The Hall-Petch strengthening, which related to the concept of dislocation pile-up against GB, and the back stress on the Frank_Read Source. For the large grains the number of dislocations in a pile-up is very large due to small the back stress on FRS. The leading dislocation feels a shear stress proportional with number dislocation in the pile-up. When the stress accumulation reaches to a certain threshold value, the leading dislocation and its tail can tear the GB pushes through the next adjacent grains. But this mechanism (dislocation sliding) is limited by the size of grains (d). Once the grain size gets smaller and reaches about 10-15 nm or less,, then the grain boundary sliding and rotation become dominant plastic deformation mechanism (Super Plasticity). Hence, the yield stress starts to decrease with the decrease in the grain size showing minus inverse SQR dependence: [ -:1/ SQR (d)].
Hall_Petch: SigmaYield= Sigma + k/ SQR(d)
Inverse Hall-Petch: SigmaYieldR=SigmaR - kR/SQR(d)
The Hall-Petch strengthening, which related to the concept of dislocation pile-up against GB, and the back stress on the Frank_Read Source. For the large grains the number of dislocations in a pile-up is very large due to small the back stress on FRS. The leading dislocation feels a shear stress proportional with number dislocation in the pile-up. When the stress accumulation reaches to a certain threshold value, the leading dislocation and its tail can tear the GB pushes through the next adjacent grains. But this mechanism (dislocation sliding) is limited by the size of grains (d). Once the grain size gets smaller and reaches about 10-15 nm or less,, then the grain boundary sliding and rotation become dominant plastic deformation mechanism (Super Plasticity). Hence, the yield stress starts to decrease with the decrease in the grain size showing minus inverse SQR dependence: [ -:1/ SQR (d)].
Hall_Petch: SigmaYield= Sigma + k/ SQR(d)
Inverse Hall-Petch: SigmaYieldR=SigmaR - kR/SQR(d)
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