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Cracks on bimaterial interfaces: elasticity and plasticity aspects
Materials Science and Engineering, A143 (1991 ) 77-90
77
Cracks on bimaterial interfaces: elasticity and plasticity aspects C. F. Shih Division of Engineering, Brown University, Providence, R102912 (U.S.A.)
Abstract Substantial progress has been made on the mechanics of interface fracture. An engineering program has emerged which allows the fracture resistance of interfaces to be measured and utilized. This recent development, assessed in an Acta-Scripta Metallurgica Proceedings, is discussed. Several results have been obtained on the plasticity aspects of interface cracks in which one (or both) of the constituent materials can deform plastically. The crack tip fields are members of a family parametrized by plastic mode mixity parameter ~. The J integral scales each member field. Analyses of finite width crack geometries loaded by remote tension show that the effects of load, ligament plasticity and geometry on the near-tip fields are adequately accounted for by the J integral. The progress is summarized.
1. Introduction For many advanced materials such as structural ceramics, ceramic and metal matrix composites and polycrystalline intermetallic alloys, interfacial and intergranular fractures are common and may in large part determine the materials' overall mechanical response. For these material systems it is the low fracture toughness that limits their use in engineering and structural components. The need to understand, quantify and improve the toughness of advanced materials has renewed interest in the elastic interface crack problem. In recent years substantial progress has been made on the mechanics of interface fracture, an account of which is given in an Acta-Scripta Metallurgica Proceedings edited by Ruhle et al. [1]. The critical ideas and developments are reviewed in Sections 2-4. The plasticity aspects of cracks on bimaterial interfaces in which one (or both) of the constituent materials can deform plastically have received some attention. Several key results are discussed in Sections 5 and 6. The small-scaleyielding near-tip fields of interface cracks are members of a family parametrized by a plastic mode mixity parameter ~. The J integral scales each member field. For "opening"-dominated load states the plastic fields bear a striking resemblance to mixed mode H R R fields for homogeneous media. However, there are surpris0921-5093/91/$3.50
ing effects. For example, the zones of large plastic strains and high triaxial stresses that develop near bimaterial interfaces are considerably larger than those that are found in similarly loaded homogeneous bodies. Solutions for finite width crack geometries loaded in remote tension showed that the effects of load, ligament plasticity and geometry on the plastic near-tip fields of interface cracks are adequately accounted for by the J integral. In general, both the J integral and the plastic mode mixity ~ are required for the specification of the near-tip plastic state.
2. Elasticity fields of interface cracks Solutions to specific problems of cracks lying along bimaterial interfaces of isotropic media, for which the characteristic stress singularity was determined by Williams [2], can be found in works of the 1960s by Cherepanov [3], England [4], Erdogan [5] and Rice and Sih [6]. Thereafter the subject received little attention. In recent years there has been an explosion of activity on interface crack mechanics. The progress is docum e n t e d in an A c t a - S c r i p t a Metallurgica Proceedings [1] and in a review article by Hutchinson and Suo [7]. The rapid pace of progress can be attributed to two concepts: (i) mode mixity and (ii) small-scale contact zone [8]. These concepts are discussed in Sections 2 and 3. © Elsevier Sequoia/Printedin The Netherlands
78
2.1. Near-tip stress fieM The near-tip stress field for an interface crack between dissimilar isotropic bimaterials is a linear combination of two types of singularities, namely a coupled oscillatory field scaled by a complex K and a non-oscillatory field scaled by a real Kin:
The two stress intensities have different dimensions: K =
[stress][length] 1/2 -it
Kil I = [stress][length] 1/2
The complex stress intensity factor K has the generic form
Re(Kr i*)
Im(Kr it) oij-(2ztr)i/2 6ij'(O;e)-I (2xr)l/20qII(o;e)
K = YTLI/2L -~ exp(@) Kill + (2~r)l/~ 6i~IIl(O)
(1)
Here r and 0 are the polar coordinates shown in Fig. 1 and the dimensionless angular functions oi](0;e), oi]l(0;e) and all(O)correspond to tractions across the interface at 0 = 0 of tensile, inplane shear and out-of-plane shear respectively. The representation in (1) is due to Rice et al. [9] and Oil( 0; e) and 0i/I( 0; E ) are given in that article; off(0) is the standard mode III angular function. The bimaterial constant is defined by [2]
1 In Ul//~l+
e 2n
(2)
,u2/,u2+lhu~]
where v is Poisson's ratio, kt is the shear modulus, u = ( 3 - v)/(1 + v) for plane stress, r = 3 - 4 v for plane strain and subscripts 1 and 2 refer to material 1 and 2 respectively. When e = 0 , tri](O;e ) and aiIl( 0; e) reduce to the standard mode I and II angular functions. The tractions at a distance r ahead of the crack tip take the form
g r it (°.' + ioyx)0=0= (2arr)l/2 Kill
(Oyz)#=ll- (2~rr)!/z
(4)
(3)
(5)
where T is a representative stress magnitude, L is a characteristic crack dimension, Y is a dimensionless real positive number and ~p is by definition the phase angle of KL it. The implications of the unusual dimensions of K are explained in Section 2.4. Dundurs [10] has observed that the solutions to plane problems of elasticity for bimaterials depend on only two non-dimensional combinations of the elastic moduli. The Dundurs parameters are (Z-- /zl(r2 + 1)-/~2(u, + 1)
l(,q + 1)+
+ 1)
/~l(~C2- 1 ) - ~2(7Cl - 1) - fli(Tc----~+~-,u2(~ l + 1)
(6)
The connection
is noted. The quantities Y and ~pin (5) depend on a and fl in general and on dimensionless specimen parameters. In other words, the calibration of a crack geometry is reduced to determining Y and ~pfor a range of crack length-to-width ratios, load ratios and a and ft. Y and ~ can be evaluated by stress analysis. To complete this subsection, two more results are provided. The displacement jumps at a distance r behind the crack tip are given by
6Y+ibx=(1 +2ie)cosh(zte) E* 6~=-~--U
Fig. 1. Interface crack tip region.
where the effective Young's modulus and shear modulus are given by
79
2
1 --
E* 2
E/ 1
-
~t* ,Ul
An equivalent definition can be given in (KI, KII , Kin) space.
1 q'
--t
E2
r
(9)
1 /*2
with E ' = E l ( 1 - v 2) for plane strain and E ' = E for plane stress. The energy release rate is related to K and KII ! by [11, 12] ~?
K. tan~p = K~
1 Igl 2 cosh2(;Trg) E*
KIll 2 ~ -2~* --
Kill COS~ = (KI2 + K n 2 + Klu2)l/2
(14)
The mixities ~p and ~bare indicated in Fig. 2(a). These definitions also apply to cracks in homogeneous materials.
(10)
The connection 1 / c o s h 2 ( ~ e ) = 1 - f12 is noted. 2.2. Change of phase angle with distance--phase index It is helpful to write the interface traction vector in the following way: t = {tl, t2, t3} ={oy x, Oyy, Oyz}= {Oyi}. For t = tz+iq we have
2.4. Mode mixity." e # 0 It is clear from Section 2.2 that tension and shear effects are inseparable near interface crack tips. A measure of the relative proportion of shear to normal tractions (or mode 1 to mode 2) requires the specification of a length quantity. For oscillatory fields the mode mixity is uniquely specified by
K r i,
t = Itl exp(i~p,)= (2ztr)l/2
(11)
The phase angle l~r is a measure of the ratio of the normal to in-plane shear tractions at a distance r ahead of the crack tip. When the distance changes from r 1 to r 2, the phase change as predicted by the K field is ~lrz--~ri=
e
lnfr21
(12)
\rl]
A convenient measure of material mismatch producing phase variation with distance is the quantity (180/z~)e ln(10)--- e*. It has the interpretation as the phase change in degrees for a decade increase in distance. As an example, for e = 0.05, as for a glass-alumina interface, the phase index e* is 6.6 °. 2.3. Mode mixity: e = 0 For e = 0 the mode mixity can be defined in the usual way. When all three modes are present, the mode mixity is fully specified by two solid angles, ~p and ~, in the space of the interface traction vector t={Ovx, o m Oyz}:
\ Oyyl r=L
(15)
cos+=(Oy// tltl},-0
The length/~ is arbitrary but must be unchanging for a material pair, i.e. L must be independent of the overall specimen size and specimen types. A length between the inelastic zone size and the specimen size is a sensible choice of L. For example, L = 100/~m is suitable for many brittle bimaterial specimens at the laboratory scale. Using the asymptotic field (1) or the tractions in (3), the mode mixities ~b and ¢ can also be defined in K space: tan ~ - Im(KLi~) Re(KL ~)
cos ¢ =(I KI
( 16 )
Kn 1 + K,n2) It2 K3
}
tanlp = ( °yx) %(Tyy l r ~O
cos+__(Oy/ tltl],-0
Ka
Im(~O')
Fig. 2. Mode mixities defined as solid angles in K space.
The mode mixities 4 and $ are indicated in Fig. 2(b). As a consequence of the oscillatory field, the traction ratio t,/t, varies slowly as r moves away from the tip. Let $, and $J~be associated with t, and i, respectively. As implied by ( 16), $, and q2 obey the translation rule
For orthotropic materials with one principal axis aligned with the interface and crack front normal to the x -y principal plane, the interface tractions ahead of the crack tip are
(20)
K,
(17) This would not be a big change for moderate variations in i. For example, for an epoxy-glass interface, E = 0.06, the mode mixity change for a decade change in distance is G2 - $~i= 7.9”. We emphasize that the choice of reference length i is arbitrary. The change from one choice of t in the definition of 4 to another simply involves the transformation rule ( 17). Nevertheless, taking i to be broadly representative of the relevant microstructural length, or perhaps the plastic zone size at fracture, is advantageous for interpreting mixed mode fracture data. This aspect is discussed in Section 4.
where 7 depends on the elastic constants. A unified concept of mode mixity based on generalized traction components, e.g. (20), has been discussed by Suo [16] and elaborated upon by Wang et al. [ 191. By taking advantage of this approach, Suo has shown that results for isotropic bimaterials can be immediately generalized to anisotropic bimaterials. Explicit formulae for the stress intensity factor, displacement jumps and the energy release rate are given in refs. 16 and 19. Similar results for dynamically propagating cracks can be found in Yang et al. [47].
2.5. Anisotropic bimaterials Interface cracks in anisotropic solids were first analyzed by Gotoh [ 131, Clements [ 141 and Willis [ 121. The structure of the interface crack tip field has been completely identified in recent works by Qu and Bassani [15], Suo [16] and Wu [17] and analytical solutions have been found to several boundary value problems. The crack tip field is a linear combination of two types of singularities, namely a coupled oscillatory field scaled by a complex K and a nonoscillatory field scaled by a real K, [ 161:
3. Small-scale contact and non-linear zones
Re( KriE) _ , u8j=
(271rjl/2
Oil
Im(Kr”) fe)+
(2xrjl/2
S,=]S] cos Ijl-tan-‘(2&)-c I
te)
Here the dimensionless angular functions also depend on elastic constants and these functions have been extracted by Choi et al. [ 181 from the near-tip solution in ref. 16 where the structure of ( 18) for a general anisotropic bimaterial is discussed. As for isotropic bimaterials, the two stress intensities have different dimensions:
K, = [stress][length]“*
3.1. Small-scale contact zone Overlapping of crack faces and contact in isotropic bimaterials does not involve the mode III field. Therefore the present discussion is confined to the coupled in-plane modes. According to the linear elasticity solution (8), interpenetration of the crack faces, S, < 0, will occur at some sufficiently small r. Writing 6 = ayy+id, and using (8), we have
_2 *ij
(18)
K = [stress][length]1’2-”
We direct attention to isotropic bimaterials.
