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The propulsion of surface flaws by elastic indentation testing
Acta metall, mater. Vol. 43, No. 3, pp. 985-99l, 1995
~
Pergamon
0956-7151(94)00313-0
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0956-7151/95 $9.50 + 0.00
THE P R O P U L S I O N OF S U R F A C E FLAWS BY ELASTIC I N D E N T A T I O N TESTING D. N. DAI, I D. A. HILLS,it P. D. WARREN2 and D. NOWELL 1 'Department of Engineering Science and 2Department of Materials, Oxford University, Parks Road, Oxford OX1 3PJ, England (Received 14 January 1994; in revised form 20 June 1994)
Abstract--An analysis is carried out of the crack tip stress intensity factor existing around the front of a semi-elliptical surface flaw, when it is propelled by a Hertzian indentation test. The influence of crack size, position, ellipticity and orientation on stress intensity factors and strain energy release rate is investigated. It is shown that generally a surface flaw will "run around" to form a shallow ring crack before propagating inwards into the material.
1. INTRODUCTION Testing ceramic materials by elastic indentation experiments is becoming an important method for the quantification of the surface flaw distribution. This will become a standard recognized enabling technology for quality control in the structural use of ceramic components in engineering. It is important, for repeatability, that a well defined indenter shape, producing an incomplete contact, is used, in order to minimize the problems associated with edge effects when flat-ended indenters are employed, and by far the most usual shape used is a sphere. It is well-known that when a sphere is pressed into a brittle material, a ring crack forms, either at the edge of the contact disk itself, or just outside [1, 2], and that if the load is increased further the crack may flare out into the frustum of a cone [3]. Before quantitative use may be made of the test data it is important to understand as fully and reliably as possible the crack tip stress intensity occurring around the crack front, with the consequent implications this has for the development of the crack's size and shape. This is essential if the material fracture toughness is to be used as a means of deducing the corresponding crack size, as we implicitly assume that fracture toughness is a true material property, with crack arrest ensuing when the crack tip stress intensity factor at a point falls below the toughness. Unfortunately, the geometries with which we are concerned in this field are very taxing to analyse, as the crack exists in a very steep stress gradient, associated with the contact loading, and the crack itself may have a complex profile. The easiest of the three stages of crack development to attack is the intermediate case, when a ring has already ?To whom all correspondence should be addressed. 985
formed, and the crack tip is propagating into a decreasing contact stress field, Fig. l(b). Even this truly axi-symmetric geometry is not without its difficulties, and it is customary to assume that the depth of the crack, c, is small compared with its radial position, d, so that a plane (two-dimensional) analysis of the problem may be used [4, 5]. If the radius of the ring crack is known, and the load at which propagation took place, it is possible to infer both the final crack depth and, from many tests, statistical information about the initial flaw distribution [6], given the fracture toughness. Conversely, if data about the final crack size are known, the fracture toughness may be deduced. The third phase of development, i.e. the formation of the resulting Hertzian cone crack, Fig. l(c), has also been analysed [7], and this, too, may be used to infer the fracture toughness if the final cone size is known. Usually, however, this is possible only for transparent materials, so that the technique has limited application as a routine test procedure. By far the most important part of the test to be treated is the development of pre-existing surface defects into the ring, and it is with this phase that the current paper is concerned, Fig. 1(a). A semi-elliptical surface flaw has been analysed by the eigenstrain method, and it is our intention to calibrate for the crack tip stress intensity factor occurring around the whole of the crack front. This will enable us both to determine the load at which a defect of known size grows, and the direction in which it advances, i.e. whether it propagates downwards into the material to produce a deeper thumbnail type crack, or whether it "runs around" to form a shallow ring, before advancing inwards into the material. In many real applications only the load at which a ring crack forms is known, but from the calibration to be provided it is possible to infer, within limits, the size (or shape) of the pre-existing defect.
DAI et al.:
986 (a)
(b)
Pre-existing defect
THE PROPULSION OF SURFACE FLAWS
(c)
Ring crack
Cone crack
Fig. 1. Three stages in the development of an indenterpropelled crack: (a) pre-existing microscopic surface defect, which grows to form; (b) ring crack, which, in turn, flares out to form: (c) conical crack.
