On the potential of recurrent neural networks for modeling path dependent plasticity

Abstract

The mathematical description of elastoplasticity is a highly complex problem due to the possible change from elastic to elasto-plastic behavior (and vice-versa) as a function of the loading path. Advanced physics-based plasticity models usually feature numerous internal variables (often of tensorial nature) along with a set of evolution equations and complementary conditions. In the present work, an attempt is made to come up with a machine-learning based model that can replicate the predictions anisotropic Yld2000-2d model with homogeneous anisotropic hardening (HAH). For this, a series of modeling problems of increasing complexity is formulated and sequentially addressed using neural network models. It is demonstrated that basic fully-connected neural network models can capture the characteristic non-linearities in the uniaxial stress-strain response such as the Bauschinger effect, permanent softening or latent hardening. A neural network with gated recurrent units (GRUs) and fully-connected layer is proposed for the modeling of plane stress plasticity for arbitrary loading paths. After training and testing the model through comparison with the Yld2000-2d/HAH model, the recurrent neural network model is also used to model the multi-axial stress-strain response of a two-dimensional foam. Here, the comparison with the results from unit cell simulations provided another validation of the proposed data-driven modeling approach.

Introduction

The availability of reliable computational models describing the large deformation behavior of solids is an essential element for the success of new material solutions. Most existing material models for finite element software have been developed using a physics-based approach: first, the governing mechanisms are identified and then described in an approximate manner using a set of algebraic and partial differential equations. Depending on the length scale at which the relevant mechanisms take place, multiple levels of homogenization are needed to come up with estimates of the stress-strain response at the macroscopic level.

In sheet metal plasticity, the main ingredients of macroscopic material models are the elastic constitutive equation, the yield function, the flow rule, and a set of evolution equations describing the hardening response. Aside from the basic isotropic Levy-von Mises model, anisotropic yield functions and flow potentials have emerged over the last six decades such as the Hill (1948, 1990) and Hershey families (e.g. Karafillis and Boyce, 1993; Barlat et al., 2003a, 2003b). Furthermore, major developments were concerned with hardening laws including the description of the Bauschinger effect (e.g. Bauschinger, 1886; Prager, 1956; Armstrong et al., 1966; Mollica et al., 2001), kinematic hardening (e.g. Chaboche, 1989), work-hardening stagnation (e.g. Yoshida and Uemori, 2002), permanent softening (e.g. Rauch et al., 2007), latent hardening (e.g. Barlat et al., 2013), distortional hardening (e.g. Barlat et al., 2011, 2014), strain rate effects (e.g. Marsh and Campbell, 1963; Gurrutxaga-Lerma et al., 2015; Balasubramanian and Anand, 2002; Nguyen et al., 2017), and thermal softening (e.g. Stainier et al., 2002; Ling and Belytschko, 2009). For practical applications, the large number of parameters of modern constitutive models are usually calibrated such as to provide an optimal fit for a wide range of experiments (e.g. Abi-Akl and Mohr, 2017; Gorji and Mohr, 2018). In other words, the analysis of the governing physical mechanisms guides the selection of suitable mathematical models, while the model parameters are identified through optimization (minimization of the error between model predictions and experimental results). Given that modern physics-based material models require the use of optimization software for identifying their parameters from experiments (e.g. Gorji et al. 2018; Pack et al., 2018), the question comes up if the direct use of data-driven neural network-based models may provide a viable alternative to physics-based models.

Artificial neural networks (ANNs) are a class of machine learning algorithms that consist of many artificial neurons (often referred to as units) that are arranged in a layered structure. As demonstrated by Cybenko (1989), any smooth non-linear function may be represented by a neural network function with one hidden layer and sigmoidal activation. In mechanics, deep feedforward networks composed of multiple Fully-Connected Neural Network (FCNN) layers have been used as metamodeling technique to offer alternatives to traditional approaches for structural applications (e.g. Kohar et al., 2016, 2017), material parameter identification (e.g. Haj-Ali et al., 2007), constitutive modeling (e.g. Al-Haik et al., 2006), control manufacturing part attributes such as springback (Cao et al., 1999.), as well as the prediction of the mechanical response of polycrystalline metals (Frankel et al., 2019).

