- Email: [email protected]
Slow crack growth in glass in combined mode I and mode II loading
Scripta METALLURGICA et MATERIALIA
Vol. 25, pp. 997-1002, 1991 Printed in the U.S.A.
Pergamon Press plc All rights reserved
VIEWPOINT SET No. 16
SLOW CRACK GROWTH IN GLASS IN COMBINED MODE I AND MODE 11 LOADING D. K. Sherry Department of Materials Science and Engineering University of Utah Salt Lake City, Utah 84112. and A. R. Rosenfield Battelle Memorial Institute Columbus, Ohio 43201 (Received December 15, 1990)
Fracture mechanics studies of slow crack growth have been conducted on glass (1-3) and several structural coramics (4-7). Researchers typically measure crack growth rate, V, as a function of a crack-driving force parameter, most commonly, mode I su~ss-imensity, KI, using a precracked test specimen, for example, a double-cantilever beam specimen. Although the relation between erack-growth rate and mode I stress intensity can be quite complex depending upon the material and the environment, crack growth rate over a signlfi~nt range of KI is often described by one of the following two empirical equations : V = V I exp (bKI )
(1)
v = A~
(2)
Equation 2 has been used more commonly in engineering applications of sulx~ritlcal crack growth for life time predictions (8). However, Equation 1 is more suitable for fundamental interpretation of slow crack growth in terms ofreaction rate theox~ (9). The use of mode I stress intensity as the crack driving force parameter in Equations 1 and 2 implies that cracks are subjected to far-field loading that produces crack surface displacements normal to the crack plane. Although mode I loading is used for convenience in laboratory characterization of slow crack growth and fracture toughness, it is not the most common type of l o ~ n g encountered for preexisting cracks in engineering structures. An arbitrarily oriented crack m an engineering structure is subjected to both normal and shear displacements. Such a crack will, in general, follow a curved or kinked path during slow growth that will eventually realign the crack to mode I loading. However, during a significant period of its slow growth the crack is subjected to combined mode loading. It is clear mat to make accm~te life time predictions one must establish a general relation between slow crack growth rate and an effective crack driving force that is a function of mode I, II and lll stress intensifies. •
,
.
.
o
~
o
•
•
While it ,s difftcult to demgn a fracture mechanics test specnnen m which a crack can be subjected to controlled mode I, mode H and mode lit loading, several s p e c i ~ n geometries are now available (10-14). However, there is a surprising paucity of slow crack growth data for c~mmics under combined mode loading. Tossell and Ashbee (15) have recently reported slow crack growth data for soda-lime glass under combined mode I and mode rll loading in a ~ . ".g humidities. The effect of a superimposed mode III loading on crack growth rate was minimal for high umidity environments. However, in dry air (< 0.01% relative humidity) superposed mode HI loading altered the log V vs KI plot. It was suggested that diffusion of moisture to the crack tip was more difficult in the presence of mode I n loading.
~
A summn_ryof the results of a study of slow crack growth in soda-lime glass under combined mode I and mode II loading are p~x~ed in this paper. Details of the experimental procedures and complete results and analyses will be published in a forthcoming paper (16).. Pure mode I, pure mode II and various combinations of mode I and mode II were achieved tu preoracked disk spectmens by loading in diametral compression at selected angles with respect to symmeuric radial cracks.
997 0056-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press plc
998
SLOW CRACK GROWTH
Vol.
25,
No. 5
Test Material And Procedures : Soda-lime glass was used in the present study because it is susceptible to pronounced suberitical crack growth in various enviomments including humid air and water. Crack-growth behavior in mode I loading, i.e, log V vs K! data, have been reported by several investigators. Inorganic glasses are model martials for fracu~ studies because of their isou~ic elastic properties. Precracking of fracture mechanics specimens is relatively easy in
glasses, particularly soda-lime glass.
Diameteral Comuression Svecimen : Awaji and Sato (17) and Yarema eL al.(18) were the first to use the diameteral compression specimen wil~ inclined center notches to assess combined mode I and mode II fracture toughness of graphite and WC-Co cermet, respectively. The dlametral compression specimen is attractive because of its simple loading and specimen geometry. Shetty et.al.(19) improved this specimen by using a chevron notch to precrack the _disk specimen prior to its use in fracture toughness measurements. The preeracked digk specimens have since been used to measure both mode I fracture toughness as well as combined mode I and mode II fracture toughness envelopes (12,14,20). In the present study, chevron-notched diameu'al-compression specimens of soda-lime glass were used to measure slow crack growth and to establish mixed-mode fracture toughness envelope in combined mode I and mode II loading. Figure 1 shows the geometry of the diamewal compression specimen. The disk specin~ns were precracked by loading the disks in pure mode I, i.e., by loading in compresston along a diameter through the chevron notch. The precracking and the stable crack extension within the chevron notch could be observed and controlled by controlling the applied load. The precracking was also promoted by placing a drop of deionized water at the notch tip. The initial precracks extended slightly out of the chevron notch. Pure mode I, pure mode H and combined mode I-mode H loadings were obtained by simply orienting the precrack at various angles relative to the loading diameter. Mode I and mode H slress intensity factors for symmetric r~ial cracks in disks subjected to _diametral compression were obtained from Atldnson et aL(21). However, a crack subjected to mode II or a combined mode I and mode .l! loading always extends at.an angle to its .original plane. This noncoplanar crack extension also occurs during subcritical growth of initially straight cracks subjected to mode H or combined mode I and mode lI loading: Analytical solutions for mode I and mode II stress intensities for a kinked crack in a diamelral compression specimen are not available. Therefore, the method of caustics was employed to experimentally evaluate stress intensifies as functions of the kinked crack length (16). The disk specimens were loaded in a universal testing machine with a spring in series. The compliance of the spring was much greater than that of the specimen and, as a result, the load was maintained constant at a fixed cross-hesd displacement during slow crack growth in the specimen. The disk specimen was placed in a chamber containing deionized water which ~ slow crack growth in glass. Slow crack growth in the disk specimens was measured by measuring the changes in the resistance of a grid of parallel gold lines depo..sitedon the s~. ~men s.urfaee by a photolithography process.. Crac.k growth rates.were calculated from the posmons of the individual grid lines and the record of step changes m r e , stance as a tuncnon o~ time (16). Figme 2 shows a disk specimen with two micmckcuit grids mounted in the vicinity of the crack tips.
