Spatial strain correlations, machine learning, and deformation history in crystal plasticity

Systems far from equilibrium respond to probes in a history-dependent manner. The prediction of the system response depends on either knowing the details of that history or being able to characterize all the current system properties. In crystal plasticity, various processing routes contribute to a history dependence that may manifest itself through complex microstructural deformation features with large strain gradients. However, the complete spatial strain correlations may provide further predictive information. In this paper, we demonstrate an explicit example where spatial strain correlations can be used in a statistical manner to infer and classify prior deformation history at various strain levels. The statistical inference is provided by machine-learning techniques. As source data, we consider uniaxially compressed crystalline thin films generated by two dimensional discrete dislocation plasticity simulations, after prior compression at various levels. Crystalline thin films at the nanoscale demonstrate yield-strength size effects with very noisy mechanical responses that produce a serious challenge to learning techniques. We discuss the influence of size effects and structural uncertainty to the ability of our approach to distinguish different plasticity regimes.

Figure 1

Schematic of experimental noninvasive test in modeling and sequence of sample loading: A schematic of the sequence of events when loading, unloading, and reloading a sample is shown, with the corresponding stress and strain graphs vs the time step of simulation. A sample is obtained from 2D-DDD simulation and it's loaded to a specific strain value—“Prior” loaded stage (stage L). Then, the sample is unloaded to zero stress and the remaining plastic strain can be calculated—“Prior” unloaded stage (stage U). Finally, the sample is reloaded to a testing deformation—“Test” reloaded stage (stage T). Even though a sample has been plastically deformed (stage L), the samples obtained from experiments can be polished, thus the surface of a sample is not able to provide information about deformation (sample at stage U can be seen in the figure as having smooth surface). Such techniques are applicable in experiments such as digital image correlation, where randomly placed tracking nanoparticles are detected optically and contribute to correlation statistics are applied to the sample. Then, as the sample is reloaded, the permanent deformation can be observed, since there are changes in the distances between tracked nanoparticles.

Figure 2

Schematic of the various loading histories and testing deformation on samples: A material has an assumed prior deformation history (stage L, red circles). We unload the sample and obtain the stage U (blue circles). How does the strain field, which characterizes stage T (green circles), reflect the prior history? The testing deformation ${\epsilon }_T-{\epsilon }_U$ is constant in all T-U cases.

Figure 3

The 2D discrete dislocation plasticity model of uniaxial compression of thin films: Slip planes (lines) span the sample, equally spaced at $d=10b$, but planes close to corners are deactivated to maintain a smooth loading boundary. Surface and bulk dislocation sources (red dots) and forest obstacles (blue dots) are spread homogeneously across the active slip planes. Initially the sample is stress and dislocation free.

Figure 4

Strain profiles captured from the 2D-DDD simulation at stage L and stage T, $w=2\mu \mathrm{m}$, double slip system: The full sample is shown in both deformed (a) or undeformed (b) coordinates. A sample is loaded to $10\%$ strain, unloaded to zero stress, and reloaded to testing deformation of $0.1\%$. (a) Strain profile for stage L at $10\%$ strain. Plastic steps are allowed to emerge on the film surface [29]. (b) A strain profile at the stage T, after subtracting the residual plastic deformation at stage U. Such strain profiles are analogous to typical DIC experimental strain profiles. The strain maps are unitless.

Figure 5

$w=2\mu \mathrm{m}$, small vs large reload stress strain curves: Samples are loaded to three different deformations levels (stage L: $▾$). For each level, the sample is then unloaded (to zero applied stress) creating stage U ($\blacksquare$). The predeformed sample is then reloaded to a testing deformation (stage T: $•$). Each color represents a particular class of prior deformed states. (a) Testing deformation $=0.1\%$. The inset figure represents a zoom-in of the circled unloading-reloading region (stages U/T at 10% strain). The dashed line represents the ideal elastic response by just considering the parent material's elastic modulus. The solid lines correspond to the unloading and reloading curves. The reloading curves slightly deviate from the ideal elastic response even in small stresses, due to the presence of inelastic precipitates and also due to the ideal form of applied uniaxial loading. (b) Testing deformation $=1\%$.

