J is the same as K. K is used only for linear elastic bodies. Once you have an elastoplastic body you have to use J. So to a certain extend is the same parameter. CTOD is related to a very specific type of test. J and K are almost everywhere in the context of fracture mechanics and help you to understand what is happening at the neighbourhood of the crack (stress configurations) with respect to the condition of the stressed body (elastic or plastic).

If you have for example a Mode 2 crack - so a shear stress acting along the crack faces - how are you going to determine your crack growth regime with CTOD?

You may have a look into some books on fracture mechanics, such as the one by Wanhill, Yanssen and Zuidema.

Both K and J are energy release rate i.e. -dU/dA. CTOD and J are related by Eq. J=m.Sf.CTOD. m may vary from material to material

Let me try to clarify your confusion without invoking mathematics. If you supply energy (read force) to a cracked body, you would anticipate that the entire energy/force is made available for extension of the crack. If that happens, you would classify the material as brittle (in relative terms) and the force that drives the crack as K (stress intensity factor). This is linear elastic fracture mechanics (LEFM). The assumption/expectation you have made here is that the entire force is utilised in crack extension and not wasted (!) in any other form.

But many engineering materials will not behave in that fashion. A portion of the force you have applied will be utilised in deforming the material around the crack and another portion will be used for crack extension (ignoring the energy loss due to heat). In other words, such materials possesses substantial plasticity and only a fraction of the applied force is expended for crack extension. Under such situations, the crack tip instead of extending right away (as in LEFM), will blunt/stretch significantly. A situation will be reached wherein the crack tip cannot blunt anymore. At that time a fresh crack will initiate from the blunted tip and the process will go on. Such materials come under the category of elastic plastic fracture mechanics (EPFM). You cannot characterise the crack tip forces in such case by K and you need a new parameter. Here you can have two options (i) since by definition, fracture toughness is the resistance of the material to crack initiation, you can monitor and measure the crack tip blunting and assign a critical value. You will be measuring an "allowable" crack extension (blunting) in millimeters. This is CTOD-crack tip opening displacement. You are measuring the displacement of the crack tip due to crack tip plasticity. A parameter popularly used by the British (visit TWI website to know the history of development), particularly to characterise the weldments. Option (ii) is to have an energy based parameter. After all you require to spend energy to create a new surface in the form of crack extension. You measure this energy and assign a critical value. This energy is measured by adopting a mathematical procedure called J integral. You will be measuring the energy per unit cross-sectional area to define the toughness of the material and call this as J. Popularly used by the Americans. Both referring to the fracture toughness of ductile materials in different perspective. It is individuals' choice to use a particular parameter. But both are related. CTOD can be derived from J and vice versa. Read ASTM E 1820 standard and some basic fracture mechanics text books (T.L. Anderson for example) to understand the procedures of evaluation and basics. Hope I have explained you enough.

Hi Uygar, I have been looking into the subject for a number of years and have a slightly different perspective on your question. Given that the elastic based fracture toughness is defined by K, the elastic-plastic fracture toughness is described in terms of CTOD and J.

CTOD is a measure of strain/displacement of the crack tip, while J is a measure of the strain energy density around the crack tip. Based on certain boundary conditions i.e. blunted crack tip and no crack propagation, Shih found a linear correlation between CTOD and J based on the HRR solution, given as CTOD=dn(J/ys). Given that the boundary conditions that governs this relationship is nearly impossible to maintain both experimentally and in numerical modelling.

As in why J is sometimes preferred is due to how the defect assessment procedures, e.g. BS 7910 were implemented. They are often assessed in terms of K regardless of the elastic-plastic behaviour, in which both CTOD and J would be used to calculate the equivalent K. Given that there are several different ways to define CTOD, one can obtain different value from another for the same specimen/model due to the definition of CTOD adopted. J is more consistent in this manner and consequently should give a more consistent defect assessment result.