Informing Mechanical Model Development Using Lower-Dimensional Descriptions of Lattice Distortion

Abstract

This paper describes a method combining in situ X-ray diffraction data and dimensionality reduction (local linear embedding) to inform the development of state variable plasticity models. The method is applied to developing a state variable plasticity model for pure nickel deformed in uniaxial tension in the small strain regime. Prior to model development, connections between state variables representing evolution of mobile dislocations and the lower-dimensional representations of the data are established. Correlations between lower-dimensional representation of data and state variable evolution motivate the introduction of new evolution equation terms to increase alignment between experiment and model. These terms capture dislocation interactions leading to hardening transients prior to steady-state plastic flow. The discussion focuses on interpreting these new evolution terms and outstanding issues associated with linking lower-dimensional representations of data to state variable evolution modeled with ordinary differential equations.

Introduction

Expressing phenomenology through the evolution of internal state variables is a widely used approach to capture and predict the mechanical response of engineering alloys during plastic deformation. These state variables may rest on a physical basis, e.g., dislocation density and lattice orientation, serving to relate the current microstructure to resultant mechanical response. The standard approach to calibrate a state variable model is to optimize state variable initial values and material parameters to best fit macroscopic loading (stress–strain) data. Ideally, each state variable is isolated using a carefully chosen series of mechanical tests, but these are not always performed or are even possible. In this process, the choice of state variable may be phenomenologically informed by microscopy observations, but generally, the model is never actually tested against data describing the underlying microstructural evolution that the model is trying to capture. This shortcoming leads to uncertainties regarding the capability of the model to accurately capture the physics of the mechanical response and, subsequently, future predictions from the model.

New data collection techniques at synchrotron light sources (and in electron microscopes) enable microstructural evolution to be monitored in real time, gathering large numbers of ‘snap-shots’ of the microstructure evolution in progress. These new data provide opportunities to develop and calibrate state variable evolution equations that reflect measured microstructural evolution, in addition to more traditional comparisons to macroscopic response. Although up to this point, use of these data for model development has been primarily limited to calibration and validation of existing constitutive models [1,2,3,4,5,6,7,8,9,10,11], and the proposition of new experimentally informed evolution equations has been limited [12]. Essentially, the data are analyzed solely through the lens of existing physical understanding of microscale material. As an example, wanting to compare elastic strain response between model and experiment drives analysis toward shifts of diffraction peaks, leaving other material information unextracted. Exploiting the full richness of these data is an ongoing challenge.

A promising new approach is to utilize machine learning dimensionality reduction techniques, distilling microstructure detail down to lower-dimensional representations. We make a point to stress that almost any type of scientific analysis is a form of dimensionality reduction (a consolidation of information). The difference for machine learning techniques is that a priori knowledge about the system is not required (albeit helpful). Instead patterns or structures within the data are found with minimal bias from the researcher. However, this in turn produces a new complication: The lower-dimensional representations cannot immediately be linked to physical processes or features with which material understanding can be built. For example, a lower-dimensional representation of time series data can indicate that something is changing at the microstructure, but provide no indication of what that something is. In the case of thermomechanical experimentation and modeling, recent work has suggested correlations between lower-dimensional representations of data discovered by various machine learning approaches and features in the microstructure [13,14,15,16,17,18,19].

Similar in respect to correlating lower-dimensional representation of data to microstructural features, a recent study [20] found correlations between dimensionality-reduced X-ray diffraction data and the evolution of state variables in a mechanical model. X-ray diffraction data are particularly sensitive to changes in the crystal lattice within an illuminated volume, particularly lattice plane spacing, lattice orientation, and distortion of the lattice. Lower-dimensional representations (embeddings) of evolving diffracted intensity measured from a plastically deforming nickel-based super alloy seemed to exhibit the same behavior as state variables (and related derivatives) representing various evolving dislocation densities that were extracted from a mechanical model fit solely to the macroscopic loading response. Specifically, lower-dimensional representations of images found using locally linear embedding (LLE) [21, 22] that captured spreading of diffraction peaks due to increasing lattice distortion during plastic deformation (averaged over the probed volume) were correlated to the evolution of the modeled mobile dislocation density and its second derivative in time, respectively. Below is a summary of the more general findings from the previous study that serve as the foundation for this work:

  1. 1.

    A correlation between state variables describing microstructure evolution and lower-dimensional representations of X-ray diffraction data was found.

  2. 2.

    Nearly identical lower-dimensional representations of X-ray diffraction data from polycrystals with different orientation distributions suggested the dimensionality reduction was capturing general material behavior that could be used to develop macroscopic models.

  3. 3.

    LLE naturally groups data points by time in this application, suggesting further association of structural features of the data with rate-dependent plastic processes.

Following upon these observations, an opportunity becomes available to guide constitutive model development. Particularly, the lower-dimensional representations of the diffraction data can be used to guide model development with the idea that reduced order state variable evolution should mimic the lattice distortion transients observed experimentally.

In this work, we use lower-dimensionality representations of diffraction (microstructural) data, embeddings found using locally linear embedding, gathered in situ using X-ray diffraction to inform the development of a state variable plasticity model for face-centered cubic (FCC) pure nickel (Ni). The state variable model includes evolving mobile and immobile dislocation populations which influence plastic strain rate and flow stress during loading. The evolution of the lower-dimensional embedding suggests the need for new terms that depend quadratically on the immobile dislocation density. With the addition of these new evolution terms and improved alignment of the model with the lower-dimensional representation of the microstructural evolution, better agreement with the macroscopic response is also achieved. Care will be taken to explore the structure of the ordinary differential equations (ODEs) describing the dislocation evolution and how the solutions to these ODE’s link to the low-dimensional embeddings found using LLE.

