3.3.1
Back stress calculation in any faceted grain
The calculations presented in the previous section have been made assuming for reason of simplicity a simple cubic crystal symmetry. Besides, computations of the stress asso- ciated to dislocations paving the facets have been simplified by considering <100> type planes. In the general case, for instance involving an FCC symmetry and dislocation of mixed character accumulated at GBs, the surface Nye’s tensor αs
∼ may contain up
to 9 non-zero components and the GB normal may be of any orientation. In order to generalize the model equations developed in the previous section, we first consider the case of FCC dislocations accumulated at the GBs of a cubic grain with facets of <100> orientations.
3.3. Faceted grain simulations 77
With such configuration, the dislocations accumulated at GB are dislocations of mixed character type. Then, the superposition of dislocations coming from different slip systems gives on each grain facets a surface Nye’s tensor with 9 components differ- ent from zero.
In this section, the procedure we developed to decompose any complex GB configu- ration is explained. This solution makes use of a decomposition of the dislocation GB facet into in simpler counterparts based on the cubic simple symmetry. This way, we treat each component of the surface Nye’s tensor as a separate GB element made of twist, tilt or epitaxial dislocation facet. Here it must be noted that such decomposition into simple cubic symmetry is the one we tested in the previous Section 3.2.2. The forms we proposed can then be used to calculate the stress associated to any facet of GB. Once calculations are made for each Nye’s tensor component, the total stress tensor induced by the real dislocations paving a GB facet can be obtained by summing the contributions of every component and common rotation operations to account for the true direction of the facet normal.
The correspondence between the Nye’s tensor components and the dislocation lines and Burgers vector symmetry we used are reported in Table3.5.
αs ∼ bs ls αsxx [100] [100] αs xy [100] [010] αs xz [100] [001] αs yx [010] [100] αs yy [010] [010] αsyz [010] [001] αszx [001] [100] αs zy [001] [010] αs zz [001] [001]
Table 3.5: Projection of the surface tensor Nye’s tensor α∼s components on
the simple cubic symmetry. The Burgers vector and dislocation line vector are indicated.
A simple example of facet calculation with the FCC crystal symmetry is shown in Figure 3.10. The size of the present facet is 10 × 10 µm. In such calculation, the facet plane is perpendicular to z-axis and contains mixed dislocations of Burgers vector parallel to [101] and line direction parallel to [110], equidistantly distributed in the plane.
The obtained surface Nye’s tensor α∼s is: αs ∼ = αs xx αsxy 0 0 0 0 αszx αszy 0 = δ(z) 6.3 6.3 0 0 0 0 6.3 6.3 0 (3.47)
where δ(z) is the Dirac distribution function. Here it must be noted that in the expres- sion of α∼, there is no component in the third column because there is no component of the dislocation line out of facet. See Figure 3.10a, the plane normal n is parallel to z-axis in the [001] direction. Following our cubic simple decomposition, this GB facet made of mixed FCC dislocations can be decomposed as follows. The component αsxx can be attributed to dislocations with bs = [100] and ls = [100], which corresponds to
screw dislocations forming a twist facet. The component αs
xy can be attributed to dis-
locations with bs = [100] and ls = [010], which correspond to edge dislocations forming
an epitaxial facet. Lastly, the two component αs
zx and αszy correspond to dislocations
with bs = [001] and ls = [100]) and also bs = [001] and ls = [010], respectively, forming
two edge dislocation sets in tilt facet configurations.
(a) (b) 0 2 4 6 8 10 z (µm) -10 0 10 20 30 40 ij (MP a) xx yy zz xy xz yz
Figure 3.10: Example of stress calculation for a facet including dislocation of one FCC slip system. The size of the present facet is 10 × 10 µm and the surface dislocation density is ρsGN D = 4.94 × 106m−1. (a) The [001] facet with mixed FCC dislocations (b parallel to [101] and line direction l parallel to [110]), (b) the full stress tensor as a function of the distance z to the interface. Open symbols refer to the stress computed with DD simulations and the dashed lines correspond to the predictions made with the simple cubic symmetry decomposition.
3.3. Faceted grain simulations 79
using the simple cubic geometry. The total stress tensor associated to the mixed dislo- cation facet is decomposed into four elementary facets. More precisely, the component αsxx induces the stress stress σxy, the component αsxy is at the origin of the three normal
stress σxx, σyy and σzz and the components αszx and αszy induce the shear stress σyz
and σxz, respectively. When comparing with the stress solution computed with the DD
simulation (shown in open samples), an excellent agreement is found for the 6 stress components. For only two stress components of the elementary epitaxial facet (σxx and
σyy), the agreement is less good at distance smaller than 3 µm from the 10 µm high
facet. Close to the dislocation facet, the maximum error with the use of the model is about 25%. We verified that such discrepancy was not related to the decomposition procedure. It was mainly caused by the fact that dislocation lines in Figure 3.10a are different in length.
3.3.2
Systematic validation tests
To extend the validation of our empirical model, we checked the simulation configu- rations with increasing complexity. DD simulations were carried out on three sets of configurations based on cubic grains of dimensions 10 µm (see Figure3.11):