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Multiple-slip crystal plasticity dislocation-density

A dislocation-density crystalline plasticity approach was used in conjunction with FEM to generate the database used in this study. The crystalline plasticity approach was developed by Zikry and Kao20 and Shanthraj and Zikry21 and uses a set of partial differential equations to describe the dislocation evolution within a unit area called dislocation density. Separate equations are used for mobile and immobile dislocation densities, ({rho }_{m}) and ({rho }_{im}), and a set of nondimensional coefficients are used to describe the sourcing, trapping, annihilation, immobilization, and recovery of dislocations21. The dislocation density evolution equations are

$$frac{d{rho }_{m}^{alpha }}{dt}=left|{dot{gamma }}^{alpha }right|left(frac{{g}_{sour}^{alpha }}{{b}^{2}}left(frac{{rho }_{im}^{alpha }}{{rho }_{m}^{alpha }}right)-{g}_{mnter-}^{alpha }-{rho }_{m}^{alpha }-frac{{g}_{immob-}^{alpha }}{b}sqrt{{rho }_{im}^{alpha }}right),$$

(1)

$$frac{d{rho }_{im}^{alpha }}{dt}=left|{dot{gamma }}^{alpha }right|left({g}_{mnter+}^{alpha }{rho }_{m}^{alpha }+frac{{g}_{immob+}^{alpha }}{b}sqrt{{rho }_{im}^{alpha }}-{g}_{recov}{rho }_{im}^{alpha }right),$$

(2)

where gsour is the coefficient pertaining to an increase in the mobile dislocation density due to dislocation sources, gmnter is the coefficient related to the trapping of mobile dislocations due to forest intersections, cross slip around obstacles, or dislocation interactions, grecov is a coefficient related to the rearrangement and annihilation of immobile dislocations, and gimmob is related to the immobilization of mobile dislocations. These coefficients, which have been nondimensionalized, are summarized in Table 1, where ({f}_{0}), and (varphi) are geometric parameters. H0 is the reference activation enthalpy, lc is the mean free path of a gliding dislocation, b is the magnitude of the Burgers vector, and s is the saturation density. It should be noted that these coefficients are functions of the immobile and mobile densities, and hence are updated as a function of the deformation mode. Shear slip rate, (dot{gamma }), is a measure of the accumulated plastic strain on a material that is related to the mobile dislocation activity in a material as

$${dot{gamma }}^{(alpha )}={rho }_{m}^{(alpha )}{b}^{left(alpha right)}{v}^{left(alpha right)},$$

(3)

where ({v}^{(alpha )}) is the average velocity of mobile dislocations on slip system (alpha).

The orientation relationships (ORs) between the (delta) hydrides examined in this work and the surrounding matrix were developed in Mohamed and Zikry22 as

$${left(0001right)}_{hcp}// {left(111right)}_{fcc} and {left[11overline{2 }1right]}_{hcp}//{left(overline{1 }10right)}_{fcc},$$

(4)

and represent the plane relationships between the h.c.p. material and the f.c.c. material.

Each simulation was developed according to its own material fingerprint and consisted of a plane strain model with displacement control for the FE model. The average mesh size was 60,000 elements and consisted of 49 zirconium h.c.p. alloy grains and approximately 50 f.c.c. hydrides, for conditions including hydride spacing and hydride length based on the fingerprint. Strain was applied uniaxially, and grain orientation with respect to the loading axis was defined using the value of the material fingerprint parameter. The strain rate was constant at 10 ({s}^{-1}). Grain orientations were defined as the angle that the zirconium alloys [0 0 1 0] axis forms with respect to the loading axis, which is uniaxial at the [0 0 1 0] global direction. Changes to this parameter rotated the grain with respect to the [0 1 0 0] normal axis. Mohamed and Zikry23 validated the material properties used in the simulations, which are presented in Table 2.

To characterize the solution space of all possible material fingerprints, a total of 210 simulations were simulated using FEM. The material fingerprints for each simulation consisted of five material parameters and were chosen according to uniform distributions bounded by the values in Table 3. The parameter values were chosen from a grid of 4 equally spaced values from within these bounds. In addition to modifying grain orientation according to the range within these bounds, grain to grain misorientation was randomized at a maximum of 10. These parameters influence dislocation activity and fracture, and the specific values used correspond to experimental values4,11,23,24. To sample from the solution space, the trajectory method used in the Elementary Effects method (implemented in SAlib) was used25,26.

To avoid numerical issues with the model input parameters being at different physical scales, the parameters were processed by centering the mean of each feature around zero and scaling to unit variance using the StandardScaler function in Scikit-Learn (Version 0.23.2)27. The critical fracture stress values were also scaled by 100MPa.

When training the models, the data was randomly split into an 85% training set and a 15% validation set. The training set was used to train model hyperparameters. The hyperparameters were randomly chosen for training within predefined ranges chosen for each model type. Typically, 50,000 iterations within the hyperparameter space were used along with a fivefold cross validation technique to reduce overfitting to the data. After the best possible model hyperparameters were found, the model was tested on the validation set which comprised 15% of the original data, and which had not been used to train the model. The goodness of fit, for this model, is the measurement of the models performance in predicting this validation set.

A linear regression using OLS was used to provide a benchmark for the other modeling methods explored in this work. Because the simulations are non-linear, there was only a small likelihood of attaining a high level of accuracy with this class of models. However, linear regression provides the most interpretable model output of any of the other modeling systems. Scikit-Learns LinearRegression function was used to produce these models27.

The random forest regression model was used to generate a model. It is comprised of an ensemble of decision trees whose outputs are averaged. The result is a general model that provides accurate predictions in high dimensional spaces28. Decision tree-based methods are also helpful because their output can be interrogated, though it may be cumbersome to do so for an ensemble of decision trees. 100 estimators were used for each regression model, corresponding to 100 decision trees, which would make this kind of interpretation difficult. Other methods exist to interpret the output of a random forest regressor, and they are implemented here. The importance of each input parameter can also be determined using methods such as recursive feature elimination. Scikit-Learns RandomForestRegressor27 was used.

A multilayer perceptron, or neural network, was also fitted to the data. While neural networks are the least interpretable method presented in this study, they have also been shown to be powerful estimators for highly dimensional data. The models presented here were comprised of 3 hidden layers with 5 neurons each. Scikit-Learns MLPRegressor function was used to train and test these estimators27.

Gaussian Process Regression (GPR) was chosen as a model type because of its built-in measure of uncertainty. A combination kernel comprised of a Matern kernel and an Exponential Sine Squared kernel were used for training. The sinusoidal attribute of this kernel was a result of the prior understanding that material properties tend to follow a sinusoidal path as a non-isotropic material is rotated. The Matern kernel additionally allowed the model to effectively capture discontinuities in the solution space. The GaussianProcessRegressor function within the SciKit-Learn package was used to train the models, and a cross validated randomized search was used to find length scale, the Matern (nu) parameter, and the periodicity parameter for the exponential sine squared kernel27.

The purpose of this study is to obtain ROMs that describe the fracture stress state of a material given its material fingerprint and strain level. This is performed by predicting the (mu) value of a Gumbel distribution trained to the 95th percentile of each data set. These (mu) values are normalized by the fracture stress to provide physically based insights. These models can provide critical microstructural fracture predictions without FEM models or experimental measurements, and is a representation of incipient fracture within the material. The fracture critical stress information predicted from these models can then be used in conjunction with other computational and experimental methods to determine the likelihood of failure. These predictions are the link between the material fingerprint and the fracture probability for that material at a certain strain level.

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Predictive machine learning approaches for the microstructural ... - Nature.com