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Strain energy release rate for interfacial cracks between dissimilar media
STRAIN ENERGY RELEASE RATE FOR INTERFACIAL CRACKS BETWEEN DISSIMILAR MEDIA
1. INTRWUCTION
Tun M problem of a semi-infinitecrnck between dimimilar materials was initiahy considered by Wiiiauts[l]. Using an eigenfunction approach, he analyzed the character of the stresses near the crack tip between two dissimilar isotropic materials joined without residual stress. He found that the resuhmg stresses possess an osc%atoty chamcter of the form Pm sin (A log r) and P*cos(A ~r),wBcrcristbe~;iistnnceftamthccrPcktipttldA isafu&iouofma&rial proper&s. This is in contrast to the hosprc&km,inwhichtheformoftheresutting suessesuearthecraektipisar-“mtypeof&guh&y. Erdogan[Z,3] and Rice and Sih[4,5] considered similar problems and obtained expressions for the stress along the bond in terms of conipkx potential functions. Their results verify the .oscillatorybehavior of the stresses previously described by Williams.In addition, they defined a pair of stress-intensity factors, K, and &, related to the Grithth-Irwin fracture theory, which are associated with symmetric (normal) and skew-symmetric (shear) stress field, respectively. There are, however, sign&ant disadvantages associated with the application of the stress intensity factor approach to the problem of interfacial crack propagation. Fit, as discussed by Rice and Sih[5], the concept of retating the normal stress and the shear stress to separate&ress intensity factors is no longer valid, since the stress intensity ftitors are related to both stress components. Second, an oscillatingstress singularityis present near the crack tip, and third, the stress intensity factorsmmrUrirl~nt.Bycomperisoa,tlaesrninenerW~ratc,F’aanof the stress and displacement along the bond as well as the mate&u properties, however, this formulation elhuinates the problems associated with the oscillating singularities.Therefore, the strain energy release rate approach is preferred for interfacial crack propagation studies. In this investigation a strain energy release rate formulation was developed and used to determine the fracture resistance of interfacial cracks. 2. THEQRETICAL ANALYSIS Problem formulation
In this section the stress field is analyzed for the piane problem of two bonded dissimilar materials with cracks along the interface. Both materials are assumed to be linearly elastic, homogeneous and isotropic. These results are used to formulate the strain energy release rate for crack propagation at the interface between the two materials. The theoretical development follows closely the methods and analysis presented in hfilne-Thomson[6]. Consider two elastic half-spaces SL and S” as shown in Fii. 1 with a common boundary, A, along the entire x axis, the positive y axis being directed normally to the x axis, and lying wholly 555 BRIVd.I,NO.3-4
D.R MULVILLEetd. Y
inS‘.LetthesegmentA’ofthexaxisboundedbytheendpoints(-I,0)and(I,O)correspondto anmterfa&boudwiththe rem&ingsegmentA”oneithersideofA’~twocracks. Assume that each half-space is extemaUyIondudby a systemof sclf-cquili~ forceswith A” tnctionfrse.under~~~tecacwigbeaolcwltMtllormdorshsprf~~~~ ~A’.In#Idition,uranrstbrtthc&rivcrtiveofthe~~ia~~aA’~ sgetlcd. Ifr(-lsfsI)~a~paRmlet~~apointonA’,tbsntbebolladuycosditbns canbeexpresaedfonnallyas fi,"(l)-~,"(t)=lc~~(t)-~~x~(~)~O
onA”
(1)
fi,‘(I)-
on A’
(2)
on A’
(3)
onA
(4)
iii$w
= fix=“(t)- iif*Yf)
(u,(t) + iut(tWL-(&r(t) f it&))‘” - d’(t)
I (~,L(r)-~,L(t))dtPX-iY30 A
where i = fl, jj’ and 6 are normal and shau stressesrespectively,u and u aredisplacanwnt componentsalongthe x and y axes respectively,X and Y arc the components of the resultant force on A and d’(t) is the derivative of the discontinuity in displacement on A’. SuperscxiptsL and R refer to the values of the functions on A as approacbd from points in S‘ and S” respectively. A function defined in S‘ is denotedby subscript1 and in S” by subscript2. Equation(1) states thatthe surfacetractionsvanish along the unbendedsection of the real axis, A”. The second boundary condition, qn (2), states that the tractionvectoris continuous acrossA!. The thii boundary condition, eqn (3). states that the strain difference is specified on A’. Finally, qn (4) states that the components of the rest&ant force on A are assumed to be zero. Stresses and displacements can be written in terms of the complex stress function, Wk(2) and w&T),as follows, E?k+j$=
w,(z)+ W&Z,
(5)
fik - EC + 2if& - ZW;(Z) + Wk(Z)
(6)
4/&b +iu~)=~‘w~(z)-z~(z)-‘w~(z) whereZESLfork=1andZESRfork=2,‘W~(Z)=IW~(Z)dZ,and’w,(Z)=Iwt(Z)dZ. In the above pr is the shear modulus, Su,= (3 - 4q) for plane strain and 3&= (3 - ti )/(l + generalized plane stress, where z+ is Poisson’s ratio.
