13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The same result was obtained on incorporation of Griffith crack tip modification as prescribed with gradient elasticity theory (GET). Another related application of such pile-up model consideration, to be developed in Section 5, is to evaluate the breakthrough of a pile-up in providing a fundamental explanation of adiabatic shear banding behavior. In such case, the rapid avalanche-like dissipation of the stored energy in pile-up provides a ready energy source for appreciable thermal heating [1]. 4. Strain Concentration in the Portevin-Le Chatelier Effect The French researchers, A. Portevin and F. Le Chatelier discovered that for certain combinations of strain rates and temperatures the stress-strain curves of Al-Cu-Mn and Al-Cu-Mn-Mg alloys exhibited serrations in the stress strain curves and that fairly well-defined bands formed in the gage length of the specimens [10]. This discovery, now eponymously known as the PLC effect, has been of keen interest in the 90 years since it was first observed and to this day controversy exists in terms of the basic mechanisms and the effect on mechanical properties such as fatigue. It was not until the advent of dislocation theory that the outline of an understanding was developed and again Prof. Cottrell played a key role in clarifying, after his work with Bilby [11], the underlying physics of the process [12]. The model that evolved was based on the concept of thermal activation. For certain régimes of strain rate and temperature there is a dynamic interaction between solute atoms (e.g. C in Fe-C alloys, Mg in Al-Mg alloys) and dislocations such that there is a repeated locking and escape mechanism attributed to the mobile dislocation density. During locking, the stress climbs for the dislocations held back by their “atmospheres” and when the dislocations escape, the stress drops. The process is repeated throughout the test. Thermal activation energies have been measured for this mechanism by numerous investigators and while one may object to some of the approaches used, it is clear that activation energies are on the order of that of diffusion of the solvent species [13,14]. Recent data for IN100, a Ni-base superalloy strengthened by the coherent L12 precipitate ’ is shown in Fig. 5 [15]. In Fig. 5 the temperature strain rate plane is divided into regions in which the PLC effect is observed and not observed. The boundary between the two regions marks the onset of the PLC effect and as such may be used to determine the activation energy which is 1.14 eV and comparing reasonably well to the activation energy for bulk diffusion of C in Ni of about 1.48 eV. However, there is an apparent contradiction in using the activation energy of an atom/dislocation pair and the large strains that are associated with the serrations and the corresponding large numbers of dislocations needed to carry that strain. In effect, currently-accepted theories describing this process appear to be incomplete. In broad outline, the deformation process is initiated by unpinning of the dislocations having the lowest effective unpinning stress. At this point, the free dislocation would encounter a more strongly pinned dislocation at a larger precipitate. However, the stress on the second dislocation would be essentially doubled since there would be a two dislocation pileup and the stress would drop. The group of two dislocations would next encounter a locked dislocation but the stress at the leading dislocation would now be three times the applied stress so, depending on the actual locking stress, the required stress to continue deformation would decrease. The process of increasing strain with decreasing stress would continue until all of the dislocations on a slip plane have been mobilized1 [1]. We thus have a picture of a decreasing stress with increasing strain for the first slip plane. The next easiest dislocation to unlock would be activated elsewhere in the crystal and as it moves across 1 This explanation is simplified and does not take into account the statistical distribution of the pinning points. This more realistic situation may be addressed through a detailed statistical analysis.
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