Figure 6

Variety of strain profiles in 2D-DDD simulations for smaller and larger testing strain: For the spatial scale of the figures see Fig. 4. A sample is loaded to a high deformation of strain (which could be either $1\%$ or $10\%$) and then unloaded. This predeformed sample is then reloaded to a testing strain. In (a) and (b) the samples are loaded to $10\%$ strain, while in (c) and (d) the samples are loaded to $1\%$ strain. (a) Small testing deformation ($0.1\%$). $w=2$ $\mu \text{m}$. Double slip system simulation. (b) Large testing deformation ($1\%$). $w=1$ $\mu \text{m}$. Single slip system simulation (c) Small testing deformation ($0.1\%$). $w=1$ $\mu \text{m}$. Single slip system simulation. (d) Large testing deformation ($1\%$). $w=2$ $\mu \text{m}$. Double slip system simulation. For description of color map see Fig. 4.

Figure 7

$w=1\mu \mathrm{m}$, 3D projection of PCA results for thin films, double slip system: $n_0,n_0$ autocorrelation. The colors follow the definition of Fig. 8. Three different clusters are shown like in Fig. E3. Introducing the third component into the PCA map does not affect the results.

Figure 8

$w=2\mu \mathrm{m}$, 2D projection of PCA results for thin films, double slip system: $n_0,n_0$ autocorrelation. (a) Projection of data set on first two principal components. Red blobs denote samples with $0.1\%$ strain (stage L), blue triangles samples with $1\%$ strain (stage L), and green squares denote samples with $10\%$ strain (stage L), respectively. (b) First principal component of PCA, shown in sample coordinates (Fig. 3, Sec. 3b). (c) Second principal component of PCA, shown in sample coordinates (Fig. 3, Sec. 3b). The color maps are unitless, showing the intensity of the PCA-transformed correlations.

Figure 9

$w=0.5\mu \mathrm{m}$, 2D projection of PCA results for thin films, double slip system: $n_0,n_0$ autocorrelations. The colors follow the definition of Fig. 8. (a) Projection of data set on first two principal components with a clustering algorithm applied to the data set, demonstrating a failure in clustering the various deformation levels. (b) Projection of data set on first two principal components without a clustering algorithm applied to the data set, justifying (a). (c) First principal component of PCA, shown in sample coordinates (Fig. 3, Sec. 3b). (d) Second principal component of PCA, shown in sample coordinates (Fig. 3, Sec. 3b). For description of color maps, see Fig. 8.

Figure 10

$w=1\mu \mathrm{m}$, the choice of the correlation domain and how it impacts the PCA maps, double slip system: $n_0,n_0$ autocorrelation. For description of color maps and colors of PCA maps, see Fig. 8. Projection of data set on first two principal components. (a) $40\times 40$ domain of correlation matrix. Highly smooth in the center and towards the boundaries of the domain. (b) $100\times 100$ domain of the correlation matrix. A highly focused area near the center of the domain is shown, where the phenomena are focused. The smoothness present in (a) is slowly removed from this domain. (c) $200\times 200$ domain of correlation matrix. We have rich phenomenology present towards the center of the correlation matrix and at the boundaries. (d) PCA maps for $40\times 40$ domain. (e) PCA map for $100\times 100$ domain. The variance of the data has changed and the projections have shifted. The information provided by (b) does not change the cluster formations, but introduces unnecessary information that has shifted the results along the PC1 and PC2 axes. (f). PCA map for $200\times 200$ domain. The variance of the data has changed even more compared to (e). The distances between the blue and green clusters have increased an order of magnitude compared to (e) and two orders of magnitude compared to (d). The information provided by (c) does not affect the clusters that are formed from our algorithm. For description of correlation domains, see Sec. 3b.

