Dynamic Fracture Toughness of a Unidirectional Graphite/Epoxy Composite C. LIU, AJ ROSAKIS ...

13 Pages·2001·1.52 MB·English

by  Cheng Liu

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Dynamic Fracture Toughness of a Unidirectional
Title:
Graphite/Epoxy Composite
Author(s): C. LIU, A.J. ROSAKIS & M.G. STOUT
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FORM 836 (10/96)
ProceedingsoftheSymposiumon“DynamicEffectsinCompositeStructures”
IMECE2001,ASME2001InternationalMechanicalEngineeringCongress&Exposition
NewYork,NY,November11-16,2001
Dynamic Fracture Toughness of a Unidirectional
Graphite/Epoxy Composite
C.LIU(cid:214),A.J.ROSAKIS(cid:215)&M.G.STOUT(cid:214)
(cid:214)MST-8,MaterialsScienceandTechnologyDivision,LosAlamosNationalLaboratory
LosAlamos,NewMexico87545,USA
(cid:215)GraduateAeronauticalLaboratories,CaliforniaInstituteofTechnology
Pasadena,California91125,USA
ABSTRACT
In this investigation, we studied the process of dynamic crack propagation in a fiber-reinforced composite
material using the optical Coherent Gradient Sensing (CGS) technique combined with high-speed photog-
raphy. The mode-I fracture toughness of the unidirectional graphite/epoxy composite, IM7/8551-7, as a
functionofthecrack-tipspeed,wasmeasuredquantitatively. ItwasfoundthatuptotheRayleighwavespeed
of the composite material, the mode-I fracture toughness is a decreasing function of the crack-tip velocity.
This behavior is similar to that observed in the dynamic crack propagation along interfaces between two
homogeneoussolids.
INTRODUCTION
In recent years, we have seen the increasing use of fiber-reinforced composite materials in aeronautical,
automotive,defenseandotherindustries. Theeverwidenappreciationoffiber-reinforcedcompositesisdue
tothefactthatthesematerialshaveaveryhighstiffnesstoweightratio. Ontheotherhand,ithasalsobeen
observedthatcrack-likedefectsdevelopeasilyinthesesolidsevenundernormalserviceloadingconditions
anddefinitelyduringlow-velocityimpactevents. Failureprocessesinthesematerialsofteninitiatefromsuch
damage. Toconclusivelyunderstandthemechanismsthatcontrolthefractureprocessinthesematerials,one
must be able to experimentally measure the fracture toughness of fiber-reinforced composites subjected to
differenttransientloadingconditions,andtodosoforbothstationaryandpropagatingcracks.
Todate,mostoftheexperimentalmeasurementsofcompositefracturetoughnessarebasedonmechan-
ical techniques, where far field loads and overall deformations are measured. The near-tip parameters that
reallycontrolthefractureeventareinferredthroughnumericalcalibrationorotherindirectanalyticalmodels
and methods [1, 2, 3, 4]. For quasi-static loading and for stationary cracks, such techniques can provide
usefulinformationregardingthefractureresistanceofthematerial. However, whentheloadingandsubse-
quentcrackgrowthtakeplaceinveryshorttimes,boundaryvaluemeasurementsareincapableofproviding
meaningful information regarding material’s dynamic fracture behavior. The only alternative in such cases
are optical methods that directly measure near-tip quantities when combined with appropriate high-speed
photographymethods.
Usingaprocedureforpreparingopticallyflatandspecularlyreflectivesurfacesoncomposites,developed
attheCaliforniaInstituteofTechnologyandattheLosAlamosNationalLaboratory,Liuetal.[5]studiedthe
quasi-staticfracturephenomenonincompositematerials. Inthisinvestigation,wewillexperimentallystudy
thedynamicfractureprocessesinafiber-reinforcedcompositematerialusingtheopticalmethodofCoherent
GradientSensing(CGS).Thedeformationfieldsurroundingthedynamicallygrowingcrackintheunidirec-
tional fiber-reinforced composite is measured using high-speed photography. Based on an elastodynamic
analysisofthedeformationfieldnearasteady-statepropagatingcracktip,inanorthotropicsolid,thevalue
ofthedynamicstress-intensityfactorcanbeextractedfromtheCGSfringepatternateachmoment. Finally,
therelationshipbetweenthemode-Idynamicfracturetoughnessandthecrack-tipvelocityisestablishedfor
thegraphite/epoxyunidirectionalfiber-reinforcedcomposite,IM7/8551-7.
