The nature of the fracture mechanism will primarily be a function of the applied stresses and strains, the temperature/strain rate and the nature of the material and microstructure. In materials where fracture is associated with limited plasticity, brittle fracture cleavage and intergranular fracture, which are primarily locally stress-controlled fracture modes. Where significant plasticity is involved, conversely, fracture is dominated by crack bunting (e.g., disloication emission from the crack tip), and fracture modes are invariably ductile. A typcial mechanism in many metallic materials is microvoid coalescence, which is a locally strain-controlled fracture but markedly enhanced by any degree of triaxial constraint.

Will be fracture be the same in the same material, etc. but under a different applied stress-state? The mechanism may be the same (but as I have noted here before, this may not always be the case) but the form of the fracture mechanism may be different due to the trajectory of the crack that created the fracture. Specifically, this trajectory will result from two completing processes: (I) a mechanical effect where the crack will try and follow the path of maximum driving force, e.g., for brittle fracture a path of maximum tangential stress, maximum strain energy release rate G, or KII = 0, and (2) a microstructural effect where the crack would like to follow the path of least microstructural resistance. Generally, where these two criteria are commensurate, the fracture toughness will be low; where they are incommensurate, high toughness can result. However, the competition between these two criteria can lead to different crack trajectories and hence a different fracture appearance in the same material when different stress-states are applied.

Yes a very interesting qn. Take a glass/epoxy or a carbon/epoxy composite for example, failed in tension. The maximum tensile stress that you see could be different based on the fibre and the volume fraction. But, the resin river patterns, cleavage patterns, fibre radial patterns, matrix cracking paterns, fibre-matrix debonding are all common. It is not that precise to frame failure theories on fracture surface morphology or modes or even with correlation at times. You are in short correlating the same set  ( fracture modes) with many things ( different stresses). This is debatable at times when we get in to a confusing domain. Quantitative fractography, which gains relevance at times, in strange cases,  is the only solace.

Indeed a good question.

Failure criteria as proposed in strength theories are  tools to evaluate failure at the macroscopic scale. There is not any consideration whatsoever of microscopic mechanisms.

Damage mechanisms are on the other hand inherently micro-/mesoscopic features. The nature and shape of fracture surfaces depend on such modes. In turn, fracture mechanisms are governed by the number and type of phases present in the material, and by the stress state at the crack front. Stress state implies the consideration of a magnitude and a direction, and it is in fact represented as a second order tensor with 6 independent values. Furthermore, the actual shape of the crack depends on the history of the fracture process, i.e. the order in which the different damage modes take place. Thus, different stress states will, in general, be related to different sequences of damage modes and, thus, to the generation of different fracture surfaces.

We can say that, if the same sequence of damage modes occur, it is reasonable to except the same fracture surface. Now, let's assume we have correlated (through some set of experiments) a number of different stress states to the same sequence of damage modes, and thus to the same fracture surface morphology. Can we except such set of stress states on that specific material to always lead to the same sequence of failure mechanisms? Well, yes on the average, but not always. Fracture is a statistical phenomenon (see the Weibull distribution). Thus, a definite stress state might activate the same sequence of damage modes for 99 times, and a different one at the 100th time. Thus, a correlation between stress state and fracture morphology could be only a statistical one.

The nature of the fracture mechanism will primarily be a function of the applied stresses and strains, the temperature/strain rate and the nature of the material and microstructure. In materials where fracture is associated with limited plasticity, brittle fracture cleavage and intergranular fracture, which are primarily locally stress-controlled fracture modes. Where significant plasticity is involved, conversely, fracture is dominated by crack bunting (e.g., disloication emission from the crack tip), and fracture modes are invariably ductile. A typcial mechanism in many metallic materials is microvoid coalescence, which is a locally strain-controlled fracture but markedly enhanced by any degree of triaxial constraint.

Will be fracture be the same in the same material, etc. but under a different applied stress-state? The mechanism may be the same (but as I have noted here before, this may not always be the case) but the form of the fracture mechanism may be different due to the trajectory of the crack that created the fracture. Specifically, this trajectory will result from two completing processes: (I) a mechanical effect where the crack will try and follow the path of maximum driving force, e.g., for brittle fracture a path of maximum tangential stress, maximum strain energy release rate G, or KII = 0, and (2) a microstructural effect where the crack would like to follow the path of least microstructural resistance. Generally, where these two criteria are commensurate, the fracture toughness will be low; where they are incommensurate, high toughness can result. However, the competition between these two criteria can lead to different crack trajectories and hence a different fracture appearance in the same material when different stress-states are applied.

