1. Chapter 10: Application to HCP Metals
Issues to Address:
• Kinetics (temperature and strain-rate dependence of slip)
• How variability amongst HCP metals and alloys (melting point, c/a ratio, slip system) affects deformation kinetics.
• The signature(s) of deformation twinning
• De-confounding effects of deformation twinning in structure evolution analyses
• Structure evolution in alloys versus that in the pure metals

2. Chapter 10: Application to HCP Metals
Take-Away Concepts:
• For pure HCP metals, in the annealed condition, a one-obstacle model is sufficient in some systems (Zn, Cd) whereas a two-obstacle model is required in others (Mg, Zr, Ti)
• The contribution of deformation twinning leads to unique signatures in the yield kinetics plot and in the plot of versus strain.
• In the three alloys studied in this chapter, saturation threshold stress values in the alloys fall on the same kinetics curve as do saturation threshold stress values in the pure metals.

3. Slip in HCP Metals
• The basal plane is a close-packed plane but slip occurs as well on the prism plane and on the pyramidal plane.
• The c / a ratio varies from HPC metal to HCP metal (1.59 for Be and 1.89 for Cd) and can be greater than or less than the ideal c / a ratio (1.632)
• Anticipate variability in deformation in HCP metals

4. Zinc Risebrough, 0.99999 and 20 mm grain size
Liu, Huang, Wu, and Zhang, and 70 mm grain size

5. Zinc – Analysis of Yield Kinetics
Single parameter model:
Risebrough, sa = 20 MPa
Liu et al, sa = 5 MPa
(consistent with difference in grain sizes)
Note: Table 10.2 in textbook lists sa = 10 MPa for the Liu et al material, but 5 MPa is the correct value.

6. Zinc – Analysis of Structure Evolution
Without prestrains and reloads, use method introduced in Chapter 9.

7. Zinc – Analysis of Structure Evolution
Analysis of the Liu et al measurement at 295 K and s-1 does not give a clear result.
The suspicion is that deformation twinning becomes active, which gives a lower rate of structure evolution from dislocation storage.
The long-dash curve may be more reflective of dislocation storage.

8. Zinc – Analysis of Structure Evolution
Plotting the saturation threshold stress as a function of test temperature and strain rate gives a familiar result.
This compares with values ~ 10x this observed in FCC and BCC metals

9. Zinc – Measured Versus Predicted Behavior
Risebrough, and 20 mm grain size

10. Cadmium – Stress Versus Strain Curves
Risebrough, and 25 mm grain size

11. Cadmium – Analysis of Yield Kinetics
Single parameter model:
Risebrough, sa = 10 MPa
A single parameter model appears to agree well with the measurements but the disagreement at the highest temperature (345 K or TH = 0.581) is odd.

12. Cadmium – Stress Versus Strain Curves
Mannan and Rodriguez, Cd and 38 mm grain size
The 77K curve exhibits linear hardening (or even a rate of strain hardening that increases with strain). Deformation twinning ?

13. Cadmium – An Alternate Analysis of Yield Kinetics
Single parameter model:
sa = 10 MPa
It is suggested that these deviations are due to deformation twinning as observed in Zr and Fe

14. Cadmium – Analysis of Structure Evolution
Without prestrains and reloads, use method introduced in Chapter 9 (and assume yield kinetics described in previous slide).
Fair amount of uncertainty in the saturation stress in face of the tensile instability.

15. Cadmium – Analysis of Structure Evolution
Analysis of the RT Mannan and Rodriguez curve shows a low initial rate of strain hardening followed by an increasing rate of hardening. Evidence of deformation twinning ?
No parameters for the evolution equation can be selected that match the experimental curve.

16. Cadmium – Analysis at 77 K is Problematic
The 77 K data point on the yield stress kinetics plot falls well below the model prediction. This is suspected to result from the contribution of deformation twinning.

17. Cadmium – Analysis at 77 K is Problematic
Assume that the (90 MPa) difference in stress does not vary with strain to proceed with analysis of evolution:
Mannan and Rodriguez, 77 K:
The evolution curve is far more linear than predicted with the evolution equation.
Could this reflect deformation twinning?

18. Cadmium – Analysis at 77 K is Problematic
The stress difference for the Risebrough measurement at 77 K is ~ 70 MPa:
Risebrough, 77 K:
Better behaved (compared to the Mannen and Rodriguez result) but a 310 MPa saturation stress seems high.

19. Cadmium – Analysis of Saturation Stress Kinetics
Solid points:
Open points:

20. Cadmium – Analysis of Saturation Stress Kinetics
Could this difference (lower temperatures are closer to zero) arise from the contribution of athermal hardening due to grain refinement from deformation twinning?

