1330 (1983). [12] M. C. Smith and R. A. Smith, Towards an understanding of mode II fatigue crack growth. A S T M STP 924, 260-280 (1988). [13] R. A. Baloch and M. W. Brown, The effect of pre-cracking history on branch crack threshold under mixed mode I/II loading. Fatigue Under Biaxial and Multiaxial Loading (Edited by K. Kussmaul) ESIS10, pp. 179-197 (1991). [14] J. Tong, J. R. Yates and M. W. Brown, The influence of precracking technique on branch crack threshold behaviour under mode I/II loading. Fatigue Fracture Engng Mater. Struct. 17, 1261-1270 (1994). [15] R. J. Hartranft and G. C. Sih, Alternating method applied to edge and surface crack problem. Methods of Analysis and Solutions of Crack Problems (Edited by G. C. Sih). Noordhoff International, Leyden (1973). [16] H. Tada, P. Paris and G. Irwin, The Stress Analysis of Cracks Handbook (2nd edn). Paris Production (and Del Research), St. Louis, Missouri, U.S.A. (1985). [17] ABAQUS version 4.7, Hibbitt, Karlsson and Sorensen, Rhode Island, U.S.A. (1988). [18] J. Tong, J. R. Yates and M. W. Brown, The significance of mean stress effects on the fatigue crack growth threshold under mixed mode I and II loading. Fatigue Fracture Engng Mater. Struct. 17, 829-838 (1994). (Recewed 14Ju~ 1994)
Pergamon
Engineering Fracture Mechanics Vol. 52. No. 4, pp. 599-611, 1995
0013-7944(95)00044-5
ElsevierScienceLtd Printed in Great Britain. 0013-7944/95 $9.50+ 0.00
A MODEL FOR SLIDING MODE CRACK CLOSURE P A R T I: T H E O R Y F O R P U R E M O D E II L O A D I N G J. TONG,~" J. R. YATES and M. W. BROWN SIRIUS, Department of Mechanical and Process Engineering, University of Sheffield, P. O. Box 600, Sheffield S! 4DU, U.K. Abstract--A physical model has been developed to quantify the sliding mode crack closure (SMCC) experienced in cyclicmode I1 loading conditions. Idealizedcrystallographic surface geometry and Coulomb friction behaviour were assumed in the model. The results reveal that faceted crack surfaces will generate a local wedging mode I component while frictional attenuation tends to reduce the effectivemode II stress intensity factor range. It is found that asperity angle and friction coefficientare two fundamental factors influencing near-threshold fatigue crack growth behaviour during the entire fatigue cycle. The local stress-strain field is significantlyaltered due to the wedging mode I component and frictional attenuation of the nominal mode II component. It is concluded that ideal mode II loading may never be achieved in practice for polycrystallinematerials due to the inevitableexistenceof local mixed mode loading conditions.
NOMENCLATURE a crack length b friction free crack length F, N friction and normal forces p, q maximum normal and tangential stresses in a linear distribution Q, P local opening and shearing forces r, ~b local coordinates R load ratio Pu/Pm~x K stress intensity factor u, v mode II and mode 1 displacements u.0 initial nominal mode II displacement u,r actual tangential displacement upon convergence U closure ratio friction angle A range of displacement or K friction coefficient 0 facet angle a, z normal and tangential stresses Subscripts
eft II lit Iw
effective value of displacement or K nominal mode II component displacement or K due to crack flank tangential stresses displacement or K due to wedging.
