13 liechti biaxial loading experiments for determining interfacial fracture toughness

7/27/2019 13 Liechti Biaxial Loading Experiments for Determining Interfacial Fracture Toughness

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K. M. LiechtiAssociate Professor,

Mem. A S M E .

Y.-S. Chai1Department ofAerospace Engineering

and Engineering M echanics,The University of Texas at Aust in,Aust in , TX 78712

Biaxial Load ing Experim ents forDeterm ining Interfacial FractureToughnessThe paper establishes the range of in-plane fracture m ode mixtures and contact zonesizes that can be obtained rom an edge-cracked bimaterial strip under biaxial applieddisplacements. The development of a suitable loading device for and the applicationof crack opening interferometry to interfacial crack initiation experiment s is described. The crack initiation process under bond-normal loading is examined indetail for a glass/epoxy interface in order to establish a hybrid optical interference/finite element analysis technique for extracting mixed-mode fracture parameters.

1 IntroductionIt has become increasingly clear that the fracture resistanceof composite materials can be strongly affected by the toughness of the interface between constituents. The reliability ofmicroelectronic devices, which may contain a large number ofdifferent interfaces, may also be compromised by their toughness. The same may also be true of structural adhesively bondedjoints and coatings subjected to hostile environments. If a crackis constrained to grow along the interface, then the growth isinherently mixed mode in nature and a suitable parameter must

be found that characterizes critical and subcritical growth overa range of mode mixes. The purpose of this paper is to describethe examination and development of amethod for providingmixtures of mode I and IIover awide range of mode mixes.Although any interfacial fracture test will, in general, involvesome mode m ix, a series of specimens loaded in different ways,a single specimen under biaxial load or a change indelami-nation shape will usually be required to determine interfacialtoughness over a range of mode mixes. The first strategy wasrecognized early by Malyshev and Salganik (1965) and Gentand Kinloch (1971) and later by Takashi et al. (1978), butfracture mode mixes were not explicitly extracted. Trantina(1972) using scarf joints, Anderson, DeVries, and Williams(1974) using cone, peel, and blister specimens, Liechti andHanson (1988) using blister specimens, Cao and Evans (1988)using symmetric and asym metric double cantilever beams, four-point flexure (Charalambides et al., 1989a) and compositecylinder (Charalambides and Evans, 1989b), and Rosenfeld etal. (1990) introducing the microindentation test all used finiteelement analyses to extract fracture m ode mixtures. A nalytical' Currently Assistant Professor, Y eungnam U niversity, Seoul, South Korea.Contr ibuted by the Applied Mechanics Division of THE AMERICAN SOCIETY

OF M E C H A N I C A L EN G I N E E R S for publication in the J O U R N A L OFA P P L I E D MEC H A N I C S .Discussion onthis paper should be addressed to the Technical Editor, Prof.Leon M. Keer, The Technological Institute, N orthweste rn University, Evansto n,IL 60208, and will be accepted until two months after f inal publication of th epaper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by theASM E Applied Mechanics Division, N ov. 10, 1989; final revision, Apr. 4, 1990.

stress intensity factor solutions were obtained for blister specimens (Arin and Erdogan, 1971), sandwich specimens (Suoand H utchinson, 1989), and brazil nut sandwiches (Wang andSuo, 1990). Single specimens under multiaxial loads were employed by Mulville et al. (1978) and Liechti and Knauss(1982a,b) and suggested by Suo and Hutchinson (1990). Finally, in the realm of thin coatings, a clever use of residualstresses has been made in determining the effect ofmode IIIon interfacial toughness by examining the shape of the delam-ination emanating from a straight cut made through the coatingto the substrate interface (Jensen et al., 1990). A simplifiedanalysis for extracting three-dimensional mode mixes fromcurved delamination fronts in thin films has recently beenpresented by Chai (1989).The approach that was chosen here for obtaining awiderange of mode mixes was touse a single specimen inconjunction with a biaxial loading device. The stress analysis ofthe specimen and loading is considered first in order to establishthe potential mode mix range and crack-face contact effects.The development of the biaxial loading device and the measurement of norma l crack opening displacements (N CO D) isthen described. Ahybrid procedure for extracting stress intensity factors based on the measured N CO D and complementary finite element analyses is then discussed with referenceto crack initiation under some initial experiments b ond-normalloading. The results of a series of experiments over a widerange of mode mixes are presented in a companion paper(Liechti and Chai, 1989).2 Specimen Geometry and Analysis

