The embedded grain problem

In order to show the possibilities offered by DD simulations on tracking the back stress in a grain and to reproduce size effect, a simple example of simulation is presented in the following. The 3D-DD simulation code and parameters used in this chapter are the same as in Chapter 2. The basic features of this code are presented in the reference (Devincre et al., 2011).

A tensile loading is applied to deform a cubic grain of copper embedded in an elastic continuum up to 0.2% plastic strain. The size of the embedded cubic grain is 10 µm. The tensile axis is chosen in order to plastically deform the grain in single slip. The final state of the dislocation microstructure is shown in Figure 3.2a. The total dislocation density in the grain can be separated into two groups (Figure 3.2b and c).

The first group of dislocations in Figure 3.2b is in the bulk of the grain; they are randomly distributed inside the grain and can be called Statistically Stored Dislocations (SSDs) as the net Burgers vector of their dislocation density equals zero. The second group of dislocations in3.2c is accumulated very close to the GBs and contains polarized dislocations (dislocations with the same Burgers vector sign). Such dislocation density can be affiliated to Geometrically Necessary Dislocations (GNDs). GNDs are accumu- lated at the GBs during plastic deformation. In our simulations, GBs are considered as impenetrable interfaces, i.e. GBs act as barriers to dislocation glide. The variation

3.2. Additional investigation with DD simulations 65

10μ

m

Figure 3.2: (a) Distribution of dislocations inside a cubic grain embedded in an elastic continuum at 0.2% of plastic strain. The grain is deformed in single slip with the slip system (¯111)[101]. (b) The dislocations in the bulk of the grain, (c) polarized dislocations (GNDs) accumulated at GBs, (d) dislocation density ρ as a function of the distance to the GBs (ρ is averaged in layers of 0.05 µm thickness parallel to the GB planes.

of the average dislocation density as a function of the distance from the GB is plotted in Figure 3.2d. For such calculation, the dislocation density is computed in layers of 0.05 µm thick parallel to the GB and up to a distance of 1 µm. The dislocation density calculated close to the GB is extremely high and then decreases rapidly to a plateau value of the density. A ratio of about 20 exists between the density found in the vicinity of the GBs and in the grain bulk. Not shown in Figure 3.2d, we checked that beyond the 1 µm region, the dislocation density remains almost constant.

The contribution of SSDs to strengthening and hardening mechanisms is relatively well known. This dislocation density generate short-range stress through the mecha- nisms of forest interaction and line tension. The mechanical properties associated with SSDs and their contribution to strain hardening was extensively studied with DD sim- ulations for FCC (Devincre, Kubin, and Hoc,2006) and BCC (Queyreau, Monnet, and Devincre, 2009) crystals during the past years. On the other hand, the detail contribu- tion of GNDs to strain hardening is still a debated question. The goal of the present study is to quantify the mechanical properties of GNDs.

Thanks to the simulation reported in Figure 3.2 as well as other simulations, the distribution of the internal stress inside a grain was investigated to get inputs regarding

the distribution of the back stress inside an embedded grain. As illustrated in Figure

3.2a, a reference plane grid (20 × 20 voxels drawn in blue) cutting the grain in its middle was defined to calculate the internal stress associated with the total density of dislocations (SSDs and GNDs) existing inside the grain. Results of this calculation are reported in Figure 3.3. One must notice here that the internal stress tensor σint is cal-

culated in each voxel by considering the dislocation microstructure under loading. In Figure3.3, the distribution of two scalar quantities is used to illustrate the variation of the internal stress, i.e. σvm, the Von Mises stress and τ , the shear stress resolved on

the active slip system. Except for the outermost layer of the grid, the amplitudes of σvm and τ are found almost constant inside the grain. Such homogeneous distribution

of the stress inside the grain can be explained by the homogeneous density of disloca- tions accumulated at the GBs. Such distribution presents strong similitude with the solutions previously discussed for the infinite dislocation walls. The stress calculated in the outermost layer is close to the GBs. Then, it is a region where the stress locally can vary rapidly and strongly as function of the relative position of the closest dislocation staying at the GBs. This is why in average the stress in the outermost layer goes to zero. In summary, one must note that the internal stress distribution was always found to be relatively homogeneous inside the tested embedded grains.

Figure 3.3: The stress state calculated at 0.2% (under load) in a grid of 20x20 voxels, normal to z-axis and cutting the middle of the embedded grain (see Figure 3.2). (a) The Von Mises stress (σvm) and (b) the resolved shear

stress τ on the active slip system are calculated from the total internal stress tensor.

In the following, we made some additional DD simulations to identify some rele- vant parameters needed to calculate the amplitude of the internal stress inside a grain assuming that we know the surface density of dislocation accumulated at the GBs.

3.2. Additional investigation with DD simulations 67