The organization of the paper is as follows: The paper begins with a description of the Ni material tested and the in situ X-ray measurements taken during the uniaxial tension of the Ni sample. An existing state variable mechanical model that evolves dislocation populations is then introduced and fit to a measured macroscopic response. The following section then presents the data processing methods used to reduce the dimensionality of the in situ X-ray diffraction data that will be used to adjust the presented mechanical model, along with the data itself. A process is then described utilizing the lower-dimensional embeddings to adjust and inform the previously introduced state variable model. The paper concludes with a discussion of an interpretation of the new material evolution terms developed with the method and outstanding issues associated with linking dimensionality-reduced data to ODEs describing physical processes.

Material and Measurement Description

Characterization and in situ study were performed at the F2 experimental station of the Cornell High Energy Synchrotron Source (CHESS). A schematic of the experimental configuration is given in Fig. 1. The sample tested was electro-discharge machined from 99.98% purity Ni rod. After machining, the sample was annealed in a vacuum furnace at 700\(^\circ \) C for 2 h and furnace cooled three times in order to reach a target grain size > 100 \(\upmu \)m and lower initial dislocation content. The geometry of the sample can be found in more detail in [23], but the cross section of the sample was 1 mm \(\times \) 1 mm. Prior to loading, grain orientation and morphology were characterized using the near-field high-energy diffraction microscopy (nf-HEDM) box-beam variant developed for measurements at CHESS (see [24, 25]). In the diffraction volume probed, 36 grains were illuminated and the mean effective grain diameter was 377 \(\upmu \)m. Figure 2 shows a reconstruction of the microstructure from nf-HEDM. The voxel spacing in the reconstruction is 5 \(\upmu \)m.

Fig. 1
figure1

Schematic of the experimental geometry. The sample was loaded and rotated about the vertical direction \({{\varvec{e}}}_y\)

Fig. 2
figure2

Grain microstructure probed during the in situ loading experiment

The sample was then continuously loaded in tension along \({{\varvec{e}}}_y\) in displacement control at a rate of 10 nm/s. The sample was loaded using the second iteration of the RAMS load frame [23] to a strain of 0.025. The measured macroscopic response is given in Fig. 3. Strain measurements were taken using digital image correlation, and load was measured using a load cell placed above the sample. Each red ‘x’ denotes a macroscopic stress–strain measurement and the beginning of a diffraction measurement.

Fig. 3
figure3

The macroscopic stress–strain response of the pure Ni sample. Points at which diffraction measurements were made are indicated with red x’s. Fit material responses from the original (blue) and updated (dashed green) models are shown with lines

During continuous loading, the sample was illuminated by a 61.332 keV X-ray beam. The incoming X-ray beam was 2 mm wide \(\times \) 1 mm tall in order to illuminate the full cross section. The sample was continuously rotated (\(0.4^\circ \)/s) about \({{\varvec{e}}}_y\) as the detector was capturing images in order to collect data from a representative number (thousands) of diffraction peaks from the illuminated grain ensembles. The diffraction peaks were measured on a GE 41RT+ area detector sitting 807 mm behind the sample. Diffraction data were collected at 0.1\(^\circ \) increments of sample rotation and then compiled into a single image in post-processing, such that the final diffraction image contained diffraction peaks from 180\(^\circ \) of rotation about the rotation axis.

Fitting of the State Variable Model

The mechanical model to be aligned to lower-dimensional embeddings of the in situ diffraction data primarily follows Estrin and Kubin [26] and Kocks [27]. The model contains two state variables: a density of currently mobile dislocations \(\rho _m\) and a density of immobile dislocations that contribute to the strength of the material \(\rho _i\). In the model, the total strain rate is additively decomposed into elastic \({\dot{\varepsilon }}_E\) and plastic \({\dot{\varepsilon }}_P\) portions

$$\begin{aligned} {\dot{\varepsilon }}={\dot{\varepsilon }}_E+{\dot{\varepsilon }}_P \end{aligned}$$
(1)

and the elastic strain rate is linearly related to the rate of change of stress \({\dot{\sigma }}\)

$$\begin{aligned} {\dot{\varepsilon }}_E={\dot{\sigma }}/E \end{aligned}$$
(2)

where E is Young’s modulus. The rate dependence of plastic deformation is captured with a power law relationship between plastic strain rate and applied stress

$$\begin{aligned} {\dot{\varepsilon }}_P=\rho _mbv_0\left( \frac{\sigma }{\sigma _Y}\right) ^{\frac{1}{m}} \end{aligned}$$
(3)

where b is the Burgers vector, \(v_0\) is the dislocation ensemble velocity, \(\sigma _Y\) is the yield strength, and m is the strain rate sensitivity. The yield strength in Eq. 3 is related to the immobile density as

$$\begin{aligned} \sigma _Y=M\alpha Gb\sqrt{\rho _i} \end{aligned}$$
(4)

where M is the Taylor factor, \(\alpha \) is a proportionality constant, and G is the elastic shear modulus. We note that a wide range of dislocation interactions can lead to a square root dependence of the dislocation population on strength [28].