2. FORMULATION Figure 2 shows the geometry to be analysed• We assume that the surface defects are in the form of semi-elliptical faws, of surface length 2b and depth c, existing normal to the surface• The defects are assumed to be sufficiently widely spaced in comparison with their size for interaction between them to be ignored, and for them to be sufficiently small by comparison with the size of the contact for their interaction with the contact problem to be negligible [8]: in practice neither of thesc assumptions is at all restrictive• We may therefore focus attention on a single defect, which is assumed to be oriented at an angle 4~ to the 0-direction, Fig. 2, and to be positioned at a radial location r = d. The problem is solved in two parts; first, we determine the state of stress induced by the contact problem alone. This is widely known [9, 10], and, in order to reduce the number of independent variables in the present analysis, it is assumed that no radial tractions arise [11], although they may play an important r6le in propelling short cracks [12]. The solution is therefore most appropriate when the indenter has the same elastic properties as the substrate, or when the interface is lubricated. The nonzero stresses arising in a cylindrical coordinate set are clearly a__,, ~r, aoo, r,:. They are usually found normalized with respect to the peak Hertzian contact pressure P0 (see [9, 10]), and as functions of cylindrical coordinates (r, z), which are normalized with respect
/ y, '~,?qF
,,!C °
T z 2be" ..,
,
- B-
"
~J'
Defect
A /~t .,4, q~
•
-
~.~
Detect
to the radius of the contact disk, a. The stress components may be transformed into a local axis set centred on the crack, Fig. 2, by application of a routine transformation. It should be noted that, unless (/) = 0, all three modes of loading of the crack front may be anticipated in general, although clearly K m must be zero at the free surface• It is not appropriate here to describe the eigenstrain procedure in full, but we will take the opportunity to outline some features of the method which render it so suitable for problems of this kind. The technique is similar in concept to a boundary element formulation. Thc underlying result which is employed is Bueckner's principle [13]. First, the tractions arising on the faces of the crack in the crack's absence are found [these are denoted by a~,, in equation (1)]. The application of equal and opposite cancelling tractions to an otherwise unloaded body then induces the same set of stress intensity factors at each point around the crack front as would be developed by the external loading. In three-dimensional problems such as this the starting point for determining a nucleus ['or a cancelling traction is the solution by Mindlin [14] for a point force within a half-space. It should be noted that any number of nuclei of this kind may be inserted within the material without violating the requirement that the surface of the half-space remain traction-fi'ee. It is customary in the eigenstrain procedure to use as the kernel of the integral equation not thc function relating the influence of an interior point force to the stress at a point, but its differential with respect to the field point. The integral expressing the requirement that the crack faces be traction-free is then a ~ " ( v " t ' ) = f ~ ,r.~,k K,,,,,(.v-?,,y-~l)b,,,(~,,q)dA
(1)
where the kernel is given in detail in Ref. [15], b,,, represents the relative displacements between the crack faces at point (~, ~7) in direction m, and a~,, are the traction components of stress. Here direction 3 is taken normal to the plane of the crack. The inversion of equation (I), i.e. its solution for the crack face displacement, h .... is not straightforward, as the kernel is singular, and varies like I/r~, where r e = (.v - ~ )2 + (y _ r/)2 and special precautions are necessary. As the integral in equation (1) cannot be evaluated explicitly, or inverted analytically, it is necessary to use a discretized representation. This is done by dividing the crack face up into a number of small triangular regions, and assuming a particular form for the displacement variation within each element• In this instance, the displacement discontinuities b,,, are assumed to be of the form
)_i
t~,,,(.r, r) "
Fig. 2. Geometry of a semi-elliptical surface-breaking defect, existing adjacent to a Hertzian indentation test.
~Z,,=
1-
b 2
-~-
c:
(2)
within each element, where the weight function, W, is fixed, as shown, for an elliptical crack, and the constants ./i,, are to be determined. This approxi-
DAI el al.: THE PROPULSION OF SURFACE FLAWS
987
X
Fig. 3. Typical mesh used in cigcnstrain calculation, composed of 126 elements, blc = 2.
mation clearly imposes the correct behaviour of displacements near the crack front, and permits the integral equation (1) to be replaced by a set of algebraic equations, thus ,v
3
Y ~ <,,,,,J",,,=
-GT,, (x'y')
~ = l, 2 . . . .
x
(3)
where N is the number of elements, and
,;{
K;,,,,= ×
(l
,
s'i 4~r(l--v)7 2v)~$,,,,,+2v($~,oa3,,+3~, .,-
+K;;,,,
f.tds (4)
with 7~ = q5x-~ - ~, 7_, = _v~ - 7, 3'~ = 0 and 7: = ( xe - ~ 2) + (3,,~ _ r/2) and where ~5,,,,,= 1 when m = n and is zero otherwise. K<,,,,, represents the correction to the Kernel required duc to the presence of the free surface and is given in full in [15]. The evaluation of the above integrals is carried out using standard Gaussian quadraturcs, for the case i ¢ j . When i = j the integral is singular and its evaluation needs special treatment which has been detailed in ReE [15]. As the classical singularity of stress near the crack tip has already been built into the solution by dint of equation (2), it is easy to abstract the crack tip stress intensity factors from the relative crack face displacement in an adjacent clement. It should be noted that the solution found may be considered most accurate at a point along the crack front nearest to the centroid of the corresponding clement. A typical mesh is shown in Fig. 3, in this case having 126 elements. It is graded uniformly to provide a concentration of elements along the crack front, as these elements provide a strong influence on the stress intensity factors, and care has been taken to ensure that the density of elements ahmg the crack front is as uniform as possible. It should bc noted that it is necessary to check the solution to ensure that the crack faces do not contact and that the traction free boundary condition expressed in equation (1) is appropriate. This is particularly important in cases where the stress field is largely compressive. The analysis of partially closed cracks is discussed in [16].
The technique outlined is extremely efficient computationally. As we require to evaluate the stress intensity lectors for geometrically similar cracks, but located at different positions, it is convenient to determine the matrix of influence coefficients relating the traction at point ( x , y ) to the displacement at (~, ~1) for a particular crack once, and then to employ it any number of times. This is done by using the appropriate left hand side of equation (1), and inverting the matrix of influence coefficients, which form a representation of the integral itself. It is difficult to estimate the computational requirements as a multiuser computer facility was used, but in the problems studied, the setting up of the matrix of coefficients then took about l min on a VAX6000/620, whilst inversion of the matrix took a further minute. A typical subsequent solution to give the crack opening and stress intensity factors took much less than 1 rain per case. This is, of course, very much faster than the computational needs of a full three-dimensional finite element analysis of the problem, to give results of comparable accuracy. Further, each new set of results for a differently positioned crack would require a full re-evaluation of the problem by FEA, whereas here, only the last phase of the solution needs to be repeated. A convergence check was first performed; the results of which are depicted in Fig. 4. These sample results relate to cases when d/a = 1.1, c/a = 0.01, and h / c = 1.0, or 3.0. The solid lines represent the solution found for K i around the crack front when the mesh of 126 elements depicted in Fig. 3 is used, whilst the individual calculation points marked are from a simpler mesh when 88 elements were used. The extreme closeness of the two sets of results may be noted, particularly away from the surface (~ ~ 0 , 1 8 0 ) . The underlying reason for the discrepancy at the surface requires some explanation. The order of the stress singularity where a crack front meets a free surface is a function of Poisson's ratio and the angle of intersection and is not necessarily r ~2 [17, 18]. Hence any method which assumes an r 1/2 singularity for near-surface points does not reflect the true form of the stress state, and the value obtained for the apparent stress intensity factor is
DAI et al.: THE PROPULSION OF SURFACE FLAWS
988 o,la F
IC~/Po4-~c
0,12
b/c=3.0 I .O
00,111
0.09 O. 0,07
b/e=l
08
~(deg.)
