Conference Agenda

The long-standing challenge of simultaneously satisfying all physical requirements for hyperelastic constitutive models, which have been widely debated over the last few decades, could be regarded as "the main open problem of the theory of material behavior"[3].

This is particularly true for neural network (NN)-based constitutive modeling of hyperelastic materials, especially for the compressible case.

Therefore, a hyperelastic constitutive model based on physics-augmented neural networks (PANNs) is presented which fulfills all common physical requirements by construction, and in particular, is applicable for compressible material behavior.

This model combines established hyperelasticity theory with the latest machine learning advancements, using an input-convex neural network (ICNN) to express the hyperelastic potential.

The presented model satisfies common physical requirements, including compatibility with the balance of angular momentum, objectivity, material symmetry, polyconvexity, and thermodynamic consistency [1,2].

To ensure that the model produces physically sensible results, analytical growth terms and normalization terms are used. These terms, which have been developed for both isotropic and transversely isotropic materials, guarantee that the undeformed state is exactly stress-free and has zero energy [1].

The non-negativity of the hyperelastic potential is numerically verified by sampling the space of admissible deformations states.

Finally, the applicability of the model is demonstrated through various examples, such as calibrating the model on data generated with analytical potentials and by applying it to finite element (FE) simulations.

Its extrapolation capability is compared to models with reduced physical background, showing excellent and physically meaningful predictions with the proposed PANN.

[1] Linden, L., Klein, D. K., Kalina, K. A., Brummund, J., Weeger, O. and Kästner, M., Neural networks meet hyperelasticity: A guide to enforcing physics, (submitted 2023).

[2] Kalina, K. A., Linden, L., Brummund, J. and Kästner, M., FEANN - An efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining, Comput. Mech. (2023).

[3] Truesdell, C. and Noll, W., The Non-Linear Field Theories of Mechanics. 3rd ed. Springer Berlin Heidelberg, 2004.

The prevailing paradigm to model the behavior of rubber-like materials is hyperelasticity. However, phenomenological constitutive modeling is prone to uncertainty and results in loss of information as data coming from experiments are not used directly in calculations. Aside, selecting an appropriate strain energy function for the problem under consideration is left to the engineer and is often based on experience.

Data-driven approaches are a promising alternative to constitutive modeling. We present a new data-adaptive approach to model hyperelastic rubber-like materials at finite strains. Our proposed modeling procedure combines the advantages of phenomenological hyperelasticity with the data-driven paradigm of directly including experimental data in calculations. Import constraints, such as thermodynamic consistency, material objectivity and frame indifference and material symmetry are satisfied a priori. In essence, we suggest formulating a finite-element-like approximation of the strain energy function as a sum of basis functions multiplied by parameters. The basis functions are expanded over the space of invariants which is, in the most generic form, formed by the principal invariants of the right Cauchy-Green tensor. Support points are distributed in the space of invariants, which are the points at which the parameters are defined. In other words, the parameters are the values of the discrete strain energy function at the support points. We consider linear Lagrangian polynomials as basis functions which boils down to (bi)linear interpolation of the parameters. The parameters are determined based on measured full-field displacements, e.g. obtained from Digital-Image-Correlation, and reaction forces by solving a non-linear optimization problem. Within this optimization problem, the 2-norm of the residual vector, which is the difference between measured and computed displacements and reaction forces, is minimized by altering the parameters. The proposed discrete approximation to the strain energy function is flexible enough to discover any admissible form of strain energy function and the fact that our approach does not rely on measured stresses is an advantage over many data-driven approaches presented to date.

We verify our approach and show that computation times are similar compared to those of phenomenological models. By numerical examples, we illustrate that only a moderate number of parameters is required to approximate well-known smooth strain energy functions sufficiently well and demonstrate the ability of our approach to re-identify an extended number of parameters. We also show the robustness of our approach against noisy experimental data.

Finite linear viscoelastic (FLV) or quasi-linear viscoelastic (QLV) models are commonly used to model the constitutive behavior of polymeric materials. However, these models are limited in their ability to accurately represent the nonlinear viscoelastic behavior of materials, particularly in capturing their strain-dependent viscous behavior. To address this issue, we have developed viscoelastic Constitutive Artificial Neural Networks (vCANNs), a novel physics-informed machine learning framework. vCANNs rely on the concept of generalized Maxwell models with nonlinear strain (rate)-dependent properties represented by neural networks. With their flexibility, vCANNs can automatically identify accurate and sparse constitutive models for a wide range of materials. To test the effectiveness of vCANNs, we trained them using stress-strain data from various synthetic and biological materials under different loading conditions, e.g., relaxation tests, cyclic tension-compression tests, and blast loads. The results show that vCANNs can learn to accurately and efficiently represent the behavior of these materials without human guidance.

Ensuring the safe and reliable operation of critical load-bearing components requires maintaining their mechanical integrity. Constitutive material models play a crucial role in analyzing mechanical integrity, and their accuracy is essential for assessing the structural integrity of load-bearing components. Notably, mechanical integrity assessments of high temperature components require constitutive models representing the highly nonlinear deformation response of alloys under various loading scenarios and across a wide temperature range. The Chaboche viscoplastic model is among the most well-known constitutive models for representing the isotropic-kinematic hardening behavior of materials. This model employs a set of differential equations to define the viscoplastic strain rate tensor as a function of the stress tensor and several scalar and tensorial internal variables. Calibrating this model for different temperature and loading conditions however requires using experimental data from various mechanical tests and determining a large number of model parameters, which is typically achieved by performing a computationally expensive inverse analysis. To address this computational challenge, we propose a new method that leverages scientific machine learning to accelerate solving the inverse problem. Specifically, we use the Physics Informed Neural Networks (PINNs) framework to incorporate the Chaboche model formulation into neural networks. In this contribution, we illustrate the framework in application to Hastelloy X, by calibrating and determining >30 model parameters based on observations from various cyclic tests at different strain rates in the temperature range of 22-1000°C.

Many materials exhibit history-dependency in their response. This is evident in the inelastic response of solid materials or in the hysteretic retention curve of multiphase porous materials. Within multiscale simulation of history-dependent materials, the underlying work focuses on testing and comparing different supervised machine learning (ML) approaches to generate suitable constitutive models. This includes the application of recurrent neural networks (RNN), the application of 1D convolutional neural network (1D CNN), and the application of the eXtreme Gradient Boosting (XGBoost) library.

The database used in the supervised learning relies on lower scale two-phase lattice Boltzmann simulations, applied to deformable and anisotropic representative volume elements (RVEs) of the porous materials as presented in [1,2]. In the training, the inputs will include the capillary pressure and its history in addition to the porosity, whereas the output will include the degree of saturation. The comparison among the different ML approaches will include the accuracy in predicting the correct saturation degree and the efficiency concerning the training.

REFERENCES

[1] Heider, Y; Suh H.S.; Sun W. (2021): An offline multi-scale unsaturated poromechanics model enabled by self-designed/self-improved neural networks. Int J Numer Anal Methods;1–26.

[2] Chaaban, M.; Heider, Y.; Markert, B. (2022): A multiscale LBM–TPM–PFM approach for modeling of multiphase fluid flow in fractured porous media. Int J Numer Anal Methods Geomech, 46, 2698-2724.