1
DEFORMATIONFIELDSURROUNDINGTHETIPOFAGROWINGCRACKINORTHOTROPICSOLID
In order to interpret the CGS fringe patterns obtained from the experiment on the unidirectional fiber-
reinforced composite, an analytic expression of the deformation field surrounding the tip of a dynamically
propagatingcrackinanorthotropicmaterialisrequired. Generalresultsregardingdynamiccrackgrowthin
anisotropicmaterialshavebeendiscussedbyWu[6]andbyYangetal.[7]. Inthissection, wesummarize
the analytic results about the deformation field surrounding the tip of a dynamically propagating crack in
anorthotropicmaterial. Specifically, theleadingtermsthatdominatethedeformationfieldsurroundingthe
cracktiparepresentedexplicitly.
ProblemDescription
Consider the problem depicted in Figure 1, where a semi-infinite straight crack propagates dynamically in
anorthotropicmaterial. Hereweonlyconsiderthatthecrackfront,oredge,isastraightlineparalleltothe
x2 (cid:141)2
l(t)
v
x3 x1 (cid:141)2 (cid:141)1
Figure1: Asemi-infinitecrackpropagatesinanorthotropicsolid.
x -axis. Thecoordinate(x , x , x )isafixedCartesiansystemandthex -axisispointinginthecrackgrowth
3 1 2 3 1
direction.
Supposethatthedeformationiseitherplanestrainorplanestress.Thedeformationfieldcanbedescribed
bythetwoin-planedisplacementcomponents,u (x , x , t)andu (x , x , t),thatarefunctionsofthetwoin-
1 1 2 2 1 2
plane coordinates x and x , and time t. Associated with this displacement field, we also have the stress
1 2
and strain fields, s and e (a, b = 1, 2). For infinitesimal deformation, the relationship between the
ab ab
displacementandstraincomponentsisgivenby
1
e (x , x , t)= u , (x , x , t)+u , (x , x , t) , (1)
ab 1 2 29 ab 1 2 ba 1 2 =
andintheabsenceofbodyforcedensity,theequationofmotionforplanardeformationcanbewrittenas
s , (x , x , t)=•u¨ (x , x , t), (2)
abb 1 2 a 1 2
where•isthemassdensityofthematerial.
In this study, we only consider a homogeneous, linearly elastic, and orthotropic material. Meanwhile,
(x , x , x )isalsotheprincipalaxesofthematerial. Forsuchamaterialundergoingplanardeformation,the
1 2 3
generalizedHooke’slawtakestheform
b
e =b s +b s , e =b s +b s , e = 66s , (3)
11 11 11 12 22 22 12 11 22 22 12 2 12
where b are the elastic compliances of the material. In terms of the elastic moduli, c , the generalized
ij ij
Hooke’slawcanberewrittenas
s =c e +c e , s =c e +c e , s =2c e . (4)
11 11 11 12 22 22 12 11 22 22 12 66 12
Now,intermsofthedisplacementcomponentsu (x , x , t)andtheelasticmodulic ,theequationofmotion
a 1 2 ij
(2)canberewrittenas
c u , +c u , +(c +c )u , =•u¨ , c u , +c u , +(c +c )u , =•u¨ . (5)
11 111 66 122 12 66 212 1 66 211 22 222 12 66 112 2
For planar deformations we may define three nondimensional material parameters in terms of b as
ij
follows:
b 2b +b 30b b +b
l= 11, r= 12 66, k= 11 22 12. (6)
b 20b b 0b b -b
22 11 22 11 22 12
2
Noticethatforisotropicmaterials,landrequal1,whilek=(3-n)/(1+n)forplane-stressandk=3-4nfor
plane-strain deformations respectively. In the degenerate case of an isotropic material, n denotes Poisson’s
ratio. Thepositivedefinitenessofthestrainenergydensityrequiresthat
b >0, b b -b2 >0, b >0.
11 11 22 12 66
Intermsofl,r,andk,theaboverequirementbecomes
(cid:236)(cid:239)(cid:239)(cid:239)(cid:239)k>1, forr>1,
l>0, r>-1, and (cid:237)(cid:239)(cid:239)(cid:239)(cid:239)1<k< 3+r, for -1<r<1.