The evolution of the mobile dislocation density is given as

$$\begin{aligned} {\dot{\rho }}_m=\left( \frac{C_1}{b^2}(\rho _i/\rho _m)-\frac{C_2}{b}\sqrt{\rho _i}\right) {\dot{\varepsilon }}_P. \end{aligned}$$
(5)

The first term on the right is a production term for mobile dislocations emitted from pinning points in the dislocation forest, while the second term corresponds to immobilization of defects in obstacle structures. The evolution of the immobile density is given by

$$\begin{aligned} {\dot{\rho }}_i=\left( \frac{C_2}{b}\sqrt{\rho _i} - C_3\rho _i\right) {\dot{\varepsilon }}_P \end{aligned}$$
(6)

where the first term on the right hand side is a corresponding increase in the immobile dislocation density from mobile dislocations becoming trapped in obstacle structures, while the second term is a standard recovery term due to annihilation of dislocations due to climb or cross-slip. These evolution equations are associated with what will be referred to as the original model.

This mechanical model was fit to the measured macroscopic response using a combination of values found in the literature and a least squares optimization process for the remaining parameters. The fit macroscopic response can be seen in Fig. 3. Parameters associated with state variable evolution equations are given in Tables 1 and 2 for values that were fixed and fitted, respectively.

Table 1 Common material parameters used in the micromechanical model simulations
Table 2 Fit parameters for the original (O) and updated (U) state variable micromechanical models

At this point, the state variable model is able to generally capture the features of the macroscopic response as seen in Fig. 3. However, the underlying evolution of the state variables \(\rho _m\) and \(\rho _i\) has not actually been compared directly to any material measurements that more directly reflect the evolution of dislocation behavior. The following sections will describe how lower-dimensional representations of evolving X-ray diffraction data, which reflect changes to the average lattice distortion within the illuminated diffraction volume, can be used to guide the determination of terms and material parameters of the underlying evolution equations.

Lower-Dimensional Representation of X-ray Diffraction Data

Before the X-ray diffraction data can be used for model development guidance, it must be changed into a form that can facilitate comparison to modeled quantities. In this section, we describe how raw diffraction images are dimensionality-reduced using the locally linear embedding technique. The features of the two lower-dimensional data embeddings which contain the most information about the structure of evolving X-ray diffraction images are then presented and described. These embeddings are of interest because they were previously shown to correlate to the evolution behavior of the mobile dislocation density in the state variable model, described in the previous section as applied to a nickel-based super alloy [20].

Dimensionality Reduction of X-ray Diffraction Data

Processing of the diffraction images to produce lower-dimensional embeddings of microstructural evolution closely follows the procedure described in [20]. For clarity, we define a ‘diffraction image’ as the image directly output from the detector in which the diffracted intensity still lies on distinct Debye rings, while a ‘diffraction peak image’ is an image in which intensity within Debye–Scherrer rings has been extracted from the diffraction image, remapped to a polar coordinate system, and then recompiled together. Figure 4a contains an example of a diffraction image and an extracted diffraction peak image from the unloaded state of the Ni sample. Figure 4b shows the diffraction peak image at the end of the experiment for comparison. Note the transition from distinct peaks in Fig. 4a to the ‘smeared’ intensity distributions in Fig. 4b. This change in diffracted signal directly reflects changes in the average lattice distortion across the polycrystal. For this study, 160\(^\circ \) arcs from the first four sets of lattice planes were extracted from each diffraction image to form the diffraction peak images. With the choice of 160\(^\circ \) arcs and the 180\(^\circ \) rotational integration range, diffraction events are collected only once from nearly all sets of the lattice planes in the first four families from all grains. Lattice planes with normals near the rotation axis are excluded as these lattice planes remain in the diffraction condition for extended rotation angles when rotating a specimen around a single axis.

Fig. 4
figure4

a Raw diffraction images collected over 180\(^\circ \) of rotation and corresponding diffraction peak image extracted from the first four Debye–Scherrer rings in the unloaded state. b Diffraction peak images from the final deformed material state

The diffraction peak images are dimensionality reduced using the LLE algorithm [21, 22] contained within the Scikit-learn package [29]. LLE was chosen because it satisfied two primary dimensionality reduction considerations [20] . The first is that diffraction peaks can rapidly change shape, particularly at the elastic–plastic transition. In order to analyze these transients, it is more appropriate to perform a local comparison of neighboring points in time. The second is that we wish to roughly maintain the relationships (relative distances and angles) between data points in the lower-dimensional representation. The distance between data points reflects how quickly the lattice distortion is changing (as evidenced in diffraction peak evolution) and the angle between data points indicates if there are changes in the character of lattice distortion, possibly due to the activation of various deformation mechanisms. However, a downside to LLE is that it does not provide a clear metric to determine how completely a data set is described. Some guidance is provided by the relative magnitudes of eigenvalues calculated during the determination of lower-dimensional representation of the data (described below), but these values are not readily interpretable. Ultimately, the researcher decides how many dimensions are necessary to understand the data set.

