Theory and implementation of inelastic Constitutive Artificial Neural Networks

TL;DR

The paper addresses thermodynamically consistent discovery of inelastic constitutive behavior from data by introducing iCANN, a framework built on a multiplicative split $\mathbf{F}=\mathbf{F}_e\mathbf{F}_i$ and two neural subnets for the Helmholtz free energy $\psi_0$ and the dissipation potential $g_0$, embedded in a co-rotated, invariant-based architecture. It demonstrates on artificially generated Maxwell-type data, a high-bond polymer (VHB 4910) under cyclic loading, and relaxation in passive skeletal muscle, achieving high fidelity with sparse training data and yielding interpretable energy/potential terms that respect objectivity and the second law of thermodynamics. The approach enforces thermodynamic consistency through a co-rotated formulation and explicit time integration, while remaining modular and adaptable to additional inelastic phenomena. It points to broad extensions to plasticity, anisotropy, and multiphysics, offering a path toward automated, physics-guided constitutive model discovery grounded in thermodynamics.

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

Figures (14)

• Figure 1: Multiplicative decomposition of the deformation gradient, $\bm{F}$, into elastic, $\bm{F}_e$, and inelastic, $\bm{F}_i$, parts as well as their arbitrarily rotated counterparts $\bm{F}_e^*$ and $\bm{F}_i^*$. Further, $\bm{F}_i=\bm{R}_i\bm{U}_i$ possesses its polar decomposition. Configurations: $rc$ -- reference configuration, $ic$ -- intermediate configuration, $ic^*$ -- arbitrarily rotated intermediate configuration, $cic$ -- co-rotated intermediate configuration, $cc$ -- current configuration.
• Figure 2: A function, $f(x)$, that is convex, non-negative, and zero-valued has the property that the sign of the derivative, $f'(x)\coloneqq {\frac{\partial f(x)}{\partial x}}$, evaluated at $x$ and $x$ itself coincide.
• Figure 3: Schematic illustration of recurrent neural network for iCANN. The history dependence of the material is taken into account by the hidden state of the recurrent neural network. Thus, the inelastic stretches, $\bm{U}_i$, as well as the previous total stretches, $\bm{C}$, are propagated through time. Figure \ref{['fig:iCANN_architecture']} illustrates the iCANN architecture in each time step.
• Figure 4: Schematic illustration of the iCANN architecture. Color code: blue -- current inputs; orange -- basic calculation; red -- feed-forward network; yellow -- hidden state variables; purple -- time discretization (Equation \ref{['eq:time_discre']}); green -- current output. It is important to note that the feed-forward networks for both $\psi_n$ and $\psi_{n+1}$ have the same weights. Thus, there is no double set of weights for the energy. The architecture of the energy network is illustrated in Figure \ref{['fig:network_psi']}, Figure \ref{['fig:network_g']} shows the architecture of the feed-forward network of the potential.
• Figure 5: Schematic illustration of the feed-forward network architecture of the elastic and compressible Helmholtz free energy, $\psi$, within the recurrent neural network. The first layer computes the invariants, the second employs the volumetric-isochoric split, the third generates the powers $(\bullet)$ and $(\bullet)^2$, and the fourth applies the custom-designed activation functions identity $(\bullet)$ and the exponential $\exp(\bullet)-1$. To satisfy polyconvexity a priori, the network is not fully connected by design.