(cid:238) 1-r
The dilatational wave speed along the x -direction, c, and the shear wave speed, c , are defined by the
1 l s
following,
c 1/2 c 1/2
c = 11 , c = 66 . (7)
l J • N s J • N
Severalparameterswillalsobeusedinthefollowingderivationsandtheyare
v2 v2 c 2 k+1 (3-k)+r(k+1)
a2(v)=1- , a2(v)=1- , h2 = l = , (8)
l c2 s c2 Jc N Jk-1N 40l
l s s
where a(v) and a (v) are functions of crack-tip speed, v, while h is a material constant and for isotropic
l s
solids,h2 =(k+1)/(k-1).
Steady-StateCrackGrowthandStrohRepresentation
Becausesteadystateconditionsprevailveryclosetothemovingcracktip,thethreevariables(x , x , t)can
1 2
bereducedtotwo,i.e.,
u (x , x , t)=u (x -vt, x ) u (x , x , t)=u (x -vt, x ). (9)
1 1 2 1 1 2 2 1 2 2 1 2
AccordingtoStroh[8],thedisplacementfieldinthemovingcoordinate(x , x , x )canbeexpressedas
1 2 3
u1 =2Re9A11f1(x1+m1x2)+A12f2(x1+m2x2)=(cid:252)(cid:239)(cid:239)(cid:253)(cid:239)(cid:239), (10)
u2 =2Re9A21f1(x1+m1x2)+A22f2(x1+m2x2)=(cid:254)
where f (z)(a=1,2)aretwoarbitraryanalyticfunctionsandA andm areundeterminedconstants. Inthe
a ab a
aboveexpression, Re{(cid:215)}representstherealpartofanycomplexquantity. ByusingStrohrepresentationfor
thedisplacementcomponents,(10),thestressfieldcanbeexpressedintermsof f (z)as
a
s11 =2Re9(c11A11+2c12m1A21)f1¢(x1+m1x2)+(c11A12+2c12m2A22)f2¢(x1+m2x2)=(cid:252)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)
ss22 ==22RRee9(cc12(AA11++22cm22Am1A)2f1¢()xf1¢(+x1m+xm1)x+2)c+((Ac12A+122+m2Ac22)mf2¢A(x22)+f2¢m(x1x+)m2x2)=(cid:253)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239). (11)
12 9 66 21 1 11 1 1 1 2 66 22 2 11 2 1 2 2 = (cid:254)
For the existence of a nontrivial solution of the constants A , we have the following restriction on the pa-
ab
rameterm (a=1, 2),
a
c +c m2 -•v2 c +c m2 -•v2 -m2(c +c )2 =0. (12)
I 11 66 a MI 66 22 a M a 12 66
Byusingthedefinitionsgivenin(6)and(8),theaboveconditionbecomes
m4+ a2(v)h2+ a2s(v) - (1+g)2 m2+ a2l(v)a2s(v) =0, g=0lh2 3-k . (13)
: l h2l h2l > l J1+kN
Wehavedroppedthesubscriptofm inwriting(13). Inamoreconvenientform,(13)canberewrittenas
a
lh4a2(v)+a2(v)-(1+g)2 a(v)a (v)
m4+2s(v)z(v)m2+z2(v)=0, s(v)= l s , z(v)= l s . (14)
20lh2a(v)a (v) 0l
l s
Onceagain,s(v)andz(v)arefunctionsofthecrack-tipspeed,aswellastheelasticconstantsoftheorthotropic
material.
3
SubsonicCrackPropagation
Whenthecrack-tipspeedvisintherangeof0 < v < c < c,thecrackissaidtopropagateinthesubsonic
s l
regime,inwhichwehave
v2 v2
a2(v)=1- >0, and a2(v)=1- >0,
l c2 s c2
l s
anda(v),a (v),s(v),andz(v)areallreal. Wealsohavez(v)>0ands(v)>-1. Onecanshowthatequation
l s
(14)hasnorealsolutionsandthecomplexsolutionsthathavepositiveimaginarypartsare
(cid:236)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)i0z(v);2s(v)2+1 –2s(v)2-1?, fors(v)>1,
ma(v)=(cid:237)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)0z(v) –21-s(v) +i21+s(v) , for -1<s(v)<1, (a=1, 2). (15)
(cid:238) ; 2 2 ?
