Interface Fracture Toughness

70 In plane strain, the energy release rate G_o of the interface crack advancing in the interface

is related to K as shown in Eqn. 10.58, and may be rewritten in the form (Malyshew and Salganik 1965) Go= [ (1− ν1) µ1 +(1− ν2) µ2 ] K ¯K 4 cosh2(πε) (10.65)

where ¯K denotes the complex conjugate of K.

71 The energy release rate G of the kinked crack (a > 0) is given by,

G = [(1− ν2) 2µ2

](K_I2+ K_II2 ) (10.66)

72 Combining this equation with Eqn. 10.62 gives

G = [(1− ν2) 2µ2

][(|c|2+|d|2)K ¯K + 2Re(cdK2a2iε)] (10.67)

4

Note the analogy between this expression and Eq. 10.13 for the SIF at the tip of a kinked crack as developed by Hussain et al. for the Maximum Energy Release Rate criterion.

K = K1+ iK2 =|K|eiεL−iε (10.68)

where by 10.59, L is the in-plane length quantity characterizing the specific interface crack problem when a = 0. From Equations 10.65, 10.67 and 10.68 and using the real angular quantity γ as the measure of the loading combination,

G = q−2Go[|c|2+|d|2+ 2Re(cd exp2i¯γ)] (10.69)

where q = [(1− β 2₎ (1 + α) ] 1/2 _(10.70) ¯ γ = γ + ε ln (a/L) (10.71) γ = tan−1(K2/K1) (10.72)

74 When ε = 0, the stress intensity factors, K_I and K_II and G are independent of a. This is

the case of similar moduli across the interface (α = β = 0).

75 From Eqn. 10.54, ε is zero when β = 0 regardless the value of α. The oscillatory behavior of

the interface crack fields and the a-dependence of G only appear when β= 0. According to He and Hutchinson (He and Hutchinson 1989), a sensible approach to gaining insight into interfacial fracture behavior, while avoiding complications associated with the oscillatory singularity, would be to focus on material combinations with β = 0.

76 The interface crack with a = 0, suffers contact between the crack faces within some small

distance from the tip, when β = 0, therefore ε = 0, as predicted by the elastic solutions of the crack fields (Comninou 1977, Rice 1988). Contact between crack faces is less likely for the kinked crack (a > 0; ω > 0) loaded such that KI and KII are positive, since this will open up

the crack at the kink. Nevertheless, contact will inevitably occur if ε= 0 when a is sufficiently small compared to L (He and Hutchinson 1989).

a L G G G G L U *

Figure 10.11: Schematic variation of energy release rate with length of kinked segment of crack for β = 0, (Hutchinson and Suo 1992)

77 The dependence of G on a for a given kink angle is shown qualitatively in Fig. 10.11

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sufficiently small, G oscillates between a maximum GU and a minimum GL, which are found

to be (He and Hutchinson 1989)

GU = q−2Go[|c| + |d|]2 (10.73)

GL = q−2Go[|c| − |d|]2 (10.74)

and which depend on K1 and K2 only through Go. For values of a/L outside the oscillatory

range G approaches GB, given by Eqn. 10.69, with ¯γ = γ, i.e.,

GB = q−2Go[|c|2+|d|2+ 2Re(cd exp2iγ)] (10.75)

GB coincides with G given by Eqn. 10.69, when ε = 0. Contact between the crack faces will invalidate the prediction for G fron Eqn. 10.69 when a/L is in the range where oscillatory behavior occurs.

78 From a physical standpoint, GB should be relevant if there exists crack-like flaws emanating

from the interface whose lengths are greater than the zone of contact. That is, GB should be relevant in testing for kinking if the fracture process zone on the interface is large compared to the contact zone of the idealized elastic solution. If it is not, then more attention must be paid to the a-dependence of G and to the consideration of the contact. In any case GB should play a prominent role in necessary conditions for a crack kinking out of an interface, because once nucleated, the kinked crack has an energy release rate which rapidly approaches GB as it lengthens (He and Hutchinson 1989).

10.4.3.1 Interface Fracture Toughness when β = 0

79 When β = 0 (and thus ε = 0) Eqns. 10.53, 10.57 and 10.58, respectively become:

(σy, τxy) = (K_1, K2) (2πr) (10.76) (δy, δx) = 4 √ 2π[ 1 ¯ E1 + 1_¯ E2 ]√r(K1, K2) (10.77) G = 1 2[ 1 ¯ E1 + 1_¯ E2 ](K₁2+ K₂2) (10.78)

80 The interface stress intensity factors K₁ and K₂ p lay p recisely the same role as their counter-

parts in elastic fracture mechanics for homogeneous, isotropic solids. The mode 1 component K1 is the amplitude of the singularity of the normal stresses ahead of the tip and the associated

normal separation of the crack flanks, while the mode 2 component, K2, governs the shear

stress on the interface and the relative shearing displacement of the flanks.

81 When β = 0, the measure of the relative amount of mode 2 to mode 1 at the crack tipis

taken as (Hutchinson and Suo 1992)

γ = tan−1(K2/K1) (10.79)

82 For the case of a finite crack in an infinite plane,

loaded in mixed mode by γ is

G = Γ(γ) (10.81)

84 Where Γ(γ) is the toughness of the interface and can be thought of as an effective surface

energy that depends on the mode of loading.

10.4.3.2 Interface Fracture Toughness when β= 0

85 When β = 0, the decoupling of the normal and shear components of stress on the interface

and associated displacements behind the crack tip within the zone dominated by the singularity, does not occur. When β= 0, the definitions of mode 1 and mode 2 require some modification. In addition, the traction-free line crack solution for the displacements implies that the crack faces interpenetrate at some point behind the tip. Both of these features have caused conceptual difficulties in the development of a mechanics of interfaces.

86 As noted by Rice(1988) (Rice 1988), a generalized interpretation of the mode measure is the

most important complication raised by the oscillatory singularity, and the approach explained here is along the lines of one of his proposals (Hutchinson and Suo 1992). First, a definition of a measure of the combination of mode is made that generalizes Eqn. 10.79.

87 Let L be a reference length whose choice will be discussed later. Noting the stress distribution

(10.53) on the interface from the K-field, define γ as γ = tan−1[Im(KL

iε₎

Re(KLiε₎] (10.82)

where K = K1+ iK2 is the complex stress intensity factor.

88 For a choice of L within the zone of dominance of the K-field, Eqn. 10.82 is equivalent to

γ = tan−1[(τxy σyy

)r=L] (10.83)

89 Moreover, the definition reduces to Eqn. 10.79 when β = 0, since Liε= 1 when ε = 0. When

ε= 0, a mode 1 crack is one with zero shear traction on the interface a distance L ahead of the tip, and a mode 2 crack has zero normal traction at that point. The measure of the proportion of ”mode 2” to ”mode 1” in the vicinity of the crack tiprequires the specification of some length quantity since the ratio of the shear traction to normal traction varies (very slowly) with distance to the tipwhen β = 0.

90 The choice of the reference length L is somewhat arbitrary, as explained by Hutchinson

(Hutchinson and Suo 1992). It is useful to distinguish between a choice based on an in-plane length L of the specimen geometry, such as crack length, and a choice based on a material length scale, such as the size of the fracture process zone or a plastic zone at fracture. The former is useful for discussing the mixed mode character of a bimaterial crack solution, independent of material fracture behavior, while the latter is advantageous in interpreting mixed mode fracture data.

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Figure 10.12: Conventions for a Crack Kinking out of an Interface, (Hutchinson and Suo 1992)