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# The Configurational Force Approach in Elasto-Plastic Fracture

New paradigms may help understanding unsolved scientific problems by looking on them from a different perspective. Or they may lead to a new unification theory of so far separate phenomena. The concept of “material” or “configurational” forces tracing back to a seminal publication of Eshelby in 1970 and significantly extended and promoted by Maugin twenty years later provides a generalised theory on the character of singularities of various kinds in continua, among which the “driving force” at a crack tip is a special case. Whereas Eshelby’s energy momentum tensor resulting in the *J*-integral is a firm constituent of fracture mechanics, the concept of configurational forces has only hesitantly been applied to fracture problems, e.g. by Kolednik, Predan, and Fischer in *Engineering Fracture Mechanics*, Vol. 77, 2010. Whether this new “look” upon *J* helped discovering anything new about it remains disputable.

Now there is a revival of this concept

K. Özenç, M. Kaliske, G. Lin, and G. Bhashyam: Evaluation of energy contributions in elasto-plastic fracture: A review of the configurational force approach. *Engineering Fracture Mechanics*, Vol. 115, 2014, pp. 137–153.

It is admittedly difficult to contribute some novel aspect to more than forty years of research on *J* in elastoplastic fracture mechanics. Though a clear perception of the nature of “path dependence” of *J* is often enough still missing in some publications to the point of the user’s manual of a major commercial FE code, there is no lack of theoretical knowledge. Background, applicability and limitations of *J* are quite clear. Those looking for deeper insight will be disappointed: The present publication just answers questions and solves problems which arose with the chosen approach of material forces.

“*The path dependency of the material force approach in elasto-plastic continua is found to be considerably depending on the so-called material body forces*.” This is well-known and trivial as the derivation of path independence of *J* is, among others, based on the absence of body forces. It does not need “*numerical examples ... to clarify the concept of path dependence nature of the crack tip domain *(?)* and effect of the material body forces*”. Correction terms re-establishing path independence have been introduced years ago, see e.g. Siegele, *Comput. Struct*., 1989.As many continuum mechanics people, the authors start with a display of fireworks introducing the general nonlinear kinematics of large deformations which can be found in every respective textbook. In the end, this impressing framework is simmered down again to “*small strain elasto-plasticity and hyperelasto-plasticity*”, whatever “*hyperelasto-plasticity” * is supposed to mean. This does not become much clearer by the statement “*the Helmholtz free energy function of finite elasto-plasticity is introduced in order to obtain geometrically nonlinear von Mises plasticity*”. Finite, i.e. Hencky-type plasticity and incremental plasticity, i.e. the von Mises, Prandtl, Reuss theory are alternative approaches, where the latter is more appropriate for describing irreversible, dissipative processes. What a “*geometrically nonlinear*” material behaviour is remains the secret of the authors. They presumably applied the so-called “deformation theory of plasticity” which actually describes hyperelastic behaviour based on the existence of a strain-energy density as stress potential. Thus “*path dependence*” should not be an issue at all as the requirements for deriving path-independence are met. The rest is numerics!

So where are the problem and its solution after all? Can “material forces” be calculated by the finite element method - who doubts? Is the implementation of this concept in a commercial FE code a major scientific achievement - who knows?

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## Comments

## Dear Prof. Brocks

Dear Prof. Brocks

I am the first author of the paper you commented on. I appreciate that you decided to learn and to study the material force approach after our talk at the IWPMEO workshop in Antalya, Turkey, 2013. You told me that you were not aware of this method. I believe that you are still missing some parts of the concept.

First of all, thank you for your interest in our paper. I also appreciate that you expected a new theory from our review paper but it is obvious from the title that it is indeed a review paper in order to evaluate the method. However, when we started to implement and to study the material force approach in small strain plasticity, the main concern was which energy contribution should be considered to derive the momentum balance equation in the material space. It had to be investigated which derivation yields the crack driving force or rather the energy release rate available for crack propagation and whether it is possible to separate them by this approach or not. Although I found some authors using different energy contributions, I could not find really a clear and comprehensive study taking all the resulting terms into account in case of plasticity neither in papers on the J-integral nor in papers on the material force method in combination with numerical studies and comparisons (if you know one, please inform me).

Then, I began to study the formulation by using different energy terms. I found that in the traditional J-integral, which is commonly used in commercial software, the total stored energy in the bulk and dissipated energy are used in the Eshelby part of the equation (which is Rice’s J-integral in this case) and the so-called material body forces are ignored where the calculation of the gradient terms is required. Thus, in the paper, we showed that when the total energy in the bulk is considered as the energy available in order to obtain the crack driving force as it is in Rice’s J-integral, there are still gradient terms, which need to be included to formulation (or material body forces). However, in software, people simply ignore those terms since they vanish when the system is under monotonic loading. Furthermore, we derived material forces for plasticity from three different energy contributions in Section 5.3 of our paper:

I) Energy in the elastic spring

II) Energy in the elastic and hardening spring (the total stored energy in the bulk)

III) Energy in the elastic and hardening spring and sliding frictional element (the total stored energy in the bulk and dissipated energy by plasticity)

and compared the results for the close field and for the far field integration to an experimental study. I believe that you missed this part in the paper and you made the criticism that it was just a software implementation. Besides, even if there is a paper which already explains the full formulation with additional terms for the different energy contributions as I explained above, numerical challenges still remain to calculate the gradient of the terms in the discretizated continuum. I believe this part of the approach still needs some further study.