(9
ulr
) for
Strainenergy rekase
ratefor interfacialarctrbctwcm dbimihr media
sl
If W,(Z) is analytically continued across the unloaded segment of the boundary A”, the equations for stress and displacement become
W,(Z)+ (Z - 2) ma
2(j$ - @, = W,(Z) -
4&(u&+i&,=Pw~(z)+
W&z) -cz-z,RGl.
(8) (9)
These equations are written in terms of a singk complex variable, W,(Z), defined in each haIf plane. imposing the stress and displacement boundary conditions on A’ yields aW,‘(t)+bW‘R(t)=4d’(t)
(10)
Where
This can be expressed as cW,L(t)+
w,“(t) -y
(11)
whuec=albisdefmedasthebi-eIaaticconstant. The dispkcement on A’, d(t), cam be expressed as fdlows d(r)=uo+iuo+qt+i&
d’(t) = q + if
(12)
(13)
Where
uo=lkJ
I.
-uo
I
00- 00”- uo=
(14) (1%
and r) = (VRL-~~R)+(~‘-t)TI)+(l)yL--t)yll).
(16)
In qns (12Hl6), q~ is the strain due to residual stresses, qr is the strain due to thermal kading, Q, is the strain due to mechanical loading, R is the radius of curvature of A’. u. and u,tmerely represent translation of A ’ in the complex pkne. As an exampk of the strain due to mechanical loading, consider the plate specimen under se&equiIibrating bending and compression loading shown in Fii. 2. For this specimen, strain along the interface can be determined from
where, as shown in Fii. 2, y L = y - h. 12is the distance from the neutral axis of the epoxy plate of depthh,,y= = y + ir./2 is the distance from the neutral axis of the aluminum plate of depth ri, R L and R JI are the radii of curvatures of the neutral axes of epoxy and ahuninum plates respectively, due to flexure alone. This geometric description for bending analysis of two bonded plates follows closely the methods used by Boley and Weiner[l for thermoelastic analysis of bending
D. R. MULVlLLE d al Y
-x
isa
-6
-_
_I
w-w
I”= WkJ2 _-WV
ha
Y8
Me
ALUYINUY
c
Fi.2.
fit 3
-
S*
Phtespechnlu6raeK~~d
comorarioalordial.
ad
buckhng of bi-metallicbeams. Evalw this expression on A’ correspondsto settbg y = 0 in the relations for yL and yR d&cd above. XX, and XXI arc uniform sw applied pa&l84 to the bond in epoxy and aluminum respectiva and 4%and & arc Young’smoduli of epoxy and ahuninum respcctivcly. Strain along the interface due to thermal loading can be determined from q-wL-‘I)TI
=(al-a2jfT.-Tp)
(181
where a, and
a2 are coefficientsof thermal expansion of the epoxy and ahuninum respectively and T, and Tp are the initial and final temperatures respectively. Strain due to residual stresses may be determined using photoelastic technkptcs or other cxpcrimcntal methods. Using (13) in (11) we have cW,“(t)t
W,“[+
tig.
WI
It now remains to solve this equation for the complex stress function and to determine the resultiry stress componenta. From this result the strain energy release rate, T, will be formulated for crack propag&on at the interface Prvbietn solufion The solution for this non-homogeneous quation is obtained by evaluating the following
where the Plemelj function x*(Z) = (2 - I)“‘(2 + 1)’ is the solution of the homogeneous Hlbert equation cXrL~f)+XIR(~)=O*
(22)
In these quatiom r = (112)+ iA and A = 00s c)12*. The stresses are then determined by substituting the value obtained for W,(Z) into the expression for the stresses (8) evaluated on A’. The stresses on A ’ are expressed as
(23)
f
-1
1
0 NORMALIZED BONOLENGTW
Fig. 3. Nard stresses alongthe bondu&r tensile badiagd
D. R. MULVILLE ei I
/ f
ii
I
Brain energy release rate for interfacialcracks betweendissimilarmedia
561
Y
s2tt1
(- 1.01
( 1.01 .
X
P2(1) J
Fii. 6. Resultantstress vectoron upperand lower half spaces.
wherep,(t) = -j$(t)
and s,(t) = -xy,(t)
and
where p&) = -fi&) and s&) = -XC&)+Equations (25)and (26)can be written in the following form and
Imposia the stress boundary condition on A’ yields iP,(?) = -i&(f)
(29)
and thus P,(t) + P&) = 0.
(30)
Consider the work, U, done by the stress vector acting along the bond (-&I), as folIows (31) Imposing the displacement boundary condition (3),and the relation between the stress vectors in (29) yields
P&)&t) dt.
