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# Discussion of fracture paper #17 - What is the second most important quantity at fracture?

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No doubt the energy release rate comes first. What comes next is proposed in a recently published study that describes a method based on a new constraint parameter *Ap*. The paper is:

The parameter *Ap* is compared with established parameters like *T*, *Q* etc. The application is to pipes with edge cracks. I would guess that it should also apply to other large structures with low crack tip constraint.

As everyone knows, linear fracture mechanics works safely only at small scales of yielding. Despite this, the approach to predict fracture by studying the energy loss at crack growth, using the stress intensity factor *K*I and its critical limit, the fracture toughness, has been an engineering success story. *K*I captures the energy release rate at crack growth. This is a well-founded concept that works for technical applications that meet the necessary requirements. The problem is that many or possibly most technical applications hardly do that. The autonomy concept in combination with *J*-integral calculations, which gives a measure of the potential energy release rate of a stationary crack, widens the range of applications. However, it is an ironin that the *J*-integral predicts the initiation of crack growth which is an event that is very difficult to observe, while global instability, which is the major concern and surely easy to detect, lacks a basic single parameter theory.

For a working concept, geometry and load case must be classified with a second parameter in addition to *K*I or *J*. The most important quantity is no doubt the energy release rate, but what is the second most important. Several successful parameters have been proposed. Most of them describe some type of crack tip constraint, such as the *T*-stress, *Q*, the stress triaxiality factor *h*, etc. A recent suggestion that, as it seems to me, have great potential is a measure of the volume exposed to high effective stress, *Ap*. It was earlier proposed by the present group GZ Wang and co-authors. *Ap* is defined as the relative size of the region in which the effective stress exceeds a certain level. As pointed out by the authors, defects in large engineering structures such as pressure pipes and vessels are often subjected to a significantly lower level of crack tip constraint than what is obtained in laboratory test specimens. The load and geometry belong to an autonomy class to speak the language of KB Broberg in his book "Fracture and Cracks". The lack of a suitable classifying parameter is covered by *Ap*.

The supporting idea is that *K*I or *J* describe the same series of events that lead to fracture both in the lab and in the application if the situations meet the same class requirements, i.e. in this case have the same *Ap*. The geometry and external loads are of course not the same, while a simpler and usually smaller geometry is the very idea of the lab test. The study goes a step further and proposes a one-parameter criterion that combines the *K*I or *J* with *Ap* by correlation with data.

The method is reinforced by several experiments that show that the method remains conservative, while still avoiding too conservative predictions. The latter of course makes it possible to avoid unnecessary disposal and replacement or repair of components. The authors' conclusions are based on experience of a particular type of application. I like the use of the parameter. I guess more needs to be done extensively map of the autonomy classes that is covered by the method. I am sure the story does not end here.

A few questions could be sent along: Like "Is it possible to describe or give name to the second most important quantity after the energy release rate?" The paper mentions that statistical size effects and loss of constraint could affect *Ap*. Would it be possible to do experiments that separates the statistical effect from the loss of constraint? Is it required or even interesting?

It would be interesting to hear from the authors or anyone else who would like to discuss or comment the paper, the proposed method, the parameter or anything related.

Per Ståhle

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## Comments

## Author’s comments

Thank Professor Per Ståhle very much for these valuable discussion, comments and suggestions for our paper entitled "Fracture assessment based on unified constraint parameter for pressurized pipes with circumferential surface cracks" which has been published in Engineering Fracture Mechanics 175 (2017), pp. 201–218.

It is well-known that the conventional fracture mechanics assumes that the single parameter

KI orJcan describe crack-tip field in laboratory specimens and engineering components at fracture, and the structural component exhibits the same fracture resistance,Kc, orJc, at the onset of unstable fracture as the laboratory specimen. However, it has been shown that the crack-tip field is affected by specimen or component geometries, crack sizes, load modes etc. This is the crack-tip constraint effect, which leads to that the fracture assessment in low constraint structural components using the conventional fracture mechanics methodologies may be excessively conservative. Thus, in addition toKI orJ, it needs a second parameter (constraint parameter) to describe accurately the crack-tip field. The constraint contains in-plane and out-of-plane constraints. The in-plane constraint is directly affected by the length of the un-cracked specimen ligament, while the out-of-plane constraint is affected by the specimen thickness. The widely-used constraint parametersT-stress andQbased on crack-tip stress field only can quantify the in-plane constraint effect. But in actual engineering structures, both in-plane and out-of-plane constraints exist simultaneously. Therefore, it is necessary to develop a new methodology of fracture assessment to incorporate both in-plane and out-of-plane constraints.It has been shown in our previous work [

