1. Chapter 4: Dislocation – Obstacle Interactions
Issues to Address:
• Resistance imposed on a dislocation as it traverses a slip plane
• Kinetics (temperature and strain-rate dependence of slip)
• Definition of the intrinsic resistance to slip at 0K
• Analyzing temperature and strain-rate dependent yield stress measurements
• Differentiating between the kinetics of slip from the kinetics of strain hardening

2. Chapter 4: Dislocation – Obstacle Interactions
Take-Away Concepts:
• A stress-versus-distance-traveled profile characterizes the interaction of a dislocation with an obstacle.
• If the applied stress is insufficient to enable the dislocation to continue its motion, the missing energy can be supplied by thermal activation.
• The time spent awaiting thermal activation leads to start-stop slip or jerky glide; the waiting time is a function of temperature and strain rate.
The combination takes on special significance.

3. Chapter 4: Dislocation – Obstacle Interactions
Take-Away Concepts (Cont.):
• Measurements of yield stress as a function of test temperature and strain rate are easily analyzed on simple coordinates to assess thermally activated slip.
• The yield stress at 0 K takes on special significance; this value can be estimated from the plots described above.
• Strain hardening demonstrates a temperature and strain rate dependence that is distinct from that observed in yield stress measurements.

4. Dislocation – Obstacle Profile
The Peierls stress was described in Chapter 4 as the intrinsic lattice resistance imposed on dislocation motion.
A simple model for the variation of stress as the dislocation moves along the slip plane is shown below.
At a stress t2 the dislocation can move, but at the lower stress t1 the dislocation will not move by virtue of the stress alone.

5. Dislocation – Obstacle Profile
Stress is force per unit area. Force x distance equals work. At a stress level of t1, the work to move the dislocation past the obstacle is the area under the curve identified as E.
In deformation, the energy required to assist the dislocation past an obstacle is thermal energy, and the assistance is termed thermal activation.

6. Thermal Energy For an ideal gas, the average energy of a molecule is
where T is the absolute temperature and k is Boltzmann’s constant.
But, some molecules have higher and some have lower energies, described by the Boltzmann equation:
where n/Ntot is the fraction of molecules with energy E.
The interest is of molecules with very high energies:

7. Thermal Energy n/Ntot versus at two temperatures
At the higher temperature, a larger fraction of molecules have high energies.

8. The Importance of Absolute Zero
At 0 K thermal energy is not available.
Dislocation motion is not possible at stresses less than t2 in this case.
The stress necessary for dislocation motion past an
obstacle – enabling deformation – at 0 K becomes an intrinsic measure of the obstacle strength.
This is referred to as the mechanical threshold or the
mechanical threshold stress and given the symbol .

9. Discrete Obstacles Discrete obstacles to dislocation motion include:
Forest dislocations
Solute atoms
Interstitial atoms
Precipitates
Spacing typically much
farther than b
Schematic shows an obstacle, characterized by t3 added to the higher density of obstacles characterized by t2.
In this case, t3 t2 which implies that the smaller obstacle will offer little resistance to dislocation motion.

10. Jerky Glide
Glide is another word for slip; when the dislocation spends time awaiting thermal activation energy to proceed, glide is “jerky”.
A common expression for jerky glide is:
where is the strain rate, is a constant, and G(s) is the stress dependent activation energy.
As the stress is lowered, G, the
thermal activation energy (identified
as E) increases, the waiting time
increases, and the strain rate
decreases.
Similarly, as the temperature increases at this stress, the
waiting time decreases and the strain rate increases.

11. Jerky Glide
Consider a dislocation awaiting thermal activation to overcome an obstacle (represented by filled-in circles).
Force on dislocation:
Work on dislocation
as it moves b:
If the total activation energy characterizing the interaction is G, then the contribution from thermal activation G is:
and

12. The Equation for Jerky Glide
Solving for s:
But, at 0 K, (the mechanical threshold stress).
Thus,
where
has special significance
Slope = k / G

13. Analysis of Experimental Data
Example: Campbell and Briggs in pure Niobium
Yield stress measurements
Campbell and Briggs, Dept. Engnr. Science Report 1091, Univ. Oxford, 1969

14. Analysis of Experimental Data
Example: Nojima in pure Iron
Yield stress measurements
Nojima,, in Impact Loading and Dynamic Behaviour of Materials, Chiem, Kunze, and Meyer, eds., Bremen, Germany, 1988

15. Analysis of Experimental Data
Example: Gray and Chen in pure 1018 steel
Yield stress measurements
Gray and Chen, Los Alamos Natl. Lab. Report LA-CP

16. Analysis of Experimental Data
Previous three examples represented measurements in annealed material.
Yield stress versus T and can also be made in deformed metals.
In this case, 4 samples deformed identically in the “prestrain”; each “reloaded” at different conditions.
Entire reload stress-strain curve shown, but for this analysis only the reload yield stress is of interest.

17. Analysis of Experimental Data
Previous chart: Prestrain: 10% at 81 s-1 and 295K
Reloads: s-1 and 77K, 180K, and 295K
and use the final stress at 10% at 81 s-1 and 295K
Actual reload yield stress measurements:
Follansbee and Kocks, Acta Met., 36, 1988, 81

18. Analysis of Experimental Data
Another example: Nickel ppm C
Prestrain: 4.8% at s-1 and 295K
and Prestrain: 29.7% at s-1 and 295K
Reloads: s-1 and 77K, 180K, and 295K
1 s-1 and 2500 s-1 at 295K
and use the final stress in the prestrains
Follansbee, Huang, and Gray, Acta Met., 38, 1990, 1241

19. Obstacle Profile in Presence of Multiple Obstacles
Very idealized obstacle profiles have been introduced.
Actual profile for a short-range (e.g., a collection of solute atoms) and a long-range (e.g., forest dislocations) may be something like that shown below.

20. Kinetics of Strain Hardening
Of interest in the reload tests were the reload yield stresses as a function or reload temperature and strain rate.
Also note in this plot that the reload stress strain curves are not collinear; there is a distinct temperature dependence to the strain hardening rate (compare 77K with 295K).

21. Kinetics of Strain Hardening
At the reload yield point, there is ~ 26 MPa between the reload yield stress at 77K versus that at 295K.
After 10% further strain, this difference in stress has grown to ~ 64 MPa, illustrating the (inverse) temperature-dependence of strain hardening.

22. Kinetics of Strain Hardening
Recall the Voce Law:
There is no universal value of ts. This quantity demonstrates a temperature and strain rate dependence – kinetics – due to the dynamic recovery mechanism. Thus
One equation for is:
where are constants.
This equation describes the kinetics of dynamic recovery due to cross slip (see Kocks, ASME J. Engnr. Matl. and Tech., 98, 1976, 76.)

23. Kinetics of Strain Hardening
Schematic showing hardening kinetics equation with possible scale axes.
At 77K is a smaller negative number than at 295K, which is consistent with the observation at ts at 77K is larger than at 295K.