(32)
From the assumed d(t) on A ’ (12) and (27) we have U=Re [i
or
I_’ i(pl(t) I
- is&))
(lo- iv0+ vf- i&)dr]
(33)
562
D. R. MULVILLE
er al.
Using the resultant foice boundary condition (4) on A’, the first integral vanishes. Using eqn (23) the second integral can be written as
rl(f+2U[)(rll-i~)X,L(l)dl -;(I
t4A')f') (qt-i&)X,L(t)dl]
(35)
and then evaluated using methods based on homogenous polynomials[8j to compute the residues, i.e. the coef%zientsof I/t, directly. This process yields the following expression for the stored elastic strain energy U = 2n(l -t4A3 Q o+b 2 The strain energy release rate,Y,
2+1’+ I’ -(l t4hq. 3R 16R2
(36)
isdefined as the decrease in stored elastic strain energy per
unit of crack increase, dnt. Thus
yApp!L.
(37)
From the geometry shown in Fig. 7, dlldm = -1 and using (36) we have ~ = 2p(1+4h3 a+b
$1 -y+$#
+4A')]
or Y=
q2t -+&(l --CL1cc2 > 2a(I t4A2) Xr+l+!&tl
t4A2)
1.
(3%)
(3%
ThisexpressionforYcanbeobteineddirtctlybycval~cqn(37)usinqtheLiebnizf~~ for differentiationof integrals provided the limits of the integral are d#erentiable and the kernal and its derivative with respect to I arecontinuous,Mast[9].The expressionfor U,eqn (35)),s&is&s these conditions, and the result obtained using this method is identical to that presented above. AIthough both methods yield identical results, there is a basic dEereace in the nature of these appn>acheS. In the h5t mdbod, the totalstrainenergy,U, is evaluated by integWug the product of the stress and displacement vector on A’. Y is then determined by d&rentiating U with respect to crack length, i.e. by determining the change in total strain energy due to a change in crack length. In the second method Y is determined by integrating the derivative of the product of the stress and displacement vector along A'with respect to crack length. This method, u&g the Liebniz
Fig. 7. Strain energyreleaserate vs crack length/initialbondlengthfor varioussurfacefinisheson specimens loadedin bending.
Strain smrgy r&case rate for itWf&&l cracks between dissimiIar media
543
formula, sums each incremental change in strain energy on A’ whereas the fhst method deals with the change in the total strain energy. Using the ~~ of % =(3-q)l(l+yt) and *=&&tz(I+a) for guneraiizod phme stress, the strain energy rekase rate formulation above can be expressed as follows
(41)
These expressions for the strain energy &ease rate are based on the plane problem of two elastic half spaces bonded alung A ‘, and can be used to analyze the following loading conditions: residual stresses, uniform thermal loading, forces applii parallelto the bond to one or both half spaces, bending loads applied to one or both half space or any combinations of these loads.
Enthe precedmg analysis, both materials were assumed to be linearly elastic, ~~~ne~s and isotropic. While these are useful assumptions from an analysis point of view, they often do not adequateiy describe the behavior of real mater&. ‘Inthe fohowing experimental section, epoxy-ah&turn bonded specimens are subjected to external toading which results in interfacial crack propagation, At room ternary most epoxy systems behave in a linear elastic manner, however, at ~~~~ unhip the post-cure ~rn~~~ the behavior becomes viscoehtstic. The preceding analysis does not describe the stress or displacement field for viscoeku&ic media, nor does the fracture mechanics analysis account for time+endent behavior, In addition to problems which may arise due to the viscoelastic nature of the materials, there are probiems due to viscous ffow and non-linear behavior in the vicinity of the crack tipThis analysis does not account for the energy absorbed by plastic gow uear tBt eraek tip. The function d(t ), defhted in qns (12)and (13),assumes that the resultingdisphreemanton A’ after bonding can be written as the difference between the displa&cmentin S”, and in S” before bonding. This is valid if the ratio of the Young’s Moduliof the materials is large, i.e. if one of the m&aids ads as a rigid body, Otherwise the resulting ~spla~rn~t on the i&ufacc mst be considered an unknown function related to the material properties. fn the ex~~~~ investigation conducted in this study, the modulus ratio was approx. 26; therefore qn (12)is a reasonable estimate of the resulting displa~e~nt on A’. An additional point with respect to the definition of d(t) is that the displacement along the bond, A’, in S’- and S” is assumed to be a linear function of position, f. This assumption may not be accmnte near the ends of the bond.