0,06q
, 20
~
~
.
.
.
.
.
80 100 120 140 . 160 180
Fig. 4. Results of convergence tests. Mode I stress intensity factor,/£i, as a function of position around crack front, 0. Geometry; d/a = 1.1, c/a =0.01, q5 =0. Solid lines represent results obtained with the mesh shown in Fig. 3 (126 elements), whilst the individual points are results obtained from a simplified mesh of 88 elements. P0 is the peak Hertzian contact pressure. therefore very sensitive to position. In the current method an r ~!2 is assumed and, since the position at which the stress intensity factor is evaluated is different from mesh to mesh by virtue of the different element size, different results will be obtained. It is important to recognize that such difficulties are a feature of all analyses where a crack front breaks a free surface. The difference between the two sets of results does not indicate a convergence problem peculiar to the eigenstrain method. 3. RESULTS We will take as the starting point of our discussion the results found for a small defect (c/a =0.01), located at a radial position close to the point of maximum tension (d/a = 1.1), and where the surface length of the crack is permitted to vary, b/c taking the values 1, 1.4, 3, 33. The last figure corresponds to a two-dimensional solution, and merits comment: previous solutions to this problem have used an axi-symmetric representation of the contact stress field, but a twodimensional solution to the crack problem, for example by using an approximate Green's function approach [4], or by using the method of distributed ?This is quite apparent when the stress intensity factors for cracks in a uniform tension field, %, are considered. In the case of a plane crack KE= aox//~a, whereas in the case of a circular crack Kl = (2/Tz)aox/~.
dislocations [12]. These methods have the advantage of simplicity, but produce somewhat artificial results (as well as being unreliable in the former case), for the following reason: it is implicitly assumed that the crack already exists as a complete ring, and that the depth of the ring is small compared with its curvature, i.e. c/d<
DAI
et al.:
989
THE PROPULSION OF SURFACE FLAWS
(a)
Kl/Po',/-~c
0.15
-31~ Kn/Po'/-ffc
xlo
0.14
0.5[- b/c=CO b/c=OO
0.13
v(deg.) 80
100
1~
160
180
0.12
"
0.11
b/c=1.(
-0.5
'--b/c=3.0 ' '
-1 0.1
b/c=l.4
b/c=l.4 -1.5
0.09
b/c-I.O -2
0.08
O.07 0.06
~o ~
~
~
1~o 1~o 1~' 1~o
o]
,;0
(c) x 10-3
b/c=3.0
-2.5
~(deg.)
(d)
K[/Po'/~c
Kiil/po'f-~c
0.09 b/c= c~
O,08
I
20
40
60
I
I
80 ~ z ~
b/c=4.0~
O.07
~(deg.) i
i
140j1~180 0,06
\--__j
~°="'
)
O.05
b/e=3.0~
04!
O.
~(deg.) 0.03
i
' 20
' 40
' 60
8'0
1~ 0
' 120
' 140
' 160
180
d/a = c/a
Fig. 5. Results obtained for cracks positioned at 1.1, and with ~b = O, as a function o f position around the crack front, ~. In (a--c) the crack size is = 0.01: (a) mode I; (b) mode II; (c) mode III; and in (d) = 0.04, mode I loading. The simple (88 element) mesh is used.
c/a
below the toughness value. It is this instability which makes the formation of a ring crack so easy to detect by acoustic methods. We now turn our attention to the influence of crack orientation (~) on growth. In Fig. 6(a-d), we show results for a semi-circular defect = 1), located at
(b/c
d/a
c/a
a radial position = 1.1, and of depth = 0.01. We plot values of the three modes of crack loading, together with the strain energy release rate, G, for three positions around the crack front (A, B, C, in Fig. 2). Figure 6(a) shows the opening mode stress intensity. It is clear that this decreases monotonically
DAI et al.: THE PROPULSION OF SURFACE FLAWS
990
at all points a r o u n d the edge o f the flaw, as the angle q~ is increased, a n d t h a t by the time 45 ° is reached closure is imminent. A t the same time, the m o d e II loading c o m p o n e n t increases significantly at points A a n d B, i.e. near the surface, Fig. 6(b), whilst the value at the b o t t o m of the crack, point C, remains constant.
The Kil values at each end of the crack are of opposite signs, and are now very c o m p a r a b l e in m a g n i t u d e to the K~ stress intensity experienced by a purely circumferential crack. T u r n i n g to the antiplane shearing mode, K m, depicted in Fig. 6(c), we see t h a t the values here are generally smaller than the other (b)
KiJpo#~C
K~tPo'f~-c
O, 12
0.1
f
0,1
0.05
0.08 "
A
O. 01i
o
B
o J
0
,o
2'o
~
;o
angle (deg.)
0.04
-0, 05
0,02
I -,o, 1
angle (deg.) -0, 02 (d)
(c) x 10 -2 1
C
x 10-2~ 1,5
Kin/po4-~c
~______ I
I
I
I
I0
20
30
40
5'o
G E(1 -v2)Po2~C
~.25
-1
B A
-2 O,7~
-3 0.51
-4 0, 25
-5 0
-6
1'0
2'o
3o
angle (deg.) -0,25
q -8 L
-0..'
