Machine Learning-Based Constitutive Modelling for Granular Materials
Authors:   , Guan Shaoheng 
Published: September 2024

As a material second only to liquids in nature, granular materials are widely used in hydraulic structures, roads, bridges etc. Dam-building granular materials are complex systems of pore structures and continuously graded rock particles. An accurate description of their mechanical properties is essential for the safety analysis of ultra-high rockﬁll dams. At the microscopic scale, granular materials are discrete elementary systems aggregated by complex internal interactions, and their microscopic mechanical structure and statistical characteristics inﬂuence the macroscopic mechanical properties; at the macroscopic scale, especially in engineering-scale computational analysis, granular materials are often regarded as continuous media and their constitutive relationship are described using non-linear or elastic-plastic theories. Yet, there is no uniﬁed theory to characterise all their constitutive properties. Constitutive modelling stands as a pivotal topic within mechanical calculations. Establishing an accurate description of the relationship between deformation and constitutive response serves as the foundation for Boundary Value Problem analysis. With the growing prominence of machine learning techniques in the data-driven realm, they are expected to enhance constitutive modelling and potentially surpass classical models based on simplifying assumptions. More and more endeavours have been dedicated to integrating machine learning into mechanical calculations and assessing its eﬃcacy. This PhD thesis focuses on the use of machine learning techniques to investigate the feasibility of developing a constitutive model for granular materials and applying it in boundary value problem calculations. The main areas of research include the following aspects:1. In Chapter 2, we introduce a deep learning model designed to reproduce the macroscopic mechanical response of granular materials across various particle size distributions (PSDs) and initial states, considering diﬀerent loading conditions. We start by extracting stress-strain data from massive DEM simulations and then proceed to capture the mechanical behaviour of these granular materials through the Long Short-Term Memory networks. The work contains three central issues: LSTM cell customisation, granular materials stress-strain sampling, and loading history pasteurisation. The validation results demonstrate that this deep learning model achieves good generalisation and a high level of prediction accuracy when tested on the true triaxial loading dataset. For the diﬀerent loading and unloading paths in the conventional triaxial simulation of the DEM, an Active Learning approach is introduced to guide the sampling (Chapter 3). Based on the positive correlation between the prediction error and the uncertainty given by activate learning method, the strain paths are evaluated without DEM simulations, from which the worst predicted paths are selected for sampling. To prevent data redundancy, points in the vicinity of one selected point will not be selected for the current resampling round. The model was trained on single-cyclic loading datasets and performed quite well under multiple-cyclic loading paths. In order to circumvent the reliance on phenomenological assumptions in boundary value problem analysis, a computational framework coupled with FEM and neural network (FEM-NN) is proposed (Chapter 4). Building on the work in Chapter 2 and 3, we further introduce FEM-DEM multiscale simulations by employing the Random Gaussian Process to generate macroscopic random loading paths to be applied to the macro-scale model. A large amount of stress-strain data on the integration points is collected. Part of them are subsequently, used to train the neural network. Material loading histories represented by encoded variables. Active learning is employed here again to assess the informativeness of the data points, according to which the points are resampled from the massive database. Two examples are provided to demonstrate the eﬀectiveness of the implemented framework which provides considerable improvements in computational eﬃciency and the ability to reproduce the mechanical response of granular materials at the macroscopic scale.4. In Chapter 5 the trained network-based constitutive model is embedded into the explicit FEM solver. In implicit FEM solvers for non-linear static problems, a global equilibrium solution is typically obtained via Newton-Raphson iteration. However, the non-linear iterations may not converge when the predicted tangential matrix is not accurate enough. Therefore, the explicit FEM solver is employed to circumvent non-linear iteration. The network is trained and investigated on data generated from two constitutive models (IME model and CSUH model) separately. The trained network is able to reproduce almost exactly the ground truth results at the macroscopic level. However, the error accumulation problem resulting from a large number of steps is an-other challenge to the prediction accuracy and robustness of the data-driven model. A check-and-revision method is proposed to iteratively optimise the model by expanding the training range and improving the network generalisation.5. Chapter 6 focuses on evaluating the capacity and performance of a network-based material cell with physics extension against boundary-value problems. The proposed material cell aims to reproduce constitutive relationships learned from datasets generated by random loading paths following random Gaussian Process. The material cell demonstrates its eﬀectiveness across three progressively complex constitutive models by incorporating physics-based basis functions as prior/assumptions. An adaptive linear transformation is introduced to mitigate the error caused by magnitude gaps between strain increments in training sets and ﬁnite element simulations. The material cell successfully reproduces constitutive relationships in FEM simulations, and its performance is comprehensively evaluated by comparing two diﬀerent material cells: the sequentially trained gated recurrent unit (GRU)-based material cell and the one-to-one trained deep network-based material cell. The GRU-based material cell can be trained without prior knowledge about the internal variables. Consequently, this enables us to directly derive the constitutive model using stress-strain data obtained from experiments.6. A universal constitutive model has been introduced, combining the recurrent machine learning structure with traditional constitutive models in Chapter 7. A dramatic drop in prediction accuracy emerges when the input strain exceeds the training space because of the poor generalisation ability of the purely data-driven method. Therefore, we introduce the widely accepted elasticity theory, yielding, hardening and plastic ﬂow as physical constraints to build a machine learning-based universal constitutive model. These constraints serve as priors/assumptions for the machine learning model. During the sample preparation stage, they alleviate the stringent demands for the completeness of data sampling. In the model calculations, they guide the model to make predictions, even for unseen loading paths. The proposed model has been calibrated and tested with FEM-DEM datasets.