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Interfacial toughness and mode mixity

When I was a graduate student, I spent several months to measure interfacial toughness between metalic (Cu and Au) films and thick substrates(Si and Polycarbonate). My methods were bulge test (blistering test) and 4-point bending test. I had many problems such as making an initial crack(pre-cracking), changing load phase angle applied to specimens, preparing/patterning thin films, constructing my own test apparatus, etc. The biggest problem was to measure the interfacial toughness over a wide range of loading phase angle. For a bimaterial with a non-zero oscillatory index(epsilon), we don't know the phase angle for a minimum interfacial toughness beforehand. Therefore, we need to measure the interfacial toughness over a wide range of phage angle. For engineering purpose, we need a minimum interfacial toughness value for reliability design because this value will lead to a conservative design of systems.

The attached file was written based on the above considerations(it's not an experimental paper, but an analysis one). It describes an interfacial toughness model which provides some useful guidelines for finding a minimum interfacial toughness. It assumes that the main contribution for the phase angle dependency of interfacial toughness is plastic dissipation. Using plastic zone size around an interface crack, it defines two stress modes which are corresponding to Mode I and II for epsilon =0, but are not orthogonal to each other for nonzero epsilon.

I think some descriptions in the paper are not good(this is one of my first papers), but it contains some useful ideas for interfacial toughness model. Any comments and advices are welcomed.


PDF icon EFM2003_625.pdf451.47 KB


Hello Jae Hyun Kim

Your research is quite related to my research. I am doing dynamic impact testing to determine the fracture toughness of the bi-material (interfacial crack). My research purpose is to determine energy dissipation during crack propagation under dynamic impact and your paper is good reference model. If you don't mind, I will read your paper and talk further in later.


Hello Chul jin,

Thanks for your interest. I'm very curious about your dynamic impact test for interfacial toughness measurement. Would you inform me of your papers or literatures on dynamic impact test? Also your any comments on my paper are welcomed.


It is an interesting approach!

I have some experience in this field of research. I investigated the interface toughness of a crack along an interface between a +-45° transversely isotropic pair of materials, which results in a 3-D analysis.

Two phase angles were defined to relate between the three different stress intensity factors. Experiments were carried out to determine the delamination toughness of a brazilian disk specimen.

You may have a look at the paper:

Fracture toughness of the +45°/ - 45° interface of a laminate composite, International Journal of Fracture, 141: 195-210 (2006).

 You might find it useful.


Hi Yuval,

Thanks for your comments. I really enjoyed your paper which you informed me of. It seems very interesting that Mode II is decoupled while Mode 1 and 3 are coupled in terms of oscillatory index in your laminate composite. I have a few questions:

- Critical stress intensity factor(K) or G is valid under small scale yielding. I'd like to know the typical plastic zone size in your experiments. Of course, this may be dependent on mode mixity. I just want to know the minimum and maximum plastic zone size.

- With considering the plastic zone size as a primary length scale at the crack tip, how do you justify the assumption that the laminate containing many fibers in epoxy matrix is homogeneous? I don't have enough experience on the experiments of composites. Please inform me of related references.



Dear Jae-Hyun,

Thank you for your comment.

A case of a delamination between a +45/-45 degrees transversely isotropic pair of materials is a unique case in which the complex stress intensity factor is associated with K1 and K3 (and no K1 and K2, as in most cases). This results from the out of plane deformation of the +45/-45 combination. You may find the derivation of the asymptotic stress and displacement fields in an earlier paper by the authors,  (A through interface crack between a +/- 45 degrees transversely isotropic pair of materials, International Journal of Fracture, 133:1-41, 2005).  

As for your questions,

  1. Unfortunately, we did not check the size of the plastic zone experimentally. With the AS4-3502 (graphite/epoxy) composite, I believe that the plastic zone is extremely small.
  2. It is clear that the 'upper' and 'lower' materials are fiber reinforced materials which are directed in the +45 and the -45 degrees, respectively. However, we investigated a delamination between a layer of the +45 degrees fibers and a layer of the -45 degrees fibers. Hence, we might look at each layer effectively,  as a transversely isotropic material. Hence, every layer is a 'homogeneous' media, which is described as a transversely isotropic media.

For more details on the mechanics of anisotropic materials in general, and transversely isotropic materials in specific, I'm directing you to a great book:

Ting, T.C.T., Anisotropic Elasticity - Theory and Applications, Oxford University Press, Oxford (1996).

This is the 'bible' for everyone who deals with anisotropic elasticity.


Dear Yuval,

Thanks for your comments.

The book you mentioned is a great one and the anisotropic elasticity is a very attractive subject for me.

As you wrote, I see that the macroscopic behavior of the composite can be modeled as that of an effective medium with a transversly isotropic properties. As we know, the K field of a crack is an asymptotic field which is only valid within a short distance around the crack tip. Apparently, at a very small distance from the crack tip, the composite is not homogeneous at al. The crack tip can interact with either the epoxy matrix or the fibers separately. I'd like to learn under what conditions the composite can be regarded as a homogeneous material, especially for the crack problem. Would you inform me of any good reference for this subject?


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