(19)
In L ( r II
(21)
The contact zone size, assuming that it is small compared to the crack size L, is estimated to be the largest r for which the opening gap S, just turns negative [81. For E > 0, r,= L exp
- [n/2 + + - tan- ‘(2c)] &
(22)
For E < 0, change E to - E and 1/, to - ly in the above equations. As an example, if L = 1 cm, 1~1 Q x/4 and I&l< 0.03, then r, < 1 A, which is smaller than all physically relevant length scales.
81 Comninou [20] and Comninou and Schmueser [21] have given, within linear elasticity theory, a more precise analysis of near-tip contact. Their contact zone sizes are comparable with the estimates by (22). Through full-field numerical studies, Aravas and Sharma [22] have shown that Comninou's asymptotic solution is valid at distances which are much smaller than the contact zone size. They also demonstrate convincingly that Comninou's solution is embedded within the "open crack" asymptotic solution (1) which is valid at distances large compared to the contact zone but still small compared to the crack size. Thus Comninou's asymptotic solution has no relevance when the contact zone size is smaller than the fracture process zone or the inelastic zone. We may conclude from the above discussion that the crack tip state is characterized by the complex K if rc/L 4. 1; Rice [8] has suggested using r~/L < 0.01. While the number 0.01 is arbitrary, the result that follows does not depend sensitively on it. Combining the above condition with (22) leads to the following estimate of the range of ~p in which K is applicable [23, 24]. For e>0, - Jr/2 + 6.6e < ~0< ~r/2 + 2e
(23)
For e < 0, - Jr/2 + 2e < ~0< Jr/2 + 6.6e
(24)
In the above equations, tan-~(2e)is approximated by 2e. The range of validity K is discussed in greater detail in refs. 23-25. 3.2. Small-scale yielding Several results have emerged from recent studies by Shih and Asaro [26, 27], Shih et al. [28] and Zywicz and Parks [29] on the plasticity of bimaterial interface cracks. They observed that the small-scale-yielding near-tip fields are members of a family parametrized by a plastic phase angle ~ defined by ~=~+eln
\oo LI
=~2+eln
(25)
where o0 is the yield strength of the weaker (lower strength) material. L is the characteristic crack dimension introduced in (5). Note that the notation in (25) is different from that used in the original publications [26-28] but is consistent with the notation used throughout this paper.
The plastic zone size rp, defined by the maximum radial extent from the tip to the elastic-plastic boundary, has the form rp = A(~)IK[2.~ o0
(26)
The dimensionless factor A(~) depends weakly on the elastic and plastic properties of the two materials. For plane strain A(~) increases from 0.15 to about 0.65 as ]~] increases from zero to :r/2. ~ may be interpreted as the phase angle of the plastic zone and is analogous to ~p, the phase angle of K L ~'. The plastic zone sizes as measured by A are rather similar to those for a homogeneous material of yield strength o 0. The latter plastic zones are parametrized by a plastic mode mixity parameter M P which has been related to the elastic mode mixity M" [30]; M ~ can be related in a simple way to ~o. These mode mixity parameters are discussed in Section 5.3. When both in-plane and out-of-plane modes are present, the size of the plastic zone is given by rp ='~(~,~ b)
IKI z + 7Ku, 2 z o0
(27)
where 7=(E*/2/~*)cosh2(Jre). The dimensionless factor ,~(~,~b) is a function of ~, defined by (25), and 4, defined by (16), and also depends on the elastic and plastic properties of the material pair. The numerical values for A(~,~) have yet to be determined. Estimates of/~, based on smallscale-yielding solutions for combined mode I, II and IlI crack problems for homogeneous materials [31], range from 0.15 to 0.65. A is a minimum for ~ = ~ = 0. 3.3. K dominance Let r~ denote the characteristic radius of the region controlled by the interface crack singularity field (1). r K will be some fraction of a relevant dimension of the crack geometry. At distances sufficiently close to the tip, the field (1) does not apply because of the presence of (i) a material non-linear zone (e.g. plasticity), (ii) a contact zone and/or (iii) small-scale heterogeneities and irregularities r i (e.g. grains, voids, microcracks, interdiffusion zones). If these zones and/ or irregularities are sufficiently small compared to a characteristic crack dimension L, then there exists an annular region rK> r ~ rp, r~, r i
(28)
82 in which(l) is dominant. In other words, if (28) is satisfied, the complex K uniquely measures the fields in an annular region surrounding the crack tip. Estimates of r K for finite width crack geometries have been obtained by O'Dowd et al. [24] and for multilayered materials by Gu and Shih [32]. Their studies show that rK -~ L / I O
(29)
where L is the shortest of the crack length, the uncracked ligament length, the layer thickness or the distance between the crack tip and the point of load application. Combining (28) and (29), we get the sandwich inequality L / l O > r~,> rp, rc, r i
(30)
Thus a conservative condition for the existence of a K annulus is L > lOOx(rp, rc, r~)
(31)
The latter provides the size requirements for a Kbased linear elastic fracture mechanics approach. Observe that the contact zone size depends on the phase of the applied load but not on its magnitude (see (22)), while the size scale of irregularities is a property of the bimaterial. In contrast, the plastic zone size is proportional to IKI 2 (see (26) and (27)), i.e. it scales with the square o f the magnitude of the applied load. If the interface is relatively tough or if one material has a low yield strength, a sizeable plastic zone can develop before the onset of fracture. In this case the use of K to correlate the onset of fracture is not justified. The plasticity aspects are taken up in Section 5. To summarize, we note that if the size requirements (31) are met, then pertinent information concerning crack geometry, material mismatch and applied loads is communicated to the crack tip through K only. Details that have no effect on K also have no effect on the near-tip field since the latter is uniquely characterized by K. Therefore K provides the boundary conditions for the inner region where the fracture processes occur, i.e. the onset of fracture can be phrased in terms of K. This aspect is elaborated upon in the next section. The above arguments are identical to those made for the relevance of the stress intensity factor in linear elastic fracture mechanics (LEFM). In LEFM methodology and in the rapidly developing subject of interface fracture mechanics the underlying idea is K dominance.
4. Interface toughness curves
It has been generally observed in experiments that cracks in isotropic, homogeneous, brittle solids seek to propagate on planes ahead of which local mode I conditions prevail. Consequently, one single parameter, K~c, can be designated to each material to quantify its resistance to fracture. By contrast, whenever planes of low fracture resistance exist, cracks may be trapped on such planes, regardless of the local mode mixity. Orthotropic materials such as composites and brittle crystals provide examples where definite weak planes exist: longitudinal planes for composites and cleavage planes for crystals. Interfaces offer another example when they are brittle compared with the substrates. A well-documented experimental fact is that fracture resistances for such weak planes depend strongly on mode mixity. Therefore it is toughness values at various mode mixities that fully characterize the fracture resistance of a weak plane.^ For a given mode mixity ¢ and % the interface fracture toughness F is defined as the energy release rate at the onset of crack growth. The fracture toughness F(~,,¢) is a property of the bimaterial interface. For a given bimaterial interface it is a surface in K space, which in principle can be determined directly by experiments. Upon loading, a crack will not propagate unless the driving force reaches the toughness surface, i.e. the mixed mode fracture condition is Y(~b,¢)= r(4,,O)
(32)
The fracture resistance is unambiguously specified by a surface F(~,¢) together with a length/~ for the definition of V). Specifically, F(~b,¢) is the critical value of the energy release rate required to advance the crack in the interface under the mode mixture ¢ and ~b, the latter being defined by the relative magnitudes of the in-plane shear to normal tractions at r = £ . This engineering approach to quantify the fracture resistance is an extension of the existing theory for isotropic solids. The conceptual basis for this approach is summarized in ref. 8 and in several articles in a volume edited by Ruhle et al. [1]. Specimen geometries suitable for interface fracture testing have been rigorously calibrated by Suo and Hutchinson [33], Charalambides et al. [34], Wang and Suo [23] and O'Dowd et al. [24] for a wide range of material combinations as characterized by Dundurs parameters (6) [10].
83
Interface fracture toughnesses of several bimaterial systems have been measured by Wang and Suo [23], Cao and Evans [35], Liechti and Chai [36] and Stout et al. [37]. Sandwich-type test specimens in which a thin layer of material is sandwiched between two layers of a second material are used in refs. 23 and 35. These specimens have the advantage that they can be calibrated easily in terms of the stress intensity factor of standard test specimens for homogeneous materials [33]. The test specimens used in refs. 34, 36 and 37 consist of two slabs of partially bonded dissimilar materials. Two test specimens are shown in Fig. 3. These crack geometries can produce near-tip fields ranging from predominantly tension to predominantly shear. Test specimens for uniaxial composites have been calibrated by Bao et al. [38] for arbitrary elastic constants. Choi et al. [18] have calibrated specimens which are suitable for cross-ply laminates. Schematic diagrams of the specimens are shown in Fig. 4. Interface toughness data from ref. 36 for an epoxy-glass interface are shown in Fig. 5(a). It can be seen that the toughness curve has a minimum at ~ ~ 20 °. In the plot the mode mixity ~ is based on [, = 12.7 mm, the thickness of the epoxy layer. Suppose /~ is taken to be two orders of magnitude smaller, say L = 100/~m. The smaller /~ lies within the K-dominant zone and is more
P
iI
@.,0"
F1 P
P
II ;; ", ..;.;.-, ~. ".... ". -. ,.. o.. ;.,
, • ". : ' : . :..'. : ;'II ~ ~ 400
A
t
A
@,,90*
Fig. 4. Specimens for the determination of mixed mode delamination toughness for cross-ply laminates [18].
i
i
i
D
g-.
I"1
Epoxy/Glass
I
30
I, = 12.7ram D
~b D
t.
(a)
I
0 -90
"°
I
I
I
-45
o
45
90
Mode Mixity
g-. I
J (a)
"-" 120
P
e~
~ I
/b/
(z) I_
,,
Epoxy/Plexiglass L = 0.1ram
80
I
2O* -r
o/
I
I
I
140
:
Fig. 3. Specimens for the determination of mixed mode interface fracture toughness: (a) Brazilian disk [231; (b) asymmetric four-point bend bar [24].
!
4O*
!
60*
8O*
Mode Mixity Fig. 5. (a) Data for an epoxy-glass interface; ~b based on L = 12.7 nun [36]. (b) Data for an epoxy-Plexiglass interface; ~bbased on/~ = 0.1 nun [23].
84
representative of the plastic or process zone. Using the translation rule (17), the new toughness curve is shifted backwards by about 16 °, resulting in a curve more nearly symmetrical about ~ = 0. The data from ref. 23 for a crack in an epoxy-Plexiglass interface are shown in Fig. 5(b). Wang and Suo's data [23] and Liechti and Chai's data [36] (adjusted for/~ = 100/~m) show a strong increase in toughness as the relative proportion of the shearing component (or mode mixity at /~ = 100/~m) increases.
5. Plastic fields
In the remaining sections of this paper, attention is directed to in-plane deformation modes and "opening"-dominated load states.