Thechoiceofthepositiveimaginarypartwillensurethattheuppercomplex(x +m x )-planecorresponds
1 a 2
tothephysicalplanewherex >0. In(15),wespelloutthefactthatm dependsonthecrack-tipspeedv.
2 a
With either solution for m (v) given in (15), the nontrivial solution for the coefficients A can be ob-
a ab
tained. Onesetofsuchsolutionsis
1 h2a2(v)+m2(v)
A =1, A =-l (v), A =- , A =1, l (v)= l a , a=1,2. (16)
11 21 1 12 l (v) 22 a (1+g)m (v)
2 a
Now,accordingto(10),thedisplacementfieldnearthemovingcracktipcanbewrittenas
1
u =2Re f (z )- f (z ) , u =2Re -l f (z )+ f (z ) , (17)
1 : 1 1 l 2 2 > 2 : 1 1 1 2 2 >
2
andaccordingto(11),thestressfieldnearthemovingcracktipcanbesimilarlywrittenas
s11 =2c66Re:Ih2-gm1l1Mf1¢(z1)- h2g--lgl2mh22lm2lf2¢(z2)> (cid:252)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)
ss22 ==22cc66RRee:I(gm--llh2)mf1¢l(z1M)f-1¢(zm12)--l2 f¢(lz2) 2 2 f2¢(z2)>(cid:253)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239), (18)
12 66 : 1 1 1 1 l 2 2 > (cid:254)
2
wherez = x +m x (a = 1,2). Inordertodeterminethedeformationfieldsurroundingthemovingcrack
a 1 a 2
tip,weneedtodeterminetheformsofthetwoanalyticalfunctions f (z )and f (z ).
1 1 2 2
Byassumingthatcracksurfaceremainstractionfree,themostgeneralsolutionfor f(z)={f (z), f (z)}¤
1 2
canbeobtainedas
f¢(z)=L-1 z-1/2a(z)+b(z) , f¢(z)=L-1 z-1/2a(z)-b(z) , (19)
9 = 9 =
wherea(z)andb(z)aretwoarbitraryentirefunctionsthatsatisfythefollowingrequirement
a(z)=a(z), b(z)=-b(z). (20)
ThematrixLisdefinedthrough
L=c66ØŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒŒºg-m1l-h2lm11l1 -g--m2lll-h22lm22l2øœœœœœœœœœœœœœœœœß.
2
The complete solution for f¢(z) can be obtained by expanding a(z) and b(z) into a Taylor series. However,
if attention is focused in the region very close to the moving crack tip, we only need to consider the most
4
singular term in the solution for f¢(z), i.e., f¢(z) = L-1a /0z, where a is a real vector. By definition, the
0 0
dynamic mode-I and mode-II stress-intensity factors, K and K , at the moving crack tip, are related to the
I II
tractionvector,{s (x , 0), s (x , 0)}aheadofthemovingcracktip,through
12 1 22 1
¤ 1 ¤
k” K , K = lim 2px s (x , 0), s (x , 0) . (21)
9 II I= xfi0+ 19 12 1 22 1 =
1
Asaresult,a = k/(202p)andfinally,themostsingularsolutionsforthetwoanalyticfunctions f¢(z )and
0 1 1
f¢(z )aredeterminedtobe
2 2
f1¢(z1)= 2c661R(v)Jml12--ll22 (cid:215) 0K2Ip - g-l1l-h2lm22l2 (cid:215) 0K2IIpN01z1(cid:252)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)
f2¢(z2)= 2clR2(v)Jml1--ll1 (cid:215) 0K2Ip - g-ll-h2lm1l1 (cid:215) 0K2IIpN01z (cid:253)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:254), (22)
66 1 2 1 2 2
whereR(v)isthegeneralizedRayleighwavefunctiongivenby
0lh2a(v)+g2a (v)
R(v)=0lh2a(v)a (v)- l s . (23)
l s 0lh2a(v)+a (v)
l s
TheRayleighwavespeedoftheorthotropicsolid,c ,issuchthatR(c )=0. Onecanshowthatforisotropic
R R
materials,R(v)degeneratestothefollowing,
D(v)