Fig. 6. Results of stress intensity factors for a semi-circular crack (h/c = 1), positioned at d/a = l.l, and of size c/a = 0.01. Results depicted are for the three positions A, B, C marked on Fig. 2, and are shown as a function of crack orientation, 4~. Values given are: (a) mode I; (b) mode II; (c) mode III, and (d) generalized crack extension force (strain energy release rate). The simple (88 element) mesh is used.
DAI et al.: THE PROPULSION OF SURFACE FLAWS modes. Indeed, at points A a n d B the values should be identically zero for all values of 4~, and the reason that finite values arise is that they are not evaluated at A and B but at the centroid of the elements closest to the surface, when t ) ~ 5,175. The eigenstrain m e t h o d is capable of evaluating stress intensity factors to within 1% for s t a n d a r d problems with a similar element density and a similar accuracy is anticipated for the current problem. It is interesting to speculate on the total energy available to drive the crack, and this is plotted in Fig. 6(d): the values have been f o u n d by using the plane strain form of the Irwin K G relationship, a n d it is assumed that if the crack laces are in contact at one or more points (which certainly occurs if /(1 < 0, but m a y occur remote from the crack front in other cases), the shear modes of loading persist unhindered, i.e. the crack faces are lubricated. It may be seen t h a t this calculation actually predicts a slighter greater tendency to p r o p a g a t e with increasing q~, which is not observed in practice. This may be a t t r i b u t a b l e to the general tendency of m o s t cracks to grow in m o d e I, whenever this is possible, a n d indicates that the stabilizing effect of the m o d e I loading ensures that a complete ring will form in preference to growth at other angles. Further, it is clear that a l t h o u g h for small cracks growth at non-zero values of 4 is energetically preferable, such growth must self arrest, as the crack is growing into a decreasing tension field at b o t h ends A and B. It should also be emphasized t h a t the faces of real cracks are not smooth, so that as crack closure occurs frictional faces between the crack faces (or even mechanical locking caused by any tessellations present), will inhibit any further growth in shear modes of loading. 4. CONCLUSIONS The calculation described has provided several insights into the initial propulsion of surface defects by contact loading. It has indicated the approxim a t i o n inherent in using a plane solution to solve for w h a t is, in reality, a three-dimensional problem, a n d permits a reliable indication of the size of flaw which may be expected to p r o p a g a t e at a given load. The tendency of the crack to " r u n a r o u n d " before propagating into the material has been highlighted, a n d the overarching effect of m o d e I loading on growth direction shown. The reason for unstable growth into a complete ring has been explained. In addition to these explications, a tool has been
991
developed which permits the i n d e n t a t i o n test to provide a reliable means of searching for flaws of a given size, provided that an assumption is m a d e a b o u t their ellipticity. Given that the majority of cracks seen in practice develop in a q~ = 0 sense, i.e. to form a ring, it would seem preferable to concentrate further attention on growth in this regime.
REFERENCES 1. F. C. Roesler, Brittle fractures near equilibrium. Proc. phys. Sot:. 69, 981 (1957). 2. K. L. Johnson, J. J. O'Connor and A. C. Woodward, The effect of indenter elasticity on the Hertzian fracture of brittle materials. Proc. roy. Sot:. A334, 95 (1973). 3. F. C. Frank and B. R. Lawn, On the theory of Hertzian fracture. Proe. roy. Soe. A299, 291 (1967). 4. R. Mouginot and D. Maugis, Fracture indentation beneath flat and spherical punches. J. Mater. Sei. 20, 4354 (1985). 5. K. Zeng, K. Breder and D. J. Rowcliffe, The Hertzian stress field and formation of cone cracks 1. Theoretical approach. Acta nwtall, mater. 40, 2595 (1992). 6. P. D. Warren, D. A. Hills and S. G. Roberts, Surface flaw distributions in brittle materials and Hertzian fracture. J. mater. Res. 9, (1994). 7. Li Yingzhi and D. A. Hills, The Hertzian cone crack. J. appl. Mech. 58, 120 (1991). 8. M. D. Bryant, G. R. Miller and L. M. Keer, A line contact between a rigid indenter and a damaged elastic body. Q. J. appl. Math. 37, 467 (1984). 9. M. T. Huber, Zur Theorie der Beruhrung fester elasticher Korper. Anna/. Physik 14, 153 (1904). 10. A. Sackfield and D. A. Hills, A note on the Hertz contact problem: a correlation of standard formulae. J. strain Anal. 18, 195 (1983). 11. D. A. Spence, The Hertz problem with finite friction. J. Elastic. 5, 297 (1975). 12. P. D. Warren and D. A. Hills, The influence of elastic mismatch between indenter and suhstrate on Hertzian Fracture. J. mater Set. 29, 2860 (I994). 13. H. F. Bueckner, The propagation of cracks and the energy of elastic deformation. J. appl. Mech. 80, 1225 (1958). 14. R. D. Mindlin, Force at a point in the interior of a semi-infinite sold. Physics 7, 195 (1936). 15. D. N. Dai, D. Nowell and D. A. Hills, Eigenstrain methods in three-dimensional crack problems: an alternative integration procedure. J. Mech. Phys. Solid~ 41, 1003 (I993). 16. D. N. Dai, D. Nowe]l and D. A. Hills, Partial closure and frictional slip of 3-D cracks. Int. J. Frac. 63, 89 (1993). 17. J. P. Benthem, State of stress at the vertex of a quarter-infinite crack in a half-space. Int. J. Solids Struct. 13, 479 (1977). 18. R. Barsoum and T.-K. Chcn, Three-dimensional surface singularity of an interface crack, h~t. J. Frae. 50, 221 (1991).
