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Some material scientists and experimentalists are generally sceptical of simulations and reproach the theoreticians with lacking knowledge of real materials. Sometimes they may just be ignorant of the mathematics behind the models but sometimes they appear to be right. An example: Introducing “damage” into a constitutive equation simply as an internal variable which obeys some evolution law, not having the foggiest notion about the specific nature of damage – whether brittle, ductile, creep, fatigue – and its micromechanical mechanisms, promotes scepticism about the benefits of modelling in general, and deservedly so.

This is not my particular point today, however, but it is related. My problem today is the handling of length scales in

H. Krull and H. Yuan: Suggestions to the cohesive traction–separation law from atomistic simulations. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 525-533.

Ductile tearing is governed by the initiation, growth and coalescence of voids in an elasto-plastic or viscoplastic material. Koplik and Needleman (1981) have been the first to perform unit-cell calculations of void growth to analyse this mechanism, and numerous studies by other authors followed varying the void shape, accounting for inclusions etc.. They helped improving and generalising constitutive equations of ductile damage like the Gurson model of porous metal plasticity. Unit-cell calculations have also been used to derive traction-separation laws for cohesive zones. The physical processes take place at length scales of micrometers to millimetres, accordant to the dimensions of the microstructure. Continuum models still apply at this length scale.

Atomistic simulations and molecular dynamics are based on models that relate binding energies or forces to spatial configurations in order to calculate accelerations of particles via Newton’s law. They describe interactions at a length scale of nanometers, which is at least three orders of magnitude below the relevant length scale of ductile tearing. What occurs at this length scale has absolutely nothing in common with plastic deformations due to dislocation motion and ductile tearing of metals, and hence the authors’ conclusions are apocryphal and unsubstantiated:

>     “The computations under mode I conditions show that crack growth even in the nano-scale single-crystal aluminum is in the form of void nucleation, growth and coalescence, which is similar to ductile fracture at meso-scale.” 

 What do the authors actually mean by vague formulations like “in the form of void nucleation …” and “similar to ductile fracture”? Voids nucleate at particles, for instance. Which particles are of atomic or sub-atomic size? What is their criterion for a process being “similar to” another?   

>   “Understanding the failure process based on atomistic simulation can provide detailed information for the cohesive law.”     

This is correct, of course, provided the correct failure process is modelled.  

>   “The relationship between atomic traction and atomic separation including elastic deformations confirms the exponential function form suggested by Needleman.”   

This is neither a miracle nor striking news, as the cohesive law proposed by Needleman in Int. J. Fracture 42 (1990) is based on the universal atomistic binding energy function proposed by Rose et al. in Phys. Rev. Letters 47 (1981), and he actually analysed “tensile decohesion along an interface” in an elastic medium.     

>  “The computations show that void nucleation and growth are controlled by tensile stress and hydrostatic stress. The Mises stress is not involved in material failure.”     

This is the final death sentence to any simulation: that it yields results which are contrary to everything that is known about the real process. Apart from this, the hydrostatic stress and the maximum tensile stress depend on each other under fully plastic conditions of plane strain. But plasticity is out of the scope of the MD simulations, anyway.

After all, the simulations could indeed provide useful information on a cohesive law for a process which is not void growth and coalescence in ductile metals, however. It is the authors’ business to present an actual decohesion mechanism following their model. 

https://imechanica.org/node/10945

Williams derived it in 1939, Irwin addressed it in 1957 as one of two parameters characterising “the influence of the test configuration, loads and crack length upon the stresses”, and Rice used it in 1974 to calculate the effects of the specimen geometry on the plastic zone in small scale yielding: the non-singular term in the series expansion of the stress field at crack tips called T-stress. It gained importance in the early 1990s in numerous investigations on constraint effects and two-parameter approaches. With the upcoming of damage mechanics, the number of publications on T-stress went down and everything seemed to be said about their significance. The few papers on T-stresses appearing in the first decade of the 21st century mostly concerned their calculation for various configurations – even for “cracks in anisotropic bimaterials” (EFM 75, 2008), which is outside of their theoretical foundation. Surprisingly, two new (partly quite similar) papers on this subject appeared recently:

J.C. Sobotka, R.H. Dodds: Steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modelling. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 343–363.J.C. Sobotka, R.H. Dodds: T-stress effects on steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modelling. Engineering Fracture Mechanics, Vol. 78, 2011, pp. 1182–1200.

