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In 1934 Sir Geoffrey Taylor, a British physicist and meteorologist, hypothesized that crystals could contain defects arising from the insertion of part of an atomic plane, as shown in Figure 5.2–4a. He called such a defect anedge dislocation.For simplicity, a simple cubic crystal has been used to illustrate the concept; however, the same principles apply to all crystal structures. Note that the crystal shown in Figure 5.2–4a is free of

FIGURE 5.2–4 (a) A 3-D representation of an edge dislocation. The dislocation is not the extra half plane of atoms that has been inserted but rather the line that runs along the bottom of the extra half plane. Parts (b) and (c) illustrate the motion of an edge dislocation in response to the application of a shear stress␶. The details of this motion are described in the text.

(a) (b) (c) Dislocation

Displacements resulting from the

motion of the dislocation

defects in all regions except along the termination of the partial plane, where there is a row of atoms with less than the ideal number of nearest neighbors. As an idealization, the row of atoms along which atomic packing is imperfect defines a linear defect.

Although the idea of the edge of an extra half plane of atoms as a model for an edge dislocation is a useful concept, it is incorrect to think of the insertion of a half plane of atoms as a mechanism of dislocation formation. Instead, dislocations are introduced into a crystal in several ways including: (1) “accidents” in the growth process during so-lidification of the crystals, (2) internal stresses associated with other defects in the crystal, and (3) interactions between existing dislocations that occur during plastic deformation.

Next we consider the mechanisms of dislocation motion. If a shear stress␶^{is applied} to the crystal as shown in Figure 5.2–4b, there is a driving force for breaking the bonds between the rows of atoms marked A and C and the formation of bonds between the atoms in rows A and B (see Figure 5.2–4c). Before the application of the shear stress, atoms in rows A and C have the correct coordination number while atoms in row B are unsatisfied (and therefore correspond to the dislocation line). After application of the stress, atoms in rows A and B have the correct CN but atoms in row C are unsatisfied. In effect, the dislocation has moved to the right by one atomic spacing.

The dislocation motion described in Figure 5.2–4b and c is known asdislocation glide.

The process of breaking and reestablishing one row of atomic bonds may continue until the dislocation passes completely out of the crystal. As shown in Figure 5.2–5, when the dislocation leaves the crystal, the top half of the crystal is permanently offset by one atomic unit relative to the bottom half.

Since the permanent deformation (i.e., irreversible motion of atoms from one equi-librium position to another) via dislocation glide was produced by breaking only one row of atomic bonds at any one time, the corresponding theoretical␶CRshould be much lower than for the previous model, where all of the bonds were broken simultaneously. In fact, detailed calculations have shown that theoretical predictions of ␶^CR for the dislocation glide model are in good agreement with the experimental values. In addition, this model is consistent with the two other experimental observations (i.e., the sensitivity to shear stress, and the crystallographic nature of slip).

Examination of Figure 5.2–5 suggests that two important geometric quantities are associated with a dislocation: the crystallographic direction in which it lies and the corresponding displacement vector (i.e., the magnitude and direction of the atomic dis-placement that results from the motion of the dislocation). Each of these characteristic can be described quantitatively.

A measure of the atomic displacement associated with the motion of a dislocation (see Figure 5.2–5c) can be obtained by making a circuit around the dislocation through

FIGURE 5.2–5 Passage of a dislocation through a crystal: (a) the dislocation is just about to enter at the left, (b) the dislocation has proceeded halfway across the crystal, and (c) the dislocation has exited on the right-hand side and the top half has shifted (been displaced) by one atomic unit.

Dislocation

defect-free material. As shown in Figure 5.2–6, the circuit is made in a clockwise sense around the dislocation. It has an equal number of atomic steps on parallel sides such that the start and end of the circuit would be coincident if it did not surround a dislocation (see Figure 5.2–6b). When a similar circuit is made around a region of material that contains a dislocation, however, the circuit does not close upon itself (see Figure 5.2–6c). A vector is defined by joining the endpoint to the starting point. Such a circuit is called aBurgers circuit,and the resulting vector is called theBurgers vector, b.

Examination of Figure 5.2–7a reveals a problem associated with our definition of a

“clockwise” Burgers circuit. One gets different Burgers vectors for the same dislocation depending on the direction in which the dislocation is viewed. This problem is solved by defining the unit tangent vector t that is locally tangent to the direction of a dislocation at the point of interest ( see Figure 5.2–7b). The Burgers circuit is then taken in a clockwise direction while looking in the direction of the unit tangent vector. While the initial choice of the positive direction for the unit tangent vector is arbitrary, the key is that the chosen

FIGURE 5.2–6 Illustration of Burgers circuit and Burgers vector for an edge dislocation. (a) A 3-D view of an edge dislocation. (b) A Burgers circuit closes upon itself when it surrounds a dislocation-free region of a crystal. (c) When the Burgers circuit surrounds a dislocation, the start and stop points are not coincident and the vector pointing from the stop point to the start point is defined to be the Burgers vector for the dislocation.