Experimental studies were u~de~ken to make q~nti~tive m~sureme~ts of the fracture parameter J for interfacial crack propagation. Using the amilysis of the gmzediq section, specimen designs were chosen and loading comities seWed for which cracks would propagate at the interface between two dissimilar materials. Specimen design studiis included phot~~ti~ observations and strain gage ~ns~mentat~n of various specimens to determine how closely the assump~~s of the theoretic analysis were satisfied. Alu~nu~~xy specimens were fabricated to conduct a series of interfacial crack propagation studies for various surface finishes on the alu~num surface of the bond. Restrained shrinkage along the bond during soli~~~n of the epoxy and ~~erenti~ thermal contraction during post-curing gave rise to a Aeldof residual stresses, whose e&t was included in the analysis. Details of the expend study are
564
D. R. MULVILLE
et al.
discussed elsewhere, Ib4ulviueflO,111,and therefore, only the results of tlmc studies will be presented here. Using the formulation of the theoretical section, the strain energy release mte 5,was calculated at various crack positions along the bond for specimens similarto that shown in Fii. 2. These specimens were loaded by applyingcompression and bending to the aluminumplates. Data shown in Fig. 8 compares the results for beading tests ou glass-peeoed, milkd and sandblasted surfaces with polished ahtminum surfaces. These values are generally greater than the value of 9rc= 0.5 in. lb/ii.’ for the buik epoxy system reported by Mostovoy and Ripling[l2].However for the polished specimens, the value of Y, is approximatelyequal to the bulk value. Results of these studies indkate that the strain energy release rate remained relatively constant as the crack propagated along the bond. Therefore, S, can be used to analyze the effects of various surface treatments and preparations on the energy required for crack propagation.
0.2
0.3
04
0.6
0.6
0.7
06
TOTAL CRACNLENGln~wTIAL 9cw WdTH
0.9
d
,II
CONCLUSIONS The purpore of this investigation was to characterize crack ptopaoation at the interface between two dissimiiar mater& using a formWon based on strain energy &case rate. To this en& a theoretical analysis was developed for the plane problem of two elastic half spaces in contact along an arc where one of the mater& acts as a relatively rigid body. The stress analysis included the cases of residual stresses, thermal loading and self equilibratingtensile or bending Ioadiugapplied to either half space. Using this analysis, the ektstic strain energy release mte, Y, was formulated as a fracture criterion for interfacial failure. Experimental studies conducted on specimens of epoxy bonded to ahuninum demonstrated that this failure criterion can be used to analyze interfacial crack growth. The technique presented in this paper provide a simple method for the study of interfacior failure in bonded materials. Results of this investigation have application in design of bonded structures, or in general problems of debonding and deiamination between dissimilarbodies. The analysis can be applied to composite materials in analysis of fiber-resin interfaces, in bonded joints or in welded structures where the crack propagates at the interface. This approach can be extended to consider the effects of crack velocity, fatigue loading, environment and various residual fields on the fracture toughness of bonded materials. A&ow&&wW--lke authors ackmwkdgc the support of tJte 0th of Naval Rmarcb. CammrsdmdTbeC&BolieUniraityotAarsriafathe~npatsdhsrein,~tadahojectNcrpkrRRM3M 4s.5451 lad SF 51-544-102-12432.
the Naval Sea Systems
REFERENCES [I] hi. L. WiHiis, The stresses around a fault or crack in dissimihr medii, B&tin ofthe&ismobgiccll Sot. Am. 4q2). mu04 (1959). [2] F. Erdogm, Stress distribution in a nmhomogcnmus elastic plate with cracks, J. App. Me& Ser. E, 5)(2), 232-236 (1963). [3] F. Erdogan, Stress distributionin bondeddissimilar materialswith cracks,I. Appl. Mech. Ser. E, 3x2). 403-410 (1965).
Strainenqy rekaxc rate forintcrfxcixlcracksbchvecndissimilarmedia
565
0. C. Sih andJ. R. Rice, The bea- of dissimilar mat&& with cmcks.I. AppLMech.Ser.E,31(3),41&482(KM). J.R. Rice xnd G. C. Sib, Pixnc pbkms of cracksin diasimii media,1. Appf. Mcch. Ser. E, 32(2), 418-423 (1965). L. M. Milne-Thomon, P&M Elastic Sys@wtr. Springer-Verlag, Berlin (1960). B. A. Boky and J. H. Weiner, Z’kcoryof Them& &ess, pp. 429-432. Wiiy, New York (l%o). P. W. Mxst and D. R. Mnivilk, The exprexsionof non-homogeneous polynomialsas venous polynominalswith appiicationsin caginccringmcchaniis, NRL. Repoti in preparation. [9] P. W. Mast, Graphsxnd tensormxnipuldon in compkx coordinxtcsxs probkm solvingtechniquesin plane-problemsof linear xnisotmpic dadcity; Doctod Dixrrtation. Nod Carolii State Univerxity (1972). [ IOj D. R. Mulvilk. C&act&tics of crack propa@& at the interfxce of two disximilxrme-&x,Do&ml Dissertation,The Catholic University of America (1974). [If] D.R.MolviMc aadR N. V&&w, Intedacixl CncL ’ , L Ad&&n 7.215-233 (1975). [121 s. Moxtovoy 8nd E. Rip@ Prxctw tol&ms of Morn, 1. Appl. Jwymer sci. 10. 1351-1371 (1966). [4] [S] [6] m [8]
