Gaussian process model improves stress prediction in quasi-brittle materials
A Gaussian Process State Space Formulation for Materials Constitutive Modeling: Application to Quasi-brittle Materials
Computational Engineering, Finance, and ScienceMachine Learning
Summary
Predicting how materials crack and break is tricky because their behavior depends on their history, which is hard to measure directly. The authors created a new model that learns from stress and strain data instead of relying on fixed formulas, making predictions more flexible and uncertain. Their model can track how materials behave under pressure and can be used in simulations to better mimic real-world responses. It even works with existing engineering software to test materials in structures.
What this means in practice
- •For structural engineers: Simulate quasi-brittle materials more flexibly in finite element software using the Gaussian process model exported as a custom material subroutine.
- •For materials testing labs: Use latent state models trained on stress-strain data to better capture history-dependent damage and loading effects for material characterization.
Authors
Houlin Xu, Chenyang Li, K. T. Ramesh, Michael D. Shields
Abstract
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 variational sparse inference. The trained model is exported into Abaqus/Explicit through a user material subroutine. Four verification cases are reported: (1) offline rollout on validation and test histories; (2) a single element under changing confinement; (3) a frozen-memory unloading-reloading extension; and (4) a multi-element structural model compared against the Karagozian and Case Concrete (KCC) model. These results show that the learned state-space update reproduces held-out RVE-scale histories in closed-loop rollout, tracks evolving confinement, supports unloading without violating the irreversibility of the learned history state, and reproduces structural-level behavior consistent with KCC.