FIGURE 5.2–7 The concept of a positive dislocation direction. (a) Burgers vector depends on direction of viewing along dislocation line. (b) The choice of a positive direction and a corresponding unit tangent vector t. When viewed in the positive direction, this dislocation has a unique Burgers vector.

direction must be used consistently throughout all related calculations associated with any single dislocation.

Note that for an edge dislocation the Burgers vector is perpendicular to the dislocation line, as represented by the unit tangent vector 共i.e., b ⊥ t兲. The slip plane for the edge dislocation contains both b and t. In cubic crystals, the Miller indices of the slip plane can be obtained by the cross product of b and t (i.e., b ⫻ t). When determining the critical resolved shear stress of a single crystal, the Burgers vector (because it specifies the slip direction) and the normal to the active slip plane must be used to determine the angular quantities in Equation 5.2–2, as illustrated in the following example.

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EXAMPLE 5.2–2

Consider a dislocation in an FCC crystal with the following characteristics: ␶CR⫽ 0.5 MPa, t⫽ 共1兾兹6兲关1 1 2兴, and b ⫽ 共a0兾2兲关1 1 0兴.

a. Determine the slip plane for this dislocation.

b. Calculate the magnitude of the applied normal stress in the关0 1 0兴 direction necessary to cause the motion of this dislocation.

c. Repeat part b for a normal stress applied in the关0 0 1兴 direction.

Solution

a. The slip plane is determined by taking the cross product of b and t. The resultant vector, n, is normal to the plane defined by b and t (i.e., the slip plane). Thus:

n⫽ b ⫻ t ⫽冉^a20冊冉兹6¹ 冊

冨

冨

where i, j, and k, are unit vectors in the x, y, and z directions. Using Miller indices, n⫽ 共a0兾兹6兲关1 1 1兴. Clearing the constant factor (recall that the magnitude of a crystal-lographic direction has no physical significance) and noting that in cubics a plane has the same indices as its normal, we see that the slip plane is共1 1 1兲.

b. Equation 5.2–2 can be used to calculate the critical normal stress required to cause plas-tic deformation (dislocation motion):

␴c⫽ ␶CR

cos␪ cos ␾

In this case␪ is the angle between the applied force direction 关0 1 0兴 and slip direction 关1 1 0兴. Using the dot product, we find:

cos␪ ⫽关0 ⫻ 共⫺1兲兴 ⫹ 共1 ⫻ 1兲 ⫹ 共0 ⫻ 0兲

共1兲共兹2兲 ⫽ 1

兹2

Similarly,␾ is the angle between the normals to the two planes involved (i.e., 关1 1 1兴 and 关0 1 0兴兲, so that:

cos␾ ⫽(1⫻ 0兲 ⫹ 共1 ⫻ 1兲 ⫹ 共1 ⫻ 0兲

共兹3兲共1兲 ⫽ 1

兹3 Substituting these values into the expression for␴cyields:

␴c⫽ 0.5 MPa

共1兾兹2兲共1兾兹3兲⫽ 1.22 MPa

c. This time␪ is the angle between 关0 0 1兴 and 关1 1 0兴 and ␾ is the angle between 关1 1 1兴 and关0 0 1兴, so that:

cos␪ ⫽关0 ⫻ 共⫺1兲 ⫹ 共0 ⫻ 1兲 ⫹ 共1 ⫻ 0兲

共1兲共兹2兲 ⫽ 0

and

cos␾ ⫽共1 ⫻ 0兲 ⫹ 共1 ⫻ 1兲 ⫹ 共1 ⫻ 0兲

共兹3兲共1兲 ⫽ 1

兹3 Substituting these values into the expression for␴cyields:

␴c⫽ 0.5MPa 共0兲共1兾兹3兲⫽ ⬁

How do we interpret this result? Since cos␪ ⫽ 0, ␪ ⫽ 90⬚. This means that the applied stress direction关0 0 1兴 is perpendicular to the slip direction 关1 1 0兴 so that the applied force has no shear component lying in the slip direction. Therefore, stress applied along 关0 0 1兴 cannot cause dislocation motion in the 关1 1 0兴共1 1 1兲 slip system.

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While edge dislocation theory can be used to explain many of the important features of plastic deformation in crystals, it is necessary to invoke the presence of other types of dislocations to explain other aspects of deformation. The generalized dislocation theory can explain essentially every known feature of plastic deformation in crystals.