R(v)= , D(v)=4a(v)a (v)- 1+a2(v) 2. (24)
1-a(v)a (v) l s 9 s =
l s
The displacement field and the stress field surrounding the moving crack tip are summarized in the
following,
u1 = c66R2(v)Re:J-ml12Jg---lll221l2-h2lm2z22p1l2-2ml112z--p1 ll-21g2-l2z1lp2-hN2KlmI21l122zp2NKII> (cid:252)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)
u2 = c66R2(v)Re:+-JJll11mgl12-l--lll-h222lm22l2z2p12-2zlp12ml-11l--2gll21-l2l-h2z2p2lmN1KlI122zp2NKII>(cid:253)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239), (25)
1 2 1 2 (cid:254)
and
ss1212 == RR11((vv))RRee::-JJmmllJ122g-----lllll1l222-h2(cid:215)(cid:215)lmhg222l-0-02l22g(cid:215)hppmh2zzm1210l1-1l21g-pm-z11mllm1l111-----lll2l1g1(cid:215)-l(cid:215)h1l2g-h0--202lmglp212mhlzp2221lzm(cid:215)22lNhK220IN-K2Igpmz22l2NKII>, (cid:252)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)
s12 = R1(v)Re:--JmlJJ112gg----llll11ll222--hh22(cid:215)llmmm022221ll2-22p(cid:215)(cid:215)zl11gm01--120-plml2zlh111p12--zm-11lll121g1(cid:215)--lm01l22g-h2-2-pllmzl12l122l-hN12Klm(cid:215)I21ml02122-(cid:215)pgzl22-0NKl2IhIp2>zm22l2NKII>(cid:253)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:239)(cid:254). (26)
5
Another crack-tip parameter is the energy release rate G. By using Irwin’s closure integral [9], the energy
releaserateGcanbeevaluatedby
1 (cid:243)a
G= lim s (x, 0)d (x, 0)+s (x, 0)d (x, 0) dx, (27)
(cid:243)afi02(cid:243)a(cid:224)0 : 12 1 22 2 >
whered (x, 0)=u (x-(cid:243)a, 0+)-u (x-(cid:243)a, 0-)(a=1, 2and06x6(cid:243)a)isthecrack-openingdisplacement.
a a a
Intermsofthetwostress-intensityfactors,K andK ,theenergyreleaserateGcanbeexpressedas
I II
G= h 1- 1+g 2 1/2 a(v)K2+0la (v)K2 . (28)
2c R(v): J0lh2a(v)+a (v)N > I l I s IIM
66 l s
Onceagain,onecanshowthatforisotropicsolids,energyreleaserateGin(28)reducesto
1-a2
G= s aK2+a K2 , D=4aa -(1+a2)2, (29)
2c DI l I s IIM l s s
66
which is the same as that given in Freund [10]. On the other hand, for a stationary crack in an orthotropic
solid,onecanshowthattheenergyreleaserateGin(28)becomes
2
1+r
G=b l-3/4K +l-1/4K . (30)
11 2 I I IIM
ThisexpressionisthesameasthatinHutchinsonandSuo[11].
CGSTECHNIQUEANDITSAPPLICATIONTODYNAMICFRACTUREOFCOMPOSITES
CGS,ortheCoherentGradientSensingmethodisafullfield,lateralshearinginterferometrictechniquewith
anon-linefilter.Thismethod,whenusedinareflectivemode,measuresthein-planegradientsofout-of-plane
surfacedisplacementsaroundthecracktip.Thebasicgoverningequationsofthemethodforareflectivesetup,
arethefollowing[12],
¶u (x , x ) n p ¶u (x , x ) n p
3 1 2 = 1 , 3 1 2 = 2 , (31)
¶x 2D ¶x 2D
1 2
whereu (x , x )istheout-of-planedisplacementofthespecimensurface. Parameters pandDarethepitch
3 1 2
and separation of the two high density gratings, and finally, n and n are the fringe orders for the x , x
1 2 1 2
gradientcontoursrespectively. Therefore, eachCGSfringeisalocusofpointsthathavethesameslopein
eitherx -orx -direction,dependingontheorientationofthetwohighdensitygratings.
