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Discussion of fracture paper #33 - The Interaction Integral

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This blog concerns an interesting review of the interaction integral methodology. It deserves to be read by everyone dealing with analyses of cracks. If one's focus is on mathematical analysis or numerics is irrelevant. The review is for all of us. The review paper is, ”Interaction integral method for computation of crack parameters K–T – A review", by Hongjun Yu and Meinhard Kuna, Engineering Fracture Mechanics 249 (2021) 107722, p. 1-34.

Already the introduction gives a thorough historic background of the incremental improvements and additions made to tackle an increasing sphere of problems. The starting point is J. Rice´s J-integral, which has served us well for more than half a century. It gives the energy release rate in the near tip region at crack growth in a homogeneous material. Inhomogeneities, bi-material interfaces, and more requires amendments. A drawback with the J-integral is that it provides the energy release rate independent of a mode mixity. When it was shown by Stern, Becker and Dunham that an auxiliary field in equilibrium, also providing path independence, added to the original field allowed decoupling of the mixed modes and their respective stress intensity factors, the interaction integral was established. Out of the large variety of other solutions to the problematic mode separation, the interaction integral seems to be the most effective and suitable for FEM implementation. 

The review in its introduction takes the reader on an odyssey through the five decades of inventive selections of auxiliary fields giving solutions to a large variety of static and dynamic problems and introducing domain integrals that improve the accuracy. 

In their paper, Yu and Kuna include the theoretical background and explain the basic amendments introduced to allow the treatment of many problems, including anisotropy, dynamics, mechanics coupled with other physical processes, etc. The inspiring reading gives a great starting point for anyone who wishes to explore new possibilities the method provides. There is also a useful section showing the implementation and example cases with data. I very much liked the declaration of advantages and especially the limitations that give a direct sense of reliability. The 351 references are also much appreciated.

The usage of the M-integral for calculations of intensity factors for thermo-elastic and piezoelectric materials (cf. L. Banks-Sills et al. 2004 and 2008) is interesting. I assume that stress-driven diffusion and other transport phenomena that are governed by Laplace's equation coupled with elastic deformation could have direct use of the M-integral. By the way, the interaction M-integral should not be confused with the M-integral that gives the energy release rate for expanding geometries (cf. L.B. Freund 1978). 

Regarding the auxiliary field applied to K-dominance problems, there is an annular ring around the crack tip in which stress is represented by a full series of r^(n/2) terms, where the n includes both negative and positive integers. Essentially only the term with n=-1 connects the remote boundary with the near tip region. Outside the annular ring, terms with n≥-1 dominate and inside the ones with n≤-1 dominate. The selection for the auxiliary field seems so far to be one of n≥ -1, -2, or -3. What I ask myself is, cannot the e.g. the Dugdale model with its exact series expansion solution including an arbitrary number of terms n≤ -1 be of use to cover elastic-plastic problems. I know that the direct superposition fails but could perhaps be given an analytically matched zone length. Perhaps I am on the wrong path. 

Please, enlighten me authors, readers, anyone. All comments, opinions or thoughts regarding the paper, the method, or anything related are encouraged. If you belong to the unfortunate that do not have an iMechanica account, please email me at and I will see what I can do.

Per Ståhle

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