5.1. Plastic near-tip field It is noted in Section 3.2 that the size of the plastic zone is controlled by the yield strength of the weaker material, e0 ((26) and (27)). This means that over length scales which are comparable to the size of the dominant plastic zone, the stress levels are set by o 0. In contrast, the asymptotic behavior of the crack tip fields is governed by the hardening characteristics of the material with the lower hardening capacity. Furthermore, as r---0, the interface crack behaves increasingly like a crack lying at the interface of a plastic solid and a rigid substrate. A detailed discussion is given by Shih and Asaro [27, 39]. Consider an interface crack between a rigid substrate and an elastic-plastic solid. Under uniaxial tension the elastic-plastic solid obeys the Ramberg-Osgood stress-strain relation o/a0 = e/eo+ a(o/ao)", where a is a material constant and e o = oo/E. The small-displacement-gradient near-tip fields for "opening"-dominated load states, - : r / 6 ~<~ ~ n/6, can be organized in the form [27, 39]
(j],/(,,+l)
°i/= °° \ aooeor]
ho(O,f; ~,n)
(33)
Here J is Rice's J integral [40] and hij is a bounded function which is slowly varying with ? ( -= r/(J/ o0) ). The dimensionless function hi/depends on the plastic mode mixity ~, i.e. the fields are members of a family parametrized by ~ which is defined in the next subsection. Each member field is scaled by the J integral. Full-field analyses
have shown that the form in (33) describes adequately the actual fields. This is elaborated upon in Section 6.1.
5.2. Separable asymptotic solutions We have investigated separable asymptotic solutions of the H R R type, satisfying vanishing tractions on the crack face (0 = n) and vanishing displacements on the bond line (0=0), corresponding to a crack lying at the interface of an elastic-plastic material and a rigid substrate. The problem was formulated in terms of a stress function and the resulting ordinary differential equations were discretized by a finite difference method discussed in ref. 30. We have also formulated the two-point boundary value problem in the weak form using a stress function and a 1D finite element discretization [41 ]. In both cases we have attempted to solve the linearized equations by the Newton-Raphson method, but the iterative solutions did not appear to converge. Gao and Lou [42] claimed to have obtained, for certain material pairs, a family ~ f separable interface crack solutions parametrized by a mode mixity parameter Np. However, by their own admission the displacements are incompatible at 0 = 0 and this is definitely an unsatisfactory aspect of their solutions. A more recent analysis by Wang [43], for an elastic-plastic material and a rigid substrate, indicates that a separable asymptotic solution exists at a single mode mixity, the value of which depends on n. Indeed, Wang's solutions for n = 3, 5 and 10 are similar to the solutions reported by Shih and Asaro [27] in their Fig. 8 for ~ = - 0.551 and - 0.027, which display features consistent with a separable mode-I-like field (with a small positive shear stress) at distances r/rp < 10 -6. However, the solution for = 0.496 in Fig. 8 as well as the solutions in Fig. 12 (for a different material pair) of ref. 27 do not resemble separable solutions. In summary, the crack tip fields ( r o 0 ) for certain load and material combinations appear to be consistent with a separable form, while others bear no resemblance to separable solutions. These are puzzling aspects indeed. Nevertheless, there is sufficient numerical evidence to conclude that the domain of validity of an HRR-like separable solution, if one exists, is vanishingly small--smaller than all length scales of physical relevance. Indeed, stress distributions consistent with a separable solution are found only at distances well within the finite strain region (see Fig. 8 of ref. 27 and Fig. 5 of ref. 39).
85 Finite deformation effects are discussed in Section 5.4.
5.3. Mode mixity The mode mixity of the plastic near-tip fields is uniquely specified by ~ evaluated at a distance/2 within the plastic zone:
tan
=
"~' (°r°(0= 0)/
(34)
If/2 is comparable to the fracture process zone size, then ~ parametrizes the relative proportion of the shear to normal interface tractions near the process zone. Numerical studies have shown that the value of ~ is relatively insensitive to distance when evaluated within the interval 2 ~
where C is less than 0.01 when /2j and /22 are within the interval 10 9rp < r < 10-1rp [27]. We may conclude that the phase change with distance within the plastic zone is weaker than the phase change in the K-dominant zone (see (17)). For example, we find no indications of crack face contact over any length scale of physical relevance. In this respect, plasticity has mitigated a pathological feature of the linear elasticity solution. A strong resemblance between the "opening"dominated plane strain interface near-tip fields and mixed mode H R R fields in homogeneous material has been demonstrated in refs. 26 and 27. The mixed mode H R R fields are members of a family parametrized by a plastic mode mixity M p defined by [30] Mp=_2 tan ~(Ooo] 7g
(36)
\ OrOl r ~0
With this definition M P = 1 designates a mode I field while M P = 0 designates a mode II field. The magnitude of the fields scales with J U(n+ 1):
(jill(n+,) - oij = oo \ aoosolr I
O,i(O;MP, n)
(37)
where 1 is an integration constant which depends on n and M P. Under small-scale-yielding conditions a one-to-one correspondence exists between M P and the elastic mode mixity M s defined by (36) but evaluated within the Kdominant zone. Mixed mode plastic fields are reviewed by Shih and Suresh [44]. The connection between (33) and (37) has been exploited in refs. 26-28 and 39 to the extent possible, in as much as the fields of interface cracks do not appear to be separable over length scales which are physically relevant.
5.4. Finite deformation effects The crack-opening profiles and the fields within the zone of finite strains have been studied using a small-scale-yielding formulation [28]. Under these conditions J and K are related by 1 -/5 2
J=
/ . ~ IK[:
(38)
Figure 6 shows the crack-opening profiles obtained by the small-scale-yielding formulation for the three load states ~ = - Jr/6, 0 and ~/6 defined by (25). The notches have opened to more than five times their initial notch width and further deformations result in nearly self-similar shapes. All three profiles are obtained at the same magnitude of K and are drawn to the same scale. It can be seen that moderate amounts of shear parallel to the interface increase the crack tip opening, and negative shear has a stronger effect (or vice versa if the lower material is the weaker material). By defining 6, as the separation where 45 ° lines intersect the deformed crack faces, a relationship of the form
at = d(aeo, n)
J o0
(39)
has been derived for homogeneous material [45]. Applying the 45 ° procedure to the notch openings in Fig. 6, we get d=0.61, 0.56 and 0.67 for = - J r / 6 , 0 and ~/6 respectively. These values are considerably larger than the mode I value of 0.51, the largest possible value for homogeneous material. These results suggest that crack tip blunting in bimaterials is more severe than that encountered in homogeneous materials loaded to the identical J value. Contours for equivalent plastic strain, gP = 0.1, 0.2 and 0.5, for the three load states are shown in the plots in the right column of Fig. 6. The rela-
86 $.0 ~ = -,r/e
4[.0'
0.1
~1.0, 0.2
Z.O, 1.0-'
°.~~...o..°°°..°0 (u) 1.0;LO3.0 xl(~l~,)
0.0
-1.0
-~LO -Z.O
-1.0
0.0
~.0" 4~.'~ 0
0~.1
Z.O. 0.1
1.0" @.0
-LOxl(.11c,O.o)O l.O 7.0
-l.O
-LO
4.0 = ,r/O
3.0
0.1
,-~ z.0
(e)
1.0 0.0 -|.0
,i~o°, .
.
-L.,
.
.
,
.
,.o
.
.
.
(f)
, .... , .... ,
1.0
z.0
x/0/%)
3.0
4.0
Fig. 6. (a), (c), (e) Crack-opening profiles for load states ~ = - n/6, 0 and n/6 at identical J value. (b), (d), (f) Contours of gP =0.1, 0.2 and 0.5 plotted on the deformed geometry. Contours for a corresponding homogeneous material under mode I are also shown in (d).
tive extents of the plastic strain contours are broadly consistent with the amounts of crack tip opening. For comparison, we include contours for gP= 0.1, 0.2 and 0.5 (drawn to the same scale) for a crack in the corresponding homogeneous material (material with properties identical to the upper material) and loaded to the same value of J. It can be seen that the spatial extent of the latter plastic strains is significantly less than those for the interface crack. To summarize, the zone of
finite deformation extends over a distance of several J 0"0 - 1 . More details about the finite deformation fields are given in ref. 28. 6. Large-scale-yielding
solutions
In this section we elucidate upon the effects of crack geometry, load and ligament plasticity on near-tip fields. This is done in the context of a row of collinear cracks lying along a bimaterial
87
;..ei
2w
-
--
2w
XX) I
..... :
-20 t zb-4-z0q--zb-4-z0-q MATERIAL
{
x
2
(Crxx)z
'¢t
qtl
J'\3o
J=8.o
.
•. .
J=30
,f=1.0
a/w=2/a.I/z.I/a, I/9.1/100 i
-8.0
i
i
1
1
.
-5.0
.
,
I
. . . .
-4.0
|
(b)
. . . .
-3.0
I
. . . .
-2.0
I
. . . .
-1.0
0.0
-8.0
log(r/a)
-,5.0
-4.0
-3.0
-2.0
-1.0
0.0
log(r/a)
Fig. 7. (a) Infinite row of coIlinear cracks. (b), (c) Stress variations near interface for etriesfor J/aoeoa= 0.I, 1.0, 6.0 and 30.0.
a/w= 2/3,
I/2, I/3, I/9 and I/I00 geom-
interface as shown in Fig. 7(a). Crack length-towidth ratios a/w differing by two orders of magnitude are considered. Throughout this section geometry refers to a/w ratio, since this is the only geometric parameter of the problem, and load refers to the remotely applied tensile stress cr~°.
media, we investigate the behavior of the smalldisplacement-gradient full-field solutions normalized by a singularity of the HRR-type:
6.1. Stresses near the bond line For the purpose of making the connection with existing solutions for cracks in homogeneous
Suppose the numerically determined stresses possess an H R R structure of the form in (37) as r--" 0. In this case the plots of the normalized solu-
o~y/o0 di/- (Jlaooe(,r),/(. +1)
(40)
88
tions, 6ij vs. r (for fixed 0), must approach the asymptotes (appropriate to the respective mode mixities) as r ~ 0. Since the mode mixities of the different a/w cases being investigated are nearly identical (as they must be for the present boundary value problem), the curves of 6oo vs. r/a, for the several a/w ratios and applied loads, must converge to a single distribution as r--, 0. On the other hand, if the numerically determined fields obey the form in (33) as r ~ 0 , then plots of 6oo vs. r/a will show a weak dependence on r. In Fig. 7(b) the stresses ahead of the tip of a Ramberg-Osgood material with n = 5 for a/w = 2/3, 1/2, 1/3, 1/9 and 1/100 are compared. Specifically, 6oo vs. log(r/a) is shown for the three load levels J/ooeoa = 0.1, 1.0 and 6.0. These load levels correspond roughly to small-scale, contained and large-scale yielding (relative to the crack length) respectively. For example, at J/ ooeoa=6.0 the ligaments of the geometries a/w = 2/3, 1/2 and 1/3 are fully yielded. It can be seen that the normalized stresses of the different geometries form three distinct distributions corresponding to the three load levels. For J/ooeoa=0.1 the plastic zone extends a distance of about 0.01a ahead of the crack tip; hence it is not unexpected that these stresses, normalized by the plastic singularity, form a distinct distribution of their own. At higher loads a definite trend can be seen. Observe that the distribution for Jlooeoa=l.O approaches that for J/ooeoa = 6.0 for r/a < 10 -5. The distributions for J/ooeoa=6 and 30 (indicated by a dashed line) are nearly indistinguishable from one another. To summarize, the comparisons based on o00 indicate that the effects of load and geometry on "opening"-dominated near-tip fields, ~ = 0, are adequately accounted for by J. Moreover, the hoop stress distribution resembles a separable form. Figure 7(c) shows the behavior of the normalized shear stress 6~o for the four load levels considered above. The curves form three distinct distributions, indicating that the a/w ratios are adequately taken into account by the J integral. We can see that the tendency for oscillatory behavior is reduced at higher values of J. It is of interest that the magnitude of the shear stress relative to the hoop stress decreases as the plastic zone size becomes comparable to the crack length (or the relevant crack dimension). In contrast, shear and hoop stresses are always of comparable magnitude under small-scale-yielding
conditions. Large-scale-yielding effects are discussed in greater detail in ref. 46.