1 2
Foranisotropicsolidsubjectedtoplanestressdeformation,theout-of-planedisplacementu (x , x )can
3 1 2
directly be related to the first stress invariant. However, for anisotropic materials, the relation is not that
simple. Fororthotropicsolid,accordingtotheconstitutiverelation,
e =b s +b s +b s , (32)
33 31 11 32 22 33 33
where b , b , and b are elastic constants. For plane stress deformation, s = 0. Also, according to
31 32 33 33
thegeneralizedplanestressconditions,theout-of-planesurfacedisplacementu (x , x )canberelatedtothe
3 1 2
in-planestresscomponentsthrough
h
u (x , x )= b s (x , x )+b s (x , x ) . (33)
3 1 2 29 31 11 1 2 32 22 1 2 =
Byusingthestressfieldnearthemovingcracktip, themostsingulartermfortheout-of-planesurfacedis-
placementcanbeexpressedintermsofthedynamicstress-intensityfactorsby
h m -l b -b m l m -l b -b m l
u = Re 2 2 (cid:215) 1 2 1 1 - 1 1 (cid:215) 1 2 2 2 K
3 2R(v) :Jl -l 02pz l -l 02pz N I
1 2 1 1 2 2
g-lh2m l b -b m l g-lh2m l b -b m l
- 2 2 (cid:215) 1 2 1 1 - 1 1 (cid:215) 1 2 2 2 K , (34)
J l -l 02pz l -l 02pz N II>
1 2 1 1 2 2
6
where b = b h2 +b g and b = b g+b lh2. Note that b and b are two material constants. The
1 31 32 2 31 32 1 2
governingequationfortheCGStechnique,(31),wouldthusbecome
p p
F (x , x )K +F (x , x )K =n , F (x , x )K +F (x , x )K =n , (35)
11 1 2 I 12 1 2 II 1J2DN 21 1 2 I 22 1 2 II 2J2DN
where
h m -l b -b m l m -l b -b m l
F =- Re 2 2 (cid:215) 1 2 1 1 - 1 1 (cid:215) 1 2 2 2 ,
11 402pR(v) :l -l (x +m x )3/2 l -l (x +m x )3/2>
1 2 1 1 2 1 2 1 2 2
h g-lh2m l b -b m l g-lh2m l b -b m l
F = Re 2 2 (cid:215) 1 2 1 1 - 1 1 (cid:215) 1 2 2 2 ,
12 402pR(v) : l -l (x +m x )3/2 l -l (x +m x )3/2>
1 2 1 1 2 1 2 1 2 2
h m (m -l ) b -b m l m (m -l ) b -b m l
F =- Re 1 2 2 (cid:215) 1 2 1 1 - 2 1 1 (cid:215) 1 2 2 2 ,
21 402pR(v) : l -l (x +m x )3/2 l -l (x +m x )3/2>
1 2 1 1 2 1 2 1 2 2
h m (g-lh2m l ) b -b m l m (g-lh2m l ) b -b m l
F = Re 1 2 2 (cid:215) 1 2 1 1 - 2 1 1 (cid:215) 1 2 2 2 .
22 402pR(v) : l -l (x +m x )3/2 l -l (x +m x )3/2>
1 2 1 1 2 1 2 1 2 2
If one knows the CGS fringe orders, n or n , and the coordinates of the point (x , x ) on the fringes, the
1 2 1 2
dynamicstress-intensityfactors,K andK ,canbecalculatedusingeitheroftheequationsin(35).
I II
MEASURINGTHEDYNAMICFRACTURETOUGHNESSOFTHEUNIDIRECTIONALFIBER-REINFORCED
IM7/8551-7GRAPHITE/EPOXYCOMPOSITE
CompositeMaterial,SpecimenPreparation,andExperimentalProcedure
A unidirectional graphite/epoxy composite, IM7/8551-7, was used in this study. The microstructure of the
unidirectionalcompositeisshowninFigure2,acollectionofopticalmicrographsshowingthethreeorthog-
onalmaterialorientations. Thediameterofthecontinuousfibersisapproximately5mmandthethicknessof
alaminateisontheorderof100mm. Thefibervolumefractionisapproximately60%;itmightbenotedthat
thematrixisarubber-toughened(meanparticlesize10–75mm)epoxy.
x
3
x
2
50„m
x
1
Figure2: Micrographsofthegraphite/epoxyunidirectionalfiber-reinforcedcomposite,IM7/8551-7.