1. I N T R O D U C T I O N NEAR-THRESHOLD fatigue crack p r o p a g a t i o n ( N T F C P ) b e h a v i o u r in m o d e I has been extensively investigated in m a n y structural materials a n d the characteristics of N T F C P b e h a v i o u r have been rationalized by using the concept o f crack closure [1-4]. A variety o f different m e c h a n i s m s can be involved in the d e v e l o p m e n t o f crack closure, a m o n g them r o u g h n e s s - i n d u c e d closure a n d oxide-induced closure have been proposed [5-7] to a c c o u n t for crack closure in the near-threshold region where plane strain c o n d i t i o n s are generally expected. I n m a n y metals a n d alloys near-threshold crack a d v a n c e takes place primarily a l o n g a single slip system, resulting in crystallographic or generally faceted fracture m o r p h o l o g i e s a n d local mixed m o d e l o a d i n g c o n d i t i o n s [2]. Oxide debris, formed as a result o f relative a b r a s i o n between faceted © Crown Copyright (1995). ~To whom all correspondence should be addressed at Dept of Materials Scienceand Technology, University of Surrey, Guildford GU2 5XH, Surrey, U.K. Eru 52/~B
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crack surfaces, can build up to 0.2 pm [7] in the threshold region, comparable to the scale of crack tip displacements. The occurrence of crystallographic plane separation or oxide built up on the fracture surface or both can lead to the creation of asperities on the fatigue fracture surfaces, which may further promote oxide-induced crack closure. The first analysis to quantify the effect of fracture surface roughness on mode I fatigue crack closure was presented by Purushothaman and Tien [8] who suggested that the closure stress intensity Kc~ could be estimated by comparing the change in fracture surface asperity height to the crack opening displacement. Suresh and Ritchie [9] proposed a simple geometric model in which the contribution of local mode II displacement was considered. A physically based model was proposed by Beevers et al. [10], who used a "local" stress intensity factor induced by symmetric concentrated point forces in deriving Kc~.The effect of oxide-induced closure was also included in the last analysis as the oxide debris may be simply treated as asperities. Displacement was used directly or indirectly to estimate Kd in almost all of the above analyses, a methodology consistent with Elber's original concept and the experimental method to obtain the load vs crack face opening displacement curve. However, if a crack is subjected to mode II loading, the situation becomes more complex as the effect of fracture surface roughness on NTFCP behaviour can be very different from the response under mode I loading, depending on the R-ratio, loading combinations, loading history and pre-crack conditions. Even when loading history and pre-crack conditions are set aside, crack closure is strongly influenced by load ratio and the mixed mode combination. Under mode II loading conditions the crack flanks will always be in contact (Fig. 1). Frictional attenuation and wedging over opposing asperities would be the leading factors in determining the real crack driving force. The term "sliding mode crack closure" (SMCC) is used here, after Tschegg [11], to specify the effect of reduction of nominal stress intensity factor because of friction, abrasion and similar dissipative processes in mixed mode loading cases, analogous to fatigue crack closure in mode I. Although experimental evidence [12, 13] seems to support the above arguments, to quantify the closure effect under mode II loading appears to be formidable, due to various uncertainties. Experimental difficulties arise in obtaining the relevant parameters, e.g. friction coefficient (assuming Coulomb friction), the extent of wear debris or even the real displacements along the crack flanks. In a study of mode II fatigue crack growth in a structural steel, Smith and Smith [12] identified three sources of crack growth attenuation: compressive residual stress ahead of the fatigue pre-crack, interlocking of fracture surface asperities and gross plastic deformation of the interlocking asperities. The unlocking response was simulated by assuming that crack flank frictional stresses obeyed a constant interfacial shear stress friction law. However, no analytical solution was proposed from the investigation to describe SMCC quantitatively. In the current study a physical model has been developed to quantify the SMCC experienced in mode II loading conditions. NTFCP behaviour is modelled by considering frictional attenuation and facet wedging as the major influential factors, The model has been evaluated for a variety of friction coefficients and inclination angles under both loading and unloading conditions. The extension of the model to mixed mode I and II cases, and the application of the model to experimental results on a structural steel BS4360 50D will be presented and discussed later in Part II of this paper.
2. P R O P O S E D M O D E L 2.1. Characteristics and assumptions
The proposed model aims at quantifying the onset condition of branch cracks under mixed mode I and II loading. However, the analysis will begin with a nominal mode II loading case without losing generality. The characteristics of the model are largely based on the current experimental conditions and assumptions are made accordingly. In experiments, the pre-crack conditions were carefully controlled and detailed elsewhere [14]. Briefly, the pre-cracks were produced using decreasing stress intensity tests at a high R-ratio (R Kmin/Kmax), terminated just above the mode I threshold so the closure induced by plasticity during the pre-cracking process would be minimized. =
A model for sliding mode crack closure. Part I
601
T
t~ .
