The subject of this blog is a fracture mechanical study of soft polymers. It is well written and technically detailed which makes the reading a good investment. The paper is:
As the title says, it is about the fracture mechanical properties of a group of polymers. The basic idea is to identify the energy release rate that is required to initiate crack growth. To distinguish between the energy required for creating crack and the energy dissipated in the surrounding continuum, the former is defined as the unstable material which has passed its largest load carrying capacity, and the remaining is the stable elastic plastic continuum. The energy required for creating crack surface is supposed to be independent of the scale of yielding.
The authors call it the essential work of fracture, as I believe was coined by Mai and Cotterell. If not the same, then this is very close to the energy dissipation in the fracture process region, as suggested by Barenblatt, and used by many others. Material instability could, of course also be the result of void or crack nucleation at irregularities of one kind or another outside the process region. How much should be included as essential work or not, could be discussed. I guess it depends on if it is a necessary requirement for fracture. The fact that it may both support and be an impediment to fracture does not make it less complicated. In the paper an FE model is successfully used to calculate the global energy release rate vis à vis the local unstable energy release in the fracture process region, modelled as a cohesive zone.
What captured my interest was the proposed two parameter cohesive zone model and its expected autonomy. With one parameter, whatever happens in the process region is determined by, e.g., K, J, G. The single parameter autonomy has its limits but more parameters can add more details and extend the autonomy and applicability. For the proposed cohesive zone, the most important parameter is the work of fracture. A second parameter is a critical stress that marks the onset of the fracture processes. In the model the critical stress is found at the tip of the cohesive zone. By using the model of the process region, the effect of different extents of plastic deformations is accounted for through the numerical calculation of the surrounding elastic plastic continuum.
The work of fracture is proportional to the product of the critical stress and the critical separation of the cohesive zone surfaces. The importance of the cohesive zone is that it provides a length scale. Without it, the process region would be represented by a point, the crack tip, with the consequence that the elastic plastic material during crack growth consumes all released energy. Nothing is let through to the crack tip.
Stationary cracks are surrounded by a crack tip field that releases energy to fracture process regions that may be small or even a singular point. If the crack is growing at steady-state very little is let through to a small fracture process region and to a singular point, nothing. In conventional thinking a large cohesive stress leads to a short cohesive zone, and by that, the available energy would be less. A variation of the critical stress is discussed in the paper. Presently, however, the two parameter model is more of a one parameter ditto, where the cohesive stress is selected just as sufficiently plausible.
What could be done to nail the most suitable critical cohesive stress? With the present range of crack length and initiation of crack growth nothing is needed. The obtained constant energy release rate fits the experimental result perfectly. Further, it is difficult to find any good reason for why the excellent result would not hold also for larger cracks. As opposed to that, small, very small or no crack at all should give crack initiation and growth at a remote stress that is close to the critical cohesive stress. As the limit result of a vanishing crack, the two stresses should be identical. I am not sure about the present polymer but in many metals the growing plastic wake requires significant increase of the remote load. Often several times rather than percentages. So letting the crack grow at least a few times the linear extent of the plastic zone, would add on requirements that may be used to optimise both cohesive parameters.
I really enjoyed reading this interesting paper. I understand that the paper is about initiation of crack growth which is excellent, but in view of the free critical cohesive stress, I wonder if the model can be extended to include very small cracks or the behaviour from initiation of crack growth to an approximate steady-state. It would be interesting if anyone would like to discuss or provide a comment or a thought, regarding the paper, the method, the autonomy, or anything related. The authors themselves perhaps.
Per Ståhle
Thank you very much for your interest in our paper and for reading it so thoroughly. I think it’s an interesting discussion and reflection. For instance, the fact that the cohesive zone has two independent parameters, the fracture energy and the maximum stress, does not really turn the model into a two-parameter model in the same sense as the K-T or the J-Q models. These other models represent more of the surrounding mechanical state than the two-parameter cohesive zone model is able to account for.
Nailing the cohesive stress, I believe, is a tricky issue. It seems to me that for ductile materials this entity lacks a clear physical interpretation, and therefore tends to play the role of a fitting parameter. Another issue that you raise is whether the present set of parameters is general enough to predict the behavior of considerably longer cracks or for short or even vanishing cracks. Although we have not investigated the predictive ability of the model for these cases, I would expect that the prediction would be far from perfect. One reason is, as you mentioned, that the cohesive zone is taken to be autonomous, that is independent of what is going on around it.
We have actually performed experiments for growing cracks as well (not published yet). These results have not been fully evaluated, but preliminary results indicate that it is, for instance, difficult to predict the behavior for long crack growth using the present approach. Furthermore, it is evident that during crack growth, the rate dependence of the fracture process is very important to capture accurately.
Finally, the question must also be raised if the cohesive zone concept is really the best way to model this kind of crack growth which exhibits a lot of ductility. In the study, we did some preliminary investigations of the fracture surfaces, but I would say that further examinations of the actual fracture process are needed. Such investigations might suggest if, for example, a continuum damage model is a better approach to model fracture of these materials.