Is this a renaissance of early fracture mechanics concepts or just a latecomer? Let us have a look on the details.

T-stress effects on stress fields at stationary cracks for small-scale yielding have been extensively investigated in the 1990s using a so-called boundary-layer model, i.e. a disk-shaped volume centred at the crack front, which is subjected to a K-field and a constant stress parallel to the ligament. It needs a particular Eulerian analysis to represent steady-state growth on a fixed mesh in a boundary-layer framework. The application of the respective “streamline integration” introduced by Dean and Hutchinson in 1980 to 3D panels is the basic achievement of the two papers, allowing to study thickness effects and variations of plastic zones, stress fields and crack opening displacement over the thickness, the first one for T = 0, the second for T ¹ 0.

These are thoroughly performed analyses yielding substantial information on the local fields. What they do not answer is the question on their relevance for actual fracture problems. In the extensive discussions on ductile tearing resistance, the T-stress has been proposed as a parameter characterising the “constraint”. This definitely works in small-scale yielding, but can steady state crack extension occur under small-scale yielding conditions? The authors argue with crack growth in thin panels of high-strength aluminium alloys as they are used in aerospace structures. They claim in the introduction that a T-L orientation of the cracked panel, “tends to favor a local ‘flat‘ mode I fracture process rather than a local ‘slanted‘ mixed-mode process. … Essentially steady conditions evolve as the crack front advances further over distances of several thicknesses, characterized by a flat-to-nearly-flat tearing resistance curve.“ Both statements are indeed essential in the context of their investigations but unfortunately, they are not substantiated by experimental evidence. And a final question: how significant is the T-stress in a cracked thin panel under tension? So what about a continuation including test data and their analysis?

This is a premiere: my first contribution to the new ESIS’ blog announced in January. Why comment on papers in a scientific journal after they have passed the review process already? Not to question their quality, of course, but animating a vital virtue of science again, namely discussion. The pressure to publish has increased so much that one may doubt whether there is enough time left to read scientific papers. This impression is substantiated by my experience as a referee. Some submitted manuscripts have to be rejected just because they treat a subject, which conclusively has been dealt years before – and the authors just don’t realise. So much to my and Stefano’s intention and motivation to start this project.

Here is my first “object of preference”:

Ehsan Barati, Younes Alizadeh, Jamshid Aghazadeh Mohandesi, “J-integral evaluation of austenitic-martensitic functionally graded steel in plates weakened by U-notches”, Engineering Fracture Mechanics, Vol. 77, Issue 16, 2010, pp. 3341-3358.

The comment

It is the concept of a finite “control” or “elementary volume” which puzzles me. It is introduced to establish “a link between the elastic strain energy E(e) and the J-integral” as the authors state. Rice’s integral introduced for homogeneous hyperelastic materials is path-independent and hence does not need anything like a characteristic volume. This is basically its favourable feature qualifying it as a fracture mechanics parameter relating the work done by external forces to the intensity of the near-tip stress and strain fields.

Fig. 2 (a) schematically presents this control volume in a homogeneous material, and the authors find that “the control volume boundary in homogeneous steel is semi-circular”. But how is it determined and what is the gain of it?

Introducing a characteristic volume for homogeneous materials undermines 40 years of fracture mechanics in my eyes..

One might argue that the introduction of this volume is necessary or beneficial for functionally graded materials (FGM). The authors state however that “comparison of the J-integral evaluated by two integration paths has shown that the path-independent property of the J-integral is valid also for FGMs”. Whether or not this is true (there are numerous publications on “correction terms” to be introduced for multi-phase materials), it questions the necessity of introducing a “control volume”. There is another point confusing me. The J-integral is a quantity of continuum mechanics knowing nothing about the microstructure of a material. The austenite and martensite phases of the FGM differ by their ultimate tensile strength and their fracture toughness. Neither of the two material parameters affects the (applied) J, only Young’s modulus does in elasticity. Hence it does not surprise that J emerged as path-independent! The authors compare J-integral values of homogeneous and FG materials for some defined stress level at the notch root in Fig. 10. The differences appear as minor. Should we seriously expect, that a comparison of the critical fracture load predicted by Jcr and the experimental results (Fig. 16) will provide more than a validation of the classical J concept for homogeneous brittle materials?

Not to forget: The authors deserve thanks that they actually present experimental data for a validation of their concept, which positively distinguishes their paper from many others!

W. Brocks

https://imechanica.org/node/9793