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

Dear Professor Per Ståhle,

Many thanks for your kind email to us. It’s a great pleasure to know that you gave such positive comments to our paper "The double-K fracture model: A state-of-the-art review” published in the international journal of Engineering Fracture Mechanics.

On the blog, you have written the following comments:

    “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.”

    This is a very interesting and challenging question. In fact, in the age of 1980s, I met the same puzzle in my early research work. My research team is still exploring the mechanism of this problem nowadays.  According to my best knowledge, I explain the reasons as follows: 

    The first reason is due to the size effect on the fracture properties. If the size of the tested specimen is too small, it is difficult to observe the stage of stable crack propagation, as measurement techniques used in the tests are not precise enough to capture the initial cracking width. In our later experiments using enough large size of specimens, the stable crack propagation was captured very well. We found that the steady state of crack propagation only occurred under certain conditions, such as when the tested specimens are in large size or when the crack tip is far away from the specimen boundaries. 

    The second reason is due to the affection of the ratio between the minimum specimen size and the maximum aggregate size. As known well, concrete is a kind of heterogeneous material. It was found in our experiments that when the ratio between the minimum specimen size and the maximum aggregate size is smaller than 6, the heterogeneity of concrete could not be ignored. The heterogeneity could result in somehow strain locality concentration that would aggravate the unstable crack propagation, especially when the size of the specimen is too small. 

    Recently, the mechanical mechanism for steady crack propagation based on energy balance is proposed in our new study. You can find more details in the paper“Occurrence condition for steady crack propagation in quasi-brittle fracture and its application in determining initial fracture toughness” published in the International Journal of Solids and Structures, authored by Yao Wu, Shilang Xu, Qinghua Li and Rena C. Yu.( Wu, Y., Xu, S., Li, Q., Yu, R.C., Occurrence condition for steady crack propagation in quasi-brittle fracture and its application in determining initial fracture toughness, International Journal of Solids and Structures (2022), doi: https://doi.org/10.1016/j.ijsolstr.2022.112094).  In this study, we found that the essence of the steady-state regime was related to the free development of the cohesive zone, the occurrence condition for the steady crack propagation was deduced based on the relative distance from the crack tip to the specimen boundaries. For example, for three-point bending beam tests the steady crack propagation occurs when (D-a)/LFPZ>1.25, where, D is the depth of specimen tested, a is the crack length, LFPZ is the length of the fracture process zone, otherwise, the unsteady crack propagation occurs. 

    With regard to the mentioned phenomenon in the numerical study in the 1980s, it could be caused by the difficulty simulating the crack propagation in concrete, which is closely related to the technique of modelling the complicated heterogeneous concrete material. Both the numerical models and the numerical approaches have been in constant development in the world. We also are making an effort to develop such a new numerical simulation tool to study the crack initiation, stable crack propagation and unstable crack failure in concrete corresponding to the double-K fracture criterion, the corresponding results will be published in a short future. 

    Indeed, as you mentioned, in the 1960s age, A Barenblatt process zone seemed ideal but it required 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. Concrete is a type of engineering material with multiscale mechanical properties and complicated heterogeneous characteristics. New approaches that are sufficiently simplified, validated and also accessible are needed to describe the structural integrity of concrete structures in engineering practice, such as concrete dams. Many researchers in the world are proposing a variety of numerical methods and making great efforts to achieve this goal. Our effort in the double-K model is to provide an engineering approach with practical simplicity and convenience.

    Due to the inconvenience of using a blog in our normal situation, I am writing this reply letter to you on behalf of all authors of this paper.

Thank you very much once more again.

Best regards，

Dr. Shilang Xu 

Professor, Zhejiang University

Director of the Institute of Advanced Structures, Zhejiang University