21. Cadmium – Analyzing the “Extra” Hardening at 77K
Consider that the difference between the MTS prediction and the measurement arises from the contribution of an increasing athermal stress due to grain refinement.

22. Cadmium – Analyzing the “Extra” Hardening at 77K
Consider the difference between the MTS prediction and the measurement:

23. Cadmium – Analyzing the “Extra” Hardening at 77K
Assume hardening from grain size refinement follows Hall-Petch behavior:
Mannan and Rodriguez measured the grain size dependence of yield:
Mannan and Rodriguez did not report the final grain size.

24. Pure Magnesium – Analysis of Yield Kinetics
Two-parameter model:
Suzuki et al, Mg
This includes measurements at TH = 0.84 but these are at only at a strain rate of 10 s-1.

25. Magnesium – Analysis of Structure Evolution
Once again, prestrain and reload tests are unavailable.
Suspect dynamic recrystallization (TH = 0.512) at 0.1 s-1

26. Magnesium Alloys AZ31
AZ31 is an alloy of magnesium with 3% Al and 1% Zn.
Takuda et al in equiaxed AZ31 with 25 mm grain size.
Strain rates between ~0.001 s-1 and ~1 s-1 were investigated.

27. AZ31 – Analysis of Yield Kinetics
Two-parameter model:
Takuda et al
Interesting that for the two-parameter analysis, the values of go for the two obstacles populations were the same in pure Mg as in the alloy. Suggests the threshold stresses are defined by chemistry.
Deformation twinning ?

28. AZ31 – Analysis of Structure Evolution
Once again, prestrain and reload tests are unavailable.
Takuda et al
Suspect dynamic recrystallization at 523 K (TH = 0.566)

29. AZ31 – Analysis of Structure Evolution
Takuda et al
The shapes of the curves at the higher strain rates suggest the contribution of deformation twinning.

30. AZ31 – Analysis of Saturation Stress Kinetics
Takuda et al
Suspect that dynamic recrystallization shunts evolution at these temperatures.

31. Magnesium – Analysis of Saturation Stress Kinetics
AZ31
Pure Magnesium
While there is a lot of scatter in this plot, it is noteworthy that evolution in pure magnesium follows the same trends as in the alloy.

32. Pure Zirconium – Analysis of Yield Kinetics
Shear stress at onset of Stage III hardening – likely close to tCRSS
Akhtar et al
Annealed single crystals

33. Pure Zirconium – Analysis of Yield Kinetics
Soo and Higgins
Arc melted single crystals

34. Pure Zirconium – Analysis of Yield Kinetics
Polycrystals
Interesting that for the two-parameter analysis, the values of go for the two obstacles populations were the same in the two single crystal data sets as in the two polycrystals.

35. Pure Zirconium – Variation of Obstacle Strength with Oxygen Concentration
Single crystal measurements.
Clearer correlation of strength with oxygen content for the Obstacle 2 population

36. Pure Zirconium – Variation of Obstacle Strength with Oxygen Concentration
Single crystals show a ~ square root dependence on Oppm.
Polycrystal (taken as ) a little high but close.

37. Zirconium – Analysis of Structure Evolution
Once again, prestrain and reload tests are unavailable.
Chen and Gray

38. Zirconium – Analysis of Saturation Stress Kinetics
Analysis of the entire Chen and Gray data set results in common trend. The filled-in data points were not included in the fit.
But, in fact the
298 K / 2800 s-1
point is in error.
Chen and Gray Data Set

39. Zirconium – Analysis of Structure Evolution
The error in analyzing the 295K / 2800 s-1 curve.
As observed earlier in cadmium, the predicted
starts at a negative number because
is too large due to deformation twinning.

40. Zirconium – Analysis of Yield Kinetics
Recall
Focus on data points at the left-hand-side that suggest deformation twinning
As the temperature rises due to adiabatic heating the difference between the model stress and the measured stress decreases.

41. Difference Between Model and Measured Stresses When Twinning is Active
To a first approximation, the difference varies linearly with temperature.
Replace
With
Recall

42. Erred Analysis in Textbook
Figure published in the textbook is shown below.
Note that a constant of 350 MPa was added rather than a value that decreased with increasing temperature.
The model parameters for the evolution equation were selected to encompass the observed hardening.

43. Erred Analysis in Textbook
The corrected Figure becomes
The corrected curve shows that the rate of evolution remains low, and that the predicted (dashed) curve is clearly an overestimate.

44. Erred Analysis in Textbook
The same error was made in creation of Figure (2800 s-1 / 76 K). The published curve is compared with the corrected analysis below.
The analysis presented in Figure where the high rate of evolution was analyzed according to grain size refinement due to deformation twinning is also in error in detail – although the continued high rate of hardening may indeed be due to grain refinement.