For analysis, each diffraction peak image is treated as an observation \({{\varvec{X}}}^{(i)}\) with dimension N. The dimension N is the total number of pixels in each diffraction peak image (16\(\times 10^7\)). In total, there are n observations, where in this case \(n=21\), corresponding to the 21 diffraction images collected after the elastic–plastic transition. First, a series of linear weights w which can be used to reconstruct an observation from neighboring observations are found. This is done by minimizing the error e between an observation and a reconstruction from neighboring observations

$$\begin{aligned} e(w_{ij})=\sum _i\left| {{\varvec{X}}}^{(i)}-w_{ij}{{\varvec{X}}}^{(j)}\right| ^2 \end{aligned}$$
(7)

and constraining the sum of the nonzero weights for each reconstruction to be equal to 1. The number of neighbors is chosen to best capture the underlying structure of the data set being dimensionality reduced, and the choice of number of neighbors used for LLE algorithm will be discussed in the next section. Generally however, including a large number of neighbors will produce a final lower-dimensional representation that has ‘averaged out’ transients in time. Including too few neighbors will make the dimensionality reduction susceptible to the influence of noise and over-fitting. To finish the process, sets of embedded coordinates, or embeddings, \({{\varvec{Y}}}_i\) of dimension n that approximately maintain relative distances between observations are found by minimizing a cost function \(\phi \)

$$\begin{aligned} \phi ({{\varvec{Y}}}_i)=\sum _i\left| {{\varvec{Y}}}_i-w_{ij}{{\varvec{Y}}}_j\right| ^2. \end{aligned}$$
(8)

The similarity between embeddings and the original data is maintained through the use of the previously determined linear weights. Equation 8 is reformulated as an eigenvalue problem and then solved to find a subset of embeddings (eigenvectors) that provide coordinates in the lower-dimensional space. The embeddings with the lowest nonzero eigenvalues are those that best maintain relationships between data points in comparison with the original high-dimensional data. We emphasize that the lower-dimensional representations of the data found using LLE cannot be used to reconstruct each data point (diffraction peak image). This can be contrasted to principal component analysis [30] where a limited number of principal components and weights can be used to approximately reconstruct a data point. Instead, LLE is providing a tractable realization that maintains relationships between data points.

Embeddings

Using the procedure described in Sect. 4, the first two lower-dimensional embeddings have been extracted from the set of diffraction peak images evolving with increasing strain. Figure 5 shows the components of the first two embeddings \({{\varvec{Y}}}_1\) (Fig. 5a) and \({{\varvec{Y}}}_2\) (Fig. 5b) calculated using different numbers of neighbors (and associated linear weights) plotted against corresponding macroscopic strain points \(\varepsilon ^{(i)}\). Each embedding is colored according to the number of neighbors used (ranging from 5 to 12), and the sign of the embedding has been chosen to facilitate later comparisons. For both embeddings, the general shape does not drastically change with number of neighbors. The first embedding \({{\varvec{Y}}}_1\) shows an inflection at strain of \(\approx 0.006\) and then a saturation of evolution by the end of loading (\(\varepsilon =0.025\)). The second embedding \({{\varvec{Y}}}_2\) shows two inflections: one immediately at the elastic–plastic transition and the other at applied strain of 0.012.

Fig. 5
figure5

The individual components (i) of the a first \(Y_1\) and b second \(Y_2\) lower-dimensional embeddings plotted against macroscopic strain values \(\varepsilon ^{(i)}\) with increasing numbers of neighbors included for reconstruction. The embeddings are colored according to the number of neighbors used in the reconstruction

Also of interest is the structure of the data without correlation to the applied strain. Figure 6 shows \({{\varvec{Y}}}_1\) plotted against \({{\varvec{Y}}}_2\) with increasing numbers of neighbors. We can see that the general shape of the path that the microstructure takes through the lower-dimensional space is relatively insensitive to varying the number of neighbors. A general ‘C’ shape can be seen in all trajectories shown. However, the path does seem to exhibit a small change of orientation as the number of neighbors is varied, with the use of eight or nine neighbors producing the largest orientation changes. This variation of embedding orientation with number of neighbors is an effect that is often observed with the use of LLE [22].

Embedding-Informed State Variable Model Development

The primary goal of this section is to elucidate a process by which dimensionality-reduced representations of raw diffraction data can be used to guide physical model development. The process may be viewed in contrast to more traditional approaches in which the mechanical constitutive model is developed and calibrated solely with macroscopic loading data (see Sect. 3). This section begins by building a correlative link between the lower-dimensional embeddings and evolving mobile dislocation density in a state variable model. The state variable evolution equations are then adjusted to improve the alignment of the mobile dislocation density evolution with the primary LLE embedding. Further comparisons between model state variable evolution and data embeddings will be explored to build confidence in the process.

Linking State Variables and Embeddings

As mentioned in the introduction, the process of linking of lower-dimensional representations of data and evolving physical quantities is still primarily performed on an ad hoc basis. For this analysis, we are guided by a previous study that posited that the population of actively mobile dislocations in the material is the primary driver of increasing lattice distortion during plastic deformation and subsequently the measured diffraction images. As such, it is expected that the embedding containing the most information about the evolution of the diffraction peak images (\({{\varvec{Y}}}_1\)) should correlate to the evolution of mobile dislocation density in a mechanical model.

After fitting the mechanical model solely to the macroscopic response, the mobile dislocation density versus time was extracted and then compared to the first embedding \({{\varvec{Y}}}_1\). Comparison of alignment between the two is shown in Fig. 7a with the embeddings scaled to facilitate comparison. In the figure, we see that there does appear to be similar trends in the general behavior of both \({{\varvec{Y}}}_1\) and \(\rho _m\). The similar falling and rising behavior between the quantities seems to indicate a correlation between the two as posited. In \(\rho _m\) and \({{\varvec{Y}}}_1\), we can see an inflection at strain of \(\approx 0.006\), which is the only minimum for both quantities. However, the alignment is not perfect with the primary difference being a saturation-type behavior at higher strain levels in the embedding as opposed to fairly linear increase in mobile density after the inflection.