(Received 28 September 1975)
1. INTRWUCTION
Tun M problem of a semi-infinitecrnck between dimimilar materials was initiahy considered by Wiiiauts[l]. Using an eigenfunction approach, he analyzed the character of the stresses near the crack tip between two dissimilar isotropic materials joined without residual stress. He found that the resuhmg stresses possess an osc%atoty chamcter of the form Pm sin (A log r) and P*cos(A ~r),wBcrcristbe~;iistnnceftamthccrPcktipttldA isafu&iouofma&rial proper&s. This is in contrast to the hosprc&km,inwhichtheformoftheresutting suessesuearthecraektipisar-“mtypeof&guh&y. Erdogan[Z,3] and Rice and Sih[4,5] considered similar problems and obtained expressions for the stress along the bond in terms of conipkx potential functions. Their results verify the .oscillatorybehavior of the stresses previously described by Williams.In addition, they defined a pair of stress-intensity factors, K, and &, related to the Grithth-Irwin fracture theory, which are associated with symmetric (normal) and skew-symmetric (shear) stress field, respectively. There are, however, sign&ant disadvantages associated with the application of the stress intensity factor approach to the problem of interfacial crack propagation. Fit, as discussed by Rice and Sih[5], the concept of retating the normal stress and the shear stress to separate&ress intensity factors is no longer valid, since the stress intensity ftitors are related to both stress components. Second, an oscillatingstress singularityis present near the crack tip, and third, the stress intensity factorsmmrUrirl~nt.Bycomperisoa,tlaesrninenerW~ratc,F’aanof the stress and displacement along the bond as well as the mate&u properties, however, this formulation elhuinates the problems associated with the oscillating singularities.Therefore, the strain energy release rate approach is preferred for interfacial crack propagation studies. In this investigation a strain energy release rate formulation was developed and used to determine the fracture resistance of interfacial cracks. 2. THEQRETICAL ANALYSIS Problem formulation
In this section the stress field is analyzed for the piane problem of two bonded dissimilar materials with cracks along the interface. Both materials are assumed to be linearly elastic, homogeneous and isotropic. These results are used to formulate the strain energy release rate for crack propagation at the interface between the two materials. The theoretical development follows closely the methods and analysis presented in hfilne-Thomson[6]. Consider two elastic half-spaces SL and S” as shown in Fii. 1 with a common boundary, A, along the entire x axis, the positive y axis being directed normally to the x axis, and lying wholly 555 BRIVd.I,NO.3-4
D.R MULVILLEetd. Y
inS‘.LetthesegmentA’ofthexaxisboundedbytheendpoints(-I,0)and(I,O)correspondto anmterfa&boudwiththe rem&ingsegmentA”oneithersideofA’~twocracks. Assume that each half-space is extemaUyIondudby a systemof sclf-cquili~ forceswith A” tnctionfrse.under~~~tecacwigbeaolcwltMtllormdorshsprf~~~~ ~A’.In#Idition,uranrstbrtthc&rivcrtiveofthe~~ia~~aA’~ sgetlcd. Ifr(-lsfsI)~a~paRmlet~~apointonA’,tbsntbebolladuycosditbns canbeexpresaedfonnallyas fi,"(l)-~,"(t)=lc~~(t)-~~x~(~)~O
onA”
(1)
fi,‘(I)-
on A’
(2)
on A’
(3)
onA
(4)
iii$w
= fix=“(t)- iif*Yf)
(u,(t) + iut(tWL-(&r(t) f it&))‘” - d’(t)
I (~,L(r)-~,L(t))dtPX-iY30 A
where i = fl, jj’ and 6 are normal and shau stressesrespectively,u and u aredisplacanwnt componentsalongthe x and y axes respectively,X and Y arc the components of the resultant force on A and d’(t) is the derivative of the discontinuity in displacement on A’. SuperscxiptsL and R refer to the values of the functions on A as approacbd from points in S‘ and S” respectively. A function defined in S‘ is denotedby subscript1 and in S” by subscript2. Equation(1) states thatthe surfacetractionsvanish along the unbendedsection of the real axis, A”. The second boundary condition, qn (2), states that the tractionvectoris continuous acrossA!. The thii boundary condition, eqn (3). states that the strain difference is specified on A’. Finally, qn (4) states that the components of the rest&ant force on A are assumed to be zero. Stresses and displacements can be written in terms of the complex stress function, Wk(2) and w&T),as follows, E?k+j$=
w,(z)+ W&Z,
(5)
fik - EC + 2if& - ZW;(Z) + Wk(Z)
(6)
4/&b +iu~)=~‘w~(z)-z~(z)-‘w~(z) whereZESLfork=1andZESRfork=2,‘W~(Z)=IW~(Z)dZ,and’w,(Z)=Iwt(Z)dZ. In the above pr is the shear modulus, Su,= (3 - 4q) for plane strain and 3&= (3 - ti )/(l + generalized plane stress, where z+ is Poisson’s ratio.
