Having known the components of the tensor [J], one can compute the components of the Almansi’s strain tensor as follows: εA 11 ¼ ∂u1 ∂x1 1 2 ∂u1 ∂x1 2 þ ∂u2 ∂x1 2 " # ð1:11Þ εA 22 ¼ ∂u2 ∂x2 1 2 ∂u1 ∂x2 2 þ ∂u2 ∂x2 2 " # ð1:12Þ εA 12 ¼ 1 2 ∂u1 ∂x2 þ ∂u2 ∂x1 ∂u1 ∂x1 ∂u1 ∂x2 þ ∂u2 ∂x1 ∂u2 ∂x2 ð 1:13Þ The above components are referred to the initial undeformed system of reference. Since the final coordinate system has a rigid body rotation, one can correct the derivatives for this rigid body rotation. In this case the corrections are negligible. In Fig. 1.6b, there are indicated the atoms 1 and 2 in whose neighborhood strain tensor components have been computed. Figure 1.13a shows again the same region of Fig. 1.6b while Fig. 1.13b shows the components of the Almansi’s strain tensor. The value of the maximum strain is e11 A ¼0.5886 and it is almost in the direction of 1120 , the rigid body rotation is 1.6 ; the minimum strain is e22 A ¼ 0.3593 and it is almost perpendicular to 1120 . The fringe analysis provides important information that is being retrieved from the 2D image produced by the electron microscope. The kinematics of the continuum has been extended to the inter-atomic space by 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. 10.5.1 of [14]). The analyzed region is located inside the atomic field. Strains in this region must be compressive in one direction because in this region extra atomic planes have been introduced during the crystal growth process. Figures 1.14a, b provide the displacement derivatives in the indicated directions; these derivatives show a spatial distribution that is very similar to the pattern corresponding to the solution of the theory of elasticity. The blue lines of equal values of the derivatives correspond to the area subjected to compression while the red lines correspond to the area subjected to tension. The yellow lines correspond to the transition from tensile to compressive stresses. These two regions have two extreme values almost equidistant from the transition line. It was also observed that at point “C” of Fig. 1.13a the lattice constant is very close to the value for the perfect crystal (a¼0.3073 nm). This indicates the passage from compression to tension in the array, a neutral region. As it has been said before, this neutral region is present in the elasticity solution of an edge dislocation and separates the region where the extra planes are located (compression region) from the region that is expanded (region that does not contain extra planes), Fig. 1.14c. The moire´ method provides the Eulerian components of the strain tensor: that is, the strains at the location in the deformed configuration. Through the above analysis it is possible to have an idea of the magnitude of the deformation caused by dislocations generated during the growth stage of a 4HSiC crystal in the region where the cooling rates have altered the regularity of the crystalline array. Fig. 1.13 (a) Principal strains at points 1 and 2; (b) Magnified view of the neighborhood of point 1 showing the principal strains of the Almansi’s strain tensor and the corresponding rigid body rotation of the principal strain directions 12 C.A. Sciammarella et al.
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