Fig. 6
figure6

The individual components (i) of the first \(Y_1\) and second \(Y_2\) embeddings extracted from the evolving diffraction peak images. The embeddings are colored according the number of neighbors used in the reconstruction

Model Guidance Through Embeddings

With the belief that the first embedding is correlated to the evolution of the mobile dislocation density (as suggested by the similar trends observed), \({{\varvec{Y}}}_1\) was used to attempt to guide the determination of mechanical model parameters to increase alignment between the two quantities. Again, the lower-dimensional embedding is reflecting increases in lattice distortion as displayed in the measured diffraction peak evolution. The fit material parameters (\(C_1\), \(C_2\), \(C_3\)) in the state variable evolution equations (Eqs. 5 and 6) of the mechanical model were adjusted in order to try to replicate the saturation type behavior observed in the embedding in the evolution of \(\rho _m\). However after trials, no combination of material parameters could be replicate the saturation. Next, a series of different terms including both linear and quadratic dependencies on \(\rho _m\) and \(\rho _i\) were then added to Eqs. 5 and 6 to try to recreate the saturation behavior. Of the terms tested, only a term with quadratic dependence on \(\rho _i\) was able to generate improved alignment of the behaviors of \(\rho _m\) and \({{\varvec{Y}}}_1\). The updated evolution equation of \(\rho _m\) is given by:

$$\begin{aligned} {\dot{\rho }}^*_m=\left( \frac{C_1}{b^2}(\rho _i/\rho _m)-\frac{C_2}{b}\sqrt{\rho _i} - C_4 b^2 \rho _i^2 \right) {\dot{\varepsilon }}_P \end{aligned}$$
(9)

The interpretation of this term will be discussed in detail later, but can be interpreted as a secondary interaction of mobile dislocations with immobile dislocations. During these interactions, a fraction \(\Phi _A\) will be annihilated, while a fraction \(\Phi _T\) will become pinned and be transferred to the immobile population. The sum of \(\Phi _A\) and \(\Phi _T\) is equal to 1. During an annihilation event, a dislocation will also be removed from the immobile population. In contrast, during a transfer event, a dislocation will be added to the dislocation population. With this interpretation, it is expected that the difference between the transferred and annihilated dislocations will subsequently contribute to the immobile population. We define a value \(\beta \) which is the difference between these two fractions:

$$\begin{aligned} \beta =\Phi _T-\Phi _A \end{aligned}$$
(10)

The updated form for the evolution of the immobile density is now given by

$$\begin{aligned} {\dot{\rho }}^*_i=\left( \frac{C_2}{b}\sqrt{\rho _i} - C_3\rho _i+\beta C_4 b^2 \rho _i^2 \right) {\dot{\varepsilon }}_P. \end{aligned}$$
(11)

As defined and expressed in Eq. 11 , \(\beta \) can range from \(-1\) to 1 which corresponds to only annihilation events and only transfer events, respectively. All values in between represent a combination of the two.

The optimized material parameters for the updated model are given in Table 2. Figure 7b shows a comparison of the evolution of \(\rho _m\) from the updated model to \({{\varvec{Y}}}_1\). We see that with the addition of the embedding-informed quadratic \(\rho _i\) term, \(\rho _m\) now satisfies the goal of saturating at the end of the experiment. In addition, the transient behavior at strain of \(\approx 0.006\) is less abrupt and better matches the shape of \({{\varvec{Y}}}_1\). With respect to the macroscopic response (Fig. 3), we were also able to simultaneously capture a better fit to the macroscopic response with the addition of the extra terms in the evolution equations.

In previous work, further support of connections between the lower-dimensional embeddings and a modeled dislocation population was found with comparisons of the second embedding \({{\varvec{Y}}}_2\) to the second derivative of the mobile dislocation density. Section 6.2 of the Discussion further explores these connections between embeddings and second-order differential descriptions of mobile dislocation density evolution. To explore the link between the modeled mobile dislocation density and the evolution of diffraction peak images, the second derivative of \(\rho _m\) with respect to time, \(\ddot{\rho }_m\), is compared to \({{\varvec{Y}}}_2\) for both the original and updated mechanical models and can be found in Fig. 8. Again, we observe in Fig. 8a a general alignment between a modeled quantity, \(\ddot{\rho }_m\), to a lower-dimensional embedding, \({{\varvec{Y}}}_2\) with the original model. Also again, the alignment is not perfect between the original mechanical model quantity \(\ddot{\rho }_m\) and the embedding. The second derivative of dislocation density has a single sharp oscillation upward and a rapid return to its initial value. In contrast, \({{\varvec{Y}}}_2\) has a distinct maximum and minimum. This is not particularly surprising as the original model had significantly sharper transients than \({{\varvec{Y}}}_1\). In comparison, \(\ddot{\rho }_m\) from the updated model appears to be more successful in capturing the behavior observed in \({{\varvec{Y}}}_2\) (Fig. 8b). The second derivative \(\ddot{\rho }_m\) from the updated model displays both a maximum and a minimum and generally a much smoother behavior that is also observed in \({{\varvec{Y}}}_2\). We note here that the improved correlation between \(\ddot{\rho }_m\) to \({{\varvec{Y}}}_2\) with the updated model was solely a by-product of attempts to improve alignment between \(\rho _m\) and \({{\varvec{Y}}}_1\).

