Energy release rate. Fracture energy

Dear Zhigang, sorry for boring you You touch my favorite topics .

Unfortunately, the identitiy σc√a=constant , predicted by Griffith, is not correct generally. It is only correct for cracks with the same sharpness (curvature or radius) at the tip of the crack.Cracks with the same length and different tips will not obey the identity. The latter explains the scattering of the toughness magnitude in tests.

Energy release rate vs. fracture toughness: mixed-mode and mode-II cases

I prefer "the fracture toughness" rather than "the fracture energy"because it is commonly used in fracture measurements such as ASTM test standards. Konstantin's view is very important because only a mathematical sharp crack tip will lead to a constant stress singularity order (-0.5) and it is shown as square root a (half crack length) in fracture toughness calculations. Unfortunately, it is very hard to make these sharp crack tips. If the crack tip is rounded or just a notch tip, the stress singularity order is not a constant then we cannot get a material constant: fracture toughness.

Griffith's model addresses a mode-I crack only. Here I'd extend our discussion to mixed mode and pure mode-II cracks under static loading. My PhD student Arun Krishnan and I are developing an efficient way to measure the pure mode-II fracture toughness of interfaces as shown in Fig. 1. We observed two fracture modes:

1) if the interfacial bonding is strong. the initial crack, although loaded in shear, kinks from the original crack path and forms a mode-I crack (its symmetrical stress field was verified by our optical techniques);and

2) a pure mode-II crack propagates along the interfacial bonding ( a self-similar crack) if the interfacial bonding is weak.

Therefore, we can obtain the correct pure mode-II fracture toughness (material resistance) is case 2). Simply because the material resistance (fracture toughness or fracture energy) for a mode-I crack case in case 1) and a mode-II crack in case 2) is very different for the same material. 

Now if we want to compare the energy release rates and fracture toughnesses in order to predict whether case 1) or case 2) will occur at first, based on Zhigang and Prof. Hutchison’s famous review paper in 1992, the energy release rate of the initial crack in Figure 1 could be easily calculated and it is a function of the potential crack path or function of θ. For the kinked mode-I crack, the energy release rate,  G(θ=θc) should exceed the mode-I fracture toughness of the polymer material ΓPMIC, i.e.

Where the crack kinking angle θc is around 70 degrees based on the Erdogan and Sih’s classical paper in 1963 (the T-stress effect was not included).  For the pure mode-II crack initiation and propagation, the energy release rate, G(θ=0) should exceed the mode-II fracture toughness of the interface ΓITIIC  , i. e.,

So the competition of the energy release rate and the fracture toughness leads to different failure modes for a mixed-mode crack, which is more complicated than the Griffith’s original problem. Here, if equation (1) is satisfied at first, a crack loaded in shear will kink as a mode-I crack. If equation (2) is satisfied at first, a crack loaded in shear will propagate as a mode-II crack along the interface. The major purpose of my note is that I found some mistakes in recent journal papers to measure the pure mode-II fracture toughness: their initial cracks kinked during experiments but they still use the crack initiation loading to characterize the “mode-II fracture toughness”.

Here are some nice movies on cracks illustrated by bubble raft.

http://homepages.cae.wisc.edu/~stone/bubble%20raft%20movies.htm