The choice of specimen geometry was motivated by a numberof factors. F irst, it was desirable to have a specimen that gaverise tocrack-length independence of fracture parameter andmode m ix. This feature simplifies da ta reduction, particularlyfor crack propagation studies and allows cracks to be initiatedand arrested by suitable control of the loading. The use of asingle specimen minimizes variations in surface preparations680 / Vol. 58, SEPTEMBER 1991 Transactions of the ASME

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'1' ^ 1 , v lf2 A"A

i H 2 , V 2

Ui (x,,h ) = (0, v

EPOXYGLASS

o)

I

J- U i ( x , , - h ) = ( u , 0 )

Table 1 Material propertiesMATERIALT Y P E

EpoxyGlass

Y O U N G ' SM O DULUS ,E(GPa)2.0768.9

P O I S S O N ' SR ATI O ,v.37.20

(MPa)34.5

HAR DEN I N GE X P O N E N T(n )5

w = 17.78 cm h = 1.27 cmFig. 1 The edge-cracked blmaterial strip specimen

a

-9 0 -10 -5 0 5 10 15 20Ratio of Applied Displacements (u 0 /v 0 )

Fig. 2 The range of mixity available under positive bond-normal displacements

which affect the intrinsic adhesion or toughness which, in turn ,control the overall toughness (A rgon et al . , 1989). It also m eansthat the specimen should be amenable to biaxial loading. Cont inu ing the desire the make measureme nts of N CO D near thecrack front in order to assess the importance of nonlinear,three-dimensional, and crack-face contact effects (Liechti andKna uss, 1982; Liechti and H ans on, 1988) required that a t leastone material be transparent. Glass was chosen for this work,but transparency need not be l imited to the visible spectrum.In view of these considerations, the specimen geometry andloading that was adopted w as the edge-cracked bima terial stripshown in F ig. 1.

The hom ogeneous strip geometry is well known for i ts linearcompliance versus crack-length relation for sufficiently longcracks (Knauss, 1966; Rice, 1967). The extension to the bimaterial case under bond-normal loading has been made byAtkinson (1977) and energy arguments yield the bond-tangential contribution so thatG = (1-21 .! ) | (l-2> 2)A*l(l - Vi ) / i 2 (l - "2)

-\ul+ XV2h r 1 n +Ml M2 (1)in the notation of F ig. 1.

F rom the analysis by Knauss (1966) we expect the steady-state solution (1) to be valid for a/h>2. This expectation wasverified by finite element analysis (Chai, 1990). However, amore important contribution of the finite element analysis wasin the extraction of the m ode m ix associated with any particular .combination of materials and 0 and v0. The definitions ofcomplex interfacial stress intensity factor K, bimaterial constant, e, etc., that was used in this work follow those given byRice (1988).The mode mix or mixity, i/-, was taken to be

*=tan i^im

Epoxy-glass Dundurs' parameter a 0.935/3 = - 0 . 188Bimaterial constant = +0.0604

Ramberg Osgood Representation:

Table 2 Energy release rates under bond-normal and tangential displacements" 0dim)

1.270

o(fim)0

1.27

G{J/m>)N UM ER IC AL

0.046690.22160

G(J/m 2)AN ALYTICAL

0.046730.22160

(DEG)-74.0316.00

The energy releasejate can also be obtained from K and itscomplex conjugate K th roughKKG = ( 1 - > i ) | (1 - ya)y-\ V- l 4 cosh ire (3)

(2)

Following a comparison (Ginsburg, 1987) of techniques forextracting mixed-mode interfacial fracture parameters basedon crack opening displacem ents (Smelser, 1979), virtual crackclosure (Raju, 1986) and a conservation integral appro ach (Y auand Wang, 1984), the latter was found to be most satisfactoryand was incorporated as a post-processing routine in the finiteelement code VISTA (Becker et al . , 1984). The auxiliary solutions required for the technique were taken from the paperby Smelser (1979), takin g into accou nt the stress intensity factordefinition in Rice (1988). The invariance of energy release rateand mixity over the range of crack lengths used in the experiments was established for unit applied displacements normaland tangential to the interface. F or the same displacement level(Table 2), the bond -norm al displacem ents give rise to an energyrelease rate that is approximately four t imes higher than thatproduced by tangential displacements. This can also be seenfrom equation (1) which differs from the finite element solutions shown by less than 1 percent.