ACartesiancoordinatesystemhasbeenchosensuchthatthex -axisisalongthefiberdirection,thex -
1 2
axisliesinthelaminateplaneandnormaltothefibers,andthex -axisisnormaltothelaminateplane. Since
3
thematerial,asshowninFigure2,issymmetricwithrespecttothethreecoordinateplanesitcanbemodeled
as an orthotropic solid, and (x , x , x ) are the principal axes of the material. The elastic constants of the
1 2 3
Table1: MaterialElasticConstants
E E E
11 22 33
Young’sModuli(GPa)
148.60 8.40 8.28
m m m
12 23 13
ShearModuli(GPa)
5.45 3.03 4.34
n n n
12 23 13
Poisson’sRatios
0.32 0.37 0.35
unidirectionalgraphite/epoxycompositehavebeenmeasuredwithrespecttotheprincipalaxes(x , x , x ),
1 2 3
andtheresultsarelistedinTable1.
7
The specimen geometry we chose for this experimental study, was the single-edge-crack specimen as
shown in Figure 3. The nominal dimensions of the specimen, shown in Figure 3, are: 2l = 150mm, b =
Specimen
P(t)
º
Gratings p
h Crack
b a Filter Lens Camera
2l
Collimator
Laser
Figure3: SchematicofCGSsetupandsingle-edge-crackspecimen.
75mm, h = 6.5mm, and a = 25mm. In this experimental study, we concentrate on the situation where
thecrackisparalleltothefibersandthespecimenplaneisparalleltothelaminates. ACartesiancoordinate
systemischosenwithitsoriginlocatedatthecracktip. Theaxesofthissystemarealignedwiththeprincipal
axesoftheorthotropicmaterialasshowninFigure2,sothatthedeformationplaneisthe(x , x )-plane,with
1 2
crack propagating along the x -axis. In this plane, the elastic compliances are listed in Table 2. Based on
1
Table2: MaterialElasticCompliance
b (GPa-1) b (GPa-1) b (GPa-1) b (GPa-1) b (GPa-1) b (GPa-1)
11 22 12 66 31 32
6.73·10-3 1.19·10-1 -2.15·10-3 1.83·10-1 -2.36·10-3 -4.40·10-2
thesevalues,thethreenondimensionalmaterialparameterscanbecalculatedtobel = 0.0565,r = 3.1676,
andk = 2.7172,respectively. Meanwhile,thedilatationalwavespeedalongthefiber(x -axis),c,theshear
1 l
wavespeedinthe(x , x )-plane,andtheRayleighwavespeedonasurfaceparalleltothe(x , x )-planeand
1 2 1 3
inthedirectionoffiber(x -axis),c ,aregiveninTable3.
1 R
Table3: CharacteristicWaveSpeeds
c (m/sec) c (m/sec) c (m/sec)
l s R
9819.84 1874.62 1852.50
A very high quality surface preparation is required in order to apply the CGS technique to composite
materials. However, because of the microscopic nature of the composite, we cannot polish the specimen
surfacetomakeitopticallyflatandreflective,norcanwedirectlydepositareflectivealuminumfilmonthe
specimenbyvacuumdeposition. Instead,thefollowingprocedurewasdevelopedandused. Anopticallyflat
glass plate is coated with a thin aluminum film, having a thickness of only several angstroms. A layer of
segregationmaterialisintentionallymaintainedbetweenthecoatingandtheglass,topreventstrongadhesion
betweenthealuminumfilmandtheglass.Thismaterialwastheresidueofliquidsoapusedtocleantheglass.
The coated glass is then affixed to the sample using an epoxy adhesive, which glues the coated surface of
theopticallyflatglasstothesample. TheepoxyadhesivewasaPC-1Bipaxofepoxyresinandadiethylen-
etriaminehardenerobtainedfromthePhotoelasticDivision,MeasurementsGroup,Inc.,Raleigh,NC.After
theepoxycured,theglasswaspeeledoff. Becausethebondingbetweenthealuminumfilmandtheglassis
ratherweak, thealuminumcoatingistransferredontothesamplesurface. Thetotalthicknessoftheepoxy
layerandthecoatingisjustacoupleofmicrons. Comparedwiththesamplethickness,thislayerisverythin
andwillnotaffectthedeformationstateinsidethespecimen.
