Recurrent neural network constitutive model for human rib cortical bone and its biological variability: Transitioning from deterministic to intrusive stochastic explicit FEM

Integrating the biological variability of human rib cortical bone into constitutive models is of interest to quantify the uncertainty of the biomechanical response at population level, e.g., in virtual automotive crash assessments, yet current constitutive models typically rely on fixed material properties. Because biological tissues are complex to model, this work proposes an Artificial Neural Network (ANN) ansatz to tackle this difficulty. The model comprises a recurrent neural network based on the extended Minimal State Cell (MSC) for elastoplasticity, augmented with transfer learning (TL) layers that account for the biological variability by modulating the input and output of the MSC. It was trained on stress–strain time series under non-monotonic multiaxial plane stress conditions for discrete values of the variability parameter and learned a smooth mapping over it, enabling interpolation to intermediate values without retraining. Self-consistency allowed training on short sequences ( ∼ 1 0 2 steps) while evaluating accurately on very long sequences ( ∼ 1 0 6 steps), as needed for online predictions inside the FEM scheme. The ANN model was deployed in a commercial explicit solver through a vectorized user material subroutine (UMAT). Moreover, a novel intrusive stochastic method is proposed in which the variability parameter follows a probability distribution within the material subroutine, propagating material uncertainty without multiple runs or solver modifications. The method allows for the estimation of the distributions of stresses and boundary forces within a single simulation. On a coupon geometry, the ANN remained stable over ∼ 1 0 6 time steps in both deterministic and stochastic simulations with uniform, normal, and bimodal distributions.