45. Pure Zirconium Summary
A two parameter (state variable) formulation works well to describe the yield kinetics of annealed material. At least one of these state variables shows a dependency on oxygen concentration.
Evolution is well-described using the temperature and strain-rate dependent saturation threshold stress equation.
Deformation twinning confounds analyses of yield and of hardening. Errors in this analysis published in the textbook are detailed and corrected figures are presented.

46. Pure Zirconium – Stress Strain Curves
Measured and predicted stress-strain curves for conditions where deformation twinning is absent show good agreement.

47. Zircaloy-2
A corrosion resistant zirconium alloy used for nuclear fuel cladding.
Stress-strain curves measured in annealed and cold-worked material:
Recall: the yield stress in pure zirconium at RT and s-1 is closer to 235 MPa.
Howe and Thomas

48. Zircaloy-2
First, for two temperatures noted, compare the yield stresses in Zircaloy-2 with those in pure zirconium.
The 2-state variable model has three parameters that could be varied to establish a kinetic model for Zircaloy.
This is the case unless one or both of the free energy values changes.

49. Zircaloy-2: One Possible Solution
Model for Chen and Gray Zr (long dash):
Zircaloy-2 (short dash)
While the short-dashed line goes right through the two data points, the combination of threshold stresses seem highly unlikely. In particular, why should decrease in the alloy?

50. Zircaloy-2: A More Probable Solution
(as for pure zirconium)
Zircaloy-2
Formation of a fine, uniform distribution of intermetallic compounds in the alloy (as reported) could lead to an increase in the athermal stress.

51. Zircaloy-2: Cold Worked Material
The short dashed line for the CW material is derived by finding the value of that forces agreement with the measured stresses (and changing nothing else).

52. Zirconium – Analysis of Saturation Stress Kinetics
Analysis of evolution of in the two Howe and Thomas curves proceeds as demonstrated in other materials:

53. Zircaloy-2 – Analysis of Saturation Stress Kinetics
The two Howe and Thomas saturation stresses can be added to the pure zirconium plot:
Fascinating that the saturation stresses in Zircaloy fall along the same curve as those for pure zirconium !

54. Evaluating Model Parameters When Chemistry or Processing Vary
Finding model parameters for Zircaloy after deriving model parameters for pure zirconium
and
Estimating in cold worked Zircaloy
These analyses demonstrate advantages of an internal state variable model, when the state variables are correctly based on microstructural characteristics.
In Chapter 8, the evaluation of irradiated nickel and the assessment of the evolution of a state variable that represented the extent of irradiation damage was described.
Evaluation of damage in irradiated Zircaloy is discussed next.

55. Irradiation Damage in Zircaloy
Reload stress-strain curves measured in annealed and cold worked zircaloy.
Recall that the RT yield stress in annealed zircaloy
~ 200 MPa, illustrating that irradiation damage leads to significant hardening.
Howe and Thomas

56. Irradiation Damage in Zircaloy
Assume all model parameters derived for Zircaloy remain the same and evaluate the irradiation induced hardening through i) an increase in sa and ii) an increase in
predicted rate of strain hardening is too high when sa is increased.
ii) predicted rate of hardening and predicted point of tensile instability is about right when is increased.
3.6 x n/cm2
RT Reload
Howe and Thomas

57. Irradiation Damage in Zircaloy
For material irradiated at the higher dose, increasing to 238 MPa enables good agreement with the strain hardening and the point of tensile instability at the two reload temperatures.
2.7 x n/cm2
Howe and Thomas

58. Irradiation Damage in Zircaloy
In cold worked material, starts at a value of 292 MPa. A value of 491 MPa gives good agreement with stress levels in the reload stress strain curves.
What is the origin of the predicted negative rate of strain hardening ?
starts at 491 MPa but at 298 K and the strain rate of the reload, is 382 MPa, so softening is predicted.
(At 553 K, is 270 MPa.)
Note, the specimen is likely undergoing tensile instability as well, but the predicts assume uniform deformation.
Cold worked material
2.7 x n/cm2

59. Titanium and Titanium Alloys
Benefits:
Low density: 4.51 g/cm3
High melting point: 1941 K
Corrosion resistance
Can be processed in thin and thick sections.
High strengths available through alloying.
Increased use in medical applications, aerospace components, and consumer products

60. Pure Titanium – Analysis of Yield Kinetics
Conrad and coworkers in annealed Ti-50A (0.5% Oeq) with 22 mm grain size:
(Other data sets included in textbook.)