Thespecimenwasmountedinadropweighttower(Dynatup-8100A)witha200kgfree-fallingweight,
whichisabletoprovideimpactsatlowtointermediateloadingrates. Aftertheimpact,intensivestresswaves
emanate from the loading point and propagate toward the stationary crack. Stress fields are thus built up
surroundingthecracktipandwhentheintensityofthestressfieldreachesacertainlevel,thestationarycrack
8
startstopropagatedynamicallyalongthefiberdirection. TorecordtheCGSimagesduringtheexperiment,a
rotatingmirrorhigh-speedcamera(Cordinmodel330A)isusedwithaSpectra-PhysicsArgon-ionpulselaser
(model 166) as the light source. The duration of the laser pulse is about 30 nsec, which is able to produce
a sharp interference pattern even for fast running cracks. The interframe time (controlled by the interval
betweenpulses)istypically1.5msec. Thelaserpulseistriggeredbyastraingagemountedtothespecimen
that senses the impact. The composite specimen is supported by resting its lower corners on two very thin
glassslides. Ithasbeendeterminedthattheseglassslidescanonlysustain20Nofloadbeforetheybreak.
This suggests that the bottom surface of the specimen can be considered traction free throughout the crack
propagationevent.
ExperimentalObservations
AsequenceofhighspeedCGSinterferogramsfromathree-pointbendtestoftheunidirectionalgraphite/epoxy
IM7/8551-7compositespecimenisshowninFigure4. Thenominalimpactspeedwas4m/secandthetime
62.63 (cid:181)sec 64.30 (cid:181)sec 65.97 (cid:181)sec 67.64 (cid:181)sec 69.31 (cid:181)sec
70.98 (cid:181)sec 72.65 (cid:181)sec 74.32 (cid:181)sec 75.99 (cid:181)sec 77.66 (cid:181)sec
Figure4: SelectedsequenceofCGSinterferogramsfromaunidirectionalcompositethree-pointbenddrop-
weighttowertest.
shown in Figure 4 is measured from the moment of impact. The interframe time for this specific test was
1.67 msec. The two high-density gratings, shown in Figure 3, were perpendicular to the crack, so that the
information regarding the quantity ¶u /¶x was obtained. The grating pitch was p = 0.025 mm and the
3 1
distancebetweenthetwohigh-densitygratingswassetatD = 46.5mm. TheangularsensitivityoftheCGS
interferometerwiththesesettingsisabout0.015º/fringe.
ThefirsttwointerferogramsshowninFigure4representtheCGSfringepatternsforthestationarycrack.
Thesizeofthefringepatternsurroundingthestationarycracktipcharacterizestheintensityofthenear-tip
stressfield. Atatimeclosetot =65.97msec,thestressfieldsurroundingthetipreachedalevelsuchthatthe
stationarycrackstartstopropagate. Sincetheunidirectionalcompositespecimenwasloadedsymmetrically
andthedeformationsurroundingthetipofthestationarycrackispredominantlymode-I,thegrowingcrack
remainsmode-Ianditpropagatesalongthefiberdirection. Thisisalsobecausethestrengthofthegraphite
fibersissohighthatthecrackisconstrainttopropagatealongthefibers. Meanwhile,asthecrackpropagates
dynamicallythroughthematerial,thesizeofthefringeloopandthenumberoffringeschangeindicatingthe
evolutionoftheintensityofthestressfieldsurroundingthegrowingcracktip. Onealsoobservesthatafter
crack initiation, stress waves emanate from the growing crack tip. They are visible in Figure 4 as discrete
kinksoftheCGSfringes. SuchdiscretekinksdonotappearintheCGSfringepatternsassociatedwiththe
stationarycrackasshowninthefirsttwointerferogramsinFigure4.
Thecrack-tiplocationateachmomentcanbeidentifiedfromtheCGSfringepattern. Consequently,the
crack-tipvelocitycanbecalculatedbasedonthevariationofcracklengthasafunctionoftime.Forthethree-
pointbendtestoftheunidirectionalgraphite/epoxycompositespecimenshowninFigure4,thevariationof
the amount of crack growth Da and the variation of crack-tip speed a˙ as functions of time t are presented
in Figure 5. After initiation, the crack tip accelerates rapidly. In a very short period of time (~ 6msec) the
9