Fig. 7
figure7

a Comparison of the evolution of dislocation density (black line) from original mechanical model fit to macroscopic data with the components (i) of first embedding \({{\varvec{Y}}}_1\). b Comparison of the evolution of dislocation density (black line) from the updated mechanical model fit to macroscopic data with the components (i) of the first embedding \({{\varvec{Y}}}_1\). The embeddings are colored according to the number of neighbors used in the reconstruction

Fig. 8
figure8

a Comparison of the evolution of the second derivative of dislocation density (black line) from original mechanical model fit to macroscopic data with the components (i) of second embedding \({{\varvec{Y}}}_2\). b Comparison of the evolution of the second derivative of dislocation density (black line) from the updated mechanical model fit to macroscopic data with the components (i) of second embedding \({{\varvec{Y}}}_2\). The embeddings are colored according to the number of neighbors used in the reconstruction

Discussion

In this work, we have proposed a novel means to utilize diffraction data that was analyzed using data dimensionality reduction (locally linear embedding) to guide the development of a state variable mechanical model. Historically, similar state variable models have been primarily constructed using series of macroscopic thermomechanical tests that may be explicitly associated with microstructural features observed from ex situ microscopy. While these models are designed to capture macroscopic behavior (and do so successfully), means of linking variables describing evolution to microstructure has been limited. The physical underpinning of the state variables is not essential if the primary goal of the model is to accurately predict mechanical response within well-bounded loading conditions. However, to predict mechanical responses outside these bounds with confidence, state variables representative of modeled mechanisms must be established. As shown, a new means to build links between state variables and measurements now exists through correlation of state variables with representations of in situ diffraction data that are directly reflecting changes in lattice distortion due to plastic deformation. By building these correlations, increased confidence can be gained that the physics of the active deformations are properly reflected in the evolution equations as opposed to a simple overfitting of measured macroscopic response.

The approach presented in this work takes this idea one step further, with the low-dimensional representation of diffraction data actually pointing to different forms of evolution equations. State variable plasticity models can be developed now to capture both macroscopic response and transients in microstructural configuration expressed in evolving diffraction peaks. Importantly, combining in situ mechanical and in situ diffraction data in fitting serves to strengthen model foundation by linking response across length scales—from reconfiguration of dislocation structure to the average state of stress. While this approach is early in development, it does provide a path forward that is worth exploring. There are outstanding issues which need further research, particularly regarding expansion of the theoretical foundations with which solutions to ordinary differential equations may be connected to lower-dimensional representations of time series data, such as that generated by LLE. Initial discussions of some of these issues are contained here.

At any given time a fraction of the total dislocation population is mobile and actively contributes to the plastic strain rate (realized as spread of diffraction peaks), while the remaining dislocation is immobile. A similar need for partitioning may be found in resistivity studies, as necessary to relate hardening to a dislocation forest [31]. The sets of dislocations contributing to these populations are transient: whether any dislocation is mobile or immobile can change depending on current loading conditions and material state. This fact, in combination with the inability to directly characterize the relative fractions of these populations at any given time, has made the use of a mobile dislocation population to describe plastic deformation contentious [32]. However, in this work we find that changes in average lattice distortion seem to correlate with the behavior of a modeled mobile dislocation population. While certainly not conclusive proof, this observation does serve to bolster the argument that separable populations of dislocations play different roles in the plastic deformation process at a given point in time. There may be other divisions of dislocation populations that may play a role in observable changes in diffracted intensity in other deformation regimes. For example, a division of dislocation populations between those that do and do not contribute to the development of size effects in the material. Although for the very small range of strain rates and minimal applied strain explored in this work, the relatively low applied strains lead us to believe that the dislocations contributing to size effects do not play a major role in diffraction peak evolution [31].

Interpretation of Additional Dislocation Density Evolution Terms

In Sect. 5.2, new terms were added to the evolution of the mobile and immobile dislocation densities, respectively. Both terms had a quadratic dependency on \(\rho _i\) whose inclusion was driven by an attempt to align the evolution behavior of the mobile dislocation density with the primary embedding \({{\varvec{Y}}}_1\), but without significant regard to a physical interpretation. We begin to rationalize these terms through an assumption that the loss of mobile dislocations, \({\dot{\rho }}_m\), to an immobile dislocation field is proportional to their density and average velocity while being inversely proportional to the mean free path \({\bar{p}}\) to the obstacles [33]

$$\begin{aligned} {\dot{\rho }}_m=-\rho _m \frac{{\bar{v}}}{{\bar{p}}}. \end{aligned}$$
(12)

where \({\bar{v}}= v_0\left( \frac{\sigma }{\sigma _Y}\right) ^{\frac{1}{m}}\). In this case, as this term was found to be dependent on \(\rho _i\), we assume the obstacles are related to the immobile density. The mean free path to an immobile dislocation is taken as

$$\begin{aligned} {\bar{p}}=\frac{1}{2l \rho _i} \end{aligned}$$
(13)

where l is a spacing between a mobile dislocation and a fixed immobile dislocation. We compute 2l as the total immobile dislocation line length in a representative volume. This total line length can then be calculated as the immobile density multiplied by a representative volume \(C_4 b^3\):

$$\begin{aligned} 2l=C_4 b^3 \rho _i \end{aligned}$$
(14)

Inserting Eqs. 13 and 14 into Eq. 12 gives:

$$\begin{aligned} {\dot{\rho }}_m=-C_4 b^2 \rho _i^2 (\rho _m b {\bar{v}}) \end{aligned}$$
(15)

which is the new interaction term introduced in Eq. 9. Following this, we interpret the quadratic dependency on \(\rho _i\) as interactions with the total line length of immobile dislocations in a representative volume. Some of those mobile dislocations are annihilated, while others are transferred to the immobile population.