Dear Robert:

why  triaxial constraint can enhance microvoid coalescence? in my opinion,the triaxiality make the plastic deformation more difficult.

Microvoid coalescence is actually a stress-state modified strain-controlled fracture mode (see Ritchie & Thompson: "On Macroscopic and Microscopic Analyses for Crack Initiation and Crack Growth Toughness in Ductile Alloys," Metall. Trans. A, vol. 16A (2), 1985, pp. 233-48). A triaxial stress field is more potent that a uniaxial field from the perspective of promoting microvoid coalescence fracture as the process involves the formation of voids, typcially around particles, which grow until they impinge. If you are pulling these voids in three directions, clearly that is a more effective means to grow the voids than simply pulling in one direction. It is for this reason that the onset of void nucleation and growth in a uniaxial tensile test (so called "cup-and-cone" fracture) generally occurs at maximum load once a neck has formed; the onset of this maximum load plastic instability creates a triaxial field when acts to promote the growth and coalescence of the voids.

Stress state (defined through triaxiality and/or constraint) will affect the mechanism of fracture, and therefore the morphology of fracture. The degree to which this happens will depend on a number of factors, including the material. Often what we perceive as a substantial difference in the state of stress, is not really that different by the time the actual fracture occurs, especially when considered at local regions.

An equally important thought to ponder on is how the state of stress may affect deformation micromechanisms.

The stress state has an influence on the fracture mechanism. In general an increasing triaxiality leads to an embrittlement of the material. The most common way to investigate this behavior are notched tensile test with different radii of the notches.

Assuming that the material is same which is getting fractured following same mechanism, it may happen that the initial stress-strain condition at the crack tip were different, but if it has failed by same mechanism probably the condition of stress/strain (local) reached at the time of fracture would also be same (for same material specimen of identical dimension). This will dictate the toughness of the material at that condition.

The fracture surface though is a result of final stress/strain condition would however capture the whole history except for the plastic deformation that took place before damage initiated (this also could have been studied by dislocation studies in TEM but I am not aware of it at present). For instance, if its a ductile fracture which started under a low constraint/triaxiality condition; after sufficient plastic deformation due to strain incompatibility at the weak points such as secondary phases, grain boundaries...voids will nucleate, then grow and coalesce. This will be followed by crack separation. However when a crack starts propagating, the smaller voids which could not grow because the others grew and met the fracture criteria/condition  will also be separated. One will see all sizes of dimples, smaller to bigger as a record of fracture damage history.

The concept that this smaller to bigger dimple fraction distribution would be self-similar with change  in magnification leads to the application of fractal space concept and a measure of fractal dimension can be made. 

So far I have seen few works on fractal dimension as a measure of toughness of material but I have found the correlation to be decreasing, increasing and to remain invariant. Also due to my limited knowledge, I would not like to comment on the relation of fractal dimension on fracture toughness.

The whole history to investigate damage accumulation perhaps can be analysed using the fracture surface but in cleavage fracture the morphology normal to the fracture surface is also required to assess the micro-crack size and distribution. This also can used in ductile fracture (the morphology of damage accumulation along the normal surface to the fracture face). 

@Robert:

Why in a uniaxial tensile test ,the crack initiates in the center of  necked section, not areas adjacent the necking where has more  triaxiality than center area?

In a uniaxial tensile test of a ductile material, the stresses should be uniform throughout the gauge length until a neck forms, invariably at maximum load, whereupon due to constraint of the adjacent un-necked regions, a triaxial stress-state is created in the necked region which actually is at a maximum at the center of the neck. If void coalescence is the mode of fracture, voids are created around particles which then preferentially grow in the triaxial field until they coalescence, principally by necking instabilities between the voids - this is the "cup" portion of the fracture. This process would continue throughout entire cross-section, but near the surfaces of the test sample, the fracture is influenced by the surfaces of maximum shear at 45 degrees to the surface. The void coalescence process can continue there but now on these 45 degree shear surfaces - this is the "cone" portion of the fracture.