(9
ulr
) for
Strainenergy rekase
ratefor interfacialarctrbctwcm dbimihr media
sl
If W,(Z) is analytically continued across the unloaded segment of the boundary A”, the equations for stress and displacement become
W,(Z)+ (Z - 2) ma
2(j$ - @, = W,(Z) -
4&(u&+i&,=Pw~(z)+
W&z) -cz-z,RGl.
(8) (9)
These equations are written in terms of a singk complex variable, W,(Z), defined in each haIf plane. imposing the stress and displacement boundary conditions on A’ yields aW,‘(t)+bW‘R(t)=4d’(t)
(10)
Where
This can be expressed as cW,L(t)+
w,“(t) -y
(11)
whuec=albisdefmedasthebi-eIaaticconstant. The dispkcement on A’, d(t), cam be expressed as fdlows d(r)=uo+iuo+qt+i&
d’(t) = q + if
(12)
(13)
Where
uo=lkJ
I.
-uo
I
00- 00”- uo=
(14) (1%
and r) = (VRL-~~R)+(~‘-t)TI)+(l)yL--t)yll).
(16)
In qns (12Hl6), q~ is the strain due to residual stresses, qr is the strain due to thermal kading, Q, is the strain due to mechanical loading, R is the radius of curvature of A’. u. and u,tmerely represent translation of A ’ in the complex pkne. As an exampk of the strain due to mechanical loading, consider the plate specimen under se&equiIibrating bending and compression loading shown in Fii. 2. For this specimen, strain along the interface can be determined from
where, as shown in Fii. 2, y L = y - h. 12is the distance from the neutral axis of the epoxy plate of depthh,,y= = y + ir./2 is the distance from the neutral axis of the aluminum plate of depth ri, R L and R JI are the radii of curvatures of the neutral axes of epoxy and ahuninum plates respectively, due to flexure alone. This geometric description for bending analysis of two bonded plates follows closely the methods used by Boley and Weiner[l for thermoelastic analysis of bending
D. R. MULVlLLE d al Y
-x
isa
-6
-_
_I
w-w
I”= WkJ2 _-WV
ha
Y8
Me
ALUYINUY
c
Fi.2.
fit 3
-
S*
Phtespechnlu6raeK~~d
comorarioalordial.
ad
buckhng of bi-metallicbeams. Evalw this expression on A’ correspondsto settbg y = 0 in the relations for yL and yR d&cd above. XX, and XXI arc uniform sw applied pa&l84 to the bond in epoxy and aluminum respectiva and 4%and & arc Young’smoduli of epoxy and ahuninum respcctivcly. Strain along the interface due to thermal loading can be determined from q-wL-‘I)TI
=(al-a2jfT.-Tp)
(181
where a, and
a2 are coefficientsof thermal expansion of the epoxy and ahuninum respectively and T, and Tp are the initial and final temperatures respectively. Strain due to residual stresses may be determined using photoelastic technkptcs or other cxpcrimcntal methods. Using (13) in (11) we have cW,“(t)t
W,“[+
tig.
WI
It now remains to solve this equation for the complex stress function and to determine the resultiry stress componenta. From this result the strain energy release rate, T, will be formulated for crack propag&on at the interface Prvbietn solufion The solution for this non-homogeneous quation is obtained by evaluating the following
where the Plemelj function x*(Z) = (2 - I)“‘(2 + 1)’ is the solution of the homogeneous Hlbert equation cXrL~f)+XIR(~)=O*
(22)
In these quatiom r = (112)+ iA and A = 00s c)12*. The stresses are then determined by substituting the value obtained for W,(Z) into the expression for the stresses (8) evaluated on A’. The stresses on A ’ are expressed as
(23)
f
-1
1
0 NORMALIZED BONOLENGTW
Fig. 3. Nard stresses alongthe bondu&r tensile badiagd
D. R. MULVILLE ei I
/ f
ii
I
Brain energy release rate for interfacialcracks betweendissimilarmedia
561
Y
s2tt1
(- 1.01
( 1.01 .
X
P2(1) J
Fii. 6. Resultantstress vectoron upperand lower half spaces.
wherep,(t) = -j$(t)
and s,(t) = -xy,(t)
and
where p&) = -fi&) and s&) = -XC&)+Equations (25)and (26)can be written in the following form and
Imposia the stress boundary condition on A’ yields iP,(?) = -i&(f)
(29)
and thus P,(t) + P&) = 0.
(30)
Consider the work, U, done by the stress vector acting along the bond (-&I), as folIows (31) Imposing the displacement boundary condition (3),and the relation between the stress vectors in (29) yields
P&)&t) dt.
(32)
From the assumed d(t) on A ’ (12) and (27) we have U=Re [i
or
I_’ i(pl(t) I
- is&))
(lo- iv0+ vf- i&)dr]
(33)
562
D. R. MULVILLE
er al.
