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

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

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!

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 .

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).

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

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

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

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.

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

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.

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

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

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.

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

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

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.

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

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.

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

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

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.