Fracture Tonehness in Combined Mode I and Mode II Loadine : ~ toughness of soda-lime glass assessed in combined n ~ l e I and mode rl loading of precracked disk specimens are summarized in Figure 3. The mode I mode II fracture toughness envelope is similar to that reported previously by Shetty et al.[12], but the absolute values of the fracture toughness are slightly lower for all loading conditions. The solid line plotted on the figure represents the 'best tic of the following empirical equation proposed independently by Palaniswamy and Knanss (22) and Richard(23) :
_ [ x%+
= 1
(3)
In Equation 3, C is an empirical constant which is the ratio of the criticalvalue of KII at fracture in pure mode II loading and mode I fracture toughness, KIC. It is reasonable to assume that the effective crack-driving force for suberiticalcrack growth should be identical in functional form to that established in mixed-moda fracturetoughness tests. Therefore, in this study, Equation 3 with C = 1.08 was taken as the basis for defining an effective stress intensity,Kerr, which is obtained by solving for KIC in Equation 3 :
Vol.
25, No.
5
SLOW CRACK GROWTH
K~f=
2
999
(4)
In combined mode I - mode II loading, fast fracture occurs when Keff > KIC. Slow Crack Growth in Mode I. Mode II and Combined Mode I and Mode II Loadin2 : Figure 4 shows a summary plot of the crack growth rates in pure mode I, pure mode II and combined mode I and mode 11 versus K2eff where Keff was defined by Equation 4. The crack velocities plotted in Figure 4 are the initial velocities corresponding to the incipient formation of the kinks at a defined initial combination of KI and KII. Crack growth rates under different loading conditions cue corw.la~ with a single effective crack-driving force parameter, K2eff. The linear plot of log V vs K2eff suggested the following general crack growth law : V = V 0 exp (B K 2 )
(5)
The optimum values of the parameters determined by least squares linear regression analysis were V0 = 9.6 x 10-9 m/s and B = 29.4 (MPa.~m) -2. Slow Growth of Kinked Cracks : Cracks subjected to mode rl or combined mode I and mode H loading abruptly deviate from their initial plane and form a kink. During continued growth kinks exhibited an initial decrease, a minimum and a subse~aent increase in the crack growth rates. Figure 5 is a representative example of crack velocity variations for an initial straight crack subjected to pure mode II which extended as a kink with vavfing KI and KII. Stress-intensity evaluations by the method of caustics indicated that KH decreased rapidly while KI increased with increasing kink length and the crack assumed a configuration of pure mode I. The solid line in Figure 5 represents the prediction of Equation 5 using K! and KII assessed from the method of caustics. The velocity variations of the kinked crack are consistent with the general slow crack growth equation. Discussion Experimental results and analyses presented in this paper have shown that slow crack growth in soda-lime glass under combined mode I and mode H loading can be adequately described by a relatively simple exponential relation with a single effective crack-driving force parameter, K2eff. Although the definition of Keff is based on an empirical equation, Equation 4, it is, in fact, an empirical representation of a cri$ical noncoplauar energy release rate criterion. It was proposed by Palaniswamy and Knanss (22) with a value C = ~/2/3 ~ 0.82 to fit their theoretical results of the critical condition for extension of a crack under combined mode I and mode II loading based on ma~dmiTAtlonof the energy release rate for noncoplanar crack extension. They also showed that critical stress for extension and directions of extension of inclined cracks in tension panels of toluene swollen polyurethane, a brittle grade of polymer, were in agreement with their theoretical calculations. Richard (23) and Shetty et aL(12) also nsed Equation 4 to analyze results on inorganic glasses. Results from tension tests were well represented by Equation 4 with C = 0.82; however, diametral compression tests gave a slightly higher C value similar to the results in this study (Figure 3). Equation 4 has since been found to represent combined mode I and mode II fracture toughness data of several polycrystalline ceramics (14) with C values in the range I to 2, the higher values reflecfinl~ increased frictional shear resistance from crack surface asperities (20). Thus, despite its empirical basis, Equallon 4 has proven to be an effective representalion of the critical condition for fracture in combined mode I and mode II loading for a variety of brittle w~__tcrials. The applicability of Equation 5 to describe subcritical crack growth under combined mode I and mode II loading suggests that the effective crack-driving force for both equilibrium and subcritical crack growth in soda-lime glass is the strain energy release rate. Further, Equation 5 is consistent with theoretical models that treat slow crack growth as a thermally-activated rate process. Lawn (24) and Tyson et ai.(25) have treated slow crack growth as a thcrmaliyactivated bond rupture process by explicitly considering periodic variation of the surface energy in a discrete lattice as originally suggested by Thompson (26). For relatively high values of the crack-driving force, Tyson et al.(25) suggested the following rate equation : AGo
wb.ere v0 is a limiting crack velocity dictated by the frequency of bond stretching, AGo is a zero sur,ss activation free
1000
SLOW
CRACK
GROWTH
Vol.
