1. Chapter 3: Contributions to Strength
Issues to Address:
• Crystallography of slip
• Contributions to the strength of metals
• Yield stress dependencies of strengthening mechanisms
• Strain hardening

2. Chapter 3: Contributions to Strength
Take-Away Concepts:
• Slip occurs on specific crystallographic planes and along specific crystallographic directions
• The stress to initiate slip in a pure, well-annealed metal is a fundamental property.
• Contributions to the strength of metals include:
Forest dislocations
The Peierls stress
Impurity atoms (intentionally added or not)
Precipitates
Grain boundaries
• Strain hardening in polycrystals can be described using a simple model

3. Strength of a Single Crystal
Stacking unit cells (of any crystal structure) so that order of the unit cell is retained throughout the volume creates a single crystal.
A single crystal consisting of 8 stacked FCC unit cells.
A 1 cm3 Cu single crystal would consist of 2.1 x 1022 unit cells!

4. Strength of a Single Crystal
Strength is characterized by the stress-strain curve. Below is a schematic stress-strain curve for a single crystal.
t1 is the initial yield strength.
Region I is the region of easy glide, where generated dislocations traverse the single crystal and generate permanent strain.
In Region II (referred to as Stage II) the rate of hardening increases as dislocations encounter other dislocations .

5. Strength of a Single Crystal
In Region III the rate of hardening decreases. reflecting a balance between dislocation generation and recovery, where the stress acts to assist dislocations past obstacles.
Single crystals (particularly those in soft, FCC metals) are unique in the pronounced Region I, which is non-existent or of very limited nature in crystals consisting of many, randomly oriented crystals (polycrystals).

6. Strength of a Single Crystal – Initial Yield
Because slip occurs along certain crystallographic planes and certain crystallographic directions, the orientation of the single crystal is important.
Take an FCC single crystal with the
orientation shown below:
Tensile Axis
For this orientation and this crystal structure, slip will occur on the (111) plane and in the [011] direction

7. Strength of a Single Crystal – Geometry
Slip direction
Normal to slip plane:
A(111) N
The stress on the slip plane is:
The force in the slip direction is:
Thus:
Schmidt Factor

8. Strength of a Single Crystal – Initial Yield
Since the stress is resolved on the operative slip plane and in an operative slip direction, t(111) has a specific physical interpretation. It is the critical resolved shear stress tCRSS to initiate slip in this system.
With one can compute the stress to initiate yield for any crystallographic orientation.
An important point: the operative temperature or strain rate were not specified for this test. One must assume room temperature and a quasi-static rate.
Metal
tCRSS (MPa)
Copper
0.48 MPa
Iron
28 MPa
Molybdenum
48 MPa

9. The Peierls Stress
The difference in the tCRSS in Cu (FCC) and Mo (BCC) reflects the difference in the ease of dislocation motion in these two crystal structures.
The resistance to the motion of a dislocation in a perfect lattice is referred to as the Peierls stress, tp.
The Peierls stress scales inversely (and exponentially) with the width of the dislocation.
A slip plane with more spacing between atoms – a smaller planar density – promotes narrow dislocations.
Lack of a close-packed plane in the BCC crystal structure implies low planar density on the operative slip plane, narrow dislocations, and a high Peierls stress.

10. The Peierls Stress
The Peierls stress leads to unique temperature and strain-rate dependencies in BCC metals when compared to FCC and even HCP metals.
Deformation models need to reflect these differences.
Note that the planar density is much higher in the Cu (111) than in the Mo (110) and that the higher order plane in Mo has a lower planar density and a closer interplanar spacing.
Metal
Crystal Structure
Slip Plane
Distance between parallel planes (nm)
Planar Density (atoms / nm2)
tCRSS (MPa) Cu
FCC
(111)
0.209
35.2
0.48 Mo
BCC
(110)
0.222
14.3 48
(112)
0.128
9.1

11. Strengthening From Stored Dislocations
Dislocation density, r, is defined:
Total dislocation line length (m)
Volume (m3)
r (m-2) =
a high r in Cu often dislocations locked up in cell walls are referred to as “stored” dislocations
a low r
Ti alloy
From Calister, Materials Science and Engineering, an Introduction, 7E, 2007, Figure. 4.6
Gray and Follansbee, Matl. Sci. and Engnr, A111, 1989
The stress to move a dislocation through a structure increases as the dislocation density increases.

12. Strengthening From Stored Dislocations
Accumulation of stored dislocations and strain hardening can result from myriad metal working operations.
Wire drawing
Rolling
Tensile test
Strength increase from dislocation accumulation:
m is the shear modulus, and b is the Burger’s vector

13. Strengthening From Grain Boundaries
Grain boundaries are found at the juncture of grains of differing crystallographic orientation.
Dislocations traversing a grain can not easily pass into a neighboring grain.
Continued deformation requires increased stress levels

14. Strengthening From Grain Boundaries
The strain (e) achieved by the motion of a r density of dislocations can be written:
where b is the burgers vector and l is the distance moved by each dislocation.
If the distance l is restricted by the grain size, d, then the dislocation density must scale with the inverse of d to achieve an equivalent strain in samples of different grain size.

15. Strengthening From Grain Boundaries
The contribution to strength from the grain size is:
The Hall-Petch equation has become the governing equation for strengthening due to grain boundaries:
Hall Petch Equation

16. Strengthening From Impurity Atoms
Impurity atoms (unlike the host atom) in substitutional or interstitial sites and vacancies create lattice distortions that impede dislocation motion.

17. Strengthening From Impurity Atoms
According to the theory proposed by Fleischer, the increase in stress scales with the square-root of the concentration
(other models suggest a linear dependence with concentration).
Niobium
Measurements of oxygen strengthening in niobium show a very strong effect. The dashed line is drawn
according to
Buch, Pure Metals Properties, ASM Intn, 1999.

18. Strengthening From Precipitates
Many alloys are strengthened by the formation of hard precipitates, which offer effective boundaries to dislocation motion.
A rough correlation for the strength contribution from these precipitates is:
where m is the shear modulus and b is the burgers vector
Dislocation line
Precipitates

19. Strengthening From Stored Dislocations – the Voce Law
The increase in strength from an increasing stored dislocation
density was introduced earlier as
Models have been proposed that balance dislocation generation with recovery (or annihilation).
The Voce hardening law – which can be derived from simple dislocation models – offers an alternate correlation.
In this differential form, the incremental increase in the stress (shear stress) with (shear strain) is the Stage II hardening rate times unity minus the ratio of the shear stress to the saturation stress – where hardening goes to zero.

20. Strengthening From Stored Dislocations – the Voce Law
for qII = 1000 MPa and ts = 200 MPa

21. Strengthening – Summary
From interaction of dislocations with:
Peierls barrier (models exist)
Forest dislocations
Impurity elements
Grain boundaries
Precipitates
(Use Voce law)

22. Strengthening – Summary
Several contributions to the strength of metals covered.
Whether these are additive or not has been an open question.
Strain hardening is considered as the increased strength due to the interaction of dislocations with stored dislocations.
The contributions to strength from the other mechanisms are considered to be invariant with strain.