The value of \(C_4\) from the fitting in Sect. 5.2 indicates a relatively large representative volume (in comparison with activation volumes after several percent strain [31]) with a characteristic dimension of \(\approx 60\) nm (approximately 230 \(\times \) 230 \(\times \) 230 unit cells). This observation is consistent with the idea that the mobile/immobile interactions being captured are associated with dislocations contained in relatively ‘open’ volumes of crystal between obstacle structures. (The material tested was heavily annealed prior to deformation.) These interactions are apart from glide interactions with features of the microstructure characterized by the \(\sqrt{\rho _i}\) term in Eqs. 5 and 9. One possibility is cross-slip fostered by attractive interactions [34] with frequency that scales with the immobile density [35, 36] and leading to the initial organization of cells [37]. In other words, the additional terms capture an early hardening transient before steady-state (Stage III) hardening and refinement of cell structures becoming dominant. In fact, the saturation of mobile density fostered by the added terms may be hypothesized to correspond with the beginning of a more standard hardening regime captured by hardening mechanisms within the \(\sqrt{\rho _i}\) terms. We imagine that in cold-worked materials, the size of these open crystal volumes is significantly smaller, shrinking the representative volume size (\(C_4\)) and thereby reducing the influence on material response of the \(\rho _i^2\) term. At this point, we again stress that the addition of these terms was primarily driven by observations of the behavior of \({{\varvec{Y}}}_1\). However, the terms that found to best align \(\rho _m\) also have reasonable physical interpretation.

Correspondence Between State Variables and Embedded Coordinates

In Sect. 5.1, we observed that the first two embedded coordinates \({{\varvec{Y}}}_1\) and \({{\varvec{Y}}}_2\) exhibited qualitatively similar behavior to the state variable \(\rho _m\) and its second derivative. This observation suggested the inclusion of the additional term in Eq. (9). As shown in Fig. 3, this revision improved the agreement between the model and macroscopic experimental measurements. Furthermore, as described in the previous section, this revision to the state variable model has a physical interpretation. However, it remains unclear why the embedded coordinates should correspond to these particular state variables. Below, we address three questions that might provide initial insight into these connections between state variables and embedded coordinates.

First, one might ask why we expect \({{\varvec{Y}}}_1\) to correspond to the mobile dislocation density. As outlined in [38], the results of LLE can be interpreted as finding the ‘articulation points’ or motion drivers for images of an evolving structure. Based on the LLE algorithm, the first embedded coordinate \({{\varvec{Y}}}_1\) should contain the most information about the observed data. In this work, one of the key assumptions made is that early after yield, the primary articulation point for the structure evolution observed in the diffraction peak data is the changes in the populations of the mobile dislocations in the alloy, and therefore, \({{\varvec{Y}}}_1\) should show correspondence to \(\rho _m\).

The second question we address is why the second embedded coordinate \({{\varvec{Y}}}_2\) bears resemblance to the second derivative of \(\rho _m\), and not, for example, its first derivative. The LLE algorithm used to derive the embedded coordinates is one example of a broad class of algorithms known as spectral methods. Previous work has shown that spectral methods perform operations on input data that are akin to a Laplacian operator [38]. As a consequence of this numerical approach, a connection between lower-dimensional representations of data and a second derivative appears reasonable.

The final question that arises is if knowledge of \(\rho _m\) provides sufficient information to determine the remaining state variables. From Eqs. 5 and 6, we see that the evolution of mobile and immobile dislocation densities take the form:

$$\begin{aligned} {\dot{\rho }}_m& = f_1 \left( \rho _m, \rho _i \right) {{\dot{\varepsilon }}}_p \end{aligned}$$
(16)
$$\begin{aligned} {\dot{\rho }}_i&= f_2 \left( \rho _i \right) {{\dot{\varepsilon }}}_p, \end{aligned}$$
(17)

If \({\dot{\rho }}_m \ne 0\), we can then write

$$\begin{aligned} \frac{d\rho _i}{d\rho _m} = \frac{f_2 \left( \rho _i \right) }{f_1 \left( \rho _m , \rho _i \right) }, \end{aligned}$$
(18)

and this differential equation can be solved to find \(\rho _i\) as a function \(\rho _m\) over a small increment in time. The remaining state variables can be found using Eqs. 16. Therefore, knowing the evolution of \(\rho _m\) or \(\rho _i\) is sufficient to determine the evolution of the remaining state variables.

Lower-Dimensional Embeddings Pointing to New State Variables

Despite the discussion in the previous section, the linking of state variable quantities to lower-dimensional embeddings is still an outstanding issue. By associating a single quantity such as \(\rho _m\) or \(\ddot{\rho }_m\) to an embedding found from LLE, there is an implicit assumption that the basis vectors of the state variables and the basis vectors in which the embeddings are expressed are parallel. The relatively good alignment between \(\rho _m\) and \({{\varvec{Y}}}_1\) and \(\ddot{\rho }_m\) and \({{\varvec{Y}}}_2\) indicates that this assumption is reasonable, but a source of discrepancy was the exact locations of transients as a function of strain. Returning to Fig. 5a, b, we note a shift in minima and maxima of embeddings with choice of number of nearest neighbors.

