Microstructural Origin of Residual Stress Relief in Aluminum

Abstract

Annealing-induced microstructural evolution and associated stress relief were investigated experimentally for various crystallographic orientations and annealing temperatures. Combinations of the site and orientation-specific X-ray plus electron diffraction were used. 2-D discrete dislocation dynamics, oriented for double slip with both dislocation glide and climb mechanisms, was employed to simulate the annealing process. Irrespective of crystal orientation, both experiments and simulations showed the highest stress relief at the intermediate annealing temperature. In the experiments, this was related to the fastest elimination of low angle grain boundaries. In the simulations, it was linked to the largest reduction in the density of pinned dislocations. The simulations also suggested that the non-monotonic temperature dependence of the stress relief, and associated substructural changes, emerged from a balance between dislocation glide and climb processes.

Appendix

A typical texture measurement scheme with 10 deg step size in tilt (ψ) and rotation (φ) is shown in Figure A1(a). By rotation and tilt, we achieved the different measurement points, shown as circles in the scheme. X-ray diffraction from a polycrystalline material captured by an area detector appeared as ring(s), shown in γ − 2θ coordinates in Figure A1(b). For any particular combination of (ψ, φ, γ, θ, ω), the part of the diffraction ring was converted to the pole figure angles (α, β) as following[71]

Fig. A1

(a) A standard pole figure measurement scheme with 10 deg step in rotation (ϕ) and tilt (ψ) angle. (b) The γ integration of a diffraction ring for (111) pole. It shows which points in the pole figure were getting calculated for a particular γ range. (c) Multiple polar angles corresponding to a single-pole figure position, which appeared to be an ambiguity of numerical solution. Such values were averaged out

$$ \alpha = \sin^{ - 1} \left| {h_{3} } \right| = \cos^{ - 1} \sqrt {h_{1}^{2} + h_{2}^{2} } $$

$$ \beta = \pm \cos^{ - 1} \frac{{h_{1} }}{{\sqrt {h_{1}^{2} + h_{2}^{2} } }}\begin{array}{*{20}c} {\beta \ge 0^{ \circ } \;{\text{if}}\;h_{ 2} \ge 0} \\ {\beta \ge 0^{ \circ } \;{\text{if}}\;h_{ 2} < 0} \\ \end{array} $$

(A1)

where h₁, h₂, and h₃ are the unit vector components along the sample coordinates,

$$ h_{1} = \sin \theta \left( {\sin \varphi \sin \psi \sin \omega + \cos \varphi \cos \omega } \right) + \cos \theta \sin \varphi \cos \psi \cos \gamma - \cos \theta \sin \gamma \left( {\sin \varphi \sin \psi \cos \omega - \cos \varphi \sin \omega } \right) $$

(A2)

$$ h_{2} = - \sin \theta \left( {\cos \varphi \sin \psi \sin \omega - \sin \varphi \cos \omega } \right) - \cos \theta \cos \varphi \cos \psi \cos \gamma - \cos \theta \sin \gamma \left( {\sin \varphi \sin \psi \cos \omega - \cos \varphi \sin \omega } \right) $$

(A3)

$$ h_{3} = \sin \theta \cos \psi \sin \omega - \cos \theta \cos \omega \cos \psi \sin \gamma - \cos \theta \cos \gamma \sin \psi $$

(A4)

Different segments of the diffraction ring (based on γ spread and 2θ) captured different measurement points of the texture scheme as shown in Figure A1(b). In this study, we followed this procedure to obtain all possible measurement points (with 10 deg step size). For all of these points, residual stress was calculated from peak shift considering d₀ as the equilibrium lattice spacing at strain-free state (obtained from annealed powder specimen). It is to be noted that this numerical scheme gave rise to multiple combinations of solutions corresponding to a single point in the texture measurement scheme (see Figure A1(c)). For such cases, the values were averaged out.