25, No.
5
energy for slow crack growth, K is the stress-intensity factor and Kc is the critical stress-intensity factor for .spon..~.~. s crac.k .p.~opagation. It should be noted that Eq.uation 6 is.a simplifie~i, special case of a.general .treatment m which the acuvauon free energy for slow crack growth is a nonlinear funcuon of the mechanical strata energy release rate (24). Tyson et a1.(25) employed Equation 6 to analyze the mode I slow crack growth data for glasses in vacuum reported by Wiederhorn et al.(27). The same formulation should be applicable for environment-assisted slow crack growth in glass in combined mode I and mode H loading if AGo is identified to be characteristic of the specific environm~t, ie. water, and K 2 is replaced by K2eff to indicate mechanical energy release rate for combined mode I and mode II loading. A comparison of Equation 6 with Equation 5 then suggests the following equalities betwe~ the expe6.mental and thcoreticsl parameters : [
AGo]
(7)
v0 =
AGo
B =
,
(8)
RTI~ Equation 8 can be used to estimate the zero - slress activation free energy, AGo, from experimental values of B and Ko For the measured values of B = 29.4 (MPaNm) -2 and Kc ~ 0.75 MPa.'~m at room temperatm'e (T - 300 K), Equation 8 gives AGo - 41 U/mole. It follows from Equation 7 and the measured value of V0 ffi9.6 x l0 "9 m/s that v0 ffi 0 . 1 5 m / s . It is useful to compare the above es~mates of the slow crack growth parameters with similar values ~
~
literature. Wiede~horn and Bolz (2) have reported a value, A G o = 109.kJ/..molefor slow crack growm m sc~..- .
glass in water. 'Foe ~ between their value and the value obtained m thi.'s study appears.to. ~ primarily..aue to the different methods cf analyses of the data and the different ranges of stress mtenm~.cov.efm m me tw.o .st~m.~. Crack growth rate measurements of Wiederbom and Bolz (2) extended to tow values oI ~ where me crac[ veiocuy is highly nonlinear with increasing slope on log V vs K plot. A reanalysls of their room tempamm~ _,~q:~"rathe ,sm,me K range as in the present study gave AGo ~ 58 kJ/mole. One can also estimate AGo from Wiederhcrn arm uolz oAt~ by first extrapolating log V vs K plots from the.high K .linear re "gime.to obtain V0 at diff.erent. temperatures. :hThe temperature dependence of V0 was analyzed vta Equauon 7 to obtain me zero stress actlvauon energy. ~uc a
gave
44 kJ/mo , thus,fu.h .d
thed crep y "m
vati
ts interesting to note that a similar discrepancy also crests m the reported a.c~..,vanon ~ energy forstow crac ~ . in vacuum. Wiederhorn et al.(27) reported AGo ~ 605 kJ/mole for soda-lime glass m vacuum. ~owever, anatyms of their data by Tyson et al.(25) via Equation 6 resulted in AGo ~ 349 k.J/mole. Pukh et al.(28) have also reported AGo ~ 338 kJhnole for slow fracture of soda-lime glass in vacuum.
1. Slow crack growth in pure mode I, pure mode H and combined mode I and mode H loading of soda-lime glass in water can be described by a simple exponential relationship with elastic strain energy release rate as the effective crack-driving force parameter. 2. The empirical formulation employed to describe the noncoplanar elastic strain energy release rate also correcdy accounts for the combined mode I and mode H fracture toughness of soda-lime glass under inert conditions. 3. Velocity variations encountered with the growth of kinked cracks can be rationalized, in ~ of. ~ e . g e n ~ slow crack growth equation provided the changes in the mode I and mode tt stress mtensmes wtm me cangmg kink geometry can be properly evaluated and accounted for in defining the elastic energy release rate during kink growth. 4. The slow crack growth equation can be interpreted in terms of theoretical models that treat subcriti.cal crack growth as a thermally-activated bond-rupture process with an activation energy dependent on me envtronment and the elastic energy release rate as the crack-driving force parameter.
Admm/td~mmn ]'be paper is based on research supported by NASA Lewis Research Cent~er under grant NAG3,789 at University of Utah, The authors are grateful to Dr. John Gyekenyesi for his support and encouragement ot tins research.
1.
S.M. Wiederhorn, Intl. J. FracL, 4, 171 (1968).
Vol.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20, 21. 22. 23. 24. 25. 26. 27. 28.
25, No.