~
Pergamon
0956-7151(94)00313-0
Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0956-7151/95 $9.50 + 0.00
THE P R O P U L S I O N OF S U R F A C E FLAWS BY ELASTIC I N D E N T A T I O N TESTING D. N. DAI, I D. A. HILLS,it P. D. WARREN2 and D. NOWELL 1 'Department of Engineering Science and 2Department of Materials, Oxford University, Parks Road, Oxford OX1 3PJ, England (Received 14 January 1994; in revised form 20 June 1994)
Abstract--An analysis is carried out of the crack tip stress intensity factor existing around the front of a semi-elliptical surface flaw, when it is propelled by a Hertzian indentation test. The influence of crack size, position, ellipticity and orientation on stress intensity factors and strain energy release rate is investigated. It is shown that generally a surface flaw will "run around" to form a shallow ring crack before propagating inwards into the material.
1. INTRODUCTION Testing ceramic materials by elastic indentation experiments is becoming an important method for the quantification of the surface flaw distribution. This will become a standard recognized enabling technology for quality control in the structural use of ceramic components in engineering. It is important, for repeatability, that a well defined indenter shape, producing an incomplete contact, is used, in order to minimize the problems associated with edge effects when flat-ended indenters are employed, and by far the most usual shape used is a sphere. It is well-known that when a sphere is pressed into a brittle material, a ring crack forms, either at the edge of the contact disk itself, or just outside [1, 2], and that if the load is increased further the crack may flare out into the frustum of a cone [3]. Before quantitative use may be made of the test data it is important to understand as fully and reliably as possible the crack tip stress intensity occurring around the crack front, with the consequent implications this has for the development of the crack's size and shape. This is essential if the material fracture toughness is to be used as a means of deducing the corresponding crack size, as we implicitly assume that fracture toughness is a true material property, with crack arrest ensuing when the crack tip stress intensity factor at a point falls below the toughness. Unfortunately, the geometries with which we are concerned in this field are very taxing to analyse, as the crack exists in a very steep stress gradient, associated with the contact loading, and the crack itself may have a complex profile. The easiest of the three stages of crack development to attack is the intermediate case, when a ring has already ?To whom all correspondence should be addressed. 985
formed, and the crack tip is propagating into a decreasing contact stress field, Fig. l(b). Even this truly axi-symmetric geometry is not without its difficulties, and it is customary to assume that the depth of the crack, c, is small compared with its radial position, d, so that a plane (two-dimensional) analysis of the problem may be used [4, 5]. If the radius of the ring crack is known, and the load at which propagation took place, it is possible to infer both the final crack depth and, from many tests, statistical information about the initial flaw distribution [6], given the fracture toughness. Conversely, if data about the final crack size are known, the fracture toughness may be deduced. The third phase of development, i.e. the formation of the resulting Hertzian cone crack, Fig. l(c), has also been analysed [7], and this, too, may be used to infer the fracture toughness if the final cone size is known. Usually, however, this is possible only for transparent materials, so that the technique has limited application as a routine test procedure. By far the most important part of the test to be treated is the development of pre-existing surface defects into the ring, and it is with this phase that the current paper is concerned, Fig. 1(a). A semi-elliptical surface flaw has been analysed by the eigenstrain method, and it is our intention to calibrate for the crack tip stress intensity factor occurring around the whole of the crack front. This will enable us both to determine the load at which a defect of known size grows, and the direction in which it advances, i.e. whether it propagates downwards into the material to produce a deeper thumbnail type crack, or whether it "runs around" to form a shallow ring, before advancing inwards into the material. In many real applications only the load at which a ring crack forms is known, but from the calibration to be provided it is possible to infer, within limits, the size (or shape) of the pre-existing defect.
DAI et al.:
986 (a)
(b)
Pre-existing defect
THE PROPULSION OF SURFACE FLAWS
(c)
Ring crack
Cone crack
Fig. 1. Three stages in the development of an indenterpropelled crack: (a) pre-existing microscopic surface defect, which grows to form; (b) ring crack, which, in turn, flares out to form: (c) conical crack.
