Mechanistic Origin of Orientation-Dependent Substructure Evolution in Aluminum and Aluminum-Magnesium Alloys

Abstract

Role of magnesium (Mg) solute and deformation temperature on the orientation-dependent substructure evolution in aluminum (Al) was investigated experimentally. The mechanistic origin of the experimental orientation dependence was then explored with numerical modelling. In experiments, the Al–Mg showed more geometrically necessary dislocation density and residual strain but had insignificant differences between hard and soft crystallographic orientations. Increased Mg-content led to the conversion of dislocation cell structures to dislocation tangles. On the other hand, an increase in deformation temperature appeared to nullify the role of solute, and irrespective of Mg content, the substructures were not orientation dependent. Molecular dynamics (MD) simulations provided temperature and solute dependence of dislocation drag coefficient and probability of cross slip. These appeared to be orientations independent. Discrete dislocation dynamics (DDD) simulations were then conducted by incorporating relevant parameters from MD and fitting DDD simulated stress-strain behavior with experimental data. Further, the solute was modelled as static obstacles to dislocation movement, hindering easy glide and short-range dislocation–dislocation interactions. Dislocation interactions at the slip plane intersections generated dynamic obstacles and sources—their ratio being determined by the probability of cross-slip. The DDD simulations indicated that evolving density of dynamic obstacles and sources determined the orientation dependence of substructure evolution.

Appendices

Appendix A: Geometrically Necessary Dislocation (GND) Density calculation from Discrete Dislocation Dynamics (DDD) Simulations

DDD-simulated GND densities, estimated a subdomain ($\omega \subseteq\Omega$), is defined^[90] as,

$${\rho }_{\rm GND}\left(\omega \right)=\frac{|\left|{\varvec{B}}\left(\omega \right)\right||}{b}=\frac{\surd {{B}_{i}B}_{i}}{b}$$

(A1)

where ${\varvec{B}}(\omega )$ is the net Burgers vector over $\omega$ and b is the material Burgers vector length. Following Eq. [A1], a network of dislocations piercing a plane with the unit normal n has a net Burgers Vector B per unit area, where,

$${B}_{i}={G}_{ij}{n}_{j}$$

(A2)

G is a tensor that quantifies the non-redundant dislocation density within the domain and can be viewed as a measure of lattice (slip plane) incompatibility. For Volterra edge dislocations under plane strain conditions, take n as the out of plane normal ${{\varvec{e}}}_{3}={{\varvec{e}}}_{1}\times {{\varvec{e}}}_{2}$ with the ${{\varvec{e}}}_{1}$ and ${{\varvec{e}}}_{2}$ the base vectors in the ${x}_{1}-{x}_{2}$ plane of deformation. On the basis of ${e}_{i}$, the components of G reduce to those of Nye’s tensor provided small transformations and neglecting elastic strains:

$${G}_{ij}=\sum_{\xi =1}^{{N}_{\rm s}}\left({\rho }_{+}^{\xi }-{\rho }_{-}^{\xi }\right){b}_{i}^{(\xi )}{n}_{j},$$

(A3)

where ${N}_{\rm s}$ denotes the number of active slip systems under the imposed loading; ${\varvec{b}}\left(\xi \right)$ is the Burgers vector common to all dislocations on the slip system $\xi$. Substituting (3) into (2) we get

$${B}_{i}=\sum_{\xi =1}^{{N}_{\rm s}}\left({\rho }_{+}^{\xi }-{\rho }_{-}^{\xi }\right){s}_{i}^{\xi }$$

(A4)

If ${\phi }^{\xi }$ denotes the orientation of the slip system measured with respect to the ${x}_{\rm 1}$-axis, GND density may be written as follows:

$${\rho }_{\rm GND}=\sqrt{{\left[\sum_{\xi =1}^{{N}_{\rm s}}\left({\rho }_{+}^{\xi }-{\rho }_{-}^{\xi }\right)\mathrm{cos}{\phi }^{\xi }\right]}^{2}+{\left[\sum_{\xi =1}^{{N}_{\rm s}}\left({\rho }_{+}^{\xi }-{\rho }_{-}^{\xi }\right)\mathrm{sin}{\phi }^{\xi }\right]}^{2}}$$

(A5)

Thus, using (5), GND density can be computed. It must, however be noted that the computation is highly dependent upon the $\omega$ chosen, which has been optimized^[90] in our study.

Appendix B: Geometrically Necessary Dislocation (GND) Density Calculation from Experimental Data

For estimating GND density, the approach of cross-correlation^[47,60] or high-resolution EBSD was used. In particular, the cross-correlation results were obtained using a shareware (Open XY™) from Brigham Young University, for details reader may refer Ruggles and Fullwood.^[47] It is to be noted that GND calculations in Open XY™ can be accomplished using three different techniques. These are termed ${{\varvec{\Lambda}}}_{3}, {{\varvec{\Lambda}}}_{5}$ and ${{\varvec{\Lambda}}}_{9}$, where $\mathrm{3,5}$ and $9$ are the number of terms in the Nye tensor used for GND calculation. In our study, ${{\varvec{\Lambda}}}_{9}$ has been used.

$${\rho }_{\rm GND}\sim \frac{3}{2b}\sum_{i}\sum_{j}{\varvec{\Lambda}},$$

(B1)

where ${\varvec{\Lambda}}$ denotes the Nye tensor and $b$ is the magnitude of the Burgers vector. Further, the Nye tensor can also be expressed in terms of lattice curvature as given by Pantleon^[91]

$${\Lambda }_{ik}={\kappa }_{ki}-{\delta }_{ki}{\kappa }_{mm}$$

(B2)

where ${\varvec{\kappa}}$ is the curvature tensor and ${\delta }_{ji}$ is the Kronecker delta. The curvature tensor can be written in terms of the disorientation vector as ${\kappa }_{kl}=\frac{\Delta {\theta }_{k}}{\Delta {x}_{l}}.$ Only 6 components of the lattice curvature tensor can then be derived. It is to be noted that in this method derivatives along the third direction (normal to the surface) cannot be accessed. This may lead to the underestimation of GND density. Nonetheless, this technique has been used for estimating the GND density from experimental microstructures in the past literature^{[60,92,93,94,95]} and also in our study. The trick is to use optimized EBSD step size^[47] for GND estimation, which has been followed in the present study as well.