Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN

We applied physics-informed neural networks to solve the constitutive relations for nonlinear, path-dependent material behavior. As a result, the trained network not only satisfies all thermodynamic constraints but also instantly provides information about the current material state (i.e., free energy, stress, and the evolution of internal variables) under any given loading scenario without requiring initial data. One advantage of this work is that it bypasses the repetitive Newton iterations needed to solve nonlinear equations in complex material models. Additionally, strategies are provided to reduce the required order of derivation for obtaining the tangent operator. The trained model can be directly used in any finite element package (or other numerical methods) as a user-defined material model. However, challenges remain in the proper definition of collocation points and in integrating several non-equality constraints that become active or non-active simultaneously. We tested this methodology on rate-independent processes such as the classical von Mises plasticity model with a nonlinear hardening law, as well as local damage models for interface cracking behavior with a nonlinear softening law. Finally, we discuss the potential and remaining challenges for future developments of this new approach

Hardness of nitric acid treated polyethylene followed by recrystallization

The hardness variation of mek crystallized polyethylene as a consequence of controlled fuming nitric exposure has been investigated using the microindentation technique. This study complements previous results obtained using other reagents (H2SO4, C1HSO3). After HNO3 exposure the microhardness of polyethylene decreases very rapidly, instead of increasing after the first hours of treatment. The hardness decrease is correlated to the volume fraction of interlamellar microvoids arising through selective acid digestion. For longer treatment times (t > 40 h) the fragility of the material increases and the sample collapses under the indenter. The hardening of the degraded material after recrystallization from the melt is followed as a function of treatment time. The results are discussed in the light of the molecular mechanisms involved. Comparison of the experimental data with hardness calculations for ideal PE lamellar structures and chain extended dicarboxylic crystals implies that the major contribution to hardening is due to electron dense gfroups attachment at the surface of a mixed lamellar structure.Peer reviewe

Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains

Physics-informed neural networks (PINNs) are a new tool for solving boundary value problems by defining loss functions of neural networks based on governing equations, boundary conditions, and initial conditions. Recent investigations have shown that when designing loss functions for many engineering problems, using first-order derivatives and combining equations from both strong and weak forms can lead to much better accuracy, especially when there are heterogeneity and variable jumps in the domain. This new approach is called the mixed formulation for PINNs, which takes ideas from the mixed finite element method. In this method, the PDE is reformulated as a system of equations where the primary unknowns are the fluxes or gradients of the solution, and the secondary unknowns are the solution itself. In this work, we propose applying the mixed formulation to solve multi-physical problems, specifically a stationary thermo-mechanically coupled system of equations. Additionally, we discuss both sequential and fully coupled unsupervised training and compare their accuracy and computational cost. To improve the accuracy of the network, we incorporate hard boundary constraints to ensure valid predictions. We then investigate how different optimizers and architectures affect accuracy and efficiency. Finally, we introduce a simple approach for parametric learning that is similar to transfer learning. This approach combines data and physics to address the limitations of PINNs regarding computational cost and improves the network's ability to predict the response of the system for unseen cases. The outcomes of this work will be useful for many other engineering applications where deep learning is employed on multiple coupled systems of equations for fast and reliable computations

A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method

Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the so-called energy form of the problem is applied to the primary variable as an additional constraint for training. The proposed approach only required up to first-order derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulation-based PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids to show the capability of deep learning for predicting the solution in a complex environment under different boundary conditions. The performance of the proposed PINN model is checked against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steady-state diffusion problem). We concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and FEM for a fast and accurate design of composite materials in future developments

A thermo-mechanical phase-field fracture model: application to hot cracking simulations in additive manufacturing

Thermal fracture is prevalent in many engineering problems and is one of the most devastating defects in metal additive manufacturing. Due to the interactive underlying physics involved, the computational simulation of such a process is challenging. In this work, we propose a thermo-mechanical phase-field fracture model, which is based on a thermodynamically consistent derivation. The influence of different coupling terms such as damage-informed thermomechanics and heat conduction and temperature-dependent fracture properties, as well as different phase-field fracture formulations, are discussed. Finally, the model is implemented in the finite element method and applied to simulate the hot cracking in additive manufacturing. Thereby not only the thermal strain but also the solidification shrinkage are considered. As for thermal history, various predicted thermal profiles, including analytical solution and numerical thermal temperature profile around the melting pool, are regarded, whereas the latter includes the influence of different process parameters. The studies reveal that the solidification shrinkage strain takes a dominant role in the formation of the circumferential crack, while the temperature gradient is mostly responsible for the central crack. Process parameter study demonstrates further that a higher laser power and slower scanning speed are favorable for keyhole mode hot cracking while a lower laser power and quicker scanning speed tend to form the conduction mode cracking. The numerical predictions of the hot cracking patterns are in good agreement with similar experimental observations, showing the capability of the model for further studies

Hybrid Modeling of Lithium-Ion Battery: Physics-Informed Neural Network for Battery State Estimation