"C
t~
ol Mode
II
Mode
I
Fig. 1. Schematic illustration of crack closure during mode II and mode I loading.
At maximum load, pre-cracks fully unlock up to the crack tips and crack surfaces slide along the whole crack length under limiting friction. However, shear displacements at both the crack mouth and tip are small compared to facet length such that slips will be confined on one facet. The SMCC effect is most active within the crystallographic area[13]. The crack tip driving force is reduced to a varying extent by frictional attenuation and wear debris, depending on the R-ratio. Among the proposed sources for frictional resistance [12], only the compressive normal stresses due to the crack surfaces wedging open over asperities are considered. The compressive residual stresses due to plasticity-induced crack closure are ignored, in view of the pre-cracking method adopted. The frictional resistance due to gross deformation of interlocking asperities is also overlooked, as it is generally regarded as a transient process of frictional attenuation. An idealized crystallographic surface geometry is assumed. The pre-crack is treated as an elastic line crack, where an idealized crystallographic surface geometry is assumed. The notch and pre-crack as a whole are idealized as a partially loaded edge crack in a semi-infinite plate (Fig. 2). 2.2. The model Assuming Coulomb friction with a friction coefficient p, an idealized zig-zag angle 0 for the crack path taken to the direction of the external shear stress. A previous study [12] based on direct crack surface displacement measurement shows that compressive normal stresses due to wedging arise as the pre-crack tip is approached [Fig. 2(a)], a fact confirmed by the present SEM analysis as rougher fracture topographies are observed right behind the pre-crack tips than at positions near the pre-crack mouths. For simplicity, a linear distribution is assumed for normal stresses tr(x) and tangential stresses r(x) [Fig. 2(b)].
J. T O N G et al.
602
j"
I_
P
"
-4
(a) Real pre-crack condition
|
O
'
•
't
"
t jj
!
L
x
i
b
(b) Idealized pre-crack condition Fig. 2. Real (a) and idealized (b) pre-crack condition (mode II), showing local stressesexerted on the upper fracture surface due to crack face contact.
Partially loaded linearly distributed normal and shear loads due to wedging and rubbing can be obtained by integrating the theoretical solution [15] for concentrated normal and shear forces on an edge crack.
(f°o(x)g(x/t,Odx K~,
= -n
~" z(x)g(x/a)dx
x/~
%//-~ __ X 2
Here a ,(x,O) =
- tr(x) =
z~,.(x,O) =
--z(X)
•
=
x-b a - b p'
x-b a--b
q,
"
(1)
A model for sliding mode crack closure. Part I
603
g(x/a) = g ( c ) = 1 + (1 - c2)(0.2945 - 0.3912c-' + 0.7685c 4 - 0.9942c 6 + 0.5094c 8)
(2)
and
which yields
(3)
K.,J
where K,w is the stress intensity factor due to wedging, K., is the attenuation in the applied stress intensity caused by the tangential stress acting along the crack flanks. Displacements at the end o f the crystallographic length (distance a - b f r o m the pre-crack tip) were derived by integrating the displacements due to the concentrated forces [16] to give: H
o'(x)dx (4)
2u = ~
"H
where
:E,.6,1_0.,84(:)°1.
(5)
F r o m eqs (4) and (5) an integrated form is obtained:
{
4 h(b/a){~} a '
2vw'~=
(6)
2u, J
where Vw and u, are displacements due to wedging and frictional attenuation, respectively, E' = E/(1 - v2), E is Y o u n g ' s m o d u l u s and v is Poisson's ratio. T h e integrated results forf(b/a) and h(b/a) are shown in Fig. 3.
1.5
1.0 ,Jc
0.5
0
.2
.4
.6
.8
1.0
b/a
Fig. 3. Geometry factors for stress intensity and displacement for a partially loaded edge crack in a semi-infinite plate.