Using the resultant foice boundary condition (4) on A’, the first integral vanishes. Using eqn (23) the second integral can be written as
rl(f+2U[)(rll-i~)X,L(l)dl -;(I
t4A')f') (qt-i&)X,L(t)dl]
(35)
and then evaluated using methods based on homogenous polynomials[8j to compute the residues, i.e. the coef%zientsof I/t, directly. This process yields the following expression for the stored elastic strain energy U = 2n(l -t4A3 Q o+b 2 The strain energy release rate,Y,
2+1’+ I’ -(l t4hq. 3R 16R2
(36)
isdefined as the decrease in stored elastic strain energy per
unit of crack increase, dnt. Thus
yApp!L.
(37)
From the geometry shown in Fig. 7, dlldm = -1 and using (36) we have ~ = 2p(1+4h3 a+b
$1 -y+$#
+4A')]
or Y=
q2t -+&(l --CL1cc2 > 2a(I t4A2) Xr+l+!&tl
t4A2)
1.
(3%)
(3%
ThisexpressionforYcanbeobteineddirtctlybycval~cqn(37)usinqtheLiebnizf~~ for differentiationof integrals provided the limits of the integral are d#erentiable and the kernal and its derivative with respect to I arecontinuous,Mast[9].The expressionfor U,eqn (35)),s&is&s these conditions, and the result obtained using this method is identical to that presented above. AIthough both methods yield identical results, there is a basic dEereace in the nature of these appn>acheS. In the h5t mdbod, the totalstrainenergy,U, is evaluated by integWug the product of the stress and displacement vector on A’. Y is then determined by d&rentiating U with respect to crack length, i.e. by determining the change in total strain energy due to a change in crack length. In the second method Y is determined by integrating the derivative of the product of the stress and displacement vector along A'with respect to crack length. This method, u&g the Liebniz
Fig. 7. Strain energyreleaserate vs crack length/initialbondlengthfor varioussurfacefinisheson specimens loadedin bending.
Strain smrgy r&case rate for itWf&&l cracks between dissimiIar media
543
formula, sums each incremental change in strain energy on A’ whereas the fhst method deals with the change in the total strain energy. Using the ~~ of % =(3-q)l(l+yt) and *=&&tz(I+a) for guneraiizod phme stress, the strain energy rekase rate formulation above can be expressed as follows
(41)
These expressions for the strain energy &ease rate are based on the plane problem of two elastic half spaces bonded alung A ‘, and can be used to analyze the following loading conditions: residual stresses, uniform thermal loading, forces applii parallelto the bond to one or both half spaces, bending loads applied to one or both half space or any combinations of these loads.
Enthe precedmg analysis, both materials were assumed to be linearly elastic, ~~~ne~s and isotropic. While these are useful assumptions from an analysis point of view, they often do not adequateiy describe the behavior of real mater&. ‘Inthe fohowing experimental section, epoxy-ah&turn bonded specimens are subjected to external toading which results in interfacial crack propagation, At room ternary most epoxy systems behave in a linear elastic manner, however, at ~~~~ unhip the post-cure ~rn~~~ the behavior becomes viscoehtstic. The preceding analysis does not describe the stress or displacement field for viscoeku&ic media, nor does the fracture mechanics analysis account for time+endent behavior, In addition to problems which may arise due to the viscoelastic nature of the materials, there are probiems due to viscous ffow and non-linear behavior in the vicinity of the crack tipThis analysis does not account for the energy absorbed by plastic gow uear tBt eraek tip. The function d(t ), defhted in qns (12)and (13),assumes that the resultingdisphreemanton A’ after bonding can be written as the difference between the displa&cmentin S”, and in S” before bonding. This is valid if the ratio of the Young’s Moduliof the materials is large, i.e. if one of the m&aids ads as a rigid body, Otherwise the resulting ~spla~rn~t on the i&ufacc mst be considered an unknown function related to the material properties. fn the ex~~~~ investigation conducted in this study, the modulus ratio was approx. 26; therefore qn (12)is a reasonable estimate of the resulting displa~e~nt on A’. An additional point with respect to the definition of d(t) is that the displacement along the bond, A’, in S’- and S” is assumed to be a linear function of position, f. This assumption may not be accmnte near the ends of the bond.