2013,36: 504-514.Fatigue & Fracture of Engineering Materials and Structures,Fatigue & Fracture of Engineering Materials and Structures,2014, 37:132–145.Engineering Fracture Mechanics,2014,115: 296-307., 2014, 190: 87-98.International Journal of Fracture2016, 39: 1461–1476.Fatigue & Fracture of Engineering Materials and Structures,, 2015, 80: 121-132] that the parameterTheoretical and Applied Fracture MechanicsApbased on crack-tip equivalent plastic strain can characterizeboth in-plane and out-of-plane constraints, and also it can be used for both brittle fracture under small-scale-yield (SSY) condition and ductile fracture under large-scale-yield (LSY) condition for steels and welded joints. In other words, the effects of geometry and load on different classes fracture in specimens and components of different materials may be captured or covered by the parameterAp. In this paper, the capability of fracture assessment based on the unified constraint parameterAphas been investigated for pressurized pipes with circumferential surface cracks. As suggested by Prof. Ståhle, the story does not end here. We will do more further work for the extensive application and verification of the parameterAp. It may include the engineering methodology of the use of the parameter, the applicability of the parameter for more specimens and components with different in-plane and out-of-plane constraint levels, different materials and welded joints, and different fracture modes (such as brittle fracture, ductile fracture initiation, ductile crack growth, fatigue and SCC crack growth, etc.).For the questions in Prof. Ståhle’s discussion, our response or comments are as follows.

(1) The second most important quantity is related to both in-plane and out-of-plane constraint. Its name may be unified constraint parameter. Based on the further studies of its physical and mechanics meanings, a more suitable name may be given.

(2) The statistical size effects and loss of constraint could affect the brittle cleavage fracture toughness of steels. The experimental investigation that separates the statistical effect from the loss of constraint on cleavage fracture toughness has been done by Rathbun et al.

[Rathbun HJ, Odette GR, Yamamoto T, Lucas GE. Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition—A single variable experiment and database. Engng Fract Mech 2006;73:134-58]. The statistical size effect also has been considered in the Master Curve approach in ASTM E1921. The fracture toughness in our paper is based on the experimental fracture toughness data, and the calculation of the parameterApis related to fractureJ-integral of the experiment data. Because the combined effects of constraint loss and statistical size have been reflected in the experimental fracture toughness data, the correlation of the fracture toughness with the parameterApmay relate to both constraint loss and statistical size effects.Best regards

G.Z.Wang

G.Z.Wang, Ph.D, Professor

School of Mechanical and Power Engineering

East China University of Science and Technology

130 Meilong Road, Mail box 402

Shanghai 200237,China

E-mail: gzwang@ecust.edu.cn

## Dear Professor Wang, It is

Dear Professor Wang, It is interesting that Ap is related to the constraint even if it is in an inverse way. I was looking (Googled) for an antonym to constrain. Of the variety of suggestions the closest may have been unleash, release, flow... probably someone already has a word for it.

If we return to your paper and the background of the Ap. I can see in eq (9) that your data indicate proportionality to Ap^0.44. It is rather close to $\sqrt{A_p}$ that is related to a length scale. If one writes $A_p^{0.44}=\sqrt{A_p} A_p^{-s}$ with the s=0.06 and also assume that $|A_p-1|<1$ one can expand around small s and small $(A_p-1)$ and obtain $A_p^{1/2-s}\approx (1+s)A_p^{1/2}-sA_p^{3/2}$ the remaining part is of the order of $s(A_p-1)^3$ and only odd exponents prevail. The series is converging very fast and I wold think that only the two first terms would be enough to replace the Ap^0.44 with good accuracy.

Could it be that there is a main a process that has a length scale $\sqrt{A_p}$ and another process that is a volume related phenomenon scaling with $A_p^{3/2}$? The ratio between the effects the respective phenomena are causing would be s. Could this help to sort out the complicated events that preceeds fracture? What are your thoughts?

I hope this is not misunderstood. I definitely think that the exponential form used in the paper is the most effective engineering approach. Per

## Author’s comments

Thank Professor Per Ståhle very much for the further discussion on our paper.

The Eq (9) in our paper is the relation between the normalized fracture toughness

KJc/Krefand constraint parameter Ap. It is established based on the experimental fracture toughness data of a specific material from specimens with different in-plane and out-of-plane constraints and finite element calculations of the parameterApfor these specimens. ThisKJc/Kref-Ap relation is usually calledmaterial toughness locus,which is similar to theJ1c-Qrelation in the early paper by R.H.Dodds et al in this area[R.H.Dodds, C.F.Shih and T.L.Anderson. Continuum and micromechanics treatment of constraint in fracture, International Journal of Fracture, 1993, 64:101-133]. TheJ1c-Qrelation mainly describe the relation between fracture toughness and in-plane constraint, and theKJc/Kref-Ap relation captures both in-plane and out-of-plane constraints. The studies in our previous paper have shown that theKJc/Kref-Ap relation and the constants in Eq.(9) depend on material and fracture mode. In other word, the relation and constants in Eq.(9) is different for brittle and ductile fracture and for different materials. For the engineering application of the parameterAp, theKJc/Kref-Ap relation needs firstly to be determined by experiments or numerical simulations and the finite element calculations of the parameterAp.The mathematical analysis made by Prof. Ståhle’s for the

KJc/Kref-Ap relation in Eq (9) in our paper may be helpful for further investigating and understanding the mechanical meaning of the parameterAp, sorting out the complicated fracture events and effective engineering application ofAp.Best regards

G.Z.Wang

G.Z.Wang, Ph.D, Professor

School of Mechanical and Power Engineering

East China University of Science and Technology

130 Meilong Road, Mail box 402

Shanghai 200237,China

E-mail: gzwang@ecust.edu.cn