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Objective Fracture Parameters and a Paradox for Interface Cracks

Bin Liu's picture

Due to the oscillatory singular stress field around a crack tip, interface fracture has some peculiar features. This paper is focused on two of them. One can be reflected by a proposed paradox that geometrically similar structures with interface cracks under similar loadings may have different failure behaviors. The other one is that the existing fracture parameters of the oscillatory singular stress field, such as a complex stress intensity factor, exhibit some nonobjectivity because their phase angle depends on an arbitrarily chosen length. In this paper, two objective and independent fracture parameters are proposed which can fully characterize the stress field near the crack tip. One parameter represents the stress intensity with classical unit of stress intensity factors. It is interesting to find that the loading mode can be characterized by a length as the other parameter, which can properly reflect the phase of the stress oscillation with respect to the distance to the crack tip. This is quite different from other crack tip fields in which the loading mode is usually expressed by a phase angle. The corresponding failure criterion for interface cracks does not include any arbitrarily chosen quantity and, therefore, is convenient for comparing and accumulating experimental results, even existing ones. The non-self-similarity of the stress field near an interface crack tip is also interpreted, which is the major reason leading to many differences between the interfacial fracture and the fracture in homogenous materials. The paper can be found at


Mike Ciavarella's picture

See eqt.6-8 of 

[PDF] psu.eduInterface cracksDA Hills, JR Barber - International journal of mechanical sciences, 1993 - ElsevierAbstract Properties of elastic solutions to interface crack problems are discussed in both the 
open formulation, in which the crack faces are assumed to be traction-free and 
interpenetration is permitted, and the unilateral formulation, where a contact zone is 

Bin Liu's picture

Hi, Mike.

Thanks for bring this paper to our attention. The authors also noted the curious features of the asymptotic field, such as the complex stress intensity factor K depending on an arbitrarily chosen length a. However, in the latter part of their paper, they still used the complex stress intensity factor K.

In our paper, after strict derivation, we suggest using two objective and independent fracture parameters to characterize the crack tip field, and completely discarding the non-objective complex stress intensity factor K. Another important conclusion is we propose a paradox which demonstrates another curious feature of an interface crack in a straightforward way.

Mike Ciavarella's picture

I prefer the Hills-Barber version. You write: "...which is against the instinct that the similar structures with different dimension under similar loadings should have the similar stress distributions."

You say that for two similar problems with the same K, the stress fields are different. But surely that is also true for all LEFM problems, not just interface cracks.
If the stress fields were the same, K_I in a Griffith crack would be lower when the body is larger!

Bin Liu's picture

Mike, I noted your comments are focused on the context related to Figure 2. Actually, as we have presented in our paper, this contradiction is from Prof. Rice’s paper in 1988. Because the corresponding expression includes the complex stress intensity factor K, the non-objectivity of K might be involved into the contradiction. Therefore, we propose a clear paradox in Figure 3 to avoid adopting any fracture parameter.

Mike Ciavarella's picture




Yes, but you haven't discussed much that in reality the oscillatory solution implies contact, and in this case what is the singularity?

Furthermore, in many practical cases, will be significantly lower than the maximum and the stresses at all points outside the process zone may be almost indistinguishable from those that would be obtained for a material pair with e = 0. In such cases, there is a strong argument for developing an approximate solution based on setting e to zero, leading to simple square-root singularities, conventional stress-intensity factors and permitting classical fracture mechanics arguments to be used [20].

[20] Z. Suo and J. W. HUTCHINSON, Steady-state cracking in brittle substrate beneath adherent films, Harvard Univ. Report No. Mech--132. 



Bin Liu's picture

Mike, Yes. You are right that the contact might affect the crack tip field. But considering that the contact region is so small if exists, one may use the outer ring region to characterize the fracture, just like the small-scale yielding situation in which the stress intensity factor K is still able to characterize the fracture. Of course, this is only our speculation. We will investigate the effect of the contact region in our future work. Thanks for pointing out this.

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