61. Pure Titanium – Analysis of Yield Kinetics
Conrad and coworkers in annealed Ti-50A (0.5% Oeq) with 22 mm grain size:

62. Pure Titanium – Analysis of Yield Kinetics
Nemat-Nasser et al in annealed Ti (0.09% Oeq) with 40 mm grain size:

63. Pure Titanium – Analysis of Yield Kinetics
Comparison of model parameters between the Conrad et al and Nemat-Nasser et al data sets:
Investigators
Grain Size mm
Purity
Oeq
Conrad et al 22 31
0.5
405
310
Nemat Nasser et al 40 23
0.09
171
200
The purer material has the lower threshold stresses (perhaps a stronger trend with .

64. Titanium – Analysis of Saturation Stress Kinetics
Once again, prestrain and reload tests are unavailable.
Conrad et al
Although not perfect, one could refer to these results as “well behaved”.

65. Titanium – Analysis of Saturation Stress Kinetics
Analysis of evolution doesn’t always yield well-behaved results.
Dynamic Strain Aging ?
Nemat-Nasser et al
Evolution law model parameters carry a great deal of uncertainty in cases such as these.

66. Titanium – Analysis of Saturation Stress Kinetics
Another odd result:
Nemat-Nasser et al
How does one evaluate evolution in face of these uncertainties?
Rely on results and
derive model parameters from results that are well behaved.
Use these parameters to predict evolution for results that are not well behaved.
This is how the dashed lines in this plot and plot on last slide were derived.

67. Titanium – Saturation Stress Kinetics
The data points along the dashed line are fairly convincing.
The open triangles off the line likely reflect dynamic strain aging.

68. Titanium – Strain Rate Dependence of the Stage II Hardening Rate
For titanium, there are sufficient results over a wide strain rate range to warrant analysis.
Consistent with trends observed in copper, nickel, AISI 1018 steel, and other materials.

69. Titanium – Dynamic Strain Aging
For the open triangles on the saturation stress kinetics plot the analysis demonstrated in vanadium was applied. Recall:
Conrad et al Ti-50A
At these temperatures and strain rates, s1 is predicted to equal 0.
Note that as in vanadium s1 begins to increase roughly linearly with at some value of

70. Titanium – Dynamic Strain Aging
For two measurements by Nemat-Nasser et al:
At these temperatures and higher strain rates, s1 is predicted to be greater than 0.
Again, s1 begins to increase roughly linearly with at some value of

71. Titanium – Stress-Strain Predictions
Comparison of three predicted stress-strain curves with measurements by Conrad et al.

72. Titanium Alloy Ti-6Al-4V
Two-phase (HCP a plus BCC b), heat treatable commercial alloy
Accounts for over 50% of titanium metal usage
Follansbee and Gray studied a 0.703% Oeq alloy with a 5 mm equiaxed grain size.
This study included a collection of prestrain and reload compression tests.
P. S. Follansbee and G. T. Gray III, Metall. Trans. A, 20A, 1989,

73. Ti-6Al-4V Analysis of Yield Kinetics
Two state variable model:
go values the same as in pure Ti.
P. S. Follansbee and G. T. Gray III, Metall. Trans. A, 20A, 1989,

74. Ti-Al Analysis of Yield Kinetics
Paton et al
Model predictions derived by varying only with go2 kept at 1.6.
Linear dependence of on Al content.
Could separate out a term representing the Al content as a third state parameter.
Wouldn’t add any to the model since the go is constant at 1.6.

75. Ti-6Al-4V Prestrain and Reload Tests
Follansbee and Gray performed a limited set of prestrain and reload tests on their Ti-6Al-4V material:
Reload Conditions
Prestrain Conditions
Temp. (K)
Strain Rate (s-1)
Strain
295
0.001
0.101
0.0015
76, ~183, 295
0.185
76, ~180, 295
0.281
76, 183, 295
495
0.100
76, 295
2500
0.084
76, ~188, 295

76. Ti-6Al-4V Prestrain and Reload Tests
Reload yield stress versus reload temperature and strain rate:
Prestrain strains
Prestrain Conditions
295K
0.001 s-1
Note: book lists incorrect values of

77. Ti-6Al-4V Prestrain and Reload Tests
Evaluation of evolution for 295 K / s-1 prestrains:

78. Ti-6Al-4V Prestrain and Reload Tests
Reload yield stress versus reload temperature and strain rate:
Prestrained at:
295K
2500 s-1
e = 0.084

79. Ti-6Al-4V Alternate Analysis of Evolution
Stress-Strain Curve
Follansbee and Gray
Note that the dashed curve (using the equation above) agrees well with the single point from the prestrain / reload tests.
Also note that is less than the value determined at s-1 (270 MPa at a strain of 0.101). Deformation twinning?

80. Ti-6Al-4V – Saturation Stress Kinetics
Analysis of prestrain and reload tests at:
295 K / s-1
495 K / s-1 and
295 K / 2500 s-1
The Ti-6Al-4V saturation stresses follow the same trend as do the pure Ti stresses.