604
J. TONG et al. LOADING
N
Q Q = Ptan (O+c¢)
(a)
UNLOADING
Q = Ptan(O-ct)
(b) Fig. 4. Normal and friction force decomposition. (a) Loading, (b) unloading.
Under limiting friction, the friction angle, ~, the normal force, N, the friction force, F, and the friction coefficient, ~, are related as p = tan ~ =
F
~.
The normal and friction forces produce a resultant force, which in turn can be decomposed into vertical P and horizontal Q components (Fig. 4). As the direction of the friction force will be reversed in going from loading to unloading, the cases of loading and unloading are treated separately in Fig. 4(a,b). For the loading case, Q = P tan(~ + 0),
(7)
Q = P tan(0-a).
(8)
and for the unloading case,
The displacement field for the current specimen was computed using the finite element program ABAQUS [17] under nominal mode II loading conditions. The local coordinate system at the crack tip is shown in Fig. 5. Considering two nodes, i and j, on opposing crack surfaces at a distance a - b from the crack tip, the shear displacement due to nominal shear stress at the end of the crystallographic length [x = - (a - b)] can be obtained as u i - uj = 2u.0. From
(9)
a simple geometrical consideration, the normal displacement induced by wedging
would be 2vw0 = 2u,0 tan 0.
(10)
A m o d e l f o r sliding m o d e c r a c k c l o s u r e . P a r t I
.I !
x
o
a-b
605
j
Fig. 5. Local coordinate system at the pre-crack tip.
The sliding displacement u.0 would be an upper bound value for the loading case as no frictional attenuation has yet been taken into account. The normal stress due to wedging may be determined from eq. (6) as ~zE'
- 2Vwo. po- 4h(b/a)a
(11)
For loading, the shear displacement attenuation due to friction would be 2U,o-
4h(b/a)a E'n q0 =
tan(~ +
O)tan(O)2u,o,
(12)
and by superposition the instantaneous effective mode II elastic displacement would be 2urn = 2uli0 - 2ut0.
(13)
This reduced displacement implies a reduction of the estimated wedging displacement to Vwl,with corresponding values for p~, qj and ut., such that 2U112 =
2UI|0 - -
2u,1.
(14)
As the actual normal and shear stresses p and q will depend on the final effective mode II displacement to give a compatible set of displacements, the above procedures are repeated so that 2Urn+ I =
2U.o - 2uti
=
2UllO- tan(~ +
O)tanO2u.,.
(15)
The formula converges to a constant value after n iterations. On convergence, 2UI|,,+ I = 2 U n ,
(16)
2u.0 1 + tan(~ + 0)tan0 '
(17)
and
2u,,, =
so the actual tangential displacement 2u~f would be tan(~ + 0)tan0 2u,. = 2u.0 - 2u.,, = 1 + tan(~ + 0)tan0 2u,0.
(18)
The attenuation in the applied stress intensity caused by the tangential stresses acting along the crack flanks is: Kilt =
F(b/a)E'£2u,.,
(19)
where the geometry factor F(b/a) =f(b/a)/2h(b/a). The mode I stress intensity factor due to wedging would be:
K.w = F(b/a)E'tan 0 N/-~--n 2(u,.0 u
u,r).
(20)
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J. TONG et al.
Ull
~~ri
~:ion locking dwell
reversed loading
~ ~ ; ~ reversed unloading
KII~
g-
Fig. 6. Schematic of variations of nominal and effective mode 11 displacements and stress intensities during a fully reversed fatigue cycle.
The effective mode II stress intensity factor can then be resolved as K~I,, = K i t - K . ,
(21)
where K. is the applied stress intensity factor at the crack tip and may be calculated from the displacement field: 8~rG 1 KH -- (2nr~)i/2 (2k + 2)
1 1
( u , - u,j) 2 ' .
