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```					MIXED-MODE STRESS INTENSITY FACTORS FOR PARTIALLY OPENED
CRACKS

Theo Fett
Forschungszentrum Karlsruhe, Institut für Materialforschung II, Postfach 3640,
D-76021 Karlsruhe, Germany
e-mail: theo.fett@imf.fzk.de

Abstract. Failure considerations under mixed-mode loading require knowledge
about the influence of friction between partially closed crack faces in the case of a
negative mode-I stress intensity factor. A simple relation is derived, which allows
to compute friction contributions to the mode-II stress intensity factor KII for the
case of negative mode-I stress intensity factors KI. The relation is exact for the
limit case of an edge crack in a half-space. It can be shown that small cracks in
finite bodies can also be described with sufficient accuracy.
Keywords: crack, stress intensity factor, friction.
1. Introduction. Under mixed-mode loading, failure of a crack-containing com-
ponent occurs, if a function f of the three stress intensity factors reaches a critical
value fc
f ( K I , K II , K III )  f c                     (1)
Several mixed-mode fracture criteria have been proposed. In most commonly used
criteria only KI and KII are included in the function f. The most popular failure
criterion is that of the coplanar energy release rate proposed by Paris and Sih
(1965). Under plane strain conditions, it reads
1
K I2  K II 
2
K III  K Ic
2
(2)
1 
This criterion makes sense only for positive KI. For KI < 0 the crack faces are un-
der compression and no singular mode-I stress field exists. Nevertheless, a mode-
II stress intensity factor can occur, caused by a superimposed shear loading. Due
to friction between the crack faces, KII has to be calculated with an effective shear
stress (Alpa (1984)).
         for   n  0

 eff      n for  n                          (3)
 0        for  n  

If the considered crack is small compared with the variation of normal stresses n
(i.e. if the effective shear stress is sufficiently constant over the crack size), it
holds simply
K I  nY a                                            (4a)

K II   eff Y a                                          (4b)

with the geometric function Y  1.98. In terms of eqs.(4a) and (4b) the friction
problem was dealt with by Alpa (1984).
For a strongly changing stress the weight function method allows to determine the
two stress intensity factors as proposed by Bückner (1979)
a
K I    n ( x ) hI ( x, a ) dx                                (5a)
0
a
K II    eff ( x ) hII ( x, a ) dx                               (5b)
0
where hI and hII are the weight functions for mode-I and mode-II loading and x is
the coordinate in crack direction. Unfortunately, eq.(3) is correct only for cracks
with completely closed crack faces. The aim of this paper is to compute the
effective stress intensity factor also for partially closed cracks.

2. Stress intensity factors. Natural surface cracks in ceramics are often
modelled as edge cracks. An edge crack at the free surface of depth a oriented in
x-direction is considered in Fig. 1.
In order to predict failure by the remaining stress intensity factor KII, it is
necessary to determine that mode-II stress intensity factor contribution which
reduces the applied stress intensity factor by crack surface friction.
Fig. 1 shows the total crack opening displacement total resulting from the
applied stress appl and the superimposed contact stress cont which prevents
penetration of the crack faces in the presence of a negative mode-I stress intensity
factor.

x                               cont
x0
total

a
Figure 1. Schematic illustration of stresses and displacements for a partially closed crack.
If a crack is partially closed, the total stress intensity factor must disappear. In the
weight function representation this condition reads
a                                a
K I , total   hI ( x, a )  n ( x ) dx   hI ( x, a )  cont ( x ) dx  0    (6)

   
0
 0                            
K I , appl                         K I , cont

with cont > 0 over the contact area and cont = 0 elsewhere. From eq.(6), it results
a
K I ,cont   hI ( x, a )  cont ( x ) dx   K I , appl              (7)
0

Due to the contact stresses cont between the crack surfaces in the presence of
additional shear loading, friction is caused, resulting in a mode-II stress intensity
factor contribution KII,frict of
a
K II , frict    hII ( x, a )  cont ( x ) dx ,            cont  0    (8)
0

where  is the friction coefficient. Then, the effective mode-II contribution is
K II , eff  K II , appl  K II , frict                    (9)

Determination of the friction stress intensity factor is relatively complicated, since
it needs the solution of an integral equation (Fett, 2001) in order to determine the
contact stress distribution. For small cracks the problem simplifies strongly.
Edge-cracked half-space: Let us use the fact that for very small cracks the mode-I
and mode-II weight functions are identical, i.e.
hI  hII           for   a /W  0                          (10)
Combining this with eqs.(7) and (8) provides the simple result of
K II , frict  K I , appl      for     K I ,appl  0             (11)

