In the beginning of the discussion on the fine particle strengthening, we have dis­tinguished between the precipitation and dispersion strengthening. If the particles are fine, stable, and incoherent, dispersion strengthening is applicable. Dispersion strengthening is a significant strengthening mechanism utilized for high — temperature alloys. Orowan bypassing mechanism at lower temperature regime is applicable here also. One example of dispersion strengthening system is thoria — dispersed nickel (T-D Ni). This tends to have much greater high temperature defor­mation resistance compared to the nickel matrix itself.

■ Example 4.7

A Li {Li (bcc): m = 32 GPa; а = 3A v = 0.29; stacking fault energy (Г) = 1250 mJ m~2; TM = 181 °C} sample exhibited an yield strength of 10 MPa.

a) Estimate the dislocation density assuming that the strengthening is all due to dislocations (strain hardening)?

Solution

b) Estimate the strengthening if the dislocation density increases by a factor of 100 following cold working (or exposure to radiation).

Solution

t — aGbyffi) — (1)(32000)(2.598 x 10~10)/1.447 x 1014 — (8.314 x 10~6)(1.203 x 107).

So, t — 100 MPa.

4.5

Summary

This chapter introduced the dislocation concept in a greater detail. Various aspects of dislocation theory, including dislocation energy, line tension, velocity, and so on, are discussed. Also, dislocation reactions in various lattices are discussed. The chapter concludes with the introduction of various strengthening (hardening) mechanisms. This understanding will lead us to better understand the effect of increased dislocation density on the properties of the irradiated materials in the subsequent chapters.

Problems

4.1 We considered a cylindrical Ni single crystal (FCC) with a diameter of 0.1 mm pulled in tension with a stress of 1000 MPa (see Figure 4.4). The loading direc­tion is along [101], while the slip system is (111)[Ї10]. We calculated the resolved shear stress along the slip direction in the slip plane.

a) What is the resolved shear stress along the slip direction in the slip plane?

b) If tCRSS for Ni is 550 MPa, would the crystal deform?

An edge dislocation (AB) is situated perpendicular to the slip direction of the slip system and pinned at two points (A and B) separated by 1000 A

c) If the Burgers vector of the dislocation is along the slip direction, what will be its direction and magnitude?

d) Determine the radius of curvature of the dislocation in part (c) above due to the applied stress?

e) Compute the minimum applied load (normal to the specimen cross section) required for the pinned dislocation (AB) to operate as a Frank-Read source.

f) If the dislocation line in the figure is a screw dislocation, what will be its Burgers vector (magnitude and representation)?

4.2 By increasing the temperature, the concentration of vacancies can be increased. Will this also increase the density of dislocations (be quantitative)?

If not, why not and how may the dislocations be multiplied?

4.3 A prismatic loop has Burgers vector perpendicular to the plane of the loop and thus can glide in that plane under an applied shear stress. True or false?

4.4 In the figure, the dislocation AB lies in the plane (111) with b along [110] (per­pendicular to the line AB).

a) Is the dislocation AB an edge or a screw or a mixed type?

b) What is the line vector of this dislocation?

c) If AB were a screw dislocation, what will be its Burger’s vector?

x

4.5 Show that a perfect dislocation, (a/2)[110], in an FCC lattice splits into two Shockley partials with Burgers vectors of (a/6)(112) type. (Write down the equation.)

a) Are the Shockley partials glissile or sessile? Why?

b) A Shockley partial (a/6)[112] reacts with a Frank partial (a/3)[111] to yield a perfect dislocation. What is the Burger’s vector of the product dislocation?

c) Show that the reaction in problem 4.3 is valid.

(This reaction occurs in irradiated stainless steel when heated. Similar reaction could also occur in quenched Al-Mg alloy from 550 °Cto —20 °C fol­lowed by heating (see Figure 5.12 in Ref. [3]). These reactions lead to unfaulting offaulted loops.)

4.6 a) Evaluate the force (magnitude and direction) acting on an edge dislocation

(b = bi, l = k) due to an external stress oxx.

b) How this dislocation may move due to this force?

c) Evaluate the force on the dislocation due to hydrostatic pressure p and comment on how the dislocation may move due to this force.

4.7 Evaluate the hardening (tc) of a Li (BCC, G = 32GPa, a = 3A) alloy with a microstructure consisting of a dislocation density of 3 x 1013m—2 (a? = 0.5) and 3 at %solutes (ac = 0.005).

180 I 4 Dislocation Theory
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Additional Reading

Meyers, M. and Chawla, K. (2009) Mechanical Behavior ofMaterials, 2nd edn, Cambridge University Press.

Reed-Hill, R. E. and Abbaschian, R. (1994) Physical Metallurgy Principles, 3rd edn, PWS Publishing, Boston.