6.2. Relations between J, 6 and o ~ The dependence of J on the remotely applied stress o ~° is of interest in fracture analysis. Plots of J/aoeoa vs. the applied stress for the n = 5 material are given in Fig. 8(a) for four crack
8.0
b-e/2 ............ b-a --.--
Periodic Periodic
Periodic
6.0-
b-2a
,/ / /
........
n=5.0
% % 4.0-
//
--3
/"
2.00.0
/ /
/ / j / ."/ i // / ..//
/ /" / I .....'
..../ //*'"./~ ~
j
,
0.5
0.0
1.0 O'°°/G0
1.5
8.0
2.0
/
Periodic : b=a/2 ......
Pe,!o c: b - , - -
Periodic : b-2a 4,0-
........
n=50
%
///
/
/
,/
/
/'/~""~"" ......
./.,,.., /
..'*"
/ ~ ~ d
2.0-
Crack
J
0.0
(b)
. . . .
I
,
0.5
0.0
8.O
. . . .
I
1.0
. . . .
I
1.5
. . . .
Gco/O'0
Periodic Periodic Periodic
e.o-
,
/ .......
, / ' / / ...."// /
% %
(a)
b-el2 ....... b-e ..... b-2e ...........
n=5.0
4,0-
2.0
/j //z /~"/'1 //~
z/~" //~ed Crack
2.0-
(c) 0.0
. . . .
oo
,
. . . .
1.o
,
. . . .
2.0
,
. . . .
30 6/
,
. . . .
,.o
,
. . . .
6.0
e.o
0a
Fig. 8. Effect of ligament spacing b relative to crack size a on relationships between J, 6 and a ~.
89
geometries. It can be seen that the curves rise rapidly for o~°/o0 greater than about unity, indicating the onset of extensive yielding. The crack-opening displacement 6 measured at the center of the crack is plotted in Fig. 8(b). The trends of the curves are similar to those for J in Fig. 8(a). Plots of J vs. 6 in Fig. 8(c) are obtained by simply replotting the curves of Figs. 8(a) and 8(b). It is remarkable that the curves of normalized J vs. normalized 6 show no dependence on the a/w ratio. Details can be found in ref. 46. 6.3. Summary of plasticity effects Under conditions pertaining to small-scale and contained yielding, "opening"-dominated load states produce fields within the weaker material which bear a strong resemblance to those of a crack in the corresponding homogeneous material of identical material properties. The plastic near-tip fields are members of a family parametrized by a plastic mode mixity parameter ~. A member field, as identified by a particular value of the mode mixity ~, is scaled by the value of the J integral. This scalability has been demonstrated by full-field numerical solutions for "opening"dominated load states. Calculations for finite width plates loaded by remote tension into fully yielded conditions, ~ = 0, show that geometry and load effects are adequately accounted for by J. These results suggest the possibility of a J-based approach, limited to "opening"-dominated states, for interface fracture. While the fields for the interface cracks are similar to those of homogeneous media, they are often more intense than those in homogeneous media when compared at comparable values of the J integral. For example, openings that develop near the tip under "tension"-dominated load states, - Jr/6 ~<~ ~< r/6, are larger than those that develop in similarly loaded homogeneous material. As a result, the finite strain zone and the plastic strains within the finite strain zone are larger than those that develop in the corresponding homogeneous material. A surprising effect of plasticity is the rapid growth of a localized zone of high hydrostatic stress in the lower strength material as the plastic deformation progresses beyond small-scale yielding. This effect is most pronounced for tensile and slightly asymmetric load states. The zone of high hydrostatic stress (defined as the region where oh > 2o0) expands and can extend to 10%
of the ligament when fully yielded conditions are reached. Furthermore, the maximum hydrostatic stress achieved can be as much as 30% higher than those that develop in homogeneous material. Thus the weaker material is subjected to large stresses as well as large strains. High stresses near the tip increase the likelihood Of brittle fracture. Mode mixity effects have been studied in refs. 27 and 28. It is important to note that although + ~ mode mixity of the near-tip fields implies similar ratios of the interracial tractions, their magnitudes may differ considerably, i.e. o~.~.(-~)~oy~,(~), though both states are loaded to the identical J value. (In contrast, the magnitudes of the interface tractions within the K-dominant zone are identical for +_ ~ mode mixities when IKI is the same.) Slightly asymmetric load states, ~<0, produce hoop and hydrostatic stresses w h o s e magnitudes exceed those stress magnitudes for ~ > 0. (The discussion pertains to an elastic-plastic material and an elastic substrate; the effects are reversed when the materials are interchanged.) For example, the maximum hydrostatic stress for ~ ~ - 0 . 1 can be as much as 20% higher than that for ~=0.1. Some evidence for the stress elevation can be seen in the plastic strain contours in Fig. 6(b) for = - :r/6. The high hydrostatic stresses ahead of the crack causes the plastic strain contours to rotate backward. In comparison, the lower hydrostatic stresses for ~ = :r/6 cause the plastic strain contours to lean forward in Fig. 6(f). Such strong effects of __ ~ mode mixity are peculiar to interface cracks. In contrast, _+~ mode mixity produces similar effects on the near-tip fields of homogeneous material. Mode mixity effects are carefully detailed in ref. 28.
Acknowledgments This work is supported by the Office of Naval Research through O N R Grant N00014-90J13800. Helpful discussions with Z. Suo are gratefully acknowledged. Several sections in this paper are shortened versions of material in a recent paper by Wang et al. [ 19].
References I M. Ruhle, A. G. Evans, M. E Ashby and J. P. Hirth (eds.), Metal-Ceramic Interfaces, Acta-Scr. Metall. Proc. Ser. 4, Pergamon, New York, 1990. 2 M. L. Williams, Bull. Seismol. Soc. Am., 49 (1959) 199-204.
90 3 G. E Cherepanov, Izv. Akad. Nauk SSSR, OTN, Mekh. Mashin., l (1962) 131-137. 4 A.H. England, Z Appl. Mech., 32 ( 1965) 400-402. 5 E Erdogan, J. Appl. Mech., 32 (1965) 403-410. 6 J. R. Rice and G. C. Sih, J. Appl. Mech., 32 (1965) 418-423. 7 J. W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials, in Advances in Applied Mechanics, Academic, New York, 1991, in the press. 8 J.R. Rice, J. Appl. Mech., 55 (1988) 98-103. 9 J. R. Rice, Z. Suo and J.-S. Wang, Metal-Ceramic Interfaces, Acta-Scr. Metall. Proc. Set. 4, Pergamon, New York, 1990, pp~ 269-294. 10 J. Dundurs, J. Appl. Mech., 36 (1969) 650-652. 11 B.M. Malyshev and R. L. Salganik, Int. J. Fract. Mech., 1 (1965) 114-128. 12 J.R. Willis, J. Mech. Phys. Solids, 19(1971) 353-368. 13 M. Gotoh, Int. J. Fract. Mech., 3(1967) 253-260. 14 D.L. Clements, Int. J. Eng. Sci., 9(1971) 257-265. 15 J. Qu and J. L. Bassani, J. Mech. Phys. Solids, 37(1989) 417-433. 16 Z. Suo, Proc. R. Soc. Lond. A, 427(1990) 331'-358. 17 K. C. Wu, Stress intensity factor and energy release rate for interfaciai cracks between dissimilar anisotropic materials, J. Appl. Mech., (1991) in the press. 18 H. C. Choi, C. E Shih and Z. Suo, Test specimens for measuring mixed mode delamination toughness for cross-ply laminates, in preparation. 19 T. C. Wang, C. E Shih and Z. Suo, Crack extension and kinking in laminates and bicrystals, Int. J. Solids Struct., ( 1991 ) in the press. 20 M. Comninou, J. Appl. Mech., 44 (1977) 631-636. 21 M. Comninou and D. Schmueser, J. Appl. Mech., 46 (1979) 345-348. 22 N. Aravas and S. M. Sharma, J. Mech. Phys. Solids, ( 1991 ) in the press. 23 J. S. Wang and Z. Suo, Acta Metall., 38 (1990) 1279-1290. 24 N. P. O'Dowd, C. E Shih and M. G. Stout, Test geometries for measuring interfacial fracture toughness, Int.~J. Solids Struct., ( 1991 ) in the press. 25 W. Yang and C. F. Shih, Fracture along an interlayer, in preparation. 26 C. E Shih and R. J. Asaro, J. Appl. Mech., 55 (1988) 299-316.
27 C. E Shih and R. J. Asaro, J. Appl. Mech., 56 (1989) 763-779. 28 C. F. Shih, R. J. Asaro and N. P. O'Dowd, Elastic-plastic analysis of cracks on bimaterial interfaces: Part III-Large scale yielding, J. Appl. Mech., (1991) in the press. 29 E. Zywicz and D, M. Parks, J. Appl. Mech., 56 (1989) 577-584. 30 C. E Shih, Fracture Analysis, ASTM Spec. Tech. Publ. 560, American Society for Testing and Materials, Philadelphia, 1974, pp. 187-210. 31 J. Pan and C. F. Shih, Elastic-plastic analysis of combined mode I, II and III crack-tip fields under small-scale yielding, to be published. 32 P. Gu and C. F. Shih, Multi-layer structure with an interface crack, in preparation. 33 Z. Suo and J. W. Hutchinson, Mater Sci. Eng. A, 107 (1989) 135-143. 34 P. G. Charalambides, J. Land, A. G. Evans and R. M. McMeeking, J. Appl. Mech., 56 (1989) 77-82. 35 H. C. Cao and A. G. Evans, Mech. Mater., 7 (1989) 295-305. 36 K. M. Liechti and Y.-S. Chai, Asymmetric shielding in interracial fracture under in-plane shear, J. Appl. Mech., (1991) in the press. 37 M. G. Stout, N. P. O'Dowd and C. F. Shih, Interfacial fracture toughness of alumina/niobium systems, in preparation. 38 G. Bao, S. Ho, B. Fan and Z. Suo, The role of material orthotropy in fracture specimens for composites, to be published. 39 C. E Shih and R. J. Asaro, Int. J. Fract., 42 (1990) 101-116. 40 J. R. Rice, J. Appl. Mech., 35(1968) 379-386. 41 M. Symington, M. Ortiz and C. F. Shih, Int. J. Fract., 45 (1990) 51-64. 42 Y. Gao and Z. Lou, Int. J. Fract., 43 (1990) 241-256. 43 T. C. Wang, Eng. Fract. Mech., 37(1990) 527-538. 44 C. E Shih and S. Suresh, Mixed-mode inelastic crack-tip fields: homogeneous solids and bimaterial interfaces, Scr. Metall. Mater., 25(1991) 1017-1022. 45 C. F. Shih, J. Mech. Phys. Solids, 29(1981) 305-326. 46 C. E Shih and R. J. Asaro, Mater. Sci. Eng. A, 107(1989) 145-157. 47 W. Yang, Z. Suo and C. F. Shih, Mechanics of dynamic debonding, Proc. Royal Soc., Series A, 1991, in the press.
