Having determined which slip system will operate first, it should now be possible to calculate the minimum force needed to cause plastic flow during the application of a stress to the crystal. The calculation proceeds as follows.

The following diagram shows the orientation of the [021] tensile axis with respect to the unit cell vectors b and c (parallel to [010] and [001] respectively). These three vectors all lie in the (100) plane.

Identification of the angles l and f. The diagram shows coplanar vectors on the (100) plane. The (001) slip plane lies horizontal and extends perpendicular to the screen.

The initial angle between the tensile axis and the slip plane normal, f₀, is

and the angle between the tensile axis and the slip plane normal, l₀, is

l₀ = 90° - f₀ = 43.3°

The Schmid factor for the [010](001) slip system is therefore

cos f₀ cos l₀ = cos 46.7° x cos 43.3° = 0.499

If the critical resolved shear stress for cadmium is 0.15 MPa, and the initial crystal diameter is 3 mm, then the force required to cause slip can be calculated:

Now consider what happens when the crystal is plastically extended. If the original length of crystal was l₀ = 50 mm, and the crystal is extended by 50 % to l₁ = 75 mm, then:

• The diameter of the crystal will decrease as volume is conserved:

  

• The geometry of the slipped crystal changes according to:

  l₀ cos f₀ = l₁ cos f₁ and l₀ sin l₀ = l₁ sin l₁

  hence:

  f₁ = 62.8° and l₁ = 27.2°

  and so the new Schmid factor for the extended crystal is:

  cos f₁ cos l₁ = 0.407

The force required to cause further deformation of the crystal in this condition can be calculated as before:

The force required to cause slip is lower after the crystal has been deformed. This phenomenon is known as geometric softening - once deformation has started, less load is required to further deform the crystal. Geometric softening depends heavily on the orientation of the tensile axis within the crystal. For some orientations, no geometric softening is observed. There are other factors that control slip, which will not be discussed here, and these can dominate over the geometric factor.

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