To reduce the sensitivity of maxima and minima positions to nearest neighbors, we consider a comparison of model quantities with linear combinations of the components n of the first two lower-dimensional embeddings, i.e.,

$$\begin{aligned} \rho _m=A_0 +A_1 Y^{(i)}_1 +A_2 Y^{(i)}_2 \end{aligned}$$
(19)

and

$$\begin{aligned} \ddot{\rho }_m=B_0 +B_1 Y^{(i)}_1 +B_2 Y^{(i)}_2, \end{aligned}$$
(20)

determined through a minimization process. Figure 9 shows both \(\rho _m\) and \(\ddot{\rho }_m\) from the original and updated micromechanical models in comparison with optimized fits of the first two embedding vectors for a range of nearest neighbors. We can see that for both models, relaxing the constraint that an embedding may be completely parallel to a single model variable improves the ability to align the embeddings to the data. The fitting of the embeddings to the updated model is still significantly closer than the fitting to the original model. Another point of note is that the relaxed constraints remove nearly all dependency on the number of neighbors and there is no longer a shift in maxima and minima positions as the number of neighbors changes. In other words, the embedded coordinates found for each number of neighbors can be realigned to produce better agreement with those found using different numbers of neighbors and with the model variables.

Fig. 9
figure9

a Fit linear combinations of \({{\varvec{Y}}}_1\) and \({{\varvec{Y}}}_2\) with varying neighbors to \(\rho _m\) of the original model. b Fit linear combinations of \({{\varvec{Y}}}_1\) and \({{\varvec{Y}}}_2\) with varying neighbors to \(\ddot{\rho }_m\) of the original model. c Fit linear combinations of \({{\varvec{Y}}}_1\) and \({{\varvec{Y}}}_2\) with varying neighbors to \(\rho _m\) of the updated model. d Fit linear combinations of \({{\varvec{Y}}}_1\) and \({{\varvec{Y}}}_2\) with varying neighbors to \(\ddot{\rho }_m\) of the updated model

As discussed, the model proposed in the present work assumes a separation between mobile and immobile dislocation densities which is bolstered by the close (but not perfect) alignment of \({{\varvec{Y}}}_1\) and \({{\varvec{Y}}}_2\) with the modeled mobile density and its second derivative. However, the fact that improved alignment can be achieved through combinations of embeddings may be indicating that other state variables (that are related to the mobile density) can provide a more appropriate description of the underlying physics. To explore this idea, we introduce new state variables \(r_1\) and \(r_2\) that are related to mobile dislocation density and its second derivative. Inverting Eqs. 19 and 20 and then associating the embedding \({{\varvec{Y}}}_1\) and \({{\varvec{Y}}}_2\) with state variables \(r_1\) and \(r_2\), respectively, lead to

$$\begin{aligned} \left\{ \begin{array}{c} r_1 \\ r_2 \end{array} \right\} =\left[ \begin{array}{cc} A_1 &{} A_2 \\ B_1 &{} B_2 \end{array} \right] ^{-1} \left\{ \begin{array}{c} \rho _m - A_0 \\ \ddot{\rho }_m -B_0 \end{array} \right\} . \end{aligned}$$
(21)

The evolution of \(r_1\) and \(r_2\) might be described by a system of first-order ordinary differential equations, similar in spirit to Eqs. 911. We note that the approach described above might have connections with previous work on using dimensionality reduction techniques to find so-called ‘effective state variables’ [39]. The point is that taken together, the transformation 21 and inter-relation of mobile and immobile density in Eq. 18 point the way to a model for evolution where dislocations are partitioned into something other than ‘mobile’ and ‘immobile.’ This is a topic for future investigation.

Summary

A novel means to develop new evolution equations for state variable plasticity models was presented that utilizes lower-dimensional representations of in situ X-ray diffraction data. The method is an advance of more traditional methods which only utilize macroscopic mechanical response to indirectly infer microstructural quantities. Open questions still remain as to how to link output from machine learning techniques which, at least initially, do not have physical interpretations to physical quantities that can be used to inform modeling, but the approach presented provides a promising path forward for increasing the accuracy of mechanical modeling efforts which contain state variables explicitly representing microstructural quantities.

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Acknowledgements

This work is based upon research conducted at the Center for High-Energy X-ray Sciences (CHEXS) which is supported by the National Science Foundation under award DMR-1829070. GHS and AJB received support through the Office of Naval Research (Contract N00014-16-1-3126). We would like to thank Dr. Edward Trigg for helpful discussions regarding orientation of the lower-dimensional embeddings. We would also like to thank Professor Matthew Miller for helpful discussions regarding this manuscript.

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Pagan, D.C., Schmidt, G.H., Borum, A.D. et al. Informing Mechanical Model Development Using Lower-Dimensional Descriptions of Lattice Distortion. Integr Mater Manuf Innov 9, 459–471 (2020). https://doi.org/10.1007/s40192-020-00196-y

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Keywords

  • Unsupervised learning
  • Plasticity
  • X-ray diffraction
  • Constitutive modeling
  • Nickel