Automated Model Discovery for Tensional Homeostasis: Constitutive Machine Learning in Growth and Remodeling

Soft biological tissues exhibit a tendency to maintain a preferred state of tensile stress, known as tensional homeostasis, which is restored even after external mechanical stimuli. This macroscopic behavior can be described using the theory of kinematic growth, where the deformation gradient is multiplicatively decomposed into an elastic part and a part related to growth and remodeling. Recently, the concept of homeostatic surfaces was introduced to define the state of homeostasis and the evolution equations for inelastic deformations. However, identifying the optimal model and material parameters to accurately capture the macroscopic behavior of inelastic materials can only be accomplished with significant expertise, is often time-consuming, and prone to error, regardless of the specific inelastic phenomenon. To address this challenge, built-in physics machine learning algorithms offer significant potential. In this work, we extend our inelastic Constitutive Artificial Neural Networks (iCANNs) by incorporating kinematic growth and homeostatic surfaces to discover the scalar model equations, namely the Helmholtz free energy and the pseudo potential. The latter describes the state of homeostasis in a smeared sense. We evaluate the ability of the proposed network to learn from experimentally obtained tissue equivalent data at the material point level, assess its predictive accuracy beyond the training regime, and discuss its current limitations when applied at the structural level. Our source code, data, examples, and an implementation of the corresponding material subroutine are made accessible to the public at https://doi.org/10.5281/zenodo.13946282.

Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks

The class of Constitutive Artificial Neural Networks (CANNs) represents a new approach of neural networks in the field of constitutive modeling. So far, CANNs have proven to be a powerful tool in predicting elastic and inelastic material behavior. However, the specification of inelastic constitutive artificial neural networks (iCANNs) to capture plasticity remains to be discussed. We present the extension and application of an iCANN to the inelastic phenomena of plasticity. This includes the prediction of a formulation for the elastic and plastic Helmholtz free energies, the inelastic flow rule, and the yield condition that defines the onset of plasticity. Thus, we learn four feed-forward networks in combination with a recurrent neural network and use the second Piola-Kirchhoff stress measure for training. The presented formulation captures both, associative and non-associative plasticity. In addition, the formulation includes kinematic hardening effects by introducing the plastic Helmholtz free energy. This opens the range of application to a wider class of materials. The capabilities of the presented framework are demonstrated by training on artificially generated data of models for perfect plasticity of von-Mises type, tension-compression asymmetry, and kinematic hardening. We observe already satisfactory results for training on one load case only while extremely precise agreement is found for an increase in load cases. In addition, the performance of the specified iCANN was validated using experimental data of X10CrMoVNb9-1 steel. Training has been performed on both, uniaxial tension and cyclic loading, separately and the predicted results are then validated on the opposing set. The results underline that the autonomously discovered material model is capable to describe and predict the underlying experimental data.

Theory and implementation of inelastic Constitutive Artificial Neural Networks

Nature has always been our inspiration in the research, design and development of materials and has driven us to gain a deep understanding of the mechanisms that characterize anisotropy and inelastic behavior. All this knowledge has been accumulated in the principles of thermodynamics. Deduced from these principles, the multiplicative decomposition combined with pseudo potentials are powerful and universal concepts. Simultaneously, the tremendous increase in computational performance enabled us to investigate and rethink our history-dependent material models to make the most of our predictions. Today, we have reached a point where materials and their models are becoming increasingly sophisticated. This raises the question: How do we find the best model that includes all inelastic effects to explain our complex data? Constitutive Artificial Neural Networks (CANN) may answer this question. Here, we extend the CANNs to inelastic materials (iCANN). Rigorous considerations of objectivity, rigid motion of the reference configuration, multiplicative decomposition and its inherent non-uniqueness, restrictions of energy and pseudo potential, and consistent evolution guide us towards the architecture of the iCANN satisfying thermodynamics per design. We combine feed-forward networks of the free energy and pseudo potential with a recurrent neural network approach to take time dependencies into account. We demonstrate that the iCANN is capable of autonomously discovering models for artificially generated data, the response of polymers for cyclic loading and the relaxation behavior of muscle data. As the design of the network is not limited to visco-elasticity, our vision is that the iCANN will reveal to us new ways to find the various inelastic phenomena hidden in the data and to understand their interaction. Our source code, data, and examples are available at doi.org/10.5281/zenodo.10066805