εA 11 ¼ ∂u1 ∂x1 1 2 ∂u1 ∂x1 2 þ ∂u2 ∂x1 2 " # ð10:20Þ εA 22 ¼ ∂u2 ∂x2 1 2 ∂u1 ∂x2 2 þ ∂u2 ∂x2 2 " # ð10:21Þ εA 12 ¼ 1 2 ∂u1 ∂x2 þ ∂u2 ∂x1 ∂u1 ∂x1 ∂u1 ∂x2 þ ∂u2 ∂x1 ∂u2 ∂x2 ð 10:22Þ Figure 10.12a shows the image corresponding to Fig. 10.6b where the two edge dislocations are represented by the two green inverted T’s. The converging yellow lines correspond to the lines that are the axis of the two added extra rows of atoms. In the neighborhood of the dislocation, the basic atomic arrangement shown in Fig. 10.8a is distorted. The region of the yellow circles is under compression and corresponds to a shortening of the parameter “a” of Fig. 10.8a. The region of the green circles is under tension as shown in Fig. 10.8b: the parameter “a” is increased. The region of the blue circles is subjected to shear as shown in Fig. 10.8c. For example, the angles of the fundamental hexagon are changed: the “a1” parameter is elongated, “a2” and “a3” are shortened. The moire´ fringe pattern analysis provides important information that is being retrieved from the 2-D image produced by the electron microscope. The kinematics of the continuum has been merged to the inter-atomic spaces via the utilization of digital moire´. The presence of fringe dislocations in the fringe patterns in the same position as physical dislocations indicates the place where there are discontinuities in the displacement field (see Chap. 10.5 of Ref. [23]). Figure 10.13a corresponds to the derivative∂ue2/∂e2 in the directione2 versor shown in Fig. 10.6a and it is obtained from the moire´ pattern of Fig. 10.10b and approximately corresponds to the upper portion of the region marked “compression” in Fig. 10.12a. In this region there is a shortening of the parameter “a”: there are the yellow circles and the corresponding level lines are reddish. The blue level lines correspond to tensile deformations (parameter “a” increasing) and correspond to the green circles in Fig. 10.12a. The greenish level lines correspond to the region of transition between the compression and tension, white circles, “0” deformation. In this region the parameter “a” keeps its undeformed value, a ¼0.3073 nm. Similar conclusions apply to Fig. 10.13b that corresponds to the versor e3 and represents the upper part of the region labeled “compression” in Fig. 10.12a. Since the rotations in the analyzed region are small, derivatives of displacements computed from the patterns of Fig. 10.10 provide close enough values of local strains in the dislocation region. It is interesting to observe that these derivatives show a spatial distribution very similar to the pattern corresponding to the theory of elasticity solution shown in Fig. 10.13c. However, while the theoretical solutions include symmetric values the patterns of Fig. 10.13a, b show larger gradients in compression than in tension. This asymmetry is connected with the asymmetry of the energy potential in the atoms illustrated in Fig. 10.1. Through the above analysis it is possible to have an idea of the magnitude of the deformation Fig. 10.12 (a) Image position of atoms in neighborhood of two edge dislocations; (b) magnified region around point 1 showing the principal strains of the Almansi’s strain tensor and corresponding rigid body rotation of principal strain directions 94 C.A. Sciammarella et al.
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