A Gaussian Process State Space Formulation for Materials Constitutive Modeling: Application to Quasi-brittle Materials

Houlin Xu, Chenyang Li, K. T. Ramesh, Michael D. Shields

Constitutive models for history-dependent, path-dependent materials typically encode this dependence in prescribed functional forms through internal variables such as damage or plastic strain that are phenomenological, model-dependent, and not directly observable. These rigid forms lack flexibility to match observations, make calibration potentially non-unique, and complicate uncertainty quantification. This paper develops a Gaussian-process state-space model (GP-SSM) that replaces the fixed form of the constitutive model with a flexible GP regression, and the prescribed internal variables with latent state variables learned directly from observable stress-strain histories. As a GP, the model is naturally endowed with a measure of uncertainty and can be constrained to satisfy physical principles. We demonstrate this framework for quasi-brittle materials under triaxial compression, where the latent variable evolution is driven by the current-state stress invariants and the increment of equivalent strain to reflect irreversible, pressure-sensitive history. A sparse multi-output GP then maps the latent state and current strain tensor to the six-component stress response, trained by…