2. FORMULATION Figure 2 shows the geometry to be analysed• We assume that the surface defects are in the form of semi-elliptical faws, of surface length 2b and depth c, existing normal to the surface• The defects are assumed to be sufficiently widely spaced in comparison with their size for interaction between them to be ignored, and for them to be sufficiently small by comparison with the size of the contact for their interaction with the contact problem to be negligible [8]: in practice neither of thesc assumptions is at all restrictive• We may therefore focus attention on a single defect, which is assumed to be oriented at an angle 4~ to the 0-direction, Fig. 2, and to be positioned at a radial location r = d. The problem is solved in two parts; first, we determine the state of stress induced by the contact problem alone. This is widely known [9, 10], and, in order to reduce the number of independent variables in the present analysis, it is assumed that no radial tractions arise [11], although they may play an important r6le in propelling short cracks [12]. The solution is therefore most appropriate when the indenter has the same elastic properties as the substrate, or when the interface is lubricated. The nonzero stresses arising in a cylindrical coordinate set are clearly a__,, ~r, aoo, r,:. They are usually found normalized with respect to the peak Hertzian contact pressure P0 (see [9, 10]), and as functions of cylindrical coordinates (r, z), which are normalized with respect
/ y, '~,?qF
,,!C °
T z 2be" ..,
,
- B-
"
~J'
Defect
A /~t .,4, q~
•
-
~.~
Detect
to the radius of the contact disk, a. The stress components may be transformed into a local axis set centred on the crack, Fig. 2, by application of a routine transformation. It should be noted that, unless (/) = 0, all three modes of loading of the crack front may be anticipated in general, although clearly K m must be zero at the free surface• It is not appropriate here to describe the eigenstrain procedure in full, but we will take the opportunity to outline some features of the method which render it so suitable for problems of this kind. The technique is similar in concept to a boundary element formulation. Thc underlying result which is employed is Bueckner's principle [13]. First, the tractions arising on the faces of the crack in the crack's absence are found [these are denoted by a~,, in equation (1)]. The application of equal and opposite cancelling tractions to an otherwise unloaded body then induces the same set of stress intensity factors at each point around the crack front as would be developed by the external loading. In three-dimensional problems such as this the starting point for determining a nucleus ['or a cancelling traction is the solution by Mindlin [14] for a point force within a half-space. It should be noted that any number of nuclei of this kind may be inserted within the material without violating the requirement that the surface of the half-space remain traction-fi'ee. It is customary in the eigenstrain procedure to use as the kernel of the integral equation not thc function relating the influence of an interior point force to the stress at a point, but its differential with respect to the field point. The integral expressing the requirement that the crack faces be traction-free is then a ~ " ( v " t ' ) = f ~ ,r.~,k K,,,,,(.v-?,,y-~l)b,,,(~,,q)dA
(1)
where the kernel is given in detail in Ref. [15], b,,, represents the relative displacements between the crack faces at point (~, ~7) in direction m, and a~,, are the traction components of stress. Here direction 3 is taken normal to the plane of the crack. The inversion of equation (I), i.e. its solution for the crack face displacement, h .... is not straightforward, as the kernel is singular, and varies like I/r~, where r e = (.v - ~ )2 + (y _ r/)2 and special precautions are necessary. As the integral in equation (1) cannot be evaluated explicitly, or inverted analytically, it is necessary to use a discretized representation. This is done by dividing the crack face up into a number of small triangular regions, and assuming a particular form for the displacement variation within each element• In this instance, the displacement discontinuities b,,, are assumed to be of the form
)_i
t~,,,(.r, r) "
Fig. 2. Geometry of a semi-elliptical surface-breaking defect, existing adjacent to a Hertzian indentation test.
~Z,,=
1-
b 2
-~-
c:
(2)
within each element, where the weight function, W, is fixed, as shown, for an elliptical crack, and the constants ./i,, are to be determined. This approxi-
DAI el al.: THE PROPULSION OF SURFACE FLAWS
987
X
Fig. 3. Typical mesh used in cigcnstrain calculation, composed of 126 elements, blc = 2.
mation clearly imposes the correct behaviour of displacements near the crack front, and permits the integral equation (1) to be replaced by a set of algebraic equations, thus ,v
3
Y ~ <,,,,,J",,,=
-GT,, (x'y')
~ = l, 2 . . . .
x
(3)
where N is the number of elements, and
,;{
K;,,,,= ×
(l
,
s'i 4~r(l--v)7 2v)~$,,,,,+2v($~,oa3,,+3~, .,-
+K;;,,,
f.tds (4)
with 7~ = q5x-~ - ~, 7_, = _v~ - 7, 3'~ = 0 and 7: = ( xe - ~ 2) + (3,,~ _ r/2) and where ~5,,,,,= 1 when m = n and is zero otherwise. K<,,,,, represents the correction to the Kernel required duc to the presence of the free surface and is given in full in [15]. The evaluation of the above integrals is carried out using standard Gaussian quadraturcs, for the case i ¢ j . When i = j the integral is singular and its evaluation needs special treatment which has been detailed in ReE [15]. As the classical singularity of stress near the crack tip has already been built into the solution by dint of equation (2), it is easy to abstract the crack tip stress intensity factors from the relative crack face displacement in an adjacent clement. It should be noted that the solution found may be considered most accurate at a point along the crack front nearest to the centroid of the corresponding clement. A typical mesh is shown in Fig. 3, in this case having 126 elements. It is graded uniformly to provide a concentration of elements along the crack front, as these elements provide a strong influence on the stress intensity factors, and care has been taken to ensure that the density of elements ahmg the crack front is as uniform as possible. It should bc noted that it is necessary to check the solution to ensure that the crack faces do not contact and that the traction free boundary condition expressed in equation (1) is appropriate. This is particularly important in cases where the stress field is largely compressive. The analysis of partially closed cracks is discussed in [16].
The technique outlined is extremely efficient computationally. As we require to evaluate the stress intensity lectors for geometrically similar cracks, but located at different positions, it is convenient to determine the matrix of influence coefficients relating the traction at point ( x , y ) to the displacement at (~, ~1) for a particular crack once, and then to employ it any number of times. This is done by using the appropriate left hand side of equation (1), and inverting the matrix of influence coefficients, which form a representation of the integral itself. It is difficult to estimate the computational requirements as a multiuser computer facility was used, but in the problems studied, the setting up of the matrix of coefficients then took about l min on a VAX6000/620, whilst inversion of the matrix took a further minute. A typical subsequent solution to give the crack opening and stress intensity factors took much less than 1 rain per case. This is, of course, very much faster than the computational needs of a full three-dimensional finite element analysis of the problem, to give results of comparable accuracy. Further, each new set of results for a differently positioned crack would require a full re-evaluation of the problem by FEA, whereas here, only the last phase of the solution needs to be repeated. A convergence check was first performed; the results of which are depicted in Fig. 4. These sample results relate to cases when d/a = 1.1, c/a = 0.01, and h / c = 1.0, or 3.0. The solid lines represent the solution found for K i around the crack front when the mesh of 126 elements depicted in Fig. 3 is used, whilst the individual calculation points marked are from a simpler mesh when 88 elements were used. The extreme closeness of the two sets of results may be noted, particularly away from the surface (~ ~ 0 , 1 8 0 ) . The underlying reason for the discrepancy at the surface requires some explanation. The order of the stress singularity where a crack front meets a free surface is a function of Poisson's ratio and the angle of intersection and is not necessarily r ~2 [17, 18]. Hence any method which assumes an r 1/2 singularity for near-surface points does not reflect the true form of the stress state, and the value obtained for the apparent stress intensity factor is
DAI et al.: THE PROPULSION OF SURFACE FLAWS
988 o,la F
IC~/Po4-~c
0,12
b/c=3.0 I .O
00,111
0.09 O. 0,07
b/e=l
08
~(deg.)