The ANN approach provides a useful framework for describing unconventional hardening behavior observed in tensile experiments. For example, Jenab et al. (2016) employed a shallow neural network to describe the stress-strain response of anisotropic aluminum 5182-O for strain rates ranging from 10⁻³ to 10³/s as a function of the material orientation, the true strain, and the true strain rate. Li et al. (2019) trained an ANN as an integral part of a Johnson-Cook type of law to capture the effects of temperature and strain rate on the hardening of dual phase (DP) steels. Pandya et al. (2019) developed a neural network model to describe the temperature-dependent plasticity and fracture of aluminum 7075 in hot-stamping processes. Gorji and Mohr (2019a) showed the potential of FCNN to characterize the hardening and softening behavior of a metals for temperatures ranging from 25°C to 1250°C and strain rates ranging from 10⁻³/s to 10³/s. Jordan et al. (2019) used a fully connected network to describe the temperature-dependent viscoplastic hardening function of polypropylene material. It is worth noting that they developed a robot-assisted automated testing system to generate the wealth of experimental data needed for machine-learning based constitutive modeling.

The complexity of the needed neural network models increases when other material model elements such as the yield function and flow rule need to be represented. In addition to the strain hardening, Ali et al. (2019) trained an ANN with 80 neurons per hidden layer to describe the behavior of single crystals subjected to uniaxial tension or shear loading. They chose the current stress and three Eulerian angles as the inputs and trained their network based on the results from crystal plasticity simulations. Greve et al. (2019) showed a substantial computational speedup when using ANN to predict the path-dependent forming limit curve (FLC) instead of Marciniak–Kuczynski (M-K) type of analysis. Lefik and Schrefler (2003) discussed the possibility of incorporating FCNN based constitutive models into an FE code for modeling the non-linear stress-strain response of composites subject to elasto-plastic hysteresis and biaxial loading. Man and Furukawa (2011) developed an ANN-based constitutive model of anisotropic carbon fiber reinforced plastics. Palau et al. (2012) used the von Mises plasticity model to generate training data for monotonic, cyclic, butterfly, and random loading paths. Subsequently, they trained an FCNN model representing incremental plasticity. Gorji and Mohr (2019b) took a similar approach to train the FCNN architecture and to describe the plane stress response of DP780 steel for monotonic loading. Rovinelli et al. (2018a, 2018b) used a semi-supervised machine learning model to predict the crack propagation of polycrystalline materials under fatigue loading conditions. Liu et al (2019) proposed a data-driven multiscale material modeling method, which incorporates analytical homogenization solutions into a neural network model. Liu and Wu (2019) extended their model for application to 3D heterogeneous materials with multiscale sources of nonlinearity such as particle-reinforced hyperelastic rubbers, polycrystalline materials, and elasto-plastic composites. While all of the above work demonstrates the high potential of neural networks as a data-driven alternative to physics-based models, prediction of the stress-strain response of elasto-plastic solids for arbitrary multi-axial loading paths remains challenging.

The family of the neural network models is constantly growing. In addition to FCNN, Convolutional Neural Network (CNN) and Recurrent Neural Network (RNN) models have been developed for applications such as image analysis and speech recognition (e.g. Goodfellow et al., 2017). A recent study by Mozaffar et al. (2019) showed their merits in predicting the behavior of composites subject to nonlinear loading. Even though RNNs have not yet been applied to the modeling of sheet metal, they seem to be particularly suitable for modeling path-dependent plasticity. Unlike FCNNs, recurrent neural networks are equipped with history-dependent internal variables that may potentially mimic the role of objects such as plastic strains and back stresses in physics-based plasticity. Training of RNNs has long been considered a challenge due to vanishing and exploding gradients in the backpropagation process (Hochreiter et al., 2001). However, Long Short-Term Memory (LSTM) models (e.g. Hochreiter and Schmidhuber, 1997) and Gated Recurrent Units (GRU) (e.g. Cho et al., 2014) are examples for successful formulations of RNNs that overcome their notorious training issues.

It is the goal of the present work to construct FCNN- as well as RNN-based constitutive models to describe the large deformation response of elasto-plastic solids. The Homogeneous Anisotropic Hardening (HAH) model with the Yld2000-2d yield function (e.g. Barlat et al., 2011; Barlat et al., 2013, 2014) will serve as a physics-based reference solution, thereby challenging the RNN model to represent complex anisotropic hardening phenomena. To gain insight into the modeling capabilities of neural networks, several constitutive modeling problems of increasing degree of difficulty are treated. Section 2 provides an overview of the corresponding problem statements. In Section 3, we recall briefly the constitutive equations for the HAH model, before detailing the formulations of the fully-connected and recurrent neural network models with gated recurrent units in Section 4. In Section 5, it is then demonstrated that FCNNs and RNNs are able to replicate the predictions of conventional plasticity models, including those of the complex HAH model for an aluminum alloy and a mild steel. A combined RNN-FCNN model is then further challenged in Section 6, where we describe the macroscopic stress-strain response of a two-dimensional metallic foam after training based on the results from unit cell simulations.