77
Cracks on bimaterial interfaces: elasticity and plasticity aspects C. F. Shih Division of Engineering, Brown University, Providence, R102912 (U.S.A.)
Abstract Substantial progress has been made on the mechanics of interface fracture. An engineering program has emerged which allows the fracture resistance of interfaces to be measured and utilized. This recent development, assessed in an Acta-Scripta Metallurgica Proceedings, is discussed. Several results have been obtained on the plasticity aspects of interface cracks in which one (or both) of the constituent materials can deform plastically. The crack tip fields are members of a family parametrized by plastic mode mixity parameter ~. The J integral scales each member field. Analyses of finite width crack geometries loaded by remote tension show that the effects of load, ligament plasticity and geometry on the near-tip fields are adequately accounted for by the J integral. The progress is summarized.
1. Introduction For many advanced materials such as structural ceramics, ceramic and metal matrix composites and polycrystalline intermetallic alloys, interfacial and intergranular fractures are common and may in large part determine the materials' overall mechanical response. For these material systems it is the low fracture toughness that limits their use in engineering and structural components. The need to understand, quantify and improve the toughness of advanced materials has renewed interest in the elastic interface crack problem. In recent years substantial progress has been made on the mechanics of interface fracture, an account of which is given in an Acta-Scripta Metallurgica Proceedings edited by Ruhle et al. [1]. The critical ideas and developments are reviewed in Sections 2-4. The plasticity aspects of cracks on bimaterial interfaces in which one (or both) of the constituent materials can deform plastically have received some attention. Several key results are discussed in Sections 5 and 6. The small-scaleyielding near-tip fields of interface cracks are members of a family parametrized by a plastic mode mixity parameter ~. The J integral scales each member field. For "opening"-dominated load states the plastic fields bear a striking resemblance to mixed mode H R R fields for homogeneous media. However, there are surpris0921-5093/91/$3.50
ing effects. For example, the zones of large plastic strains and high triaxial stresses that develop near bimaterial interfaces are considerably larger than those that are found in similarly loaded homogeneous bodies. Solutions for finite width crack geometries loaded in remote tension showed that the effects of load, ligament plasticity and geometry on the plastic near-tip fields of interface cracks are adequately accounted for by the J integral. In general, both the J integral and the plastic mode mixity ~ are required for the specification of the near-tip plastic state.
2. Elasticity fields of interface cracks Solutions to specific problems of cracks lying along bimaterial interfaces of isotropic media, for which the characteristic stress singularity was determined by Williams [2], can be found in works of the 1960s by Cherepanov [3], England [4], Erdogan [5] and Rice and Sih [6]. Thereafter the subject received little attention. In recent years there has been an explosion of activity on interface crack mechanics. The progress is docum e n t e d in an A c t a - S c r i p t a Metallurgica Proceedings [1] and in a review article by Hutchinson and Suo [7]. The rapid pace of progress can be attributed to two concepts: (i) mode mixity and (ii) small-scale contact zone [8]. These concepts are discussed in Sections 2 and 3. © Elsevier Sequoia/Printedin The Netherlands
78
2.1. Near-tip stress fieM The near-tip stress field for an interface crack between dissimilar isotropic bimaterials is a linear combination of two types of singularities, namely a coupled oscillatory field scaled by a complex K and a non-oscillatory field scaled by a real Kin:
The two stress intensities have different dimensions: K =
[stress][length] 1/2 -it
Kil I = [stress][length] 1/2
The complex stress intensity factor K has the generic form
Re(Kr i*)
Im(Kr it) oij-(2ztr)i/2 6ij'(O;e)-I (2xr)l/20qII(o;e)
K = YTLI/2L -~ exp(@) Kill + (2~r)l/~ 6i~IIl(O)
(1)
Here r and 0 are the polar coordinates shown in Fig. 1 and the dimensionless angular functions oi](0;e), oi]l(0;e) and all(O)correspond to tractions across the interface at 0 = 0 of tensile, inplane shear and out-of-plane shear respectively. The representation in (1) is due to Rice et al. [9] and Oil( 0; e) and 0i/I( 0; E ) are given in that article; off(0) is the standard mode III angular function. The bimaterial constant is defined by [2]
1 In Ul//~l+
e 2n
(2)
,u2/,u2+lhu~]
where v is Poisson's ratio, kt is the shear modulus, u = ( 3 - v)/(1 + v) for plane stress, r = 3 - 4 v for plane strain and subscripts 1 and 2 refer to material 1 and 2 respectively. When e = 0 , tri](O;e ) and aiIl( 0; e) reduce to the standard mode I and II angular functions. The tractions at a distance r ahead of the crack tip take the form
g r it (°.' + ioyx)0=0= (2arr)l/2 Kill
(Oyz)#=ll- (2~rr)!/z
(4)
(3)
(5)
where T is a representative stress magnitude, L is a characteristic crack dimension, Y is a dimensionless real positive number and ~p is by definition the phase angle of KL it. The implications of the unusual dimensions of K are explained in Section 2.4. Dundurs [10] has observed that the solutions to plane problems of elasticity for bimaterials depend on only two non-dimensional combinations of the elastic moduli. The Dundurs parameters are (Z-- /zl(r2 + 1)-/~2(u, + 1)
l(,q + 1)+
+ 1)
/~l(~C2- 1 ) - ~2(7Cl - 1) - fli(Tc----~+~-,u2(~ l + 1)
(6)
The connection
is noted. The quantities Y and ~pin (5) depend on a and fl in general and on dimensionless specimen parameters. In other words, the calibration of a crack geometry is reduced to determining Y and ~pfor a range of crack length-to-width ratios, load ratios and a and ft. Y and ~ can be evaluated by stress analysis. To complete this subsection, two more results are provided. The displacement jumps at a distance r behind the crack tip are given by
6Y+ibx=(1 +2ie)cosh(zte) E* 6~=-~--U
Fig. 1. Interface crack tip region.
where the effective Young's modulus and shear modulus are given by
79
2
1 --
E* 2
E/ 1
-
~t* ,Ul
An equivalent definition can be given in (KI, KII , Kin) space.
1 q'
--t
E2
r
(9)
1 /*2
with E ' = E l ( 1 - v 2) for plane strain and E ' = E for plane stress. The energy release rate is related to K and KII ! by [11, 12] ~?
K. tan~p = K~
1 Igl 2 cosh2(;Trg) E*
KIll 2 ~ -2~* --
Kill COS~ = (KI2 + K n 2 + Klu2)l/2
(14)
The mixities ~p and ~bare indicated in Fig. 2(a). These definitions also apply to cracks in homogeneous materials.
(10)
The connection 1 / c o s h 2 ( ~ e ) = 1 - f12 is noted. 2.2. Change of phase angle with distance--phase index It is helpful to write the interface traction vector in the following way: t = {tl, t2, t3} ={oy x, Oyy, Oyz}= {Oyi}. For t = tz+iq we have
2.4. Mode mixity." e # 0 It is clear from Section 2.2 that tension and shear effects are inseparable near interface crack tips. A measure of the relative proportion of shear to normal tractions (or mode 1 to mode 2) requires the specification of a length quantity. For oscillatory fields the mode mixity is uniquely specified by
K r i,
t = Itl exp(i~p,)= (2ztr)l/2
(11)
The phase angle l~r is a measure of the ratio of the normal to in-plane shear tractions at a distance r ahead of the crack tip. When the distance changes from r 1 to r 2, the phase change as predicted by the K field is ~lrz--~ri=
e
lnfr21
(12)
\rl]
A convenient measure of material mismatch producing phase variation with distance is the quantity (180/z~)e ln(10)--- e*. It has the interpretation as the phase change in degrees for a decade increase in distance. As an example, for e = 0.05, as for a glass-alumina interface, the phase index e* is 6.6 °. 2.3. Mode mixity: e = 0 For e = 0 the mode mixity can be defined in the usual way. When all three modes are present, the mode mixity is fully specified by two solid angles, ~p and ~, in the space of the interface traction vector t={Ovx, o m Oyz}:
\ Oyyl r=L
(15)
cos+=(Oy// tltl},-0
The length/~ is arbitrary but must be unchanging for a material pair, i.e. L must be independent of the overall specimen size and specimen types. A length between the inelastic zone size and the specimen size is a sensible choice of L. For example, L = 100/~m is suitable for many brittle bimaterial specimens at the laboratory scale. Using the asymptotic field (1) or the tractions in (3), the mode mixities ~b and ¢ can also be defined in K space: tan ~ - Im(KLi~) Re(KL ~)
cos ¢ =(I KI
( 16 )
Kn 1 + K,n2) It2 K3
}
tanlp = ( °yx) %(Tyy l r ~O
cos+__(Oy/ tltl],-0
Ka
Im(~O')
Fig. 2. Mode mixities defined as solid angles in K space.
The mode mixities 4 and $ are indicated in Fig. 2(b). As a consequence of the oscillatory field, the traction ratio t,/t, varies slowly as r moves away from the tip. Let $, and $J~be associated with t, and i, respectively. As implied by ( 16), $, and q2 obey the translation rule
For orthotropic materials with one principal axis aligned with the interface and crack front normal to the x -y principal plane, the interface tractions ahead of the crack tip are
(20)
K,
(17) This would not be a big change for moderate variations in i. For example, for an epoxy-glass interface, E = 0.06, the mode mixity change for a decade change in distance is G2 - $~i= 7.9”. We emphasize that the choice of reference length i is arbitrary. The change from one choice of t in the definition of 4 to another simply involves the transformation rule ( 17). Nevertheless, taking i to be broadly representative of the relevant microstructural length, or perhaps the plastic zone size at fracture, is advantageous for interpreting mixed mode fracture data. This aspect is discussed in Section 4.
where 7 depends on the elastic constants. A unified concept of mode mixity based on generalized traction components, e.g. (20), has been discussed by Suo [16] and elaborated upon by Wang et al. [ 191. By taking advantage of this approach, Suo has shown that results for isotropic bimaterials can be immediately generalized to anisotropic bimaterials. Explicit formulae for the stress intensity factor, displacement jumps and the energy release rate are given in refs. 16 and 19. Similar results for dynamically propagating cracks can be found in Yang et al. [47].
2.5. Anisotropic bimaterials Interface cracks in anisotropic solids were first analyzed by Gotoh [ 131, Clements [ 141 and Willis [ 121. The structure of the interface crack tip field has been completely identified in recent works by Qu and Bassani [15], Suo [16] and Wu [17] and analytical solutions have been found to several boundary value problems. The crack tip field is a linear combination of two types of singularities, namely a coupled oscillatory field scaled by a complex K and a nonoscillatory field scaled by a real K, [ 161:
3. Small-scale contact and non-linear zones
Re( KriE) _ , u8j=
(271rjl/2
Oil
Im(Kr”) fe)+
(2xrjl/2
S,=]S] cos Ijl-tan-‘(2&)-c I
te)
Here the dimensionless angular functions also depend on elastic constants and these functions have been extracted by Choi et al. [ 181 from the near-tip solution in ref. 16 where the structure of ( 18) for a general anisotropic bimaterial is discussed. As for isotropic bimaterials, the two stress intensities have different dimensions:
K, = [stress][length]“*
3.1. Small-scale contact zone Overlapping of crack faces and contact in isotropic bimaterials does not involve the mode III field. Therefore the present discussion is confined to the coupled in-plane modes. According to the linear elasticity solution (8), interpenetration of the crack faces, S, < 0, will occur at some sufficiently small r. Writing 6 = ayy+id, and using (8), we have
_2 *ij
(18)
K = [stress][length]1’2-”
We direct attention to isotropic bimaterials.