5
SLOW CRACK GROWTH
i001
S.M. Wiederhom and L. H. Bolz, J. Am. Ceram. Sot., 53, 543 (1970). S.M. Wiederhorn, p. 613 in Fracture M~chanics of Ceramics. Vol.2. Edited by R. C. Bradt, D. P. H. Hasselman and F. F. Lange. Plenum Press, New York, 1974. A. G, Evans, J. Mat. Sci., 7, 1137 (1972). A.G. Evans and F. F. Lange, J. Mat. Sci., 10, 1659 (1975). A.G. Evans and S. M. Wiederhorn, J. Mat. Sci., 9, 270 (1974). P.F. Becher, J. Mater. Sci., 21, 297 (1984). J.E. Ritter, Jr. and J. A. Meisel, J. Am. C.cram. Soc., 59, 478 (1976). S.M. Wiederhom, E. R. Fuller and R. Thomson, Metal Science, 14, 450 (1980). J. J. Petrovic, J. Am. Ceram. Soc., 68, 348 (1985). D. B. Marshall, J. Am. C.cram. Soc., 67, 110 (1984). D. K. Shetty, A. R. Roscnfield and W. H. Duckworth, Engineering Fracture Mechanics, 26, 825 (1987). S. Suresh and E. K. Tschegg, J. Am. C.eram. Sot., 70, 726 (1987). D. Singh and D. K. Shetty, J. Am. C.cram. Soc., 72, 78 (1989). D. A. Tossell andK. H. G. Ashbee, J. Am. Ceram. Soc.,71, C - 138(1988). D. Singh and D. IC Shetty, J. Am. Ceram. Soc. (In Press) (1990) H. Awaji and S. Sato, J. Eng. Mater. Technol., 100, 175 (1978). S. Ya. Yamma, G. S. Ivanitskaya, A. L. Mais~enko and A. I. Zbromirskii, Probl. Prochn., 16, 112 (1984). D. K. Shetty, A. IL Rosenfield and W. H. Duckworth, J. Am. Ceram. Soc., 68, C-325 (1985) A. IL Rosenfield, p.501 in Fracture Behavior and Desit, n of Materials and Structures. Edited by D. F'trrao. EMAS, Warley, UK (1990) C. Atkinson, R. E. Smelser and J. Sanchez, Int. J. Fract., 18, 279 (1982). K. Palaniswamy and W. G. Knanss, Mechanics Today, 4, 87 (1978) H. A. Richard, VDI - Forschnngsh., 631, 1 (1985) B. R. Lawn, J. of Mater. Sci., 10, 469 (1975) W. R. Tyson, H. M. Cckirge and A. S. Krausz, J. of Mater. Sci., 11, 780 (1976) R. M. Thompson, Annual Rev. of Mat. Science, 3, 31 (1973) S. M. Wiederhorn, H. Johnson, A. M. Diness and A. H. Heuer, J. of Am. Ceram. Soc., 57, 336 (1974) V. P. Pukh, S. A. Laterner and V. N. Ingal, Soy. Phys. Solid State, 12, 881 (1970)
A
'
B-'I
_
A
I'-
_
Section AA
FIG. 1. Geometry of the Chevron-Notched Disk Specimens Used in Slow Crack Growth Tests.
1002
SLOW CRACK GROWTH
Vol.
25, No. 5
A OT
e=
,
,
,
,
,
~
=<:~1="~ 0.6 ~ -.
0.5
u z )
0.4
~ o°
o
~
• o
_~z O3 ,.,,"" c=:: 02
*
•
" " °/e-o 2
o
..o/e.oso K
o°
i(
,
SODA-LINEGLASS DIANETRAL-CONPRESSI-Ol
\
S
2
\
o, o,
I 0.1
0.0
c.,.
I 0.2
I
0 .~
,
,
0,4
0.5
°
I
\
0~0~
0.6
0.7
.. .
I
0.8
0.9
MODETT STRESS INTENSITY, K]z(MPomi/2)
FIG. 2. A Center-Cracked Disk Specimen of SodaLime Glass With Two Microcixcuit Grids Deposited on the Surface.
io-3
A v
i
l
SODA-LIMEGLASS / DIAMETRAL"COMPRESSION 25C,O.I. WATER o PUREMODEI io-4 • , , PURE MODE1Z n : KZI+K.IZ 1,0.45MOeml/2 • , RZI+K.EI.O.55MPemI/2
/
i
i
0 oo 0 o
I0"5 Do o
.-I k*J
,
I
,/
io~ I--
FIG. 3. Fracture Toughness of Soda-Lime Glass in Combined Mode I and Mode 11Loading.
,
,
o ' K~I.O.40~IPo./i , PREDICTION
E i0 -lIt o °
N
o
10"6 o
I0 "I
I0' 0.0
io o.o
I
o.i
0'.2
0'3
0.05
Ot,I
KINK LENGTH, ~/e
0.15
i
o4
STRESS INTENSITY, K~ff (MPomt/2)2 HG. 4. Subcridcal Crack Growth Rates in Soda-Lime Glass in Pure Mode I, Pure Mode II and Combined Mode I and Mode n Loading.
HG. 5. Comparison of Measured and Predicted Crack Growth Rates For Kinked C~tcks in Soda-Lime Glass.
