# 7.12: Summary


This TLP has covered the following points:

1.New dislocations are generated from a Frank-Read source. The minimum shear stress required to operate a Frank-Read source, which is found by balancing the force on the dislocation and the line tension, is:

$\tau = 2 \frac{Gb}{d}$

Frank-Read source is important as it is the mechanism by which dislocation density increases as a material is work-hardened.

2.A dislocation can interact with either another dislocation or a solute atom. When dislocations interact with each other, they can either repel (if they have the same sign) or annihilate (if they have opposite signs). On the other hand, when a dislocation interacts with a solute atom, an energetically favorable arrangement is formed, leading to solute-solution strengthening.

3.A Lomer lock is a type of sessile dislocation formed when the plane which contains the line and Burgers of the edge dislocation is not a close-packed slip plane of the system.

4.Climb and cross-slip are the two processes by which dislocations can become organised and annihilate. Climb is the mechanism of moving an edge dislocation from one slip plane to another through the incorporation of vacancies or atoms. Cross-slip is the movement of a screw dislocation from one allowable slip plane to another. Cross-slip is favoured in metals with a high stacking-fault energy.

5.Jogs are formed by dislocation intersections. In particular, the intersection between two screw dislocations is crucial to work hardening.

6.The two stages of the plastic deformation of a single crystal are discussed. It is important to note that stage l only exists in single crystals, as only one slip system is operated (Poly-crystals do not exhibit stage l as multiple slips are initiated from the start). Stage ll, where the hardening rate is constant, is governed by forest hardening. The flow stress at this stage is given by the Taylor equation:

$\tau=\alpha b G \sqrt{\rho}$

7.Forest hardening arises when the active dislocations in the primary slip system are impeded by the forest dislocations in the secondary slip system. The pinning points, which are created by the intersections between the active and forest dislocations, are either jogs or Lomer locks.

8.The typical value of the hardening rate is G/200. The expression for the hardening rate is found by considering the movement of a dislocation segment:

$\frac{\mathrm{d} \tau}{\mathrm{d} \gamma}=\frac{\alpha G}{2 \lambda \sqrt{\rho}}$

9.Poly-crystals have higher yield stresses than single-crystals as each grain needs to undergo a shape change which is consistent with those of their neighbor, requiring multiple slips from the start. In addition, the yield stress of a poly-crystal is related to the grain size by the Hall-Petch relationship:

$\sigma_{y}=\sigma_{0}+\frac{k}{\sqrt{D}}$

## Going further

### Books

The following books contain extensive information about Frank-Read source, jogs formation, Lomer locks and single-crystals deformation.

R.W.K. Honeycombe, The Plastic Deformation of Metals, Second Edition, 1984, ISBN: 0-7131-3468-2

W.F. Hosford, Mechanical Behavior of Materials, Second Edition, 2010, ISBN: 978-0-521-19569-0

G.E. Dieter, Mechanical Metallurgy, SI Metric Edition, 1988, ISBN: 0-07-100406-8

For a more detailed and mathematical description of forest hardening and single-crystals deformation, consult:
A.S.Argon, Strengthening mechanisms in crystal plasticity, 2008, ISBN: 978-0-19-851600-2

### Other resources

A.D. Rollett, U.F. Kocks, A Review of the Stages of Work Hardening, 35-36 (1993) pp 1-18, 10.4028/www.scientific.net/SSP.35-36.1

U.F.Kocks, A Statistical Theory of Flow Stress and Work-hardening (1965)

Both contain a detailed discussion of forest hardening and single-crystals deformation.

7.12: Summary is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Dissemination of IT for the Promotion of Materials Science (DoITPoMS).