Accurate forecasting of the lifetime and degradation mechanisms of lithium-ion batteries is crucial for their optimization, management, and safety while preventing latent failures. However, the typical state estimations are challenging due to complex and dynamic cell parameters and wide variations in usage conditions. Physics-based models need a tradeoff between accuracy and complexity due to vast parameter requirements, while machine-learning models require large training datasets and may fail when generalized to unseen scenarios. To address this issue, this paper aims to integrate the physics-based battery model and the machine learning model to leverage their respective strengths. This is achieved by applying the deep learning framework called physics-informed neural networks (PINN) to electrochemical battery modeling. The state of charge and state of health of lithium-ion cells are predicted by integrating the partial differential equation of Fick’s law of diffusion from a single particle model into the neural network training process. The results indicate that PINN can estimate the state of charge with a root mean square error in the range of 0.014% to 0.2%, while the state of health has a range of 1.1% to 2.3%, even with limited training data. Compared to conventional approaches, PINN is less complex while still incorporating the laws of physics into the training process, resulting in adequate predictions, even for unseen situations

Micro-mechanical testing of transition metal (oxy)nitride coatings

Transition metal (oxy)nitride coatings are used in polymer forming operations for a combination of outstanding wear resistance and chemical compatibility with the polymer materials. Varying the chemical composition and deposition parameters for the coatings will optimise mechanical properties by a combination of chemistry and microstructural optimisation. By developing a representative model for these materials, these materials can be rapidly and efficiently prototyped and improved. However, as both chemistry and microstructure play a role in the material properties, both of these variables must be taken account of in this model. This work demonstrates the first steps in linking quantum-mechanics, micro-mechanics, and meso-scale finite element models together in order to fully understand the behaviour of these coatings. Please click Additional Files below to see the full abstract

Multiscale modelling of the damage and fracture behaviour in nanostructured hard coatings

The requirements for manufacturing and production tools are constantly increasing. As a result, a quest for achieving and designing high-performance new materials increases as well. An effective way to design new materials with desired mechanical properties is to study their microstructures. In other words, materials macroscopic behaviors such as plasticity and damage can be traced back to smaller scales (e.g. dislocation movements, void growth, intergranular and transgranular cracking). Performing microstructural examinations on various materials with different compositions is a tedious task. Therefore, developing accurate numerical tools to efficiently compute various microstructures is of great importance for material design. The present cumulative dissertation focuses on developing numerical models that will serve for a better design of hard coating by increasing their lifetime and performance. Different tools and elements ranging from high-tech industries to daily life products, need to be coated for reasons of wear resistance and better performance. Recently, the high power pulse magnetron sputtering (HPPMS) technique has received much attention which has several advantages compared to other well-known coating processes. Hard coatings produced by such a process usually consist of rather fine columnar grain morphology and are about just a few micrometers thick. To increase the coatings’ lifetime and durability, numerical models are a useful tool to predict the coatings’ system behavior under different conditions. The results of the simulation are then used to develop a strategy for the identification of the optimum choice of parameters. According to much experimental evidence, the grain boundaries play a major role in determining the mechanical properties (such as fracture toughness), since the main source of damage, fracture, and plasticity in these coatings are intergranular fracture and grain boundary sliding. The main contribution of this thesis is also to develop interface models to capture the previously mentioned complexities at the grain boundary and study the influence of various parameters on the overall behavior of the coatings. In the first two works, the mechanical behavior of coating systems is studied utilizing a standard cohesive zone (CZ) model. First, the focus is on the implementation of a standard cohesive zone element and improving its numerical stability to be able to perform advanced simulations with complex cracking such as nanoindentation test. Using the proposed computational strategy, it is possible to study the influence of various parameters such as different grain morphologies, residual stresses and interface parameters to name a few. Later on, these results are compared to a very similar nanoindentation test regarding the obtained crack patterns as well as the indentor reaction force. Qualitative and quantitative comparisons show the predictive ability of the proposed model. The interface properties are usually expressed in terms of a traction-separation relation which is very crucial to be determined. In the third work, it was shown that by comparing the results of the numerical model with the one from micro-cantilever fracture toughness tests, one can obtain the interface properties for titanium and vanadium aluminum nitride and oxynitride coatings deposited by HPPMS. To understand and to capture every detail via the current experimental tests is not a trivial task. Therefore, to gain a deeper insight into the behavior of different grain boundaries, molecular dynamics (MD) simulations are utilized for mode I and mode II loadings. The MD investigations motivate a model that accounts for anisotropic plasticity and damage within the grain boundary. Therefore, a two-surface formulation is utilized in which damage and plasticity at the interface are coupled in a thermodynamically consistent way. The parameters for the introduced interface model are determined using the MD simulations. Finally, a microstructural volume element is selected which depicts a point in an arbitrary nanocrystalline material at the macroscale. The results of these studies reveal how important the role of grain morphology (architecture) and grain boundary sliding is in improving the damage resistance. In the last part of the thesis, a new and simple method to capture the complex mechanical behavior at an arbitrary interface is proposed. The main motivation is to explain the physics of the interface and also be thermodynamically consistent and satisfy the basic balance laws. The new nonlocal formulation is based on introducing a new quantity called "traction density". The traction-separation relation is computed by integrating the traction density over the interface. Although the mathematical representation of the traction density is relatively simple, after integrating this quantity, the complex behavior of the traction at the interface is captured automatically. Via the current formulation, basic balance laws are satisfied although an anisotropic relation for the interface traction is considered. When it comes to the grain boundary behavior, the proposed methodology can represent not only intergranular fracture but also grain boundary sliding. For calibration and verification of the model, molecular dynamics (MD) simulations for aluminum grain boundary (GB) are utilized. Interestingly, the calculations from current MD simulations show size-dependent behavior for the GB. By introducing a healing parameter in the new interface model, it is now possible to explain and predict possible GB size-dependent behavior. It has to be mentioned that the outcomes of this work can also be applied in various applications at different scales especially when it comes to interface dominant materials