0.2
Vol. 25, pp. 997-1002, 1991 Printed in the U.S.A.
Pergamon Press plc All rights reserved
VIEWPOINT SET No. 16
SLOW CRACK GROWTH IN GLASS IN COMBINED MODE I AND MODE 11 LOADING D. K. Sherry Department of Materials Science and Engineering University of Utah Salt Lake City, Utah 84112. and A. R. Rosenfield Battelle Memorial Institute Columbus, Ohio 43201 (Received December 15, 1990)
Fracture mechanics studies of slow crack growth have been conducted on glass (1-3) and several structural coramics (4-7). Researchers typically measure crack growth rate, V, as a function of a crack-driving force parameter, most commonly, mode I su~ss-imensity, KI, using a precracked test specimen, for example, a double-cantilever beam specimen. Although the relation between erack-growth rate and mode I stress intensity can be quite complex depending upon the material and the environment, crack growth rate over a signlfi~nt range of KI is often described by one of the following two empirical equations : V = V I exp (bKI )
(1)
v = A~
(2)
Equation 2 has been used more commonly in engineering applications of sulx~ritlcal crack growth for life time predictions (8). However, Equation 1 is more suitable for fundamental interpretation of slow crack growth in terms ofreaction rate theox~ (9). The use of mode I stress intensity as the crack driving force parameter in Equations 1 and 2 implies that cracks are subjected to far-field loading that produces crack surface displacements normal to the crack plane. Although mode I loading is used for convenience in laboratory characterization of slow crack growth and fracture toughness, it is not the most common type of l o ~ n g encountered for preexisting cracks in engineering structures. An arbitrarily oriented crack m an engineering structure is subjected to both normal and shear displacements. Such a crack will, in general, follow a curved or kinked path during slow growth that will eventually realign the crack to mode I loading. However, during a significant period of its slow growth the crack is subjected to combined mode loading. It is clear mat to make accm~te life time predictions one must establish a general relation between slow crack growth rate and an effective crack driving force that is a function of mode I, II and lll stress intensifies. •
,
.
.
o
~
o
•
•
While it ,s difftcult to demgn a fracture mechanics test specnnen m which a crack can be subjected to controlled mode I, mode H and mode lit loading, several s p e c i ~ n geometries are now available (10-14). However, there is a surprising paucity of slow crack growth data for c~mmics under combined mode loading. Tossell and Ashbee (15) have recently reported slow crack growth data for soda-lime glass under combined mode I and mode rll loading in a ~ . ".g humidities. The effect of a superimposed mode III loading on crack growth rate was minimal for high umidity environments. However, in dry air (< 0.01% relative humidity) superposed mode HI loading altered the log V vs KI plot. It was suggested that diffusion of moisture to the crack tip was more difficult in the presence of mode I n loading.
~
A summn_ryof the results of a study of slow crack growth in soda-lime glass under combined mode I and mode II loading are p~x~ed in this paper. Details of the experimental procedures and complete results and analyses will be published in a forthcoming paper (16).. Pure mode I, pure mode II and various combinations of mode I and mode II were achieved tu preoracked disk spectmens by loading in diametral compression at selected angles with respect to symmeuric radial cracks.
997 0056-9748/91 $3.00 + .00 Copyright (c) 1991 Pergamon Press plc
998
SLOW CRACK GROWTH
Vol.
25,
No. 5
Test Material And Procedures : Soda-lime glass was used in the present study because it is susceptible to pronounced suberitical crack growth in various enviomments including humid air and water. Crack-growth behavior in mode I loading, i.e, log V vs K! data, have been reported by several investigators. Inorganic glasses are model martials for fracu~ studies because of their isou~ic elastic properties. Precracking of fracture mechanics specimens is relatively easy in
glasses, particularly soda-lime glass.
Diameteral Comuression Svecimen : Awaji and Sato (17) and Yarema eL al.(18) were the first to use the diameteral compression specimen wil~ inclined center notches to assess combined mode I and mode II fracture toughness of graphite and WC-Co cermet, respectively. The dlametral compression specimen is attractive because of its simple loading and specimen geometry. Shetty et.al.(19) improved this specimen by using a chevron notch to precrack the _disk specimen prior to its use in fracture toughness measurements. The preeracked digk specimens have since been used to measure both mode I fracture toughness as well as combined mode I and mode II fracture toughness envelopes (12,14,20). In the present study, chevron-notched diameu'al-compression specimens of soda-lime glass were used to measure slow crack growth and to establish mixed-mode fracture toughness envelope in combined mode I and mode II loading. Figure 1 shows the geometry of the diamewal compression specimen. The disk specin~ns were precracked by loading the disks in pure mode I, i.e., by loading in compresston along a diameter through the chevron notch. The precracking and the stable crack extension within the chevron notch could be observed and controlled by controlling the applied load. The precracking was also promoted by placing a drop of deionized water at the notch tip. The initial precracks extended slightly out of the chevron notch. Pure mode I, pure mode H and combined mode I-mode H loadings were obtained by simply orienting the precrack at various angles relative to the loading diameter. Mode I and mode H slress intensity factors for symmetric r~ial cracks in disks subjected to _diametral compression were obtained from Atldnson et aL(21). However, a crack subjected to mode II or a combined mode I and mode .l! loading always extends at.an angle to its .original plane. This noncoplanar crack extension also occurs during subcritical growth of initially straight cracks subjected to mode H or combined mode I and mode lI loading: Analytical solutions for mode I and mode II stress intensities for a kinked crack in a diamelral compression specimen are not available. Therefore, the method of caustics was employed to experimentally evaluate stress intensifies as functions of the kinked crack length (16). The disk specimens were loaded in a universal testing machine with a spring in series. The compliance of the spring was much greater than that of the specimen and, as a result, the load was maintained constant at a fixed cross-hesd displacement during slow crack growth in the specimen. The disk specimen was placed in a chamber containing deionized water which ~ slow crack growth in glass. Slow crack growth in the disk specimens was measured by measuring the changes in the resistance of a grid of parallel gold lines depo..sitedon the s~. ~men s.urfaee by a photolithography process.. Crac.k growth rates.were calculated from the posmons of the individual grid lines and the record of step changes m r e , stance as a tuncnon o~ time (16). Figme 2 shows a disk specimen with two micmckcuit grids mounted in the vicinity of the crack tips.