0,06q
, 20
~
~
.
.
.
.
.
80 100 120 140 . 160 180
Fig. 4. Results of convergence tests. Mode I stress intensity factor,/£i, as a function of position around crack front, 0. Geometry; d/a = 1.1, c/a =0.01, q5 =0. Solid lines represent results obtained with the mesh shown in Fig. 3 (126 elements), whilst the individual points are results obtained from a simplified mesh of 88 elements. P0 is the peak Hertzian contact pressure. therefore very sensitive to position. In the current method an r ~!2 is assumed and, since the position at which the stress intensity factor is evaluated is different from mesh to mesh by virtue of the different element size, different results will be obtained. It is important to recognize that such difficulties are a feature of all analyses where a crack front breaks a free surface. The difference between the two sets of results does not indicate a convergence problem peculiar to the eigenstrain method. 3. RESULTS We will take as the starting point of our discussion the results found for a small defect (c/a =0.01), located at a radial position close to the point of maximum tension (d/a = 1.1), and where the surface length of the crack is permitted to vary, b/c taking the values 1, 1.4, 3, 33. The last figure corresponds to a two-dimensional solution, and merits comment: previous solutions to this problem have used an axi-symmetric representation of the contact stress field, but a twodimensional solution to the crack problem, for example by using an approximate Green's function approach [4], or by using the method of distributed ?This is quite apparent when the stress intensity factors for cracks in a uniform tension field, %, are considered. In the case of a plane crack KE= aox//~a, whereas in the case of a circular crack Kl = (2/Tz)aox/~.
dislocations [12]. These methods have the advantage of simplicity, but produce somewhat artificial results (as well as being unreliable in the former case), for the following reason: it is implicitly assumed that the crack already exists as a complete ring, and that the depth of the ring is small compared with its curvature, i.e. c/d<
DAI
et al.:
989
THE PROPULSION OF SURFACE FLAWS
(a)
Kl/Po',/-~c
0.15
-31~ Kn/Po'/-ffc
xlo
0.14
0.5[- b/c=CO b/c=OO
0.13
v(deg.) 80
100
1~
160
180
0.12
"
0.11
b/c=1.(
-0.5
'--b/c=3.0 ' '
-1 0.1
b/c=l.4
b/c=l.4 -1.5
0.09
b/c-I.O -2
0.08
O.07 0.06
~o ~
~
~
1~o 1~o 1~' 1~o
o]
,;0
(c) x 10-3
b/c=3.0
-2.5
~(deg.)
(d)
K[/Po'/~c
Kiil/po'f-~c
0.09 b/c= c~
O,08
I
20
40
60
I
I
80 ~ z ~
b/c=4.0~
O.07
~(deg.) i
i
140j1~180 0,06
\--__j
~°="'
)
O.05
b/e=3.0~
04!
O.
~(deg.) 0.03
i
' 20
' 40
' 60
8'0
1~ 0
' 120
' 140
' 160
180
d/a = c/a
Fig. 5. Results obtained for cracks positioned at 1.1, and with ~b = O, as a function o f position around the crack front, ~. In (a--c) the crack size is = 0.01: (a) mode I; (b) mode II; (c) mode III; and in (d) = 0.04, mode I loading. The simple (88 element) mesh is used.
c/a
below the toughness value. It is this instability which makes the formation of a ring crack so easy to detect by acoustic methods. We now turn our attention to the influence of crack orientation (~) on growth. In Fig. 6(a-d), we show results for a semi-circular defect = 1), located at
(b/c
d/a
c/a
a radial position = 1.1, and of depth = 0.01. We plot values of the three modes of crack loading, together with the strain energy release rate, G, for three positions around the crack front (A, B, C, in Fig. 2). Figure 6(a) shows the opening mode stress intensity. It is clear that this decreases monotonically
DAI et al.: THE PROPULSION OF SURFACE FLAWS
990
at all points a r o u n d the edge o f the flaw, as the angle q~ is increased, a n d t h a t by the time 45 ° is reached closure is imminent. A t the same time, the m o d e II loading c o m p o n e n t increases significantly at points A a n d B, i.e. near the surface, Fig. 6(b), whilst the value at the b o t t o m of the crack, point C, remains constant.
The Kil values at each end of the crack are of opposite signs, and are now very c o m p a r a b l e in m a g n i t u d e to the K~ stress intensity experienced by a purely circumferential crack. T u r n i n g to the antiplane shearing mode, K m, depicted in Fig. 6(c), we see t h a t the values here are generally smaller than the other (b)
KiJpo#~C
K~tPo'f~-c
O, 12
0.1
f
0,1
0.05
0.08 "
A
O. 01i
o
B
o J
0
,o
2'o
~
;o
angle (deg.)
0.04
-0, 05
0,02
I -,o, 1
angle (deg.) -0, 02 (d)
(c) x 10 -2 1
C
x 10-2~ 1,5
Kin/po4-~c
~______ I
I
I
I
I0
20
30
40
5'o
G E(1 -v2)Po2~C
~.25
-1
B A
-2 O,7~
-3 0.51
-4 0, 25
-5 0
-6
1'0
2'o
3o
angle (deg.) -0,25
q -8 L
-0..'
