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

Abstract: 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.