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Discussion of fracture paper #13 - Cohesive properties at ductile tearing

esis's picture

In this review of particularly readworthy papers in EFM, I have selected a paper about the tearing of large ductile plates, namely:

”Cohesive zone modeling and calibration for mode I tearing of large ductile plates”  by P.B. Woelke, M.D. Shields, J.W. Hutchinson, Engineering Fracture Mechanics, 147 (2015) 293-305.

The paper begins with a very nice review of the failure processes for plates with thicknesses from thick to thin, from plane strain fracture, via increasing amounts of strain localisation and failure along shear planes, to the thinnest foils that fail by pure strain localisation.

The plates in the title have in common that they contain a blunt notch and are subjected to monotonically increasing load. They are too thin to exclusively fracture and too thick to fail through pure plastic yielding. Instead the failure process is necking, followed by fracture along a worn-out slip plane in the necking region. Macroscopically it is mode I but on a microscale the final failure along a slip plane have the kinetics of mixed mode I and III and, I guess, also mode II. 

A numerical solution of the problem resolving the details of the fracture process, should perhaps be conceivable but highly unpractical for engineering purposes. Instead, the necking region, which includes the strain localisation process and subsequent shear failure is a region of macroscopically unstable material and is modelled by a cohesive zone. The remaining plate is modelled as a power-law hardening continuum based on true stress and logarithmic strains.

The analysis is divided into two parts. First a cross-section perpendicular to the stretching of the cohesive zone is treated as a plane strain section. This is the cross section with a shape in which the parable with a neck becomes obvious. Here the relation between the contributions to the cohesive energy from strain localisation and from shear failure is obtained. A Gurson material model is used. Second, the structural scale model reveals the division of the tearing energy into the cohesive energy and the plastic dissipation outside the cohesive zone. The cohesive zone model accounts for a position dependent cohesive tearing energy and experimental results of B.C. Simonsen, R. Törnqvist, Marine Structures, vol. 17, pp. 1-27, 2004 are used to calibrate the cohesive energy.

It is found that the calibrated cohesive energy is low directly after initiation of crack growth, and later assumes a considerably higher steady state value. The latter is attained when the crack has propagated a distance of a few plate thicknesses away from the initial crack tip position. Calculations are continued until the crack has transversed around a third of the plate width.

I can understand that the situation during the initial crack growth is complex, as remarked by the investigators. I guess they would also agree that it would be better if the lower initial cohesive energy could be correlated to a property of the mechanical state instead of position. As the situation is, the position dependence seems to be the correct choise until it is figured out what happens in a real necking region

I wonder if the investigators continued computing the cohesive energy until the crack completely transversed the plate. That would provide an opportunity to test hypothesises both at initiation of crack growth and at the completed breaking of the plate. The situations that have some similarities but are still different would put the consistency of any hypothesis regarding dependencies of mechanical state to the test. 

I am here taking the liberty to suggest other characteristics that may vary with the distance to the original crack tip position.

The strains across the cohesive zone are supposed to be large compared to the strains along it. This is the motivation for doing the plane strain calculations of the necking process. Could it be different in the region close to the original blunt crack tip where the situation is closer to plane stress than plane strain? The question is of course, if that influences the cohesive energy a distance of several plate thicknesses ahead of the initial crack position.

Another hypothesis could be that the compressive residual stress along the crack surface that develops as the crack propagate, influence the mechanical behaviour ahead of the crack tip. For very short necking regions the stress may even reach the yield limit in a thin region along the crack surface. Possibly that can have an effect on the stresses and strains in the necking region that affects the failure processes.

My final candidate for a hypothesis is the rotation that is very large at the crack tip before initiation of crack growth. In a linear elastic model and a small strain theory, rotation becomes unbounded before crack growth is initiated. A similar phenomenon has been reported by Lau, Kinloch, Williams and coworkers. The observation is that the severe rotation of the material adjacent to a bi-material adhesive lead to erroneous calibration of the cohesive energy. Could this be related to the lower cohesion energy? I guess that would mean that the resolution is insufficient in the area around the original crack tip position.

Are there any other ideas, or, even better, does anyone already have the answer to why the cohesive energy is very small  immediately after initiation of crack growth?

Per Ståhle


There are many interesting aspects of the cohesive energy dependence on the crack tip position. Before discussing the details, it is important to note that the scale of a considered plate prohibits use of detailed numerical analysis with fine scale resolution and advanced constitutive models that would allow accurate representation of the fracture process. Generally, large structures must be analyzed using plate/shell elements with characteristic in-plane lengths larger than the plate thickness. Standard shell elements cannot capture the details of necking localization and subsequent micro-mechanical damage and fracture. The complicated behavior beyond the onset of necking is addressed by means of a cohesive zone which must represent the sequence of failure processes, including the R-curve behavior.

