[Paper Review] Theory and implementation of inelastic Constitutive Artificial Neural Networks

Hagen Holthusen, Lukas Lamm+3 more|arXiv (Cornell University)|Nov 2, 2023

Elasticity and Material ModelingEngineering107 references9 citations

TL;DR

This paper extends Constitutive Artificial Neural Networks (CANNs) to inelastic materials (iCANNs) by enforcing thermodynamics-driven architecture with multiplicative decomposition and pseudo potentials, combining energy and potential networks within a recurrent framework to learn inelastic behavior.

ABSTRACT

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

Motivation & Objective

• Incorporate thermodynamic principles (objectivity, rigid motion of the reference configuration, and non-uniqueness of decomposition) into a neural network framework for inelasticity.
• Develop a general iCANN architecture that uses a multiplicative decomposition of deformation gradient to capture finite deformations and rates.
• Learn Helmholtz free energy and dissipation potential with dedicated neural networks to enable data-driven discovery of inelastic models.
• Ensure the learned models satisfy thermodynamic consistency a priori via energy and pseudo potential formulations.
• Provide a modular framework that can specialize to visco-elasticity and be extended to other inelastic phenomena.

Proposed method

• Use a multiplicative decomposition F = Fe Fi to separate elastic and inelastic parts under objectivity and reference-frame considerations.
• Represent the Helmholtz free energy ψ0 as an isotropic function of elastic and inelastic stretch tensors, incorporating a volumetric-isochoric split and polyconvexity considerations.
• Introduce a co-rotated formulation to handle non-uniqueness and enable algorithmic differentiation of ψ0 and the pseudo potential.
• Employ two feed-forward neural networks to learn ψ0 and the dissipation potential g0, with a recurrent network integrating temporal evolution via an evolution equation Di = γ ∂g0/∂Γ.
• Specialize the framework to visco-elasticity to demonstrate autonomous model discovery from artificial data, cyclic loading, and relaxation data.
• Ensure thermodynamic admissibility by constraining ψ0 and g0 to be convex, non-negative, and zero-valued at the origin, with a structure that enforces Γ : Di ≥ 0.

Experimental results

Research questions

• RQ1 Can a thermodynamically consistent iCANN framework discover and represent inelastic material behavior from data?
• RQ2 How can multiplicative decomposition, objectivity, and non-unique intermediate configurations be incorporated into a neural network without breaking training or physical admissibility?
• RQ3 Can the iCANN architecture autonomously identify visco-elastic and related inelastic phenomena from synthetic and experimental-like data?
• RQ4 What network structures and activation designs are required to enforce isotropy, polyconvexity, and dissipation constraints in learned energies and potentials?

Key findings

• The iCANN framework extends CANNs to inelastic materials by integrating a learned Helmholtz energy with a learned dissipation potential within a recurrent setup.
• Thermodynamic constraints are satisfied a priori through co-rotated formulations and carefully designed energy/potential networks, ensuring non-negativity and convexity.
• The energy ψ0 is modeled as an isotropic function of elastic/inelastic stretch tensors with a volumetric-isochoric split to maintain mechanical stability.
• The pseudo potential g0 is expressed in invariants of the co-rotated driving force, with activation functions enforcing convexity and non-negativity to guarantee dissipation.
• When specialized to visco-elasticity, the iCANN can autonomously discover models from synthetic data and reproduce polymer cycle loading and muscle relaxation behaviors.
• The modular architecture supports extension to other inelastic phenomena beyond visco-elasticity and aims to reveal interactions among inelastic effects from data.