Fracture Tonehness in Combined Mode I and Mode II Loadine : ~ toughness of soda-lime glass assessed in combined n ~ l e I and mode rl loading of precracked disk specimens are summarized in Figure 3. The mode I mode II fracture toughness envelope is similar to that reported previously by Shetty et al.[12], but the absolute values of the fracture toughness are slightly lower for all loading conditions. The solid line plotted on the figure represents the 'best tic of the following empirical equation proposed independently by Palaniswamy and Knanss (22) and Richard(23) :
_ [ x%+
= 1
(3)
In Equation 3, C is an empirical constant which is the ratio of the criticalvalue of KII at fracture in pure mode II loading and mode I fracture toughness, KIC. It is reasonable to assume that the effective crack-driving force for suberiticalcrack growth should be identical in functional form to that established in mixed-moda fracturetoughness tests. Therefore, in this study, Equation 3 with C = 1.08 was taken as the basis for defining an effective stress intensity,Kerr, which is obtained by solving for KIC in Equation 3 :
Vol.
25, No.
5
SLOW CRACK GROWTH
K~f=
2
999
(4)
In combined mode I - mode II loading, fast fracture occurs when Keff > KIC. Slow Crack Growth in Mode I. Mode II and Combined Mode I and Mode II Loadin2 : Figure 4 shows a summary plot of the crack growth rates in pure mode I, pure mode II and combined mode I and mode 11 versus K2eff where Keff was defined by Equation 4. The crack velocities plotted in Figure 4 are the initial velocities corresponding to the incipient formation of the kinks at a defined initial combination of KI and KII. Crack growth rates under different loading conditions cue corw.la~ with a single effective crack-driving force parameter, K2eff. The linear plot of log V vs K2eff suggested the following general crack growth law : V = V 0 exp (B K 2 )
(5)
The optimum values of the parameters determined by least squares linear regression analysis were V0 = 9.6 x 10-9 m/s and B = 29.4 (MPa.~m) -2. Slow Growth of Kinked Cracks : Cracks subjected to mode rl or combined mode I and mode H loading abruptly deviate from their initial plane and form a kink. During continued growth kinks exhibited an initial decrease, a minimum and a subse~aent increase in the crack growth rates. Figure 5 is a representative example of crack velocity variations for an initial straight crack subjected to pure mode II which extended as a kink with vavfing KI and KII. Stress-intensity evaluations by the method of caustics indicated that KH decreased rapidly while KI increased with increasing kink length and the crack assumed a configuration of pure mode I. The solid line in Figure 5 represents the prediction of Equation 5 using K! and KII assessed from the method of caustics. The velocity variations of the kinked crack are consistent with the general slow crack growth equation. Discussion Experimental results and analyses presented in this paper have shown that slow crack growth in soda-lime glass under combined mode I and mode H loading can be adequately described by a relatively simple exponential relation with a single effective crack-driving force parameter, K2eff. Although the definition of Keff is based on an empirical equation, Equation 4, it is, in fact, an empirical representation of a cri$ical noncoplauar energy release rate criterion. It was proposed by Palaniswamy and Knanss (22) with a value C = ~/2/3 ~ 0.82 to fit their theoretical results of the critical condition for extension of a crack under combined mode I and mode II loading based on ma~dmiTAtlonof the energy release rate for noncoplanar crack extension. They also showed that critical stress for extension and directions of extension of inclined cracks in tension panels of toluene swollen polyurethane, a brittle grade of polymer, were in agreement with their theoretical calculations. Richard (23) and Shetty et aL(12) also nsed Equation 4 to analyze results on inorganic glasses. Results from tension tests were well represented by Equation 4 with C = 0.82; however, diametral compression tests gave a slightly higher C value similar to the results in this study (Figure 3). Equation 4 has since been found to represent combined mode I and mode II fracture toughness data of several polycrystalline ceramics (14) with C values in the range I to 2, the higher values reflecfinl~ increased frictional shear resistance from crack surface asperities (20). Thus, despite its empirical basis, Equallon 4 has proven to be an effective representalion of the critical condition for fracture in combined mode I and mode II loading for a variety of brittle w~__tcrials. The applicability of Equation 5 to describe subcritical crack growth under combined mode I and mode II loading suggests that the effective crack-driving force for both equilibrium and subcritical crack growth in soda-lime glass is the strain energy release rate. Further, Equation 5 is consistent with theoretical models that treat slow crack growth as a thermally-activated rate process. Lawn (24) and Tyson et ai.(25) have treated slow crack growth as a thcrmaliyactivated bond rupture process by explicitly considering periodic variation of the surface energy in a discrete lattice as originally suggested by Thompson (26). For relatively high values of the crack-driving force, Tyson et al.(25) suggested the following rate equation : AGo
wb.ere v0 is a limiting crack velocity dictated by the frequency of bond stretching, AGo is a zero sur,ss activation free
1000
SLOW
CRACK
GROWTH
Vol.