Figure 11

$w=2\mu \mathrm{m}$, strain profiles and 2D projection of PCA results for thin films, single slip system: $n_0,n_0$ autocorrelation. The colors follow the definition of Fig. 8. Projection of data set on first two principal components. (a) Different strain profiles are seen for a single slip system of $w=2\mu \text{m}$. The top figure is a sample's strain profile with stage L at $0.1\%$ strain. The middle figure is a sample's strain profile obtained from stage L at $1\%$ strain. Finally, the bottom figure is a sample's strain profile from stage L at $10\%$ strain. Strain localization are formed from the quantity ${\epsilon }_{TU}$ (see Sec. 2b). For the spatial scale of the figures see Fig. 4. PCA projection of our results for samples of $w=2\mu \text{m}$. The similarity between strain profiles at 1% and $10\%$ strain does not affect the formation of separate clusters for samples that were initially loaded at these strains. For description of color map see Fig. 4.

Figure 12

Efficiency of ML compared to visual inspection of results, $w=2\mu \mathrm{m}$, double slip system: $n_0,n_0$ autocorrelation. The colors follow the definition of Fig. 8. (a) Strain profiles obtained through our model. The top figure corresponds to strain profiles for a sample with stage L at $0.1\%$ strain. The middle figure is a sample strain profile obtained from stage L at $1\%$ strain. Finally, the bottom figure is a sample strain profile from stage L at $10\%$ strain. All strain profiles show the strain localizations formed from the quantity ${\epsilon }_{TU}$ (see Sec. 2b). For the spatial scale of the figures see Fig. 4. (b) PCA map for samples that have similar strain profiles as in (a). Three distinct clusters are formed. The projection is upon the first and second principal components. For description of color map see Fig. 4.

Figure 13

$w=2\mu \mathrm{m}$, comparison of principal components among double and single slip systems: Components shown correspond to the analysis of Figs. 12 and 11. (a) First principal component of double slip system. (b) First principal component of single slip system. (c) Second principal component of double slip system. (d) Second principal component of single slip system. For description of color maps, see Fig. 8.

Figure 14

$w=2\mu \mathrm{m}$, 2D projection of PCA results for thin films, double slip system, validation: $n_0,n_0$ autocorrelation. Red blobs denote samples with $0.1\%$ strain (stage L), blue triangles samples with $1\%$ strain (stage L), and green squares denote samples with $10\%$ strain (stage L), respectively. Red stars depict testing samples of $0.1\%$ strain (stage L), blue stars testing samples of $1\%$ strain (stage L), and green stars testing samples of $10\%$ strain (stage L). Validated-split data set. Projection on first two principal components.

Figure 15

Measures of success for classification of samples, $0.1\%$ testing strain: $n_0,n_0$ autocorrelations. (a) Accuracy score for the samples. Maximum value 1 means that all the samples have been correctly classified. (b) $F_1$ score of our three clusters that are formed. The line with the squares represents the cluster with samples at stage $\mathrm{L}=10\%$ strain, while the line with the triangles is for the cluster with samples at stage $\mathrm{L}=1\%$ strain. Finally, the line with the circles is for the cluster with samples at stage $\mathrm{L}=0.1\%$ strain. For smaller sized systems we have observed that most of the samples are classified as belonging in the “square” cluster, hence the scored value for that cluster only. Since the algorithm correctly classifies the samples that were initially loaded to $10\%$ strain, but also classifies more samples as belonging to that cluster, then the score does not have the maximum value of 1 but lower. (c) $F_2$ score of our three clusters that have formed. The definition of the colored lines follows (b). Since for $F_2$ score we have increased weight of the recall, the 0.7 maximum value is expected for the square cluster. (d) $F_{0.5}$ score of our three clusters. The color definitions follow (b). Since we have reduced weight of the precision, for lower sample widths it is expected to have lower score than $F_1$ for the square cluster.