(19)
In L ( r II
(21)
The contact zone size, assuming that it is small compared to the crack size L, is estimated to be the largest r for which the opening gap S, just turns negative [81. For E > 0, r,= L exp
- [n/2 + + - tan- ‘(2c)] &
(22)
For E < 0, change E to - E and 1/, to - ly in the above equations. As an example, if L = 1 cm, 1~1 Q x/4 and I&l< 0.03, then r, < 1 A, which is smaller than all physically relevant length scales.
81 Comninou [20] and Comninou and Schmueser [21] have given, within linear elasticity theory, a more precise analysis of near-tip contact. Their contact zone sizes are comparable with the estimates by (22). Through full-field numerical studies, Aravas and Sharma [22] have shown that Comninou's asymptotic solution is valid at distances which are much smaller than the contact zone size. They also demonstrate convincingly that Comninou's solution is embedded within the "open crack" asymptotic solution (1) which is valid at distances large compared to the contact zone but still small compared to the crack size. Thus Comninou's asymptotic solution has no relevance when the contact zone size is smaller than the fracture process zone or the inelastic zone. We may conclude from the above discussion that the crack tip state is characterized by the complex K if rc/L 4. 1; Rice [8] has suggested using r~/L < 0.01. While the number 0.01 is arbitrary, the result that follows does not depend sensitively on it. Combining the above condition with (22) leads to the following estimate of the range of ~p in which K is applicable [23, 24]. For e>0, - Jr/2 + 6.6e < ~0< ~r/2 + 2e
(23)
For e < 0, - Jr/2 + 2e < ~0< Jr/2 + 6.6e
(24)
In the above equations, tan-~(2e)is approximated by 2e. The range of validity K is discussed in greater detail in refs. 23-25. 3.2. Small-scale yielding Several results have emerged from recent studies by Shih and Asaro [26, 27], Shih et al. [28] and Zywicz and Parks [29] on the plasticity of bimaterial interface cracks. They observed that the small-scale-yielding near-tip fields are members of a family parametrized by a plastic phase angle ~ defined by ~=~+eln
\oo LI
=~2+eln
(25)
where o0 is the yield strength of the weaker (lower strength) material. L is the characteristic crack dimension introduced in (5). Note that the notation in (25) is different from that used in the original publications [26-28] but is consistent with the notation used throughout this paper.
The plastic zone size rp, defined by the maximum radial extent from the tip to the elastic-plastic boundary, has the form rp = A(~)IK[2.~ o0
(26)
The dimensionless factor A(~) depends weakly on the elastic and plastic properties of the two materials. For plane strain A(~) increases from 0.15 to about 0.65 as ]~] increases from zero to :r/2. ~ may be interpreted as the phase angle of the plastic zone and is analogous to ~p, the phase angle of K L ~'. The plastic zone sizes as measured by A are rather similar to those for a homogeneous material of yield strength o 0. The latter plastic zones are parametrized by a plastic mode mixity parameter M P which has been related to the elastic mode mixity M" [30]; M ~ can be related in a simple way to ~o. These mode mixity parameters are discussed in Section 5.3. When both in-plane and out-of-plane modes are present, the size of the plastic zone is given by rp ='~(~,~ b)
IKI z + 7Ku, 2 z o0
(27)
where 7=(E*/2/~*)cosh2(Jre). The dimensionless factor ,~(~,~b) is a function of ~, defined by (25), and 4, defined by (16), and also depends on the elastic and plastic properties of the material pair. The numerical values for A(~,~) have yet to be determined. Estimates of/~, based on smallscale-yielding solutions for combined mode I, II and IlI crack problems for homogeneous materials [31], range from 0.15 to 0.65. A is a minimum for ~ = ~ = 0. 3.3. K dominance Let r~ denote the characteristic radius of the region controlled by the interface crack singularity field (1). r K will be some fraction of a relevant dimension of the crack geometry. At distances sufficiently close to the tip, the field (1) does not apply because of the presence of (i) a material non-linear zone (e.g. plasticity), (ii) a contact zone and/or (iii) small-scale heterogeneities and irregularities r i (e.g. grains, voids, microcracks, interdiffusion zones). If these zones and/ or irregularities are sufficiently small compared to a characteristic crack dimension L, then there exists an annular region rK> r ~ rp, r~, r i
(28)
82 in which(l) is dominant. In other words, if (28) is satisfied, the complex K uniquely measures the fields in an annular region surrounding the crack tip. Estimates of r K for finite width crack geometries have been obtained by O'Dowd et al. [24] and for multilayered materials by Gu and Shih [32]. Their studies show that rK -~ L / I O
(29)
where L is the shortest of the crack length, the uncracked ligament length, the layer thickness or the distance between the crack tip and the point of load application. Combining (28) and (29), we get the sandwich inequality L / l O > r~,> rp, rc, r i
(30)
Thus a conservative condition for the existence of a K annulus is L > lOOx(rp, rc, r~)
(31)
The latter provides the size requirements for a Kbased linear elastic fracture mechanics approach. Observe that the contact zone size depends on the phase of the applied load but not on its magnitude (see (22)), while the size scale of irregularities is a property of the bimaterial. In contrast, the plastic zone size is proportional to IKI 2 (see (26) and (27)), i.e. it scales with the square o f the magnitude of the applied load. If the interface is relatively tough or if one material has a low yield strength, a sizeable plastic zone can develop before the onset of fracture. In this case the use of K to correlate the onset of fracture is not justified. The plasticity aspects are taken up in Section 5. To summarize, we note that if the size requirements (31) are met, then pertinent information concerning crack geometry, material mismatch and applied loads is communicated to the crack tip through K only. Details that have no effect on K also have no effect on the near-tip field since the latter is uniquely characterized by K. Therefore K provides the boundary conditions for the inner region where the fracture processes occur, i.e. the onset of fracture can be phrased in terms of K. This aspect is elaborated upon in the next section. The above arguments are identical to those made for the relevance of the stress intensity factor in linear elastic fracture mechanics (LEFM). In LEFM methodology and in the rapidly developing subject of interface fracture mechanics the underlying idea is K dominance.
4. Interface toughness curves
It has been generally observed in experiments that cracks in isotropic, homogeneous, brittle solids seek to propagate on planes ahead of which local mode I conditions prevail. Consequently, one single parameter, K~c, can be designated to each material to quantify its resistance to fracture. By contrast, whenever planes of low fracture resistance exist, cracks may be trapped on such planes, regardless of the local mode mixity. Orthotropic materials such as composites and brittle crystals provide examples where definite weak planes exist: longitudinal planes for composites and cleavage planes for crystals. Interfaces offer another example when they are brittle compared with the substrates. A well-documented experimental fact is that fracture resistances for such weak planes depend strongly on mode mixity. Therefore it is toughness values at various mode mixities that fully characterize the fracture resistance of a weak plane.^ For a given mode mixity ¢ and % the interface fracture toughness F is defined as the energy release rate at the onset of crack growth. The fracture toughness F(~,,¢) is a property of the bimaterial interface. For a given bimaterial interface it is a surface in K space, which in principle can be determined directly by experiments. Upon loading, a crack will not propagate unless the driving force reaches the toughness surface, i.e. the mixed mode fracture condition is Y(~b,¢)= r(4,,O)
(32)
The fracture resistance is unambiguously specified by a surface F(~,¢) together with a length/~ for the definition of V). Specifically, F(~b,¢) is the critical value of the energy release rate required to advance the crack in the interface under the mode mixture ¢ and ~b, the latter being defined by the relative magnitudes of the in-plane shear to normal tractions at r = £ . This engineering approach to quantify the fracture resistance is an extension of the existing theory for isotropic solids. The conceptual basis for this approach is summarized in ref. 8 and in several articles in a volume edited by Ruhle et al. [1]. Specimen geometries suitable for interface fracture testing have been rigorously calibrated by Suo and Hutchinson [33], Charalambides et al. [34], Wang and Suo [23] and O'Dowd et al. [24] for a wide range of material combinations as characterized by Dundurs parameters (6) [10].
83
Interface fracture toughnesses of several bimaterial systems have been measured by Wang and Suo [23], Cao and Evans [35], Liechti and Chai [36] and Stout et al. [37]. Sandwich-type test specimens in which a thin layer of material is sandwiched between two layers of a second material are used in refs. 23 and 35. These specimens have the advantage that they can be calibrated easily in terms of the stress intensity factor of standard test specimens for homogeneous materials [33]. The test specimens used in refs. 34, 36 and 37 consist of two slabs of partially bonded dissimilar materials. Two test specimens are shown in Fig. 3. These crack geometries can produce near-tip fields ranging from predominantly tension to predominantly shear. Test specimens for uniaxial composites have been calibrated by Bao et al. [38] for arbitrary elastic constants. Choi et al. [18] have calibrated specimens which are suitable for cross-ply laminates. Schematic diagrams of the specimens are shown in Fig. 4. Interface toughness data from ref. 36 for an epoxy-glass interface are shown in Fig. 5(a). It can be seen that the toughness curve has a minimum at ~ ~ 20 °. In the plot the mode mixity ~ is based on [, = 12.7 mm, the thickness of the epoxy layer. Suppose /~ is taken to be two orders of magnitude smaller, say L = 100/~m. The smaller /~ lies within the K-dominant zone and is more
P
iI
@.,0"
F1 P
P
II ;; ", ..;.;.-, ~. ".... ". -. ,.. o.. ;.,
, • ". : ' : . :..'. : ;'II ~ ~ 400
A
t
A
@,,90*
Fig. 4. Specimens for the determination of mixed mode delamination toughness for cross-ply laminates [18].
i
i
i
D
g-.
I"1
Epoxy/Glass
I
30
I, = 12.7ram D
~b D
t.
(a)
I
0 -90
"°
I
I
I
-45
o
45
90
Mode Mixity
g-. I
J (a)
"-" 120
P
e~
~ I
/b/
(z) I_
,,
Epoxy/Plexiglass L = 0.1ram
80
I
2O* -r
o/
I
I
I
140
:
Fig. 3. Specimens for the determination of mixed mode interface fracture toughness: (a) Brazilian disk [231; (b) asymmetric four-point bend bar [24].
!
4O*
!
60*
8O*
Mode Mixity Fig. 5. (a) Data for an epoxy-glass interface; ~b based on L = 12.7 nun [36]. (b) Data for an epoxy-Plexiglass interface; ~bbased on/~ = 0.1 nun [23].
84
representative of the plastic or process zone. Using the translation rule (17), the new toughness curve is shifted backwards by about 16 °, resulting in a curve more nearly symmetrical about ~ = 0. The data from ref. 23 for a crack in an epoxy-Plexiglass interface are shown in Fig. 5(b). Wang and Suo's data [23] and Liechti and Chai's data [36] (adjusted for/~ = 100/~m) show a strong increase in toughness as the relative proportion of the shearing component (or mode mixity at /~ = 100/~m) increases.
5. Plastic fields
In the remaining sections of this paper, attention is directed to in-plane deformation modes and "opening"-dominated load states.
