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Discussion of fracture paper #31 - Toughness of a rigid foam
A most readworthy paper, "Static and dynamic mode I fracture toughness of rigid PUR foams under room and cryogenic temperatures" by E. Linul, L. Marşavina, C. Vălean, R. Bănică, Engineering Fracture Mechanics, 225, 15 February 2020, 106274, 1-10, is selected for this ESIS blog. It has received a lot of attention and was for an extended period of time one of the most read papers in EFM. The attention is earned because of the clear and concise writing about an intricate material that did not yet get as much focus as it deserves.
As the title says, the paper concerns fracture mechanical testing of a solid polyurethane foam. The material has a closed pore structure. It is frequently used in the transport sector for its low density. It also has desirable performance at compression, giving a continuous and almost constant mechanical resistance. The beneficial properties are taken advantage of in applications such as sandwich composites, shock absorbers, packaging materials etc.
I have no professional experience of the material but I have come across it a few times and I recognise its character. The excellent description in the introductions confirms the feeling of something that I am familiar with, i.e., the crushing under compressive load and the brittle fracture in tension. Judging from the listed yield stresses given in the paper, I guess that one can manually make an indent e.g. with a finger.
Before fracture the material may be treated as linear elastic with the elastic limit reached only in a small region at the crack tip, which is controlled by the stress intensity factor KI. The linear extent of the non-linear region is supposed to be below at most a tenth, or so, of the crack length. The exact limit depends of course on the specific geometry.
The ASTM convention described in STP 410 by Brown and Srawley in 1966, claims for structural steels that ligaments, thickness, and crack length should not be less than 2.5(KIc/yield stress)2. It is not mentioned in the paper but the results show that the specimen in all cases fulfill these requirements with an almost four-folded safety, i.e. ligament, thickness and crack length exceed 9.6(KIc/yield stress)2.
The validity of the obtained toughnesses KIc becomes important when it is applied to real structures with cracks that could be too small. This is not within the scope of the present paper. When is a crack too short for linear fracture mechanics? It may not be the most urgent thing to study, but I guess that it has to be checked before the results are put into general use. I am particularly excited over how it compares with the STP 410 recommendations.
When the scale of yielding or damage becomes excessive the fracture process region generally loses its KI autonomy. It happens when the shielding of the fracture process region increases which leads to an increased energy release rate required for crack growth. An analysis would require a more elaborate continuum mechanical model in combination with a box or line model of the fracture process region. The material model would be a challenge I guess.
I did a minor literature search for both establishing the limits of linear fracture mechanics and the application of non-linear models beyond these limits for solid foam materials but didn't find anything definite. I could have missed some. Who knows?
Per Ståhle
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Dear Per,
Dear Per,
The first of the two questions you bring up, i.e., if anyone found a relation like the Brown and Srawley ASTM convention that the crack length should be larger than 2.5(KIc/yield stress)^2 for linear elastic fracture mechanics. I am not sure if it applies to other materials than structural steel and possibly other metals. But I know that it has been used with some success also for other materials. I am a steel guy but I vaguely know that there are other standards for other materials...
Answer1: On our studies regarding the Size effect in fracture of PUR materials we found that for specimens big enough the plane strain condition a, B >= 2.5(KIc/yield stress)^2 is applicable, Marsavina, L., et al, Refinements on fracture toughness of PUR foams, Engineering Fracture Mechanics, Vol.129, 2014, pp. 54-66. However, for the smallest size specimens this condition did not apply. The plane strain condition is required for validation of the fracture toughness tests for polymeric materials ASTM 5045-2014. Unfortunately, there is no standard methodology for fracture toughness determination of cellular materials, and the ASTM 5045-2014 is often used.
The second question is if you or if you know of anyone who modelled the PUR foam or similar with a plasticity or damage model. Spontaneously I would guess that it is damage rather than plasticity. Perhaps not very close to metal plasticity.
Answer2: We applied the CRUSHABLE FOAM models for compression of un-notched foam specimens and the simulations provide good results comparing with experiments and with Thermographic measurements performed during tests. more details about this can be found in
L Marsavina et al 2016 IOP Conf. Ser.: Mater. Sci. Eng. 123 012060. However, for the behavior in tensile of notched PUR foam specimens the theory of critical distance was successfully applied due to their quasi-brittle behavior in tensile in presence of cracks or notches. References: Voiconi, T., Negru, R., Linul, E., Marsavina, L. and Filipescu, H. (2014) “The notch effect on fracture of polyurethane materials”, Frattura ed Integrità Strutturale, 8(30), pp. 101–108;
R. Negru et al., Application of TCD for brittle fracture of notched PUR materials, Theoretical and Applied Fracture Mechanics, Vol. 80, Part A, 2015, pp 87-95.
Prof. Dr. Eng. Liviu MARSAVINA
Dear Liviu,
Dear Liviu,
I have read your publications with great interest, including the editorial for the IOP conference series. I did not find the conference proceedings themselves. If they are available online, it would be exciting to browse them.
We are on the same track regarding the ASTM condition a, B, t,...≥2.5(KIc/yield stress)^2 requiring minimum sizes of crack, width, thickness, etc. It has served us well but undeniably we can agree that it cannot be universal. Extreme geometries in one way or another or a significantly different non-linear material behaviour require an adjustment of the coefficient 2.5. Such a study will probably be time-consuming and require a lot of specimens. Perhaps you already have an idea based on your vast collection of test results of how small the test specimens can be and still produce accurate toughnesses.
There are, of course, methods for nonlinear crack analysis, but the convention base on the paramount material length scale (KIc/yield stress)^2 is the first and most important step because it specifies the circumstances for which more complicated calculations are completely unnecessary.
Per
ASTM E399
The latest version of ASTM E399 recommends that W-a be at least least equal to 2.5*(KIc/yield stress)^2, where W is the width of the specimen and a the crack length. In his book on fracture mechanics, Knott explains that the experimental data on aluminum alloys and maraging steel are indicative that the critical thickness is roughly equal to the crack length restriction (I.e. B = a). Specimen width is usually taken to be twice the thickness (W=2B). Considering all these, the condition W-a be at least equal to 2.5*(KIc/yield stress)^2 becomes equivalent to B being at least equal to 2.5*(KIc/yield stress)^2. The question is whether the later is also applicable to alloys other than aluminum and maraging steel, although for preliminary specimen sizing, the condition B being at least equal to 2.5*(KIc/yield stress)^2 may be ok to use. Remember that ASTM E399 is a highly iterative plane strain fracture toughness assessment method. Therefore, B = 2.5*(Kic/yield stress)^2 may serve as a parameter in one of its iterations.