Active Slip System Identification in Polycrystalline Metals by Digital Image Correlation (DIC)

Abstract

In this paper, a novel approach was proposed to increase the confidence of active slip system identification in polycrystalline metals. The approach takes advantage of microscale deformation tracking via Digital Image Correlation (DIC) combined with scanning electron microscopy (SEM). The experimentally-obtained high-resolution deformation fields were mapped to an undeformed configuration, which gives slip traces suitable for comparison with undeformed crystal orientation data. A metric, named herein as the ‘relative displacement ratio’ (RDR), is calculated from the displacement fields near slip traces to characterize the localized deformation due to slip. In validation cases, the experimentally-measured RDRs matched well with RDRs theoretically-calculated from active slip systems. In test cases, active slip system identification by incorporating RDR as an additional constraint was demonstrated to be preferable to using Schmid factor alone as a constraint. The proposed approach supplements existing techniques for slip system identification with increased confidence.

Appendices

Appendix A

This appendix describes the procedure of calculating the predicted slip trace direction and theoretical RDR values for a given slip system based on crystal orientation data.

• (1) Express the slip plane normal and slip direction as a unit vector in a Cartesian coordinate system. For example, for basal slip system (0 0 0 1) [2–1 -1 0], the slip plane normal is expressed as a vector n’ = (0, 0, 1), and the slip direction is expressed as a vector m’ = (1, 0, 0). This Cartesian coordinate system is referred to as the ‘lattice coordinate system’.

• (2) Generate the transformation matrix (Q) between the lattice coordinate system (x’-y’-z’) to the sample coordinate system (x-y-z). Here, the sample coordinate system is a Cartesian coordinate system, with the x, y, and z axis along the longitudinal (loading direction), transverse, and normal direction of the sample. The transformation matrix Q between two coordinate systems, × ₁ ’-× ₂ ’-× ₃ ’ and × ₁ -× ₂ -× ₃ , is defined in a way such that the element on the ith row and jth column of Q is written as Q _ij  = cos(x _i ’, x _j ), where cos(x _i ’, x _j ) is the cosine of the angle between the x _i ’-axis and the x _j -axis. Based on this definition, a vector v ’ = [v ₁’, v ₂’, v ₃’] in the x’-y’-z’ coordinate system can be expressed as v = v’Q in the x-y-z coordinate system. The crystal orientation is determined by EBSD and can be described by a set of Euler angles which represents a sequence of rotations about the coordinate axes, and through which the sample coordinate system becomes aligned with the lattice coordinate system. Therefore, for a given crystal orientation, a transformation matrix Q can be constructed.

• (3) Convert the slip plane normal and slip direction into the sample coordinate system, using the relationship n = n’Q, and m = m’Q.

• (4) The theoretical slip trace direction is calculated by taking the cross product of the slip plane normal n and the sample surface normal (0, 0, 1).

• (5) Let m = (m _x , m _y , m _z ), the theoretical RDR is calculated as the ratio: m _x /m _y .

Appendix B

This appendix provides an analysis on the RDR measurement error due to lattice rotation.

As shown in Fig. 10, assume the black solid line is the slip trace line with an inclination of α degrees (we define the ‘inclination’ of a straight line to be the angle between x-direction and this line). The blue solid line is one of the lines along which the eleven data points were taken for RDR measurement. Because the blue line is perpendicular to the slip trace line, its inclination is (α˗90°). For simplicity, consider only the two data points at the ends of the blue line, and take one of them, in this case the left point, as the reference point, so its coordinate is (0, 0). Assume the length of the data line is L, then the coordinate of the right point is (L·cos(α˗90°), L·sin(α˗90°)). Next, assume slip occurs, and the displacement of the left point is (0, 0). Then, the displacement of the right point should be written in the form (u, u/r), where r is the theoretical RDR value:

$$ RDR\_ theoretical=r. $$

Fig. 10

An illustration for the effect of lattice rotation on RDR measurement. The black solid line represents the slip trace line. The blue solid line represents one of the lines along which data points were taken for RDR measurement. The green and red lines represent the position of the data points after slip and lattice rotation, respectively

Therefore, the coordinate of the right point is (L·cos(α˗90°) + u, L·sin(α˗90°) + u/r) after slip. Next, assume there is a lattice rotation of θ degrees around the left point, the final position of the right point can be calculated as (× ₁, y ₁) = (L·cos(α˗90°) + u, L·sin(α˗90°) + u/r) G ( θ ), where G is the transformation matrix and is a function of θ :

$$ G=\left[\begin{array}{cc}\hfill \cos \left(\theta \right)\hfill & \hfill \cos \left({90}^{\circ }-\theta \right)\hfill \\ {}\hfill \cos \left({90}^{\circ }+\theta \right)\hfill & \hfill \cos \left(\theta \right)\hfill \end{array}\right] $$

Therefore, the experimentally-measured RDR value is determined from the relative displacement between the right point and the left point, which is (× ₁, y ₁) – (L·cos(α˗90°), L·sin(α˗90°)). So,

$$ RDR\_\mathrm{measured}\kern0.5em =\kern0.5em \left({x}_1\hbox{--} L\kern0.5em \cos \left(\alpha -90\circ \right)\right)\kern0.5em /\kern0.5em \left({y}_1\hbox{--} L\cdotp \mathit{\sin}\left(\alpha -90\circ \right)\right) $$

It can be seen from this analysis that, even under such simple assumption, the measured RDR value affected by lattice rotation is a function of several variables instead of the lattice rotation alone. These variables include the: (1) theoretical RDR value; (2) inclination of the slip trace line, α; (3) lattice rotation angle, θ; (4) relative u-displacement, u. In general, θ and u are correlated, and both of them increase with increasing strain. For the current study, a rough approximation with u = 2θ could be used based on observation, where θ is in degrees and u is in pixels; and (5) the length of the data line, L. However, in our current study this value does not significantly change, and can be approximated as a constant, e.g., 20 pixels. As an example, quantitative assessment was performed on the slip traces in grains 125 and 136 to illustrate how the RDR measurement was affected by the aforementioned parameters, and the result is shown in Fig. 11. The rotation of each labeled slip trace, measured by tracking the displacement of the data points on the slip trace, was used to estimate the lattice rotation. The relative u-displacement was estimated by the range of centered-u values on the centered-u versus centered-v plot, such as the plot in Fig. 6d.

Fig. 11

The theoretical RDR values, the RDR values after considering lattice rotation, and the estimated lattice rotation for the three slip traces in grains 125 and 136, at different global tensile strain levels. Typically, with increasing global tensile strain, the lattice rotation increases, and the difference between the RDR values predicted with and without considering lattice rotation also increases

The simple model here provides an evaluation of the effect of lattice rotation. However, it should be noted that the experimentally-measured RDR is affected by many sources of error. Even limited to the effect of rotation, there are additional factors. For example, the sample could have rigid body rotation that is independent of slip activity. In addition, lattice rotation actually occurs in 3D, which is not considered in the above evaluation.