(22)
3
(2k + 3, sin(~ 4~,)+ sm(~ ~b,) where k = 3 - 4v for plane strain, G is the modulus of rigidity, r, q~refer to the local coordinate system (Fig. 5) at the crack tip, u,i, u,j are the displacements in the vicinity of the crack tip, symmetrically situated with respect to the y-axis. Note here that K . , Kl,, u, and Vware strongly interdependent via p, 0 and b due to contact pressure and friction on opposing fracture surfaces. These effects are incorporated in the terms p and q, which govern the mode I and II responses, respectively. For the unloading case, the frictional force will reverse its direction. Under the limiting friction condition, the displacement induced by the tangential stress acting along the crack flanks would be 2Utf =
tan(0
-
~)tan
1 +- t--~(-O • ~
0
0 2u.0,
(23)
and the corresponding stress intensities can be expressed in the same forms as in eqs (19)-(21). The more general case would be to consider a fully reversed loading cycle, including reversed loading and unloading phases, as well as loading and unloading phases. Figure 6 shows a schematic of variations of nominal and effective mode II displacements and stress intensities during an entire fatigue cycle at R = - 1. During loading the pre-crack is unlocked to the crack tip and the mechanical behaviour from a to b may be depicted by eqs (18)-(22). Load reversal at maximum load locks the pre-crack. Unlocking would not take place until the reduction in the applied load is such that the limiting friction condition is achieved and the behaviour from c to d may be described by eqs (19)-(23). Further reversed loading causes the opposite facets to make contact and the
A model for sliding mode crack closure. Part I
607
behaviour would repeat itself and the mechanical responses from d to e and from f to g may be modelled again by eqs (18)-(22) and eqs (19)-(23) accordingly. Generalization of these results to cover both loading and unloading mechanical behaviour under limiting friction gives: tan(0 + ~ ) t a n 0 1 + tan(0 + ~ ) t a n 0 2u.0,
(24)
tan 0 K~w = F(b/a)E' / :- 1 + tan(0 + ~ ) t a n 0 2u,0, va
(25)
KII,
F(b/a)E, f-ff
4a
where ~ = 1 for loading, ~ = - 1 for unloading and Klleff =
KII - - K . , .
(26)
3. M O D E L B E H A V I O U R
From the above analysis, it is apparent that the friction angle and angle of inclination of facets on the crack profile are two fundamental factors which significantly influence the behaviour of the model. The variation of these factors can lead to substantial changes of the local tangential force and normal wedging force, which therefore change the crack tip stress-strain field. The influence of friction angle ~ and facet angle 0 on SMCC have been investigated numerically, and the results were characterized by the parameters xK, t and xK~w, where
Z-
F(b/a)
E'2u,,o "
In Fig. 7 the attenuation of the mode II stress intensity factor due to friction is presented for loading and unloading cases, respectively. The dependence of the wedging force on the angle of friction for loading and unloading cases is shown in Fig. 8. An examination of the curves in Figs 7 and 8 reveals that the effects of both friction and asperity angle are significant. The local tangential force will increase with the friction and asperity angles on loading. Upon unloading the tangential force will again see an increase for the cases of ~ > 0, no change for ~ = 0 and a decrease for ~ < 0, as predicted from eqs (19) and (23) and illustrated in Fig. 7(b). On the other hand, the normal wedging force will decrease with the friction angle on loading, and increase upon unloading, leading to a reduced wedging stress intensity factor range with friction angle. In the pulsating loading cases studied currently, however, the tangential and wedging forces during the loading course are much more sensitive to the frictional attenuation compared to the unloading process, due to the differences in initial external displacements 2u,0. This may be appreciated from Figs 9 and 10, where the predicted variations of the crack flank displacements 2U.ogK.~ and 2u.,,zK,~tan 0 are plotted as a function of the friction angle for the angle of inclination of 20 ~, 2u.o = 1 F~m and R = 0.1. For zero inclination angle all local stress intensity factors are zero. This corresponds to the case of a perfectly flat crack face. The combination of large friction and asperity angles will tend to lock the crack flanks and fatigue crack propagation becomes impossible. 4. DISCUSSION The results presented here suggest that the contributions of crystallographic features of the crack faces and the frictional attenuation to the alternating stress-strain field in the vicinity of the crack tip are substantial. The faceted crack surface tends to generate a local wedging mode I displacement, while frictional attenuation tends to reduce the mode II displacement, reducing the mode II stress intensity factor and limiting the local mode I stress intensity factor at the same time. The local tangential and wedging forces are shown to be functions of the external force and the crack morphology leads to a change of loading combinations at the crack tip. As wedging seems to
608
J. TONG
et al.