Under some general combinat ion of appl ied bond-normaland bond-tangential displacements, the real and imaginaryparts of the complex stress intensity factor can be written asKi = aK\"o ) + b K\ vo ) (4)K2 = aK iU0 ) + b Ki V0) (5)

where t6" o) an d A^"o), / '= 1,2 are the base stress intensity factorsdue to unit applied displacements tangential and normal to theinterface, respectively, and the coefficients a and b are loadfactors. In view of the crack length invariance of K and (4)and (5), only two finite element analyses are required in orderto map ou t the spectrum of mixities that can be obtained fromthe geometry and loading show n in F ig. 1.

The range of mixities that can be obtained for i> 0>0 areshown in Fig . 2 . Pure bond-no rmal d isp lacements (wo = 0) giverise to a mixity of 16 deg, bringing out the mismatch betweenthe glass and epoxy elastic properties (Table 1). A 1:1 ra tio ofbond-tangential to bond-normal displacements is required toproduce i* = 0 deg, whereas a - 7 : 1 ra tio gives rise to i< = 90deg. The mixity does not drop much below - 6 0 deg for w0/t>o>20. Thus, for positive bond-normal displacements, therange of mix it ies i s essent ial ly - 6 0 d e g < ^ < 9 0 deg .Journal of Applied Mechanics SEPTEM BER 1991, Vo l . 58 / 681

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BOND - NORMAL LOADING (v >0 )

0.2

0.2

0.4

0.6

0.8

B O N D - T AN GE N T I AL LO ADI N G ( n . > 0 )

AUj/Juol

-\ V- \ ^

" ' " " J ^ - - _0 0,2 0,4 0.6 0,8

r/aBOND - TANGENTIAL LOADING (u , < 0)

Aiii/luol (Interpenetration)

- e - C o n t a c t- a - Inicrpcnelration

0 0.01 0.02 0.03 0.04 0.05r/a

- e - Contact-B-Inteipenctration

j 0 0.01 0.02 0.03 0.04 0.05r/a

Fig. 3 Crack opening displacements under various loadings

Another interesting aspect of the proposed specimen geometry and loading is the extent of crack-face contact. Fortwo semi-finite b odies with a central interface crack, Comninou(1978) found that, under a shear load, frictionless contact couldoccur over as much as 33 percent of the crack length. Evenlarger contact zones are possible for compression and shear,although complete contact can never occur. Experimental evidence of these trends has also been provided (Liechti andKnauss, 1982).The stress analysis for this portion of the work was conducted with the ABAQUS finite element code3, making use ofspecial gap elements to eliminate interpenetration of crackfaces. The response of the glass and epoxy was considered to ,be linearly elastic using the properties noted in Table 1. Thesize of the smallest elements surrounding the crack tip was2 x l (T 4 /!. The components, Auh of the displacement jumpacross the crack faces were taken to be

A, = H-"P> (6)3 The permission to use ABAQUS under academic license, granted by Hibbit,Karlsson, and Sorensen, Inc., is gratefully acknowledged.

where the superscripts (1) and (2) refer to the epoxy and glass,respectively. Under bond-normal loading (Fig. 3), Aw2 w a salways positive, implying no crack-face contact within the resolution of the mesh. However, the tangential crack openingwas negative over a small region (r/a < 0.02). Positive bond-tangential displacements gave rise to some contact (F ig. 3) nearthe crack tip and mouth (r/a = 1). The near-tip contact zonewas 0.007r/a. When the constraint was removed to allow interpenetration of crack faces, the near-tip interpenetration region was more than double the contact zone . Comninou (1978)also noticed that contact zones were smaller than interpenetration zones; For negative bond-tangential loading, Fig. 3 andits insert indicate that there was some opening a t the crack tipbut the crack faces were in contact over most of the cracklength (96 percent). When interpenetration was allowed, theopen region was again crack length (96 percent). When interpenetration was allowed, the open region was again smaller(r/a< 0.02).The noted differences between sizes of the contact zones andinterpenetration regions did not give rise to any variations inenergy release rate values. This is probably due to the as-

682 / Vol . 58, SEPT EM BER 1991 Transactions of the ASMEDownloaded 17 Jun 2008 to 146.6.102.215. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm


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STEPPER MOTOR

PRELOADED BALLSCREW AND NUT

SPECIMEN MOUNTINGAND ILLUMINATION

LASERBEAM

GLASSMIRROR

Fig. 4 Biaxial loading device

(1) Yo measurement (2) "o measurement

A A

Fig. 5 Measurements of applied and normal crack opening dlsplace-ments

sumption of frictionless contact, although an analysis of cohesivemode II cracks in an adhesive layer did not reveal muchchange in energy release rate when frictional contact was allowed (Liechti and Freda, 1989).Moreover, energyrelease ratevaluescalculated using the conservation integral approach andcrack-opening displacements (VISTA) and virtual crack extension (ABAQUS) were all within 1 percent of the valuesobtained from (1). A positive bond-normal applied displacement yielded positive K 1 and K2 values. Surprisingly, the conservation integral calculation indicated that K 1>or positivebond-tangential displacements, in spite of the local crack-tipclosure (Fig. 3). The positive K, value may have been due tothe fact that the contours were evaluated in regions where thecrack was opening. The K2 value was negative under uo>O,which seems reasonable. All crack initiation experiments described later involved combinations of Uo and Vo that gave rise,to a total K 1(4) that was positive at initiation. Under combinedtension and shear, Comninou and Schmueser (1979) foundthat K2 was a nonlinear function of load ratio due to variationsin contact lengths. As a result, one would think that the superpositions in (4) and (5) are invalid. However, the nonlinearity did not appear in the edge-cracked bimaterial strip inthe sense that the energy release rate was linearly proportionalto ( u ~ + even when crack contact was allowed.Journal of Applied Mechanics SEPTEMBER 1991, Vol. 58/683

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Time (sec)100 108.5 109 .5 110.5 111.5 112.5 113.5

G -^ -

60

40

12Sz

Ztc

Pla

Iy~^s";a'55cS

15

10

o ' ' ' ' ' ' u0 0.2 0.4 0.6 0.8 1.0-3Crack Propaga tion (10 da/a)

Fig. 8 Crack extension under bond-normal displaceme nts

critical value of applied displacement (defined later) and somewhat later as the crack propagated steadily. In both cases, theoriginal slope of 0.52 was retained well away from the crackfront. N ear the crack front the slopes reduced to 0.4 and 0.38at the critical and post critical applied displacements, respectively. These lower slopes are indicative of some inelastic response but do not yield the value of - = 0.167 that would7 7 + 1be expected from the power-law hardening exponent of n = 5for the epoxy (Table 1) and HRR singular fields. However,some crack extension had occurred at the times that theseanalyses were conducted and the singularities are really thoseof a growing, rather than stationary crack. Shih and Asaro(1988) recently showed tha t the asym ptotic fields of a stationarycrack between a power-law ha rdening and a rigid one are nearlysimilar to the HRR fields that arise in a cracked, hom ogeneous,power-law hardening material under mixed-mode loading. O nthe other h and, experimental analyses (Epstein, 1989) have notrevealed HRR fields on the specimen surface near the tip ofan interface crack.The extent of the plastic zone behind the crack front wastaken to be at the intersection of the lines representing theelastic and inelastic response in Fig. 7(a). The plastic zonesize at crack initiation (u0 = t>oc) was therefore found to be 49.3/mi. Considering that the specimen thickness was 5.97 mm,the yielding was small scale in nature, thus permitting a previously employed hybrid approach (Liechti et al., 1987) forextracting mixed-mode fracture parameters from N CO D measurements to be considered here. The basis for the approach isthe comparison between measured N CO D and linear elasticfinite element solutions of the corresponding geometry andloadings, an example of which is shown in Fig. 1(b). Theinitial profile was matched by applying a suitable bon d-norm aldisplacement in the finite element analysis. The subsequentlyapplied displacements were then added to the initial displacements so that measured and predicted N CO D could be properlycompared. T he experimental and numerical results for variousapplied bo nd norma l displacements levels up to the critical onewere in close agreement, thus permitting the finite elementsolution to be used for extracting mixed-mode fracture parameters. The favorable comparison also indicates that plane-strain conditions prevail at the center of the specimen. Allvalues of fracture parameters subsequently reported were obtained by matching N CO D well outside any regions of inelasticresponse.