The path dependency study in the paper is a kind of prove of the equilibrium of the material momentum balance equation. In other words, the system must satisfy the balance equation and the path independency is the result of vanishing of nodal material forces in the part of the body, where singularity (or inhomogeneity) does not exist (in far field). Therefore, path dependency study is a verification of the concept of the material momentum balance and a verification of the implementation especially in the case that the calculation of the gradient terms is required in the formulation.

Therefore, we made a path dependency study on the material force and the Rice’s J-integral. As a result, we did not get path dependency for the material force approach even in the fracture process zone (in plastic regime) which has not been illustrated numerically in a comparative study yet according to my knowledge. As you know, the limitations of the J-integral in plasticity have always been a question. Therefore, even though it is a review paper, I believe that we presented something new by using various energy terms and showing the results numerically. Moreover, another contribution of the paper is to validate the computations against published test data, which is not available according to our knowledge. The reviewers and editor agreed with us and decided to publish the work.

To your other questions.

“the authors start with a display of fireworks introducing the general nonlinear kinematics of large deformations which can be found in every respective textbook. In the end, this impressing framework is simmered down again to “small strain elasto-plasticity and hyperelasto-plasticity”, whatever “hyperelasto-plasticity”is supposed to mean.”What you consider as "fireworks", are just the required fundamentals of the general description of the plasticity concept and it must be introduced to clarify the notation and to derive the momentum balance equation in both spaces. This procedure is standard in deriving thoroughly results in continuum mechanics. However, I believe that it is not possible to claim that Eq. (23) in the paper, which is obtained from the gradient of the Helmholtz energy function, is a momentum balance equation in material space, unless we describe the difference between material (un-deformed) space and spatial (deformed) space and the mappings between them. For this point, I would like to draw your attention as well to the following papers

Steinmann, P. (2000). International Journal of Solids and Structures 37, 7371–7391.Steinmann et al. (2001). International Journal of Solids and Structures 38, 5509–5526.Maugin et al. (1992). Acta Mechanica 94, 1–28.Menzel et al. (2004). Computer Methods in Applied Mechanics and Engineering 193, 5411–5428.Naeser et al. (2007). Computational Mechanics 40, 1005–1013.By the way, “

geometrically nonlinear“ stands for a geometrically exact description without restrictions which means that the theory is not reduced to small strains or any type of simplifying assumption. I believe it is clear to the readers. Moreover, the term hyperelasto-plasticity is not used by us for the first time. It is a standard terminology in continuum mechanics. You can find more information on it athttp://scholar.google.com/scholar?hl=tr&as_sdt=0,5&q=hyper+elasto+plastic .

Your other question:

“So where are the problem and its solution after all?Problem:

What is the main difference between Rice’s J-integral , Kishimoto’s J-integral and the material force approach in plasticity and what is the role of the different energy contributions in material force and material body forces?

Answer:

The main differences are the energy contributions and additional terms, which we call material body forces. It seems that all three cases, that we studied, give the equivalent of the crack driving force for monotonic loading with the correction terms in far field calculation and they are path independent. For further studies, one may study arbitrary loading cases.

Your other question:

“Can “material forces” be calculated by the finite element method - who doubts? Is the implementation of this concept in a commercial FE code a major scientific achievement - who knows?”Answer:

As I mentioned that although it is a review paper to show how people can use the approach and interpret the difference with the classical J-Integral, it gives a further study on energy contributions to clarify the difference between the methods by numerical studies. Does it solve all the questions on the approach? Of course not, but I believe that it is a step forward. Moreover, I do not know from which part of the paper you came to the conclusion that deformation plasticity is used in our study but we indeed used incremental plasticity. If we used deformation plasticity then material body forces would vanish which has already been shown in the following paper

Simha et al. Journal of the Mechanics and Physics of Solids 56 (2008) 2876-2895,which I definitely recommend you to read.

Last but not least, I am really surprised about your style to comment on a scientific work. You could have approached us directly (email is given in the paper) in order to have an open and honest scientific discussion. Due to the fact that it is not the case, we just can speculate on your motivation putting these comments somewhere in the internet without any information given to us.

Regards,

Kaan Ozenc

## Post Processing Method for Material Force

Could you guys please share the steps involved in calculating the 'material force' in abaqus? I need to understand the crack propagation study of tires.

Regards

Ujjwal

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