be unavoidable in real cases, it is not unreasonable to suspect that it is possible, after all, to achieve pure m o d e II case in practice. T o various degrees, a mixed m o d e loading condition will always exist near the crack tip, whatever the external loading might be. The branch crack direction seems to be less sensitive [18] to the change of the local stress field, mainly because the angles measured were actually global average values. As soon as the branch crack has initiated, the crack tip condition will tend to be d o m i n a n t l y controlled by the external m o d e I opening c o m p o n e n t and the local forces between facets will become negligible. 0.8
(a)
0.6
0= 35
==
,¢
0.4
0.2
0
10
20
30
40
50
40
50
Friction angle a (degrees)
(b)
0.4
~~--35 0.2
0
-0.2 0
10
20
30
Friction angle a (degrees)
Fig. 7. (a) Variation of frictional attenuation of shear stress intensity factor, zK., with friction angle ct (loading). (b) Variation of frictional attenuation of shear stress intensity factor, ;(K,, with friction angle ct (unloading).
A model for sliding mode crack closure. Part I
(a)
609
0.5 O= 35
0.4
O=20
0.3
o=15 0.2 a=10
e=5
0.1
e L
0
10
20
30
40
50
Friction angle a (degrees)
(b)
0.8
0.6
e=3
~
0.4
0 =20 J
0=15
------
0=10
0.2
e=5
0
.
0
.
.
.
'
10
.
.
.
.
'
20
.
.
.
.
.
.
.
30
'
'
40
.
.
.
.
50
FdclJon angle = (degrees) Fig. 8. (a) Variation of the induced opening mode stress intensity factor zK~ , with friction angle :t (loading). (b) Variation of the induced opening mode stress intensity factor, zK~, with friction angle ~t (unloading).
The proposed model principally serves to quantitatively illustrate the marked importance of crystallographic features and friction force in promoting crack closure in shear mode loading for the near-threshold regime. The physical insight provided by the model is essential to understand the underlying mechanisms of branch crack threshold behaviour under mode II loading. However, some considerations should be given with respect to the limitations of the model. Firstly, the proposed model is two-dimensional and no account was taken of crack front shape; secondly, the operative mechanism may be a fretting process which is far more complex than that depicted by the model assuming the simple friction law; the actual profiles o f p and q can be very different from the linear
610
J. TONG et al. 0.5 Inclination angle 0 = 20 °
0.3 E: ¢
==
v, O4
0.1
unloading
-0.1 0
10
20
30
40
50
FdclJon angle a (degrees)
Fig. 9. Variation of 2uHozK.,with friction angle ct. simplification; and most of all, although crystallographic features 0 may be estimated from the knowledge of microstructure, the friction coefficient # may have to remain as a fitting parameter, which certainly limits the model's prediction capability. Nevertheless, the proposed model, as a preliminary attempt towards quantitatively modelling the SMCC behaviour, does provide a physical insight into the mechanisms of crack closure. It is believed that the proposed model may be extended to cover mixed mode I and II loading cases, where pre-crack face interference is expected under predominantly mode II loading conditions.
Inclination angle O = 20 °
0.15
E=
~
loading
0.10
O4
0.05
unloading
0
10
20
30
40
50
Friction angle ct (degrees)
Fig. 10. Variation of the induced opening displacement. 2u.,,zK,,, tan O. with friction angle a.
