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Interfacial Fracture

Zhigang Suo's picture

These notes belong to a course on fracture mechanics

A body consists of two materials bonded at an interface. On the interface there is a crack. The body is subject to a load, causing the two faces of the crack to open and slide relative to each other. When the load reaches a critical level, the crack either extends along the interface, or kinks out of the interface.

A crack on the interface between two materials is similar to a mixed-mode crack in a homogeneous material. There is a significant difference, however. When a mixed-mode crack in a homogeneous material reaches a critical condition, the crack kinks out of its plane. By contrast, when a crack on an interface between two materials reaches a critical condition, the crack can extend along the interface, provided the interface is sufficiently weak compared to either material.

For the crack extending on the interface, the energy release rate is still defined as the reduction in the potential energy of the body associated with the crack advancing by unit area. The energy release rate characterizes the amplitude of the load. The critical condition for the extension of the crack also depends on the mode of the load. To characterize the mode of the load, we need the field around the tip of the crack.

Williams (1959) discovered that the singular field around the tip of a crack on an interface is not square-root singular, but takes a new form. The Williams field makes puzzling predictions: when the tip of the crack is approached, the stresses oscillate, and the faces of the crack interpenetrate. Interpenetration is clearly a wrong prediction. Is the Williams field useful?

In late 1980s, applications involving thin films and composites motivated Evans, Hutchinson, Rice and others to develop interfacial fracture mechanics. As pointed out by Rice (1988), so long as the Williams field is wrong only in a small zone around the tip of the crack, an annulus exits, within which the Williams field correctly predicts the field. The situation is analogous to using the square-root singular field under the small-scale yielding condition. Today the interfacial fracture mechanics on the basis of the Williams field is practiced routinely in industries.

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Dear Prof. Zhigang,

I have been your big 'fan' and have following your lecture notes for some time. They really help me a lot in my research. Thanks for your effort.

In your note, I am confused about the difference between the phase angle Φ and mode angle ψ. They are the same in what way?

If the angle ω is related to elastic mismatch between the film and the substrate, then what is ε? I thought that the elastic mismatch is already taken care of by ε.

I really appreciate if you could clear my doubt.

Zhigang Suo's picture

Dear Khong Wui:  Thank you very much for your kind note.  My responses to your questions follow.

Phase angle vs. mode angle.   For an interfaical crack, the stress intensity is complex-valued.  Following the mathematics of complex numbers, we can write the stress intensity factor in terms of its amplitude and phase angle.  But the stress intensity factor also has a strange dimension, involving iε.  Following Rice (1988), we write the stress intensity factor in terms of its amplitude and mode angle, by assigning an arbitrary length.  The comparision of the phase angle and the mode angle was made on p. 6 of the lecture notes.

A thin film on a substrate. ε is defined on p.2 of the lecture notes.  Thin film is discussed on p. 9, perhaps too briefly.   ε does not take care of elastic mismatch completely.  For details of a thin film on a substrate, see the original paper:

Z. Suo and J. W. Hutchinson, Interface crack between two elastic layers. Int. J. Fracture, 43, 1-18 (1990).

Kejie Zhao's picture

Dear Zhigang,

On Page 10 of the lecture notes the expression of stress intensity factor, is ω supposed to phase angle Ψ? What does elastic mismatch refer to? Thanks


Zhigang Suo's picture

Dear Kejie: ω is not the phase angle of the stress intensity factor.  In the lecture notes, I downplayed ω, because it is rather small.  If you would like to know its precise meaning, see the original paper:

Z. Suo and J. W. Hutchinson, Sandwich specimens for measuring interface crack toughness. Materials Science and Engineering A107, 135-143 (1989).

In this paper, ω is defined by Equation (16a). 

Elastic mismatch means the difference in elastic constants of dissimilar materials.

Hi Zhigang,

From my understanding, the physical meaning of phase angle Φ indicates the ratio of KI/KII. But we can also write the K intensity factor in terms of its amplitude and mode angle ψ, by assigning an arbitrary length.

So, am I right to say that the mode angle ψ alone doesn't have any physical meaning or meaningless unless we specify a length to it? And from your note, the the phase angle Φ corresponds to the mode angle ψ associated with a special choice of the length l=1. Am I right?

Zhigang Suo's picture

Dear Khong Wui:  If you have any further doubt about the idea of the mode angle, you may wish to look at Rice's original paper, where he introduced the idea, and carefully discussed it.  The paper is available online:

J. R. Rice, "Elastic Fracture Mechanics Concepts for Interfacial Cracks", Journal of Applied Mechanics, 55, l988, pp. 98-103.

surajmravindran's picture

How this mismach parameter is getting the expression that mentioned in notes.

Zhigang Suo's picture

The expression for epsilon was determined by solving the eigenvalue problem of an interfacial crack.  The expression was first obtained by Williams.  See J. R. Rice, "Elastic
Fracture Mechanics Concepts for Interfacial Cracks
", Journal of
Applied Mechanics, 55, l988, pp. 98-103.

surajmravindran's picture

Dear Sir

i think for understanding the problem of crack in bimaterila interface, i need to go for the Zak and Williams paper that is published in 1963. If possible can you send the link for that paper also. I didnt get that paper from net..

surajmravindran's picture

Dear sir,

Also i have doubt with the expression for crack deflection or penetration in your notes, how that expression is coming.Gd/Gp


Thanking you

Zhigang Suo's picture

See He, N.Y., Hutchinson, J.W., " Crack Deflection at an Interface Between Dissimilar Elastic Materials ." Int. J. Solids Structures, 25, 1053-1067 (1989).

surajmravindran's picture

Thank u sir..

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