PSI - Issue 14

Amit Singh et al. / Procedia Structural Integrity 14 (2019) 78–88 Amit Singh et al./ Structural Integrity Procedia 00 (2018) 000 – 000

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Aronson (1972), Man et al. (1967) and Kashyap and Tangri (1995) that these power laws are not good enough to describe entire flow behavior adequately for several metal and alloys with low stacking fault energy (SFE). So, Ludwigson proposed a new equation which can better describe the flow behavior from initial stage of strain hardening: = 3 3 + ( 4 + 4 ) (4) where ͵ ƒ†  ͵ are the same as per equation (1) and n, respectively and 4 and  Ͷ are additional constants. In cases where materials show a saturation stress at large strain, a new form of stress versus strain was proposed by Voce and the flow behavior equation can be expressed as = − ( − ) (− 5 ) (5) where • is saturation stress extrapolated to zero strain hardening rate, – is threshold stress at which homogeneous plastic deformation becomes appreciable and  ͷ is the parameter that controls the shape of true stress – true plastic strain plot ( in the true sense, it is the proportional characteristics to the slope of strain hardening rate vs. true stress plot). Experimental true stress true plastic strain data were fitted to the above mentioned relationships using Levenberg Marquardt least square algorithm, with the unknown constants as free parameters. The goodness fit of the equations were represented by correlation coefficient R 2 . Higher the correlation coefficient better is the fit. The plastic deformation in materials at temperatures below 0.4 T MP is controlled by the motion of dislocations in the lattice. Orowan (1940), pointed out that, the strain rate can be related to mobile dislocation density and average dislocation velocity ̇ = ̅ (6) where m is the mobile dislocation density, is the burgers vector and ̅ is the average dislocation velocity. As the plastic deformation proceeds, the density of dislocations increases due to the interaction of mobile dislocations with the increasing obstacles e.g. forest dislocations, grain boundaries, kinks and jogs etc. Strain hardening can be understood by the competition between accumulation and annihilation of dislocations in the crystal, i.e. = − ℎ (7) The accumulation of the dislocations is due to the concurrent multiplication (e.g. Frank – Read sources and generation of new dislocations from grain boundaries etc.) and trapping of dislocations (e.g. dislocations debris). While annihilation of dislocations involves cancelation of strain fields of opposite dislocations by glide, climb and cross slip (depends upon the SFE of materials). At the saturation of flow stress, the accumulation and annihilation of dislocations terms balance. Mecking and Kocks (1981), described this strain hardening model, called Kocks Mecking model. At higher strain (e.g. well known as stage III), the strain hardening behavior of metals has been observed to follow linear relationship between stain hardening rate and true stress which is described as = / = − (8) where is strain hardening rate (MPa), is true stress (MPa), p is true plastic strain, o is initial strain hardening rate (an athermal constant that reflects the dislocation storage rate) and is a unit-less parameter (defines slope of the linear regime governed by annihilation term). The Kocks-Mecking plot can divided in to two basic regions; the elastic to plastic transition region and the fully plastic region. It is noted that the elastic to plastic transition region 2.2. Strain hardening behavior