A model for sliding mode crack closure. Part I
611
5. CONCLUSIONS
The general conclusions are summarized as follows: (1) A physical model has been proposed to quantify the sliding mode crack closure experienced in the mode II loading condition. The results show that the asperity angle and friction coefficient are two crucial factors influencing near-threshold fatigue crack propagation during the entire fatigue cycle. (2) The model reveals that the faceted crack surface tends to generate a local wedging mode I displacement, while frictional attenuation tends to decrease the mode II displacements, consequently reducing both external mode II and local mode I stress intensity factors at the same time. (3) In view of the real crack profiles, an ideal mode II loading may never be achieved in practice due to the inevitable local mixed mode loading conditions. Acknowledgement--The financial support of the Science and Engineering Research Council of the U.K. is gratefully acknowledged.
REFERENCES [I] S. Suresh and R. O. Ritchie, A perspective on the role of crack closure. Fatigue Crack Growth ThreshoM Concepts, 227-261 (1984). [2] K. Minakawa and A. J. McEvily, On crack closure in the near-threshold region. Scripta Metall. 15, 633-636 (1981). [3] P.K. Liaw, Overview of crack closure at near-threshold fatigue crack closure. Mechanics of Fatigue Crack Closure, A STM STP 982, 62-92 (1988). [4] J. Schijve, Fatigue closure: observations and technical significance. Mechanics c?f Fatigue Crack Closure, A S T M STP 982, 5-34. [5] R. O. Ritchie and S. Suresh, Some considerations on fatigue crack closure at near-threshold stress intensities due to fracture surface morphology. Metall. Trans. 13A, 937-940 (1982). [6] G. T. Gray, III, J. C. Williams and A. W. Thompson, Roughness-induced crack closure: an explanation for micro structurally sensitive fatigue crack growth. Metall. Trans. 14A, 421~,33 (1983). [7] S. Suresh, G. F. Zamiski and R. O. Ritchie, Oxide-induced crack closure: an explanation for near-threshold corrosion fatigue crack growth behaviour. Metall. Trans. 12A, 1435-1443 (1981). [8] S. Purushothaman and J. K.Tien, Strength of metals and alloys. Proc. ICSMA5 Conf. Vol. 2, pp. 1267-1276 (1979). [9] S. Suresh and R. O. Ritchie, A geometric model for fatigue crack closure induced by fracture surface roughness. Metall. Trans. 13A, 1627-1631 (1982). [10] C.J. Beevers, K. Bell, R. L. Carlson and E. A. Starke, A model for fatigue crack closure. Engng Fracture Mech. 2, 93-100 (1984). [11] E. K. Tschegg, Sliding mode closure and mode III fatigue crack growth in mild steel. Acta Metall. 31, 1323-1330 (1983). [12] M. C. Smith and R. A. Smith, Towards an understanding of mode II fatigue crack growth. A S T M STP 924, 260-280 (1988). [13] R. A. Baloch and M. W. Brown, The effect of pre-cracking history on branch crack threshold under mixed mode I/II loading. Fatigue Under Biaxial and Multiaxial Loading (Edited by K. Kussmaul) ESIS10, pp. 179-197 (1991). [14] J. Tong, J. R. Yates and M. W. Brown, The influence of precracking technique on branch crack threshold behaviour under mode I/II loading. Fatigue Fracture Engng Mater. Struct. 17, 1261-1270 (1994). [15] R. J. Hartranft and G. C. Sih, Alternating method applied to edge and surface crack problem. Methods of Analysis and Solutions of Crack Problems (Edited by G. C. Sih). Noordhoff International, Leyden (1973). [16] H. Tada, P. Paris and G. Irwin, The Stress Analysis of Cracks Handbook (2nd edn). Paris Production (and Del Research), St. Louis, Missouri, U.S.A. (1985). [17] ABAQUS version 4.7, Hibbitt, Karlsson and Sorensen, Rhode Island, U.S.A. (1988). [18] J. Tong, J. R. Yates and M. W. Brown, The significance of mean stress effects on the fatigue crack growth threshold under mixed mode I and II loading. Fatigue Fracture Engng Mater. Struct. 17, 829-838 (1994). (Recewed 14Ju~ 1994)