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NEEP 541 – Hardening

NEEP 541 – Hardening. Fall 2002 Jake Blanchard. Outline. Hardening. Radiation Hardening. Radiation tends to increase the strength of metals Point defects Impurity atoms Depleted zones Dislocation loops Line dislocations Voids precipitates. negligible. Two Mechanisms.

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NEEP 541 – Hardening

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  1. NEEP 541 – Hardening Fall 2002 Jake Blanchard

  2. Outline • Hardening

  3. Radiation Hardening • Radiation tends to increase the strength of metals • Point defects • Impurity atoms • Depleted zones • Dislocation loops • Line dislocations • Voids • precipitates negligible

  4. Two Mechanisms • Increase stress needed to start dislocation motion (source hardening) • Impede dislocation motion (friction hardening)

  5. Source Hardening • Stress to initiate dislocation motion is associated with unpinning of Frank-Read source • This source increases dislocation density as a result of deformation

  6. DislocationSource =pinning point

  7. Frank-Read Source

  8. Animation

  9. Frank-Read Source Si

  10. What stress is required to activate source? • Shear stress acting on the dislocation, which is pinned by defects, distorts dislocation • We can estimate the stress needed to bend the dislocation beyond the critical strain needed to activate the source and create a new loop

  11. Force on a Dislocation s  R

  12. Model for critical shear stress

  13. Critical Stress • Critical point is when radius is half the distance between pinning points (dislocation is semi-circular) • Decreasing distance between pinning points increases stress needed to initiate motion

  14. Friction Hardening • Defects impede dislocation motion • 2 sources of resistive force • Long range forces from interaction with other dislocations • Short range forces from obstacles

  15. Long Range Stresses • Dislocations repel each other because of stress fields associated with interruption of lattice structure • Model dislocation as an ordered array of defects

  16. Dislocation Network Model

  17. Select a Unit Cell Dislocation loop L • Find force on loop from network of line dislocations • L determined by dislocation density

  18. Modeling • Let =total length of dislocations in cube/cube volume (dislocation density) • =(12/4)L/L3=3/L2 • (each dislocation shared by 4 unit cells) • L=(3/)1/2 • Loop is only affected by parallel dislocations (4 top, 4 bottom) • Approximate force by force only on parallel dislocations

  19. Modeling Fy Fx  y

  20. Modeling • Maximum force (Fx) is at angle where fx is a maximum • Differentiate fx and set to 0 • Maximum angle is 22.5 degrees • Maximum value of fx is 0.25 • Let poisson’s ratio=1/2 • Y=L/2

  21. Modeling • Applied stress must overcome this force to move dislocation • Increasing dislocation density increases this friction stress

  22. Short Range Forces • Short range stresses are due to obstacles lying in the slip plane • Force is exerted at point of contact • Two types: • Athermal=bowing around obstacle • Thermal=climbing over or cutting through barrier (energy is supplied by thermal activation) • Friction stress depends on distance between obstacles

  23. Obstacles Area=A Radius=r L

  24. Modeling • N=particle density • Slab volume is 2rA • Number of particles in slab=2rAN • Average distance between particles=L • 2rANL2 =A More defects implies higher strength

  25. Hardening by Depleted Zones • Significant at low fluence and low temperatures • Mechanism is thermally activated friction hardening • Thermal activation allows dislocation to cut through or jump over obstacle • Dislocation is moved by short range stress

  26. Picture of Model Lo Lo h R Lo=distance between pinning points L=distance between obstacles Lo>L Lo

  27. Model So the dislocation line adjusts its position until Lo satisfies this equation

  28. Diagram If La<Lo, then dislocation cuts through so that Lo is the pinning point distance Lo La

  29. Diagram Lo If La>Lo, then dislocation does not cut through and La becomes the pinning point distance La

  30. Strain Rate • Strain is determined by step size, which is b • Shear strain is b/a

  31. Modeling • Assume N1 loops in a volume V • Assume each loop grows by amount dA • N1adA=dV • 1/a=N1dA/dV

  32. Modeling • R=loop radius • V=dislocation glide velocity

  33. Glide Velocity • Velocity depends on T, activation energy, and thermal vibration frequency • Increasing temperature increases strain rate because it becomes easier to overcome obstacles

  34. Shearing Obstacles • Slicing a sphere is easier off the diameter • Obstacle radius about 10 angstroms • Average radius is r’

  35. Stress to penetrate obstacle • The stress needed to cut a model can be approximated as: • r=obstacle size • N=obstacle density • b, G =material properties

  36. Temperature Effects

  37. Temperature Dependence Plot

  38. Fluence Dependence • According to the model, the strength is proportional to the square root of the fluence • But saturation occurs • The theory is that as depleted zones get too close, their hardening effect is diminished

  39. Saturation Modeling Destruction rate # zones per collision Collision rate per unit volume V=volume around depleted zone that is unavailable for cascade production

  40. Saturation Modeling

  41. Overcoming Obstacles

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