Figure 16

Measures of success for classification of samples, $0.1\%$ testing strain: $n_0,n_0$ autocorrelations. The $x$ axis of each graph is percentage of samples tested for classification. (a) Accuracy score for samples of $w=2\mu \text{m}$ (stars) and $w=1\mu \text{m}$ (disks). Maximum value 1 means that all the samples have been correctly classified. (b) Averaged $F_1$ score across the three clusters that have formed for samples of $w=2\mu \text{m}$ (stars) and $w=1\mu \text{m}$ (disks). It is obvious that we have good agreement for the classified samples even when we test less than $30\%$ of the total number of samples.

Figure 17

Large-reload vs small-reload testing; example of PCA projection results for thin films of $w=0.5\mu \mathrm{m}$: $n_0,n_0$ autocorrelations. The colors follow the definition of Fig. 8. Strain profiles are created from the quantity ${\epsilon }_{TU}$ (a) Sample with stage $\mathrm{L}=10\%$ strain is unloaded to zero stress and then reloaded to small testing deformation (stage $\mathrm{T}=0.1\%$). (b) Stage T at small testing deformation ($0.1\%$), without a clustering algorithm applied to data set. Projection on two principal components. Actual representation of the data set, with some mixing of the samples. The clusters have shifted closer to one another but not indistinguishable. (c) Sample with stage $\mathrm{L}=10\%$ strain is unloaded to zero stress and then reloaded to large testing deformation (stage $\mathrm{T}=1\%$). (d) Stage T at large testing deformation ($1\%$), without a clustering algorithm applied to data set. Actual representation of the data set. For the higher testing deformation of $1\%$, we can see that there is much more mixing of the samples. Reloading to higher strain values adds plastic memory to the samples, rendering our process inapplicable for these cases. For description of color map see Fig. 4.

Figure 18

First and second principal component of PCA application on thin films of $w=0.5\mu \mathrm{m}$ shown in sample coordinates: (a) First principal component, stage $\mathrm{T}=1\%$ (b) Second principal component, stage $\mathrm{T}=1\%$. For description of color maps, see Fig. 8.

Figure 19

Autocorrelations vs cross correlations for preprocessing; example of PCA projection maps for $w=2\mu \mathrm{m}$: The colors follow the definition of Fig. 8. $w=2\mu \text{m}$. (a) $n_0,n_1$ cross correlations. (b) $n_1,n_1$ autocorrelations.

Figure 20

Effect of strain invariant type for preprocessing; examples of PCA projection maps: The colors follow the definition of Fig. 8. (a) $n_0,n_0$ autocorrelations. $w=2\mu \text{m}$. Deviatoric $\varphi$ invariant. (b) $n_0,n_0$ autocorrelations. $w=2\mu \text{m}$. $J_2$ invariant. (c) $n_1,n_1$ autocorrelations. $w=1\mu \text{m}$. Deviatoric $\varphi$ invariant. (d) $n_1,n_1$ autocorrelations. $w=1\mu \text{m}$. $J_2$ invariant.

Figure 21

Total vs Residual or Plastic strain for pre-processing; examples of PCA projection maps: The colors follow the definition of Fig. 8. (a) Plastic strain. $n_0,n_0$ autocorrelations. $w=2\mu \text{m}$ (b) Total strain. $n_0,n_0$ autocorrelations. $w=2\mu \text{m}.$

Figure 22

Effect of discretization schemes on preprocessing; examples of PCA projection maps for $w=1\mu \mathrm{m}$: The colors follow the definition of Fig. 8. $n_0,n_0$ autocorrelations. (a) Two local states. (b) Three local states. (c) Four local states. (d) Five local states.

Figure 23

Comparison of different dimensionality reduction methods for $w=2\mu \mathrm{m}$: The colors follow the definition of Fig. 8. $n_1,n_1$ autocorrelations. (a) PCA. (b) TSVD. (c) IPCA.