Stress representations for tensor basis neural networks: alternative formulations to Finger-Rivlin-Ericksen

Data-driven constitutive modeling frameworks based on neural networks and classical representation theorems have recently gained considerable attention due to their ability to easily incorporate constitutive constraints and their excellent generalization performance. In these models, the stress prediction follows from a linear combination of invariant-dependent coefficient functions and known tensor basis generators. However, thus far the formulations have been limited to stress representations based on the classical Rivlin and Ericksen form, while the performance of alternative representations has yet to be investigated. In this work, we survey a variety of tensor basis neural network models for modeling hyperelastic materials in a finite deformation context, including a number of so far unexplored formulations which use theoretically equivalent invariants and generators to Finger-Rivlin-Ericksen. Furthermore, we compare potential-based and coefficient-based approaches, as well as different calibration techniques. Nine variants are tested against both noisy and noiseless datasets for three different materials. Theoretical and practical insights into the performance of each formulation are given.Comment: 32 pages, 20 figures, 4 appendice

Learning hyperelastic anisotropy from data via a tensor basis neural network

Anisotropy in the mechanical response of materials with microstructure is common and yet is difficult to assess and model. To construct accurate response models given only stress-strain data, we employ classical representation theory, novel neural network layers, and L1 regularization. The proposed tensor-basis neural network can discover both the type and orientation of the anisotropy and provide an accurate model of the stress response. The method is demonstrated with data from hyperelastic materials with off-axis transverse isotropy and orthotropy, as well as materials with less well-defined symmetries induced by fibers or spherical inclusions. Both plain feed-forward neural networks and input-convex neural network formulations are developed and tested. Using the latter, a polyconvex potential can be established, which, by satisfying the growth condition can guarantee the existence of boundary value problem solutions.Comment: 36 pages, 20 figure

Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques

Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of $\mathrm{CO_2}$ sequestration). Here, we present a non-intrusive reduced order model of natural convection in porous media employing deep convolutional autoencoders for the compression and reconstruction and either radial basis function (RBF) interpolation or artificial neural networks (ANNs) for mapping parameters of partial differential equations (PDEs) on the corresponding nonlinear manifolds. To benchmark our approach, we also describe linear compression and reconstruction processes relying on proper orthogonal decomposition (POD) and ANNs. We present comprehensive comparisons among different models through three benchmark problems. The reduced order models, linear and nonlinear approaches, are much faster than the finite element model, obtaining a maximum speed-up of $7 \times 10^{6}$ because our framework is not bound by the Courant-Friedrichs-Lewy condition; hence, it could deliver quantities of interest at any given time contrary to the finite element model. Our model's accuracy still lies within a mean squared error of 0.07 (two-order of magnitude lower than the maximum value of the finite element results) in the worst-case scenario. We illustrate that, in specific settings, the nonlinear approach outperforms its linear counterpart and vice versa. We hypothesize that a visual comparison between principal component analysis (PCA) or t-Distributed Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform better prior to employing any specific compression strategy

Establishing the relationship between generalized crystallographic texture and macroscopic yield surfaces using partial input convex neural networks

In this study, we present a methodology to predict the macroscopic yield surface of metals and metallic alloys with general crystallographic textures. In previous work, we have established the use of partially input convex neural networks (pICNN) as macroscopic yield functions of crystal plasticity simulations. However, this work was performed with an over-abundance of data, and on limited crystallographic textures. Here, we extend this study to approach more realistic material states (i.e., complex crystallographic textures), and consider data-availability as a major driver for our approach. We present our modified framework capable of handling generalized material states and demonstrate its effectiveness on samples with multi-modal textures deformed under plane stress conditions. We further describe an adaptive algorithm for the generation of training data as informed by the shape of yield surfaces to reduce the time for both the generation of training data as well as pICNN training. Finally, we will discuss errors in both training and test datasets, limitations, and future extensibility.Comment: 29 pages, 16 figure

A review on data-driven constitutive laws for solids

This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiments, verification, and validation.Comment: 57 pages, 7 Figure

Local approximate Gaussian process regression for data-driven constitutive models: development and comparison with neural networks

Hierarchical computational methods for multiscale mechanics such as the FE2 and FE-FFT methods are generally accompanied by high computational costs. Data-driven approaches are able to speed the process up significantly by enabling to incorporate the effective micromechanical response in macroscale simulations without the need of performing additional computations at each Gauss point explicitly. Traditionally artificial neural networks (ANNs) have been the surrogate modeling technique of choice in the solid mechanics community. However they suffer from severe drawbacks due to their parametric nature and suboptimal training and inference properties for the investigated datasets in a three dimensional setting. These problems can be avoided using local approximate Gaussian process regression (laGPR). This method can allow the prediction of stress outputs at particular strain space locations by training local regression models based on Gaussian processes, using only a subset of the data for each local model, offering better and more reliable accuracy than ANNs. A modified Newton-Raphson approach specific to laGPR is proposed to accommodate for the local nature of the laGPR approximation when solving the global structural problem in a FE setting. Hence, the presented work offers a complete and general framework enabling multiscale calculations combining a data-driven constitutive prediction using laGPR, and macroscopic calculations using an FE scheme that we test for finite-strain three-dimensional hyperelastic problems. (C) 2021 Elsevier B.V. All rights reserved

Fiber plasticity and loss of ellipticity in soft composites under non-monotonic loading

In this work, we relate fiber plasticity in soft composites to the loss of ellipticity of the governing equations of equilibrium of a composite under non-monotonic uniaxial loading. The loss of ellipticity strongly indicates the emergence of localization phenomena in the composite, reminiscent of the emergence of kinking instabilities in tendon, which occur as a response to tendon “overload” without requiring any macroscopic compressive loading. We examine soft composites where both fibers and matrix can be highly extensible and plastic deformations are present in the fiber phase. We first propose a transversely isotropic constitutive model for the fibers allowing for plastic deformations, taking into account a single slip direction, consistent with the microstructure of hierarchically assembled collagen fibers. Following, we propose a simple hyperelastic model for the matrix and combine the two following the Voigt assumption. We then formulate a general loss of ellipticity criterion for an elastoplastic material subjected to finite deformations. We use this criterion to indicate critical conditions for loss of ellipticity in the soft composite and individually in the fiber phase, under various loading–unloading paths. Results show that plastic deformation of the fiber phase during tensile loading can lead to ellipticity breakdown during elastic unloading while, macroscopically, the material is still in tension, indicating the possible onset of an instability. © 2022 Elsevier Lt