The relatively high degree of magnification that was usedto resolve the interference fringe patterns me ant that very small

amounts of crack extension (Aa/a 2x 10~ 5) could be resolved. The question arose as to what degree of crack extensionconstituted "initiation." The procedure that was adopted isnow described with reference to Fig. 8, where a number ofparameters are presented as a function of crack extension.F irst, it can be seen that the energy release rate increased withload level and crack extension until the crack attained a steadyvelocity. The elapsed times from load initiation are noted forvarious values of crack extension and indicate the energy release rate peaked just prior to dropping off slightly to a constant value.as steady propagation occurred. If the load washeld constant during the time when th e energy release rate wasincreasing, then crack extension would stop. Thus, on thisscale, the relatively b rittle crack initiation process un der bond-normal displacements as judged by the maximum G value of18.4 J/m 1 displays a response that is reminiscent of stable crackinitiation in very tough materials. T he critical value of energyrelease rate was taken to be the constant value associated withsteady crack extension and all quoted values of critical applieddisplacements were likewise associated with the attainment ofconstant crack velocity. Since the G values in F ig. 8 wereessentially derived from N CO D profiles, the results indicatethat steady crack propa gation isassociated with a fixed N CO Dprofile.

The other pa rameters no ted in F ig. 8 were derived fromlogarithmic plots of the type shown in Fig. 7(a). The valuesnoted above and below the resistance curve at va rious degreesof crack extension correspond respectively to the slopes of thelines in the regions of elastic and inelastic response. Thus, itcan be seen that the exponent in the elastic region was consistently the 0.52 value noted in Fig. 7(a) and that exponentvariations occurred in the inelastic region, depending on thedegree of crack extension, until steady crack propagation occurred. The plastic zone sizes, rp, were also recorded as afunction of crack extension. An increase in rp was noted duringstable crack extension but it was then followed by a sharpdecrease to a co nstant value which was associated with steadycrack grow th. The smaller plastic zone size during steady crackextension is presumably due to rate effects. The synchronization of changes in energy release rate values, inelastic exponents and plastic zone sizes were all very consistent and givea picture of blunting (on a very small scale) prior to steadypropagation.

A series of experiments under bond-normal applied displacements were conducted on a single specimen by unloadingvery quickly once steady crack propagation was well established. The arrested crack became the starter crack for the nextexperiment. This procedure gave rise to starter cracks that weresharp and not influenced by the previous experiment. The samecrack extension behavior noted above was observed in all experiments and the critical value of energy release rate a ssociatedwith steady extension was found to be 17 J/m 2 with a coefficient of variation of 8.3 percent, indicating reasonable reproducibility within one specimen.5 Conclusions

The paper has described the analysis of a single specimenwhich, when used with a specially developed biaxial loadingdevice, should be capable of providing a wide range of mixturesof mode I and mode II. A stress analysis revealed that, forpositive bond-no rmal applied displacements, the mixity rangedfrom - 60 deg to 90 deg for ra tios of applied bond-tangentialdisplacement to bond-normal displacements of 10 to -7.5,respectively. The degree of crack face contact near the cracktip was relatively small (


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crack propagation under all loading directions. Crack extension accompanied by near-tip crack face c ontact was observedusing optical interferometry to m easure N CO D. In a series ofexperiments under bond-normal applied displacements, themeasured N CO D revealed that, for the relatively weak bondbetween epoxy and very smooth glass, crack initiation wasaccompanied by small-scale blunting whose extent was tracedas a function of crack velocity. Energy release rates were extracted from linear elastic finite element solutions that matchedthe measured N CO D in regions of elastic response. Due to thehigh resolution incrack extension measurements, the energyrelease rates were found to increase with increasing crack extension until steady propagation occurred. The constant Gvalue corresponding to steady propag ation was taken to be thecritical value and was found to be 17 J/m 2 for ^ = 16 deg. Theextension of the analyses and the procedures developed hereto determine and examine the increase in G c with positive andnegative mixities is described in an accompanying paper (Liechtiand Chai, 1989).Acknowledgments

The authors would like to acknowledge the support of theN ational Science F oundation through Grant N umber MSM-8813822. The digital image analysis system and some of theloading device components w ere provided und er the UniversityResearch Instrumentation Prog ram throug h the O ffice of N aval Research (Grant N umber N 00014-84-G-0175). We wouldalso like to thank Jan Shrode for timely preparation of themanuscript.References

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