Experimental studies were u~de~ken to make q~nti~tive m~sureme~ts of the fracture parameter J for interfacial crack propagation. Using the amilysis of the gmzediq section, specimen designs were chosen and loading comities seWed for which cracks would propagate at the interface between two dissimilar materials. Specimen design studiis included phot~~ti~ observations and strain gage ~ns~mentat~n of various specimens to determine how closely the assump~~s of the theoretic analysis were satisfied. Alu~nu~~xy specimens were fabricated to conduct a series of interfacial crack propagation studies for various surface finishes on the alu~num surface of the bond. Restrained shrinkage along the bond during soli~~~n of the epoxy and ~~erenti~ thermal contraction during post-curing gave rise to a Aeldof residual stresses, whose e&t was included in the analysis. Details of the expend study are
564
D. R. MULVILLE
et al.
discussed elsewhere, Ib4ulviueflO,111,and therefore, only the results of tlmc studies will be presented here. Using the formulation of the theoretical section, the strain energy release mte 5,was calculated at various crack positions along the bond for specimens similarto that shown in Fii. 2. These specimens were loaded by applyingcompression and bending to the aluminumplates. Data shown in Fig. 8 compares the results for beading tests ou glass-peeoed, milkd and sandblasted surfaces with polished ahtminum surfaces. These values are generally greater than the value of 9rc= 0.5 in. lb/ii.’ for the buik epoxy system reported by Mostovoy and Ripling[l2].However for the polished specimens, the value of Y, is approximatelyequal to the bulk value. Results of these studies indkate that the strain energy release rate remained relatively constant as the crack propagated along the bond. Therefore, S, can be used to analyze the effects of various surface treatments and preparations on the energy required for crack propagation.
0.2
0.3
04
0.6
0.6
0.7
06
TOTAL CRACNLENGln~wTIAL 9cw WdTH
0.9
d
,II
CONCLUSIONS The purpore of this investigation was to characterize crack ptopaoation at the interface between two dissimiiar mater& using a formWon based on strain energy &case rate. To this en& a theoretical analysis was developed for the plane problem of two elastic half spaces in contact along an arc where one of the mater& acts as a relatively rigid body. The stress analysis included the cases of residual stresses, thermal loading and self equilibratingtensile or bending Ioadiugapplied to either half space. Using this analysis, the ektstic strain energy release mte, Y, was formulated as a fracture criterion for interfacial failure. Experimental studies conducted on specimens of epoxy bonded to ahuninum demonstrated that this failure criterion can be used to analyze interfacial crack growth. The technique presented in this paper provide a simple method for the study of interfacior failure in bonded materials. Results of this investigation have application in design of bonded structures, or in general problems of debonding and deiamination between dissimilarbodies. The analysis can be applied to composite materials in analysis of fiber-resin interfaces, in bonded joints or in welded structures where the crack propagates at the interface. This approach can be extended to consider the effects of crack velocity, fatigue loading, environment and various residual fields on the fracture toughness of bonded materials. A&ow&&wW--lke authors ackmwkdgc the support of tJte 0th of Naval Rmarcb. CammrsdmdTbeC&BolieUniraityotAarsriafathe~npatsdhsrein,~tadahojectNcrpkrRRM3M 4s.5451 lad SF 51-544-102-12432.
the Naval Sea Systems
REFERENCES [I] hi. L. WiHiis, The stresses around a fault or crack in dissimihr medii, B&tin ofthe&ismobgiccll Sot. Am. 4q2). mu04 (1959). [2] F. Erdogm, Stress distribution in a nmhomogcnmus elastic plate with cracks, J. App. Me& Ser. E, 5)(2), 232-236 (1963). [3] F. Erdogan, Stress distributionin bondeddissimilar materialswith cracks,I. Appl. Mech. Ser. E, 3x2). 403-410 (1965).
Strainenqy rekaxc rate forintcrfxcixlcracksbchvecndissimilarmedia
565
0. C. Sih andJ. R. Rice, The bea- of dissimilar mat&& with cmcks.I. AppLMech.Ser.E,31(3),41&482(KM). J.R. Rice xnd G. C. Sib, Pixnc pbkms of cracksin diasimii media,1. Appf. Mcch. Ser. E, 32(2), 418-423 (1965). L. M. Milne-Thomon, P&M Elastic Sys@wtr. Springer-Verlag, Berlin (1960). B. A. Boky and J. H. Weiner, Z’kcoryof Them& &ess, pp. 429-432. Wiiy, New York (l%o). P. W. Mxst and D. R. Mnivilk, The exprexsionof non-homogeneous polynomialsas venous polynominalswith appiicationsin caginccringmcchaniis, NRL. Repoti in preparation. [9] P. W. Mast, Graphsxnd tensormxnipuldon in compkx coordinxtcsxs probkm solvingtechniquesin plane-problemsof linear xnisotmpic dadcity; Doctod Dixrrtation. Nod Carolii State Univerxity (1972). [ IOj D. R. Mulvilk. C&act&tics of crack propa@& at the interfxce of two disximilxrme-&x,Do&ml Dissertation,The Catholic University of America (1974). [If] D.R.MolviMc aadR N. V&&w, Intedacixl CncL ’ , L Ad&&n 7.215-233 (1975). [121 s. Moxtovoy 8nd E. Rip@ Prxctw tol&ms of Morn, 1. Appl. Jwymer sci. 10. 1351-1371 (1966). [4] [S] [6] m [8]
(Received 28 September 1975)