2D Dislocation Dynamics

Simulation and observation of dislocation pattern evolution in the early stages of fatigue in a copper single crystal

J. Yang, Y. Li, S. Li, C. Ma, G. Li

Materials Science & Engineering A, 2001

• Experimental and numerical study of single-crystal copper oriented for single-slip under constant plastic strain amplitude of 2e-4 at a frequency of 0.2 Hz
• Only modeled parallel edge dislocations of magnitude +/- b
• Initially a random distribution of 20 + and 20 - dislocations
• Only consider multiplication; annihilation is ignored
• Climb and cross-slip are ignored
• simulation area of 3.6 x 3 $\mu m^{2}$ with PBCs in x&y
• using the typical singular elastic stress fields
  • If distance between dislocations is <10b, a distance of 10b is used in the stress field calculation to avoid singularity since they don't consider annihilation.
• Mobility: $v=v_{o}\left({\frac {\tau }{\tau _{o}}}\right)^{m}$
• It appears that if they had an initial dislocation density on the order of $10^{10}m^{2}$ that the dislocations would randomly arrange after cycling? If they used an initial density of $10^{12} m^2</math> the dislocation patterns would emerge - this is why they chose 40 initial dislocations. (It's hard to tell because this section was poorly written
• State they use triangular waveform, but appear to indicate it's the applied shear stress. (confusing)
• During cyclic deformation the maximum stress increases until saturation at about 28 MPa.
• Only consider 5 values of resolved shear stress (10, 15, 20, 25, 28 MPa) to speed up calculations?
• If strain rate is <2e-4 when the resolved shear stress is 28 MPa, then dislocation multiplication occurs
  • 20 to 200 dislocations with an equal distribution of +/- b are added based on the how far below the predicted strain rate the simulation was.
  • Makes no mention of where these new dislocations are located
• Dislocations initially pattern themselves into "matrix walls" (not the same as PSB walls), which become veins with continued cycling.
  • vertical walls are comprised of dislocations with the same Burgers vector
  • Dislcations of opposite sign form walls oriented at 45 degrees.
  • Says this is an equilibrium distribution according to classical dislocation theory, but cites a Chinese book I can't find.
• claim that cross-slip of screw dislocations is unimportant in the early stages of fatigue prior to PSB emergence.
• claim that screw dislocations spanning the matrix walls cause them to fragment and form into veins.
  • In the simulation they select 6-8 cutting positions randomly, but I can't understand what they were doing because it is poorly written.

Evolution of persistent slip bands and simulation of its stress field in a fatigued copper single crystal

J. Yang, Y. Li, Z. Cai, S. Li, C. Ma, E. Han, W. Ke

Materials Science & Engineering A, 2003

• claims to be a 3D "discrete dislocation method", but the edge and screw dislocations are treated as perfectly straight
  • Never clearly explained if they are using dynamics at all
  • No mention of how their structures are generated and if they're even stable
• constant plastic strain amplitude of 1e-3
• only edge dislocations of +/-b in the matrix veins and PSB walls; only screw dislocations in the channels
• equal distribution of left/right screw and +/- edge
• Simulate a volume of 6 x 5 x 4 $\mu m^{3}$ with PBCs
• edge dislocations are distributed randomly in the veins/walls and screw dislocations were distributed randomly in the channels, "according to an actual experimental photograph"
• Edge dislocation density of ~3e14 and screw dislocation density on the order of $10^{1}2$
• again using the singular elastic stress fields, but with a critical distance of 3b this time
• DD is used to calculate the internal stress distributions and FEM is used to calculate the external stress distributions
  • In the FEM mesh the veins and walls has a yield stress of 100 MPa and a Young's modulus of 110 GPa, while the matrix between veins has 60 MPa and a modulus of 108 GPa and the matrix between PSB walls is only 56 MPa and a modulus of 106 GPa.
  • Claim PSB channels are softer due to vacancies