5.1. Plastic near-tip field It is noted in Section 3.2 that the size of the plastic zone is controlled by the yield strength of the weaker material, e0 ((26) and (27)). This means that over length scales which are comparable to the size of the dominant plastic zone, the stress levels are set by o 0. In contrast, the asymptotic behavior of the crack tip fields is governed by the hardening characteristics of the material with the lower hardening capacity. Furthermore, as r---0, the interface crack behaves increasingly like a crack lying at the interface of a plastic solid and a rigid substrate. A detailed discussion is given by Shih and Asaro [27, 39]. Consider an interface crack between a rigid substrate and an elastic-plastic solid. Under uniaxial tension the elastic-plastic solid obeys the Ramberg-Osgood stress-strain relation o/a0 = e/eo+ a(o/ao)", where a is a material constant and e o = oo/E. The small-displacement-gradient near-tip fields for "opening"-dominated load states, - : r / 6 ~<~ ~ n/6, can be organized in the form [27, 39]
(j],/(,,+l)
°i/= °° \ aooeor]
ho(O,f; ~,n)
(33)
Here J is Rice's J integral [40] and hij is a bounded function which is slowly varying with ? ( -= r/(J/ o0) ). The dimensionless function hi/depends on the plastic mode mixity ~, i.e. the fields are members of a family parametrized by ~ which is defined in the next subsection. Each member field is scaled by the J integral. Full-field analyses
have shown that the form in (33) describes adequately the actual fields. This is elaborated upon in Section 6.1.
5.2. Separable asymptotic solutions We have investigated separable asymptotic solutions of the H R R type, satisfying vanishing tractions on the crack face (0 = n) and vanishing displacements on the bond line (0=0), corresponding to a crack lying at the interface of an elastic-plastic material and a rigid substrate. The problem was formulated in terms of a stress function and the resulting ordinary differential equations were discretized by a finite difference method discussed in ref. 30. We have also formulated the two-point boundary value problem in the weak form using a stress function and a 1D finite element discretization [41 ]. In both cases we have attempted to solve the linearized equations by the Newton-Raphson method, but the iterative solutions did not appear to converge. Gao and Lou [42] claimed to have obtained, for certain material pairs, a family ~ f separable interface crack solutions parametrized by a mode mixity parameter Np. However, by their own admission the displacements are incompatible at 0 = 0 and this is definitely an unsatisfactory aspect of their solutions. A more recent analysis by Wang [43], for an elastic-plastic material and a rigid substrate, indicates that a separable asymptotic solution exists at a single mode mixity, the value of which depends on n. Indeed, Wang's solutions for n = 3, 5 and 10 are similar to the solutions reported by Shih and Asaro [27] in their Fig. 8 for ~ = - 0.551 and - 0.027, which display features consistent with a separable mode-I-like field (with a small positive shear stress) at distances r/rp < 10 -6. However, the solution for = 0.496 in Fig. 8 as well as the solutions in Fig. 12 (for a different material pair) of ref. 27 do not resemble separable solutions. In summary, the crack tip fields ( r o 0 ) for certain load and material combinations appear to be consistent with a separable form, while others bear no resemblance to separable solutions. These are puzzling aspects indeed. Nevertheless, there is sufficient numerical evidence to conclude that the domain of validity of an HRR-like separable solution, if one exists, is vanishingly small--smaller than all length scales of physical relevance. Indeed, stress distributions consistent with a separable solution are found only at distances well within the finite strain region (see Fig. 8 of ref. 27 and Fig. 5 of ref. 39).
85 Finite deformation effects are discussed in Section 5.4.
5.3. Mode mixity The mode mixity of the plastic near-tip fields is uniquely specified by ~ evaluated at a distance/2 within the plastic zone:
tan
=
"~' (°r°(0= 0)/
(34)
If/2 is comparable to the fracture process zone size, then ~ parametrizes the relative proportion of the shear to normal interface tractions near the process zone. Numerical studies have shown that the value of ~ is relatively insensitive to distance when evaluated within the interval 2 ~
where C is less than 0.01 when /2j and /22 are within the interval 10 9rp < r < 10-1rp [27]. We may conclude that the phase change with distance within the plastic zone is weaker than the phase change in the K-dominant zone (see (17)). For example, we find no indications of crack face contact over any length scale of physical relevance. In this respect, plasticity has mitigated a pathological feature of the linear elasticity solution. A strong resemblance between the "opening"dominated plane strain interface near-tip fields and mixed mode H R R fields in homogeneous material has been demonstrated in refs. 26 and 27. The mixed mode H R R fields are members of a family parametrized by a plastic mode mixity M p defined by [30] Mp=_2 tan ~(Ooo] 7g
(36)
\ OrOl r ~0
With this definition M P = 1 designates a mode I field while M P = 0 designates a mode II field. The magnitude of the fields scales with J U(n+ 1):
(jill(n+,) - oij = oo \ aoosolr I
O,i(O;MP, n)
(37)
where 1 is an integration constant which depends on n and M P. Under small-scale-yielding conditions a one-to-one correspondence exists between M P and the elastic mode mixity M s defined by (36) but evaluated within the Kdominant zone. Mixed mode plastic fields are reviewed by Shih and Suresh [44]. The connection between (33) and (37) has been exploited in refs. 26-28 and 39 to the extent possible, in as much as the fields of interface cracks do not appear to be separable over length scales which are physically relevant.
5.4. Finite deformation effects The crack-opening profiles and the fields within the zone of finite strains have been studied using a small-scale-yielding formulation [28]. Under these conditions J and K are related by 1 -/5 2
J=
/ . ~ IK[:
(38)
Figure 6 shows the crack-opening profiles obtained by the small-scale-yielding formulation for the three load states ~ = - Jr/6, 0 and ~/6 defined by (25). The notches have opened to more than five times their initial notch width and further deformations result in nearly self-similar shapes. All three profiles are obtained at the same magnitude of K and are drawn to the same scale. It can be seen that moderate amounts of shear parallel to the interface increase the crack tip opening, and negative shear has a stronger effect (or vice versa if the lower material is the weaker material). By defining 6, as the separation where 45 ° lines intersect the deformed crack faces, a relationship of the form
at = d(aeo, n)
J o0
(39)
has been derived for homogeneous material [45]. Applying the 45 ° procedure to the notch openings in Fig. 6, we get d=0.61, 0.56 and 0.67 for = - J r / 6 , 0 and ~/6 respectively. These values are considerably larger than the mode I value of 0.51, the largest possible value for homogeneous material. These results suggest that crack tip blunting in bimaterials is more severe than that encountered in homogeneous materials loaded to the identical J value. Contours for equivalent plastic strain, gP = 0.1, 0.2 and 0.5, for the three load states are shown in the plots in the right column of Fig. 6. The rela-
86 $.0 ~ = -,r/e
4[.0'
0.1
~1.0, 0.2
Z.O, 1.0-'
°.~~...o..°°°..°0 (u) 1.0;LO3.0 xl(~l~,)
0.0
-1.0
-~LO -Z.O
-1.0
0.0
~.0" 4~.'~ 0
0~.1
Z.O. 0.1
1.0" @.0
-LOxl(.11c,O.o)O l.O 7.0
-l.O
-LO
4.0 = ,r/O
3.0
0.1
,-~ z.0
(e)
1.0 0.0 -|.0
,i~o°, .
.
-L.,
.
.
,
.
,.o
.
.
.
(f)
, .... , .... ,
1.0
z.0
x/0/%)
3.0
4.0
Fig. 6. (a), (c), (e) Crack-opening profiles for load states ~ = - n/6, 0 and n/6 at identical J value. (b), (d), (f) Contours of gP =0.1, 0.2 and 0.5 plotted on the deformed geometry. Contours for a corresponding homogeneous material under mode I are also shown in (d).
tive extents of the plastic strain contours are broadly consistent with the amounts of crack tip opening. For comparison, we include contours for gP= 0.1, 0.2 and 0.5 (drawn to the same scale) for a crack in the corresponding homogeneous material (material with properties identical to the upper material) and loaded to the same value of J. It can be seen that the spatial extent of the latter plastic strains is significantly less than those for the interface crack. To summarize, the zone of
finite deformation extends over a distance of several J 0"0 - 1 . More details about the finite deformation fields are given in ref. 28. 6. Large-scale-yielding
solutions
In this section we elucidate upon the effects of crack geometry, load and ligament plasticity on near-tip fields. This is done in the context of a row of collinear cracks lying along a bimaterial
87
;..ei
2w
-
--
2w
XX) I
..... :
-20 t zb-4-z0q--zb-4-z0-q MATERIAL
{
x
2
(Crxx)z
'¢t
qtl
J'\3o
J=8.o
.
•. .
J=30
,f=1.0
a/w=2/a.I/z.I/a, I/9.1/100 i
-8.0
i
i
1
1
.
-5.0
.
,
I
. . . .
-4.0
|
(b)
. . . .
-3.0
I
. . . .
-2.0
I
. . . .
-1.0
0.0
-8.0
log(r/a)
-,5.0
-4.0
-3.0
-2.0
-1.0
0.0
log(r/a)
Fig. 7. (a) Infinite row of coIlinear cracks. (b), (c) Stress variations near interface for etriesfor J/aoeoa= 0.I, 1.0, 6.0 and 30.0.
a/w= 2/3,
I/2, I/3, I/9 and I/I00 geom-
interface as shown in Fig. 7(a). Crack length-towidth ratios a/w differing by two orders of magnitude are considered. Throughout this section geometry refers to a/w ratio, since this is the only geometric parameter of the problem, and load refers to the remotely applied tensile stress cr~°.
media, we investigate the behavior of the smalldisplacement-gradient full-field solutions normalized by a singularity of the HRR-type:
6.1. Stresses near the bond line For the purpose of making the connection with existing solutions for cracks in homogeneous
Suppose the numerically determined stresses possess an H R R structure of the form in (37) as r--" 0. In this case the plots of the normalized solu-
o~y/o0 di/- (Jlaooe(,r),/(. +1)
(40)
88
tions, 6ij vs. r (for fixed 0), must approach the asymptotes (appropriate to the respective mode mixities) as r ~ 0. Since the mode mixities of the different a/w cases being investigated are nearly identical (as they must be for the present boundary value problem), the curves of 6oo vs. r/a, for the several a/w ratios and applied loads, must converge to a single distribution as r--, 0. On the other hand, if the numerically determined fields obey the form in (33) as r ~ 0 , then plots of 6oo vs. r/a will show a weak dependence on r. In Fig. 7(b) the stresses ahead of the tip of a Ramberg-Osgood material with n = 5 for a/w = 2/3, 1/2, 1/3, 1/9 and 1/100 are compared. Specifically, 6oo vs. log(r/a) is shown for the three load levels J/ooeoa = 0.1, 1.0 and 6.0. These load levels correspond roughly to small-scale, contained and large-scale yielding (relative to the crack length) respectively. For example, at J/ ooeoa=6.0 the ligaments of the geometries a/w = 2/3, 1/2 and 1/3 are fully yielded. It can be seen that the normalized stresses of the different geometries form three distinct distributions corresponding to the three load levels. For J/ooeoa=0.1 the plastic zone extends a distance of about 0.01a ahead of the crack tip; hence it is not unexpected that these stresses, normalized by the plastic singularity, form a distinct distribution of their own. At higher loads a definite trend can be seen. Observe that the distribution for Jlooeoa=l.O approaches that for J/ooeoa = 6.0 for r/a < 10 -5. The distributions for J/ooeoa=6 and 30 (indicated by a dashed line) are nearly indistinguishable from one another. To summarize, the comparisons based on o00 indicate that the effects of load and geometry on "opening"-dominated near-tip fields, ~ = 0, are adequately accounted for by J. Moreover, the hoop stress distribution resembles a separable form. Figure 7(c) shows the behavior of the normalized shear stress 6~o for the four load levels considered above. The curves form three distinct distributions, indicating that the a/w ratios are adequately taken into account by the J integral. We can see that the tendency for oscillatory behavior is reduced at higher values of J. It is of interest that the magnitude of the shear stress relative to the hoop stress decreases as the plastic zone size becomes comparable to the crack length (or the relevant crack dimension). In contrast, shear and hoop stresses are always of comparable magnitude under small-scale-yielding
conditions. Large-scale-yielding effects are discussed in greater detail in ref. 46.