Fig. 6. Results of stress intensity factors for a semi-circular crack (h/c = 1), positioned at d/a = l.l, and of size c/a = 0.01. Results depicted are for the three positions A, B, C marked on Fig. 2, and are shown as a function of crack orientation, 4~. Values given are: (a) mode I; (b) mode II; (c) mode III, and (d) generalized crack extension force (strain energy release rate). The simple (88 element) mesh is used.
DAI et al.: THE PROPULSION OF SURFACE FLAWS modes. Indeed, at points A a n d B the values should be identically zero for all values of 4~, and the reason that finite values arise is that they are not evaluated at A and B but at the centroid of the elements closest to the surface, when t ) ~ 5,175. The eigenstrain m e t h o d is capable of evaluating stress intensity factors to within 1% for s t a n d a r d problems with a similar element density and a similar accuracy is anticipated for the current problem. It is interesting to speculate on the total energy available to drive the crack, and this is plotted in Fig. 6(d): the values have been f o u n d by using the plane strain form of the Irwin K G relationship, a n d it is assumed that if the crack laces are in contact at one or more points (which certainly occurs if /(1 < 0, but m a y occur remote from the crack front in other cases), the shear modes of loading persist unhindered, i.e. the crack faces are lubricated. It may be seen t h a t this calculation actually predicts a slighter greater tendency to p r o p a g a t e with increasing q~, which is not observed in practice. This may be a t t r i b u t a b l e to the general tendency of m o s t cracks to grow in m o d e I, whenever this is possible, a n d indicates that the stabilizing effect of the m o d e I loading ensures that a complete ring will form in preference to growth at other angles. Further, it is clear that a l t h o u g h for small cracks growth at non-zero values of 4 is energetically preferable, such growth must self arrest, as the crack is growing into a decreasing tension field at b o t h ends A and B. It should also be emphasized t h a t the faces of real cracks are not smooth, so that as crack closure occurs frictional faces between the crack faces (or even mechanical locking caused by any tessellations present), will inhibit any further growth in shear modes of loading. 4. CONCLUSIONS The calculation described has provided several insights into the initial propulsion of surface defects by contact loading. It has indicated the approxim a t i o n inherent in using a plane solution to solve for w h a t is, in reality, a three-dimensional problem, a n d permits a reliable indication of the size of flaw which may be expected to p r o p a g a t e at a given load. The tendency of the crack to " r u n a r o u n d " before propagating into the material has been highlighted, a n d the overarching effect of m o d e I loading on growth direction shown. The reason for unstable growth into a complete ring has been explained. In addition to these explications, a tool has been
991
developed which permits the i n d e n t a t i o n test to provide a reliable means of searching for flaws of a given size, provided that an assumption is m a d e a b o u t their ellipticity. Given that the majority of cracks seen in practice develop in a q~ = 0 sense, i.e. to form a ring, it would seem preferable to concentrate further attention on growth in this regime.
REFERENCES 1. F. C. Roesler, Brittle fractures near equilibrium. Proc. phys. Sot:. 69, 981 (1957). 2. K. L. Johnson, J. J. O'Connor and A. C. Woodward, The effect of indenter elasticity on the Hertzian fracture of brittle materials. Proc. roy. Sot:. A334, 95 (1973). 3. F. C. Frank and B. R. Lawn, On the theory of Hertzian fracture. Proe. roy. Soe. A299, 291 (1967). 4. R. Mouginot and D. Maugis, Fracture indentation beneath flat and spherical punches. J. Mater. Sei. 20, 4354 (1985). 5. K. Zeng, K. Breder and D. J. Rowcliffe, The Hertzian stress field and formation of cone cracks 1. Theoretical approach. Acta nwtall, mater. 40, 2595 (1992). 6. P. D. Warren, D. A. Hills and S. G. Roberts, Surface flaw distributions in brittle materials and Hertzian fracture. J. mater. Res. 9, (1994). 7. Li Yingzhi and D. A. Hills, The Hertzian cone crack. J. appl. Mech. 58, 120 (1991). 8. M. D. Bryant, G. R. Miller and L. M. Keer, A line contact between a rigid indenter and a damaged elastic body. Q. J. appl. Math. 37, 467 (1984). 9. M. T. Huber, Zur Theorie der Beruhrung fester elasticher Korper. Anna/. Physik 14, 153 (1904). 10. A. Sackfield and D. A. Hills, A note on the Hertz contact problem: a correlation of standard formulae. J. strain Anal. 18, 195 (1983). 11. D. A. Spence, The Hertz problem with finite friction. J. Elastic. 5, 297 (1975). 12. P. D. Warren and D. A. Hills, The influence of elastic mismatch between indenter and suhstrate on Hertzian Fracture. J. mater Set. 29, 2860 (I994). 13. H. F. Bueckner, The propagation of cracks and the energy of elastic deformation. J. appl. Mech. 80, 1225 (1958). 14. R. D. Mindlin, Force at a point in the interior of a semi-infinite sold. Physics 7, 195 (1936). 15. D. N. Dai, D. Nowell and D. A. Hills, Eigenstrain methods in three-dimensional crack problems: an alternative integration procedure. J. Mech. Phys. Solid~ 41, 1003 (I993). 16. D. N. Dai, D. Nowe]l and D. A. Hills, Partial closure and frictional slip of 3-D cracks. Int. J. Frac. 63, 89 (1993). 17. J. P. Benthem, State of stress at the vertex of a quarter-infinite crack in a half-space. Int. J. Solids Struct. 13, 479 (1977). 18. R. Barsoum and T.-K. Chcn, Three-dimensional surface singularity of an interface crack, h~t. J. Frae. 50, 221 (1991).