In the case of the plate considered in the paper (t=10mm, Al5083-H116) a high level of constraint in the thickness direction at the initial notch tip prevents a full neck development limiting plastic dissipation. The thickness effect cannot be resolved by shell elements since they are in plane stress state, while in reality the notch tip is approximately in a state of plane strain (for this plate). The thickness effect at the initial notch tip can only be accounted for by reducing the initial cohesive energy. For a considered plate the calibrated minimum cohesive energy approaches the nominal plane strain toughness of the material (33kJ/m2 vs. 27kJ/m2 ). Thus, the reduction of the cohesive energy at the initial tip is caused by the thickness direction constraint which cannot be resolved at the level of discretization used for a large structural component.

From the point of view of the cohesive zone calibration, the fact that the minimum cohesive energy is approximately equal to the nominal material plane strain toughness provokes a question: can the initial cohesive energy be correlated to plane strain toughness (as asked above by Per)? The initial (minimum) cohesive energy will differ depending on the tip geometry and plate thickness in reference to the plastic zone size Rp. For a thinner plate the minimum cohesive energy would likely be higher than the material plane strain toughness. This indicates that plane strain toughness cannot be universally used as a minimum cohesive energy irrespective of the plate and notch geometry. However, plane strain toughness does provide a lower bound for the initial cohesive energy.  

The initial stage of crack advance is highly three-dimensional with the crack typically advancing faster at the center where the constraint is highest. After the crack advances through this initial stage, a full neck develops with a shear localization resulting in a slant crack. This corresponds to attaining the steady-state toughness which can be much larger than the plane strain toughness (as is the case for considered plate) due to the large plastic dissipation in the neck. Our analysis, which builds on previous work of Kim Nielsen and John Hutchinson (K.L. Nielsen, J.W. Hutchinson, Cohesive traction–separation laws for tearing of ductile metal plates; Int. J. of Impact Engineering; 48, (2012), p. 15–23) indicates that the steady-state toughness can be estimated without consideration of the complexity of the initial phase of the crack advance. 

The suggestion that compressive residual stresses may affect the material behavior ahead of the crack tip is interesting. In the large scale calculation of the plate response the steady-state cohesive zone parameters were calibrated based on the fine scale calculations performed by Kim Nielsen and John Hutchinson (see the ref. above) without any consideration of residual stresses. This suggests that the residual stress effect on steady-state toughness is not significant. It is difficult to dismiss that effect entirely in the case of the initial stage of crack advance, i.e. before reaching the steady-state. I do however believe that this effect is negligible, at least macroscopically.

I believe that the large rotation problem is especially pertinent for a double cantilever (DCB) mode I test for interlaminar toughness in composites. Typically, the DCB arms are relatively flexible which causes their large deformation during the test and large rotations at the tip. This causes significant errors in toughness measurements. In the case of a large plate in mode I tension, such as the one analyzed in the paper, this effect does not play a significant role. The plate ‘arms’ are very large and their deformation around the initial notch, before crack advances is relatively small. Also, the R-curve behavior in composites is typically attributed to fiber bridging from one side of the DCB to the other, across the advancing crack, which is a very different mechanism that increasing plastic dissipation in large ductile plates.   

On a separate note, we should mention that while simulation of the details of the fracture process was not the objective of our paper and we showed that these details do not matter for the macroscopic behavior, it is possible to observe and even simulate these details, including the mechanism of ‘crack flipping’. This has recently been investigated by Kim Nielsen of DTU (S.A. El-Naaman, K.L. Nielsen, Observations on Mode I ductile tearing in sheet metals; European Journal of Mechanics A/Solids 42 (2013) 54-62; K. L. Nielsen and C. Gundlach, Crack Tip Flipping under Mode I Tearing: Investigated by X-Ray Tomography – under review). The phenomenon of crack flipping is also being investigated numerically by Ken Nahshon and Jessica Dibelka of the US Naval Surface Warfare Center Carderock Division.

esis's picture

Dear Pawel,

Thank you for your comments. Very instructive. It is interesting that so little energy is needed during the first few centimeters of crack growth, and that this is a phenomenon soon to be explained. When I wrote my doctoral thesis in the mid 80’s I used a cohesive zone model in an elastic plastic environment to see how the process zone interacts with the surrounding continuum to give stable crack growth. At that time I envisioned the shielding of the process zone to be caused only by the energy dissipation during plastic work. Obviously there is much more to it. 

The frequent crack flipping is also interesting. I see that it happens after some 10t, a distance that more or less equals the length of the initial stage preceding steady state. It is interesting to learn that there are several ongoing projects and even some that presently are under review. I am looking forward to see the result. I will update this blog with correct references when they appear.

``I see that it happens after some 10t

I am looking forward to see the result.

I will update this blog with a correct references when they appear.


So will I.


Thank you, Europeans, Americans, Africans, the World!





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