25, No.
5
energy for slow crack growth, K is the stress-intensity factor and Kc is the critical stress-intensity factor for .spon..~.~. s crac.k .p.~opagation. It should be noted that Eq.uation 6 is.a simplifie~i, special case of a.general .treatment m which the acuvauon free energy for slow crack growth is a nonlinear funcuon of the mechanical strata energy release rate (24). Tyson et a1.(25) employed Equation 6 to analyze the mode I slow crack growth data for glasses in vacuum reported by Wiederhorn et al.(27). The same formulation should be applicable for environment-assisted slow crack growth in glass in combined mode I and mode H loading if AGo is identified to be characteristic of the specific environm~t, ie. water, and K 2 is replaced by K2eff to indicate mechanical energy release rate for combined mode I and mode II loading. A comparison of Equation 6 with Equation 5 then suggests the following equalities betwe~ the expe6.mental and thcoreticsl parameters : [
AGo]
(7)
v0 =
AGo
B =
,
(8)
RTI~ Equation 8 can be used to estimate the zero - slress activation free energy, AGo, from experimental values of B and Ko For the measured values of B = 29.4 (MPaNm) -2 and Kc ~ 0.75 MPa.'~m at room temperatm'e (T - 300 K), Equation 8 gives AGo - 41 U/mole. It follows from Equation 7 and the measured value of V0 ffi9.6 x l0 "9 m/s that v0 ffi 0 . 1 5 m / s . It is useful to compare the above es~mates of the slow crack growth parameters with similar values ~
~
literature. Wiede~horn and Bolz (2) have reported a value, A G o = 109.kJ/..molefor slow crack growm m sc~..- .
glass in water. 'Foe ~ between their value and the value obtained m thi.'s study appears.to. ~ primarily..aue to the different methods cf analyses of the data and the different ranges of stress mtenm~.cov.efm m me tw.o .st~m.~. Crack growth rate measurements of Wiederbom and Bolz (2) extended to tow values oI ~ where me crac[ veiocuy is highly nonlinear with increasing slope on log V vs K plot. A reanalysls of their room tempamm~ _,~q:~"rathe ,sm,me K range as in the present study gave AGo ~ 58 kJ/mole. One can also estimate AGo from Wiederhcrn arm uolz oAt~ by first extrapolating log V vs K plots from the.high K .linear re "gime.to obtain V0 at diff.erent. temperatures. :hThe temperature dependence of V0 was analyzed vta Equauon 7 to obtain me zero stress actlvauon energy. ~uc a
gave
44 kJ/mo , thus,fu.h .d
thed crep y "m
vati
ts interesting to note that a similar discrepancy also crests m the reported a.c~..,vanon ~ energy forstow crac ~ . in vacuum. Wiederhorn et al.(27) reported AGo ~ 605 kJ/mole for soda-lime glass m vacuum. ~owever, anatyms of their data by Tyson et al.(25) via Equation 6 resulted in AGo ~ 349 k.J/mole. Pukh et al.(28) have also reported AGo ~ 338 kJhnole for slow fracture of soda-lime glass in vacuum.
1. Slow crack growth in pure mode I, pure mode H and combined mode I and mode H loading of soda-lime glass in water can be described by a simple exponential relationship with elastic strain energy release rate as the effective crack-driving force parameter. 2. The empirical formulation employed to describe the noncoplanar elastic strain energy release rate also correcdy accounts for the combined mode I and mode H fracture toughness of soda-lime glass under inert conditions. 3. Velocity variations encountered with the growth of kinked cracks can be rationalized, in ~ of. ~ e . g e n ~ slow crack growth equation provided the changes in the mode I and mode tt stress mtensmes wtm me cangmg kink geometry can be properly evaluated and accounted for in defining the elastic energy release rate during kink growth. 4. The slow crack growth equation can be interpreted in terms of theoretical models that treat subcriti.cal crack growth as a thermally-activated bond-rupture process with an activation energy dependent on me envtronment and the elastic energy release rate as the crack-driving force parameter.
Admm/td~mmn ]'be paper is based on research supported by NASA Lewis Research Cent~er under grant NAG3,789 at University of Utah, The authors are grateful to Dr. John Gyekenyesi for his support and encouragement ot tins research.
1.
S.M. Wiederhorn, Intl. J. FracL, 4, 171 (1968).
Vol.
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20, 21. 22. 23. 24. 25. 26. 27. 28.
25, No.
