Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

GTBL042-08 GTBL042-Callister-v3 October 4, 2007 11:51

2nd Revised Pages

252 • Chapter 8 / Deformation and Strengthening Mechanisms

Figure 8.9 Slip in a zinc single crystal. (From

C. F. Elam,The Distortion of Metal Crystals,

Oxford University Press, London, 1935.)

EXAMPLE PROBLEM 8.1

Resolved Shear Stress and Stress-to-Initiate-Yielding

Computations

Consider a single crystal of BCC iron oriented so that a tensile stress is applied

along a [010] direction.

(a)Compute the resolved shear stress along a (110) plane and in a [111] direc-

tion when a tensile stress of 52 MPa (7500 psi) is applied.

(b)If slip occurs on a (110) plane and in a [111] direction and the critical

resolved shear stress is 30 MPa (4350 psi), calculate the magnitude of the

applied tensile stress necessary to initiate yielding.

Solution

(a)A BCC unit cell along with the slip direction and plane as well as the

direction of the applied stress are shown in the accompanying diagram.

In order to solve this problem we must use Equation 8.2. However, it is

first necessary to determine values forφandλ, where, from the diagram,

φis the angle between the normal to the (110) slip plane (i.e., the [110]

direction) and the [010] direction, andλrepresents the angle between [111]

and [010] directions. In general, for cubic unit cells, an angleθbetween

directions 1 and 2, represented by [u 1 v 1 w 1 ] and [u 2 v 2 w 2 ], respectively, is

equal to

θ=cos−^1

⎡

⎣√( u^1 u^2 +v^1 v^2 +w^1 w^2

u^21 +v^21 +w^21

)(

u^22 +v^22 +w^22

)

⎤

⎦ (8.6)