6.2. Relations between J, 6 and o ~ The dependence of J on the remotely applied stress o ~° is of interest in fracture analysis. Plots of J/aoeoa vs. the applied stress for the n = 5 material are given in Fig. 8(a) for four crack
8.0
b-e/2 ............ b-a --.--
Periodic Periodic
Periodic
6.0-
b-2a
,/ / /
........
n=5.0
% % 4.0-
//
--3
/"
2.00.0
/ /
/ / j / ."/ i // / ..//
/ /" / I .....'
..../ //*'"./~ ~
j
,
0.5
0.0
1.0 O'°°/G0
1.5
8.0
2.0
/
Periodic : b=a/2 ......
Pe,!o c: b - , - -
Periodic : b-2a 4,0-
........
n=50
%
///
/
/
,/
/
/'/~""~"" ......
./.,,.., /
..'*"
/ ~ ~ d
2.0-
Crack
J
0.0
(b)
. . . .
I
,
0.5
0.0
8.O
. . . .
I
1.0
. . . .
I
1.5
. . . .
Gco/O'0
Periodic Periodic Periodic
e.o-
,
/ .......
, / ' / / ...."// /
% %
(a)
b-el2 ....... b-e ..... b-2e ...........
n=5.0
4,0-
2.0
/j //z /~"/'1 //~
z/~" //~ed Crack
2.0-
(c) 0.0
. . . .
oo
,
. . . .
1.o
,
. . . .
2.0
,
. . . .
30 6/
,
. . . .
,.o
,
. . . .
6.0
e.o
0a
Fig. 8. Effect of ligament spacing b relative to crack size a on relationships between J, 6 and a ~.
89
geometries. It can be seen that the curves rise rapidly for o~°/o0 greater than about unity, indicating the onset of extensive yielding. The crack-opening displacement 6 measured at the center of the crack is plotted in Fig. 8(b). The trends of the curves are similar to those for J in Fig. 8(a). Plots of J vs. 6 in Fig. 8(c) are obtained by simply replotting the curves of Figs. 8(a) and 8(b). It is remarkable that the curves of normalized J vs. normalized 6 show no dependence on the a/w ratio. Details can be found in ref. 46. 6.3. Summary of plasticity effects Under conditions pertaining to small-scale and contained yielding, "opening"-dominated load states produce fields within the weaker material which bear a strong resemblance to those of a crack in the corresponding homogeneous material of identical material properties. The plastic near-tip fields are members of a family parametrized by a plastic mode mixity parameter ~. A member field, as identified by a particular value of the mode mixity ~, is scaled by the value of the J integral. This scalability has been demonstrated by full-field numerical solutions for "opening"dominated load states. Calculations for finite width plates loaded by remote tension into fully yielded conditions, ~ = 0, show that geometry and load effects are adequately accounted for by J. These results suggest the possibility of a J-based approach, limited to "opening"-dominated states, for interface fracture. While the fields for the interface cracks are similar to those of homogeneous media, they are often more intense than those in homogeneous media when compared at comparable values of the J integral. For example, openings that develop near the tip under "tension"-dominated load states, - Jr/6 ~<~ ~< r/6, are larger than those that develop in similarly loaded homogeneous material. As a result, the finite strain zone and the plastic strains within the finite strain zone are larger than those that develop in the corresponding homogeneous material. A surprising effect of plasticity is the rapid growth of a localized zone of high hydrostatic stress in the lower strength material as the plastic deformation progresses beyond small-scale yielding. This effect is most pronounced for tensile and slightly asymmetric load states. The zone of high hydrostatic stress (defined as the region where oh > 2o0) expands and can extend to 10%
of the ligament when fully yielded conditions are reached. Furthermore, the maximum hydrostatic stress achieved can be as much as 30% higher than those that develop in homogeneous material. Thus the weaker material is subjected to large stresses as well as large strains. High stresses near the tip increase the likelihood Of brittle fracture. Mode mixity effects have been studied in refs. 27 and 28. It is important to note that although + ~ mode mixity of the near-tip fields implies similar ratios of the interracial tractions, their magnitudes may differ considerably, i.e. o~.~.(-~)~oy~,(~), though both states are loaded to the identical J value. (In contrast, the magnitudes of the interface tractions within the K-dominant zone are identical for +_ ~ mode mixities when IKI is the same.) Slightly asymmetric load states, ~<0, produce hoop and hydrostatic stresses w h o s e magnitudes exceed those stress magnitudes for ~ > 0. (The discussion pertains to an elastic-plastic material and an elastic substrate; the effects are reversed when the materials are interchanged.) For example, the maximum hydrostatic stress for ~ ~ - 0 . 1 can be as much as 20% higher than that for ~=0.1. Some evidence for the stress elevation can be seen in the plastic strain contours in Fig. 6(b) for = - :r/6. The high hydrostatic stresses ahead of the crack causes the plastic strain contours to rotate backward. In comparison, the lower hydrostatic stresses for ~ = :r/6 cause the plastic strain contours to lean forward in Fig. 6(f). Such strong effects of __ ~ mode mixity are peculiar to interface cracks. In contrast, _+~ mode mixity produces similar effects on the near-tip fields of homogeneous material. Mode mixity effects are carefully detailed in ref. 28.
Acknowledgments This work is supported by the Office of Naval Research through O N R Grant N00014-90J13800. Helpful discussions with Z. Suo are gratefully acknowledged. Several sections in this paper are shortened versions of material in a recent paper by Wang et al. [ 19].
References I M. Ruhle, A. G. Evans, M. E Ashby and J. P. Hirth (eds.), Metal-Ceramic Interfaces, Acta-Scr. Metall. Proc. Ser. 4, Pergamon, New York, 1990. 2 M. L. Williams, Bull. Seismol. Soc. Am., 49 (1959) 199-204.
90 3 G. E Cherepanov, Izv. Akad. Nauk SSSR, OTN, Mekh. Mashin., l (1962) 131-137. 4 A.H. England, Z Appl. Mech., 32 ( 1965) 400-402. 5 E Erdogan, J. Appl. Mech., 32 (1965) 403-410. 6 J. R. Rice and G. C. Sih, J. Appl. Mech., 32 (1965) 418-423. 7 J. W. Hutchinson and Z. Suo, Mixed mode cracking in layered materials, in Advances in Applied Mechanics, Academic, New York, 1991, in the press. 8 J.R. Rice, J. Appl. Mech., 55 (1988) 98-103. 9 J. R. Rice, Z. Suo and J.-S. Wang, Metal-Ceramic Interfaces, Acta-Scr. Metall. Proc. Set. 4, Pergamon, New York, 1990, pp~ 269-294. 10 J. Dundurs, J. Appl. Mech., 36 (1969) 650-652. 11 B.M. Malyshev and R. L. Salganik, Int. J. Fract. Mech., 1 (1965) 114-128. 12 J.R. Willis, J. Mech. Phys. Solids, 19(1971) 353-368. 13 M. Gotoh, Int. J. Fract. Mech., 3(1967) 253-260. 14 D.L. Clements, Int. J. Eng. Sci., 9(1971) 257-265. 15 J. Qu and J. L. Bassani, J. Mech. Phys. Solids, 37(1989) 417-433. 16 Z. Suo, Proc. R. Soc. Lond. A, 427(1990) 331'-358. 17 K. C. Wu, Stress intensity factor and energy release rate for interfaciai cracks between dissimilar anisotropic materials, J. Appl. Mech., (1991) in the press. 18 H. C. Choi, C. E Shih and Z. Suo, Test specimens for measuring mixed mode delamination toughness for cross-ply laminates, in preparation. 19 T. C. Wang, C. E Shih and Z. Suo, Crack extension and kinking in laminates and bicrystals, Int. J. Solids Struct., ( 1991 ) in the press. 20 M. Comninou, J. Appl. Mech., 44 (1977) 631-636. 21 M. Comninou and D. Schmueser, J. Appl. Mech., 46 (1979) 345-348. 22 N. Aravas and S. M. Sharma, J. Mech. Phys. Solids, ( 1991 ) in the press. 23 J. S. Wang and Z. Suo, Acta Metall., 38 (1990) 1279-1290. 24 N. P. O'Dowd, C. E Shih and M. G. Stout, Test geometries for measuring interfacial fracture toughness, Int.~J. Solids Struct., ( 1991 ) in the press. 25 W. Yang and C. F. Shih, Fracture along an interlayer, in preparation. 26 C. E Shih and R. J. Asaro, J. Appl. Mech., 55 (1988) 299-316.
27 C. E Shih and R. J. Asaro, J. Appl. Mech., 56 (1989) 763-779. 28 C. F. Shih, R. J. Asaro and N. P. O'Dowd, Elastic-plastic analysis of cracks on bimaterial interfaces: Part III-Large scale yielding, J. Appl. Mech., (1991) in the press. 29 E. Zywicz and D, M. Parks, J. Appl. Mech., 56 (1989) 577-584. 30 C. E Shih, Fracture Analysis, ASTM Spec. Tech. Publ. 560, American Society for Testing and Materials, Philadelphia, 1974, pp. 187-210. 31 J. Pan and C. F. Shih, Elastic-plastic analysis of combined mode I, II and III crack-tip fields under small-scale yielding, to be published. 32 P. Gu and C. F. Shih, Multi-layer structure with an interface crack, in preparation. 33 Z. Suo and J. W. Hutchinson, Mater Sci. Eng. A, 107 (1989) 135-143. 34 P. G. Charalambides, J. Land, A. G. Evans and R. M. McMeeking, J. Appl. Mech., 56 (1989) 77-82. 35 H. C. Cao and A. G. Evans, Mech. Mater., 7 (1989) 295-305. 36 K. M. Liechti and Y.-S. Chai, Asymmetric shielding in interracial fracture under in-plane shear, J. Appl. Mech., (1991) in the press. 37 M. G. Stout, N. P. O'Dowd and C. F. Shih, Interfacial fracture toughness of alumina/niobium systems, in preparation. 38 G. Bao, S. Ho, B. Fan and Z. Suo, The role of material orthotropy in fracture specimens for composites, to be published. 39 C. E Shih and R. J. Asaro, Int. J. Fract., 42 (1990) 101-116. 40 J. R. Rice, J. Appl. Mech., 35(1968) 379-386. 41 M. Symington, M. Ortiz and C. F. Shih, Int. J. Fract., 45 (1990) 51-64. 42 Y. Gao and Z. Lou, Int. J. Fract., 43 (1990) 241-256. 43 T. C. Wang, Eng. Fract. Mech., 37(1990) 527-538. 44 C. E Shih and S. Suresh, Mixed-mode inelastic crack-tip fields: homogeneous solids and bimaterial interfaces, Scr. Metall. Mater., 25(1991) 1017-1022. 45 C. F. Shih, J. Mech. Phys. Solids, 29(1981) 305-326. 46 C. E Shih and R. J. Asaro, Mater. Sci. Eng. A, 107(1989) 145-157. 47 W. Yang, Z. Suo and C. F. Shih, Mechanics of dynamic debonding, Proc. Royal Soc., Series A, 1991, in the press.