Generation of training data for Machine Learning flow rules¶
Machine Learning (ML) algorithms provide a great flexibility to describe aribitrary mathematical functions. At the same time they offer the possibility to handle large data sets and multi-dimensional features as input. Hence, using ML algorithms as constitutive rules for plastic material behavior offers the possibility to explicitly take into account microstructural information of the material in the constitutive modeling.
The present example is using Support Vector Classification (SVC) as yield function. The SVC algorithm is trained by using full tensorial stresses as input data, including normal and shear components, and the information whether a given stress state leads to purely elastic or rather to elastic-plastic deformation of the material as result data. In this way, a ML yield function is obtained, which can determine whether a given stress state lies inside or outside of the elastic regime of the material. Furthermore, the yield locus, i.e., the hyperplane in stress space on which plastic deformation occurs, can be reconstructed from the SVC, and the gradient on this yield locus can be conveniently calculated. Therefore, the standard formulations of continuum plasticity, as the return mapping algorithm, can be applied in Finite Element Analysis (FEA) in the usual way. Thus, it is demonstrated that the new ML yield function can replace conventional FEA yield functions. In a next step, microstructural information will be considered directly as feature. Machine learning algorithms have been adopted from the scikit-learn platform (https://scikit-learn.org/stable/).
This package demonstrates how suited data sets for the training of ML flow rules in FEA can be generated. These data sets represent the ground truth that is approximated by the trained ML model. in form of a simple example with data synthetically produced from standard flow rules, like isotropic J2 (von Mises) and Hill-type anisotropic plasticity. It uses the pyLabFEM module. This notebook uses the matplotlib (https://matplotlib.org/) library for the visualization of results and NumPy (http://www.numpy.org) for mathematical operations with arrays.
The scientific background of this work is desribed in mode detail in the article by A. Hartmaier "Data-Oriented Constitutive Modeling of Plasticity in Metals" Materials 2020, 13(7), 1600. This open access article is available here.
Author: Alexander Hartmaier, ICAMS, Ruhr-Universtität Bochum, Germany, December 2021
This work is licensed under a Creative
Commons Attribution-NonCommercial-ShareAlike 4.0 International License
(CC-BY-NC-SA)
The pyLabFEA package comes with ABSOLUTELY NO WARRANTY. This is free software, and you are welcome to redistribute it under the conditions of the GNU General Public License (GPLv3)
Theoretical background¶
The notation of the Voigt Stress tensor is \begin{equation} \pmb \sigma = \left( \sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5, \sigma_6 \right) \end{equation}
The yield function of a material is defined as \begin{equation} f(\pmb \sigma) = \sigma_{eq}(\pmb \sigma) -\sigma_y \end{equation} where plastic deformation sets in at $f=0$, i.e. when the the equivalent stress $\sigma_{eq}$ equals the yield strength $\sigma_y$ of the material. The equivalent stress used here follows the definition of Hill for anisotropic materials.
\begin{equation} \begin{aligned} \sigma_{eq}(\pmb \sigma) = \frac{1}{\sqrt{2}} \Big[ \Big. & H_1 \left(\sigma_1-\sigma_2\right)^2 + H_2\left(\sigma_2-\sigma_3\right)^2 + H_3\left(\sigma_3-\sigma_1\right)^2 \\ & \Big. {} + 6H_4\sigma_4{}^2 + 6H_5\sigma_5{}^2 + 6H_6\sigma_6{}^2 \Big]^{1/2} \end{aligned} \end{equation}In this yield function, the anisotropy of the material's flow behavior is described in a Hill-like approach, where the anisotropic flow behavior of the material is controlled by the parameters $H_1$, ..., $H_6$. Note that these paramaters correspond to the parameters used in the conventional Hill yield criterion
\begin{equation} F\left(\sigma_2 -\sigma_3 \right)^2 + G\left(\sigma_3 -\sigma_1 \right)^2 + H\left(\sigma_1 -\sigma_2 \right)^2 + 2L\sigma_4^2 + 2M\sigma_5^2 + 2N\sigma_6^2 = 1 \end{equation}when $H=H_1\sigma_y^2, F=H_2\sigma_y^2, G=H_3\sigma_y^2, L=3H_4\sigma_y^2, M=3H_5\sigma_y^2, N=3H_6\sigma_y^2$. The special case $H_1 = H_2 =...= H_6 = 1$ describes isotropic plastic behavior, where the equivalent stress defined here is identical to the von Mises (J2) equivalent stress.
The gradient of the yield function with respect to the stress components is needed for calculating the plastic strain increments in the return mapping algorithm of continuum plasticity, and can be evaluated analytically as \begin{eqnarray} \frac{\partial f}{\partial \sigma_1} &=& \frac{\partial \sigma_{eq}}{\partial \sigma_1} = \frac{\left(H_1+H_3\right) \sigma_1 - H_1 \sigma_2 - H_3 \sigma_3}{2\sigma_{eq}} \\ \frac{\partial f}{\partial \sigma_2} &=& \frac{\partial \sigma_{eq}}{\partial \sigma_2} = \frac{\left(H_2+H_1\right) \sigma_2 - H_1 \sigma_1 - H_2 \sigma_3}{2\sigma_{eq}} \\ \frac{\partial f}{\partial \sigma_3} &=& \frac{\partial \sigma_{eq}}{\partial \sigma_3} = \frac{\left(H_3+H_2\right) \sigma_3 - H_3 \sigma_1 - H_2 \sigma_2}{2\sigma_{eq}} \\ \frac{\partial f}{\partial \sigma_4} &=& \frac{\partial \sigma_{eq}}{\partial \sigma_4} = 3 H_4 \frac{\sigma_4}{\sigma_{eq}} \\ \frac{\partial f}{\partial \sigma_5} &=& \frac{\partial \sigma_{eq}}{\partial \sigma_5} = 3 H_5 \frac{\sigma_5}{\sigma_{eq}} \\ \frac{\partial f}{\partial \sigma_6} &=& \frac{\partial \sigma_{eq}}{\partial \sigma_6} = 3 H_6 \frac{\sigma_6}{\sigma_{eq}} \\ \end{eqnarray}
Note that in the case of isotropic plasticity, i.e. $H_1=H_2=...=H_6=1$, the gradient takes the simple form \begin{equation} \frac{\partial f}{\partial \pmb \sigma} = \frac{3}{2} \frac{\pmb \sigma^\mathrm{dev}}{\sigma_{eq}} \hspace{2em} , \end{equation} where $\pmb \sigma^\mathrm{dev}=\left( \sigma_1 -p, \sigma_2-p, \sigma_3-p, \sigma_4, \sigma_5, \sigma_6 \right)$ is the deviatoric stress and $p = \mbox{Tr}(\sigma)/3$ is the hydrostatic stress.
Material definition¶
In the following, the Python class Material is invoked to be used as a material card in FEA, demonstrating the application of standard flow rules and machine learning (ML) flow rules. This class Material contains the attributes and methods to defining the elastic and plastic material behavior and to evaluate the materials constitutive behavior. Furthermore, all necessary subroutines for plotting the results are defined.
Two identical materials are defined,mat_h will be used to apply the standard Hill-like flow rules in FEA and to generate synthetical data that is used to train a ML flow rule in mat_ml. Consequently,mat_h and mat_ml should reveal identical properties, which is verified in simple FEA demonstrations for 2D plane stress cases. Here, we consider ideal plasticity, i.e. no work hardening, and no dependence on the hydrostatic stress.
