Figure Captions:
FIG. 1. Schematic of main scenario investigated in this work. A polycrystalline sample with an unknown mechanical response is being uniaxially loaded under tension (a) and im- aged on one of the surfaces through digital image correlation (DIC) (b). The total strain image output of DIC serves as the basis of an inquiry of yielding signatures in DIC correla- tions, as more images are collected. In this work, we utilize two novel tools based on principal component analysis (PCA) and discrete wavelet transforms (DWT). We consider various yield behaviors in this work by phenomenologically control- ling the loading rate and rate hardening exponent (γ ̇, n).
FIG. 2.
\caption{{\bf Polycrystalline texture and mechanical response}. (a) Polycrystalline texture of samples studied in this work on 512x512x2 { 2D} and 64x64x64 { (3D)} grids, with 256 grains in each microstructure, (b) Grain orientation distribution through loading-axis projection (given the representation of angles through Bunge Euler angles the projection on the x-axis is cos$\phi_1$cos$\phi_2$-sin$\phi_1$sin$\phi_2$cos$\Phi$) and (c) Von Mises stress values on the microstructure of the case $(\dot\gamma,n)=\{5\times10^{-2}/s,10\}$ { for the 2D grid}, at total strain $0.25\%$. { In (d) we show the mechanical response (Average stress along loading x-direction Vs. loading strain) for all studied samples, with loading along y-direction with various loading rates and rate sensitivity exponents $(\dot\gamma,n)=$ in the 2D: $\varhexagon:\{5\times10^{-4}/s,5\}$, $\star:\{5\times10^{-4}/s,10\}$, $\bullet:\{5\times10^{-3}/s,5\}$, $\triangleright:\{5\times10^{-3}/s,10\}$, $\triangleleft:\{5\times10^{-4}/s,10\}$ and also, 3D: $\square:\{5\times10^{-4}/s,10\}$. The 3D sample has identical loading and strain-rate sensitivity as the 2D case $\triangleleft:\{5\times10^{-4}/s,10\}$.}}
FIG. 3
\caption{{\bf Von Mises Stress, Plastic, and Total distortion at the yield point and post-yield strain}. { (a,d,g) Von Mises stress, (b,e,h) Plastic distortion defined as $|F_p-I|$, (c, f, i) }Total distortion, defined as $|F-I|$. (a,b,c) At yield point (0.12\% total strain), { for 2D samples and} loading rate and rate sensitivity exponent $(\dot\gamma,n)=$ $\{5\times10^{-4}/s,10\}$, (d,e,f) at $0.25\%$ total strain, for loading rate and rate sensitivity exponent $(\dot\gamma,n)=$ $\{5\times10^{-4}/s,10\}$ (g,h,i) at $0.25\%$ total strain, for loading rate and rate sensitivity exponent $\{5\times10^{-2}/s,10\}$ }
FIG. 4.
\caption{
{\bf PCA of Total Strain Profiles and Order Parameters for Plasticity}: Principal component analysis is applied on properly normalized total strain ($|F-I|$) maps. { (a) Total strain map for strain $0.25\%$ and $(\dot\gamma,n)=$ $:\{5\times10^{-3}/s,10\}$, similar to the cases shown in Fig.~\ref{fig:3}(c,f,i) for examples), after subtracting the mean, taking the absolute value and then, dividing with the sample strain variance.
(b) First and (c) Second PCA component for $(\dot\gamma,n)=\{5\times10^{-4}/s,10\}$
(d) Principal component cumulative variance of the components, showing a saturation to more than 99\% of the observed variability by just utilizing 2 components,
(e) The projection of the first ($P^{(k)}_1$) and second ($P^{(k)}_2$) components on the strain map samples, after normalizing with $\sqrt{\sigma_i}$ where $\sigma_i$ is the corresponding singular case, for the various material cases discussed alongside Fig.~\ref{fig:2}, with symbols/colors corresponding to
{ $(\dot\gamma,n)=$ in the 2D: $\varhexagon:\{5\times10^{-4}/s,5\}$, $\star:\{5\times10^{-4}/s,10\}$, $\bullet:\{5\times10^{-3}/s,5\}$, $\triangleright:\{5\times10^{-3}/s,10\}$, $\triangleleft:\{5\times10^{-4}/s,10\}$ and also, 3D: $\square:\{5\times10^{-4}/s,10\}$. The 3D sample has identical loading and strain-rate sensitivity as the 2D case $\triangleleft:\{5\times10^{-4}/s,10\}$.}
(f) { The first component projection, $P^{(k)}_1$,} and in the background the stress-strain curves from Fig.~\ref{fig:2} are shown. A clear correlation with stress-strain behavior, is seen in the peak of the second PCA component and the decrease of the first PCA component, signifying the onset of the elastic-plastic transition.
(g) { The second component projection, $P^{(k)}_2$,} defined in text, as function of the applied strain,
(h) The PCA-predicted yield point Vs, the actually --from the stress-strain curve-- found yield point
}}
FIG. 5.
\caption{{\bf PCA for surfaces of 3D samples}. A 64x64x64 simulation was performed at $(\dot\gamma,n)=$ $:\{5\times10^{-4}/s,10\}$ and the same process as Fig.~\ref{fig:4} leads to analogous results. { A comparison of the overlap in 2D and 3D shows good agreement, with no qualitative differences.}}
FIG. 6.
\caption{{\bf Wavelet Coefficients and the onset of yielding}. { (a) A subset of de Daubechies coefficients in the strain direction, as function of the applied strain for the case $(\dot\gamma,n)=$ $:\{5\times10^{-4}/s,10\}$, (b) Average, spatially, wavelet coefficients as function of strain for all cases with color/symbol codes analogous to the other figures: { $(\dot\gamma,n)=$ in the 2D: $\varhexagon:\{5\times10^{-4}/s,5\}$, $\star:\{5\times10^{-4}/s,10\}$, $\bullet:\{5\times10^{-3}/s,5\}$, $\triangleright:\{5\times10^{-3}/s,10\}$, $\triangleleft:\{5\times10^{-4}/s,10\}$ and also, 3D: $\square:\{5\times10^{-4}/s,10\}$.}, The average onset of de Daubechies coefficients coincides with the yield point $\epsilon_y$, detected through measuring applied loads. (c) $\epsilon^{Wavelet}_Y$ is defined through the point of the first non-zero value in (b). A natural comparison between PCA and wavelet predictions is shown, for both the yield strain $\epsilon_Y$ and yield stress $\sigma_Y$, where the projected yield stress is just the yield strain multiplied with the elastic modulus. Lines are guide to the eye.}}