5
SLOW CRACK GROWTH
i001
S.M. Wiederhom and L. H. Bolz, J. Am. Ceram. Sot., 53, 543 (1970). S.M. Wiederhorn, p. 613 in Fracture M~chanics of Ceramics. Vol.2. Edited by R. C. Bradt, D. P. H. Hasselman and F. F. Lange. Plenum Press, New York, 1974. A. G, Evans, J. Mat. Sci., 7, 1137 (1972). A.G. Evans and F. F. Lange, J. Mat. Sci., 10, 1659 (1975). A.G. Evans and S. M. Wiederhorn, J. Mat. Sci., 9, 270 (1974). P.F. Becher, J. Mater. Sci., 21, 297 (1984). J.E. Ritter, Jr. and J. A. Meisel, J. Am. C.cram. Soc., 59, 478 (1976). S.M. Wiederhom, E. R. Fuller and R. Thomson, Metal Science, 14, 450 (1980). J. J. Petrovic, J. Am. Ceram. Soc., 68, 348 (1985). D. B. Marshall, J. Am. C.cram. Soc., 67, 110 (1984). D. K. Shetty, A. R. Roscnfield and W. H. Duckworth, Engineering Fracture Mechanics, 26, 825 (1987). S. Suresh and E. K. Tschegg, J. Am. C.eram. Sot., 70, 726 (1987). D. Singh and D. K. Shetty, J. Am. C.cram. Soc., 72, 78 (1989). D. A. Tossell andK. H. G. Ashbee, J. Am. Ceram. Soc.,71, C - 138(1988). D. Singh and D. IC Shetty, J. Am. Ceram. Soc. (In Press) (1990) H. Awaji and S. Sato, J. Eng. Mater. Technol., 100, 175 (1978). S. Ya. Yamma, G. S. Ivanitskaya, A. L. Mais~enko and A. I. Zbromirskii, Probl. Prochn., 16, 112 (1984). D. K. Shetty, A. IL Rosenfield and W. H. Duckworth, J. Am. Ceram. Soc., 68, C-325 (1985) A. IL Rosenfield, p.501 in Fracture Behavior and Desit, n of Materials and Structures. Edited by D. F'trrao. EMAS, Warley, UK (1990) C. Atkinson, R. E. Smelser and J. Sanchez, Int. J. Fract., 18, 279 (1982). K. Palaniswamy and W. G. Knanss, Mechanics Today, 4, 87 (1978) H. A. Richard, VDI - Forschnngsh., 631, 1 (1985) B. R. Lawn, J. of Mater. Sci., 10, 469 (1975) W. R. Tyson, H. M. Cckirge and A. S. Krausz, J. of Mater. Sci., 11, 780 (1976) R. M. Thompson, Annual Rev. of Mat. Science, 3, 31 (1973) S. M. Wiederhorn, H. Johnson, A. M. Diness and A. H. Heuer, J. of Am. Ceram. Soc., 57, 336 (1974) V. P. Pukh, S. A. Laterner and V. N. Ingal, Soy. Phys. Solid State, 12, 881 (1970)
A
'
B-'I
_
A
I'-
_
Section AA
FIG. 1. Geometry of the Chevron-Notched Disk Specimens Used in Slow Crack Growth Tests.
1002
SLOW CRACK GROWTH
Vol.
25, No. 5
A OT
e=
,
,
,
,
,
~
=<:~1="~ 0.6 ~ -.
0.5
u z )
0.4
~ o°
o
~
• o
_~z O3 ,.,,"" c=:: 02
*
•
" " °/e-o 2
o
..o/e.oso K
o°
i(
,
SODA-LINEGLASS DIANETRAL-CONPRESSI-Ol
\
S
2
\
o, o,
I 0.1
0.0
c.,.
I 0.2
I
0 .~
,
,
0,4
0.5
°
I
\
0~0~
0.6
0.7
.. .
I
0.8
0.9
MODETT STRESS INTENSITY, K]z(MPomi/2)
FIG. 2. A Center-Cracked Disk Specimen of SodaLime Glass With Two Microcixcuit Grids Deposited on the Surface.
io-3
A v
i
l
SODA-LIMEGLASS / DIAMETRAL"COMPRESSION 25C,O.I. WATER o PUREMODEI io-4 • , , PURE MODE1Z n : KZI+K.IZ 1,0.45MOeml/2 • , RZI+K.EI.O.55MPemI/2
/
i
i
0 oo 0 o
I0"5 Do o
.-I k*J
,
I
,/
io~ I--
FIG. 3. Fracture Toughness of Soda-Lime Glass in Combined Mode I and Mode 11Loading.
,
,
o ' K~I.O.40~IPo./i , PREDICTION
E i0 -lIt o °
N
o
10"6 o
I0 "I
I0' 0.0
io o.o
I
o.i
0'.2
0'3
0.05
Ot,I
KINK LENGTH, ~/e
0.15
i
o4
STRESS INTENSITY, K~ff (MPomt/2)2 HG. 4. Subcridcal Crack Growth Rates in Soda-Lime Glass in Pure Mode I, Pure Mode II and Combined Mode I and Mode n Loading.
HG. 5. Comparison of Measured and Predicted Crack Growth Rates For Kinked C~tcks in Soda-Lime Glass.
0.2