Small edge cracks in a finite body: In order to estimate the errors made by appli-
cation of eq.(11) to larger cracks, one has to look for the deviations between hI and
hII. In Fig. 2a the two weight functions proposed by Fett and Munz (1997) are
plotted for several relative crack depths a/W. Figure 2b shows the ratio
  hI / hII                               (12)
The crack depths of natural cracks in ceramic materials are of the order of 50
m, the widths of commonly used test specimens are > 3mm in most cases.
Hence, the relative crack size for standard tests is a/W < 0.02. For cracks of such
relative depths the maximum deviations between the two weight functions are less
than 2%. The maximum deviations of the stress intensity factors are, of course,
less than the maximum deviations of the weight functions. This is due to the inte-
gration of the weight function over a positive stress, by which the curves in Fig.
2b are averaged as a consequence of the mean value theorem for integrals. This
can be concluded e.g. from eq.(8)
a                                              a
1                       1                   K I ,appl
K II , frict                   hI  cont dx    hI  cont dx                                      (13)
0
( x / a )                  ' 0                  '

where ’ is the weight function ratio at a certain location 0  x’ a.
Equation (13) allows a simple computation of the friction stress intensity
factor from the applied stresses appl present in the uncracked body, namely,
                 ' '
a                               a
K I ,appl
  hI  appl dx 
' 
K II , frict                                            hII appl dx                          (14)
'      ' 0                   0

where ’’ is  at 0  x’’ a which may differ from x’. The ratio ’’/’ is closer to
“1” than ’’ or ’. This allows one to write, finally,
a
K II , frict   hII appl dx                                         (15)
0

20                                                   1.3
hI
hW                          hII
hI/hII
15
a)                                        1.2                                   b)
a/W=0.025
10                                                                       a/W=0.1

0.05                       1.1
0.1
5                                                                      0.05
0.025
0
1
0
0      0.2    0.4     0.6     0.8       1            0       0.2        0.4   0.6   0.8   1
x/a                                                       x/a
Figure 2. Comparison of the mode-I and mode-II weight functions.

Semi-circular cracks: Natural cracks are often modelled as half-penny shaped
surface cracks. For such cracks the stress intensity factor varies along the crack
front. In most practical cases the stress intensity factor for the deepest point of the
semi-circle (point A in Fig. 3) is considered. For this point the applied stress inten-
sity factor results from
K I( ,A)   hI( A) ( x, y ) appl ( x, y ) dxdy
appl                                                        (16)
S

where S is the total crack area S = Sopen + Scont and h I( A ) is the two-dimensional
weight function for the specially chosen crack front point A. The stress intensity
factor due to the contact stresses is
K I( ,A)     h            ( x, y )  cont ( x, y ) dxdy
( A)
cont           I                                            (17)
Scont

and we would obtain the same result as for edge cracks
K IIAeff  K IIAappl  K I( ,A)
( )
,
( )
,              appl                         (18)
(
if the weight functions for shear loading (xz), hIIA) ( x, y ) and normal tractions
(n), hI( A) ( x, y ) , would be identical. This is exactly the case for of 0 and
a/W0. Unfortunately, the mode-II weight function depends slightly on Poisson’s
ratio  in the two-dimensional case. The error in KII,frict obtained by setting hI = hII
is of the order of /(2-), e.g. for  = 0.2 about 11%.
y
Sopen

x                                    Scont                          xz

A

Figure 3. Semi-circular surface crack under shear loading for KI < 0 at point A.

3. Conclusions
A simple relation is derived, which allows to compute friction contributions to the
mode-II stress intensity factor KII for the case of negative mode-I stress intensity
factors KI. The relation, exact for the limit case of very small cracks with a/W0,
is valid for natural cracks in ceramics, modelled as edge cracks, with a maximum
error of less than 2%. In the case of semi-circular surface cracks, the errors made
by this simple analysis may increase up to 11%.
References
Alpa, G. (1984). On a statistical approach to brittle rupture for multiaxial states of stress. Engineering
Fracture Mechanics 19, 881–901.
Bückner, H. (1979). A novel principle for the computation of stress intensity factors, Zeitschrift für
Angewandte Mathematik und Mechanik ZAMM 50, 529–546
Fett, T. (2001). Effective mode-II stress intensity factor for partially opened natural cracks under mixed-mode
loading, to be presented at the International Conference on Fracture ICF10, December 2001, Honolulu.
Fett, T., Munz, D. (1997). Stress Intensity Factors and Weight Functions, Computational Mechanics
Publications, Southampton, UK
Paris, P.C., Sih, G.C. (1965). Stress analysis of cracks. ASTM STP 381, 30–80.

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