User login


You are here

Discussion of fracture paper #36 - The Double-K Fracture Model

ESIS's picture

The fracture of concrete and other semi-brittle materials offers some simplifications that simplify the analytical analysis. The simple check that reveals if something broken requires an elastic or an elastic-plastic fracture mechanical analysis by just trying to fit the pieces together sometimes fails. The suggestion is that if they do not fit together, we have an elastic-plastic fracture and if they do we have an elastic fracture. We may jump to the false conclusion that linear elastic fracture mechanics can be applied. The fracture processes are confined to a narrow zone stretching ahead of the crack tip for concrete and similar materials. A Barenblatt process zone seems ideal but it requires knowledge of how the cohesive capacity decays with increasing stretch across the crack plane. The version proposed by Dugdale* is intended for plastic necking in thin sheets and requires only yield stress and sheet thickness. Out of a variety of other proposals, the double-K model seems to have achieved widespread attention and appreciation because of its engineering approach providing practical simplicity. The review paper,

"The double-K fracture model: A state-of-the-art review", by Xing Yin, Qinghua Li, Qingmin Wang, Hans-Wolf Reinhardt, Shilang Xu, Engineering Fracture Mechanics 277 (2023) 108988, p. 1-42,

gives a thorough overview including the theoretical background of the method. It is approved by the Chinese organisation of standards and the international organisation for construction materials experts RILEM for fracture mechanical testing of a restricted group of materials. 

The method is based on two critical stress intensities, one for initiation of crack growth and a second for the switch to fast uncontrollable crack growth. A large number of experimental techniques and numerical methods to improve measurements and their evaluation accuracies are nicely organised into a large number of subsections. The review is a rewarding reading that gave me great pleasure and introduced me to the difficulties and advances in numerical techniques to approach the fracture mechanics of one of the most important groups of materials. The nearly three decades of history put much into perspective. 

One thing that puzzled me regarding the unstable crack growth considering observations during the 1980s when it was discovered that small cracks are prone to jump the stable crack growth part. Instead, unstable crack growth was initiated earlier than what was expected from linear fracture mechanics analyses. In modelling the event using cohesive zones replacing the plastic deformation and the fracture processes, the tip of the already growing cohesive zone tip becomes unstable while the crack length is unchanged. The increasing load resulted in unstable crack growth shortly thereafter. The larger the crack the shorter the time gap between the initiation of unstable growth of the tip of the cohesive zone and that of the crack tip.  

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

The link that leads to the paper is presently not fully open access paper but it will be within a couple of days.

Per Ståhle

*D.S. Dugdale's paper from 1960 was published the year after G.I. Barenblatt's original Russian paper from 1959, which was published in English in 1963.

Subscribe to Comments for "Discussion of fracture paper #36 - The Double-K Fracture Model"

Recent comments

More comments


Subscribe to Syndicate