A Framework for Data-driven Constitutive Modelling with Internal Variables and its Integration in the Finite Element Method

  • Dettmer, Wulf (Swansea University)
  • Alhayki, Eman (Swansea University)
  • Muttio, Eugenio (Three Cliffs Geomechanical Analysis)
  • Perić, Djordje (Swansea University)

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Data-driven constitutive modelling for solid materials, using machine learning and stress-strain data ob- tained from physical experiments or from numerical homogenisation models, circumvents the need for complex mathematical constitutive equations and their integration in numerical strategies. This is particularly attractive for emerging composite materials as it bridges the gap between Materials Engineering and Structural Analysis efficiently and quickly. While much progress has been made in data-driven modelling of elastic and hyper-elastic materials, it has proven more challenging to develop generic data- driven strategies for history dependent materials. In this work, we propose a concept based on recursive artificial neural networks and hidden internal variables. The nature of the internal variables is not defined but emerges during the training process. The methodology is based on [1] but addresses the challenge to ensure consistency with respect to small strain increments, i. e., by design, the stress and the material state do not change if the strain increment is zero. While this requirement is easily and a priori satisfied by classic numerical constitutive equations, it is typically subject to “learning” in data-driven constitutive modelling. The network training is based on Adam optimisation with exact gradients. The proposed con- cept can be integrated in standard finite element procedures for solid materials, and a consistent tangent operator can be constructed. We present applications to plain strain elasto-plastic model problems and comment on numerical efficiency and on the requirements for training and validation data. REFERENCES [1] Dettmer, W. G., et al., A framework for neural network based constitutive modelling of inelastic materials, Computer Methods in Applied Mechanics and Engineering, Vol. 420, pp. 116672, 2024.