Applications of Geometrical Singular Perturbation Theory in Hyperplasticity Accelerated Ratcheting Models

EPSRC Project Title: Applications of Geometrical Singular Perturbation Theory in Hyperplasticity Accelerated Ratcheting Models
An approach for studying lateral loading for monopile designs is the Hyperplastic Accelerated Racheting Model (HARM) framework developed in collaboration between Orsted and Oxford University [1-3]. Using HARM one can study the loading response of the foundation as a function of stresses. Solving for displacement as a function of lateral loading is a problem in which many cycles of different amplitude are applied during the lifetime of the structure (due to waves etc), and because of a bias in loading from predominant wind directions, ratcheting is often observed [1].

The objective of the proposed PhD thesis is to significantly reduce simulation time for modelling cyclic loading such that extended series of unloading/reloading cycles can be introduced in the optimization phase of the pile design. The method used is targeted to be some variation of Geometrical Singular Perturbation Theory (GSPT) [4-6].

In my Master Thesis, we utilize the GSPT framework to prove the existence of a relevant slow-fast dynamical system equivalent to a continuous version of the HARM model, where the hyperbolicity of the critical manifold align with unloading and reloading parts of each cycle. This result may address direct solutions to some of the currently faced issues with long-term ratcheting and expected hysteresis analysis [7]. For the 0-D macro modelling approach of HARM two main directions are heavily motivated by the found slow-fast dynamical system in [7]:
- First, a study in the performance of GSPT as a standalone tool to obtain integrable (thereby analytical) and non-conservative upper bounds for experienced ratcheting along pile depth. If such results are in alignment with gathered experiments/simulations, they can be used as immediate design parameters.
- Secondly, a study on utilizing the knowledge of exactly where the HARM model becomes stiff to current numerical solvers. An idea is to model the fast and slow parts separately & glue them together - avoiding issues with machine number precision. This could lead to not only faster results but also more accurate.
In [7] we have only shown existence of a slow-fast system for the 0-D macro model with plasticity surfaces. To align with currently used numerical solutions and improve complexity, a major part of the PhD will also be to extend the theoretical results to higher dimensions (or to new constitutive models such as HySand). As soil modelling is in general complex, it is expected that we reach a boundary point for the theoretical work. The 'until then' reach results are likely to motivate new numerical methods, also expected to be covered within the PhD thesis. Lastly, both the theoretical and numerical work will be validated by comprehensive studies on real-world data from Oxford's test pilings at Yorkshire, Kent and Dunkirk.
Areas of investment & support:
This project falls within the 'EPSRC Ground Engineering research area'.

References
[1] G. Houlsby and A. Puzrin, "A thermomechanical framework for constitutive models for rate-independent dissipative materials," International Journal of Plasticity, vol. 16, no. 9, pp. 1017-1047, 2000. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S074964199900073X
[2] T. Balaam, "Development and calibration of cyclic loading models for monopile foundations in clays," Ph.D. dissertation, Oxford University, 2020.
[3] C. Abadie, "Cyclic lateral loading of monopile foundations in cohesionless soils," Ph.D. dissertation, Oxford University, 01 2015.
[4] K. U. Kristiansen, "A review of multiple time scale dynamics: Fundamental phenomena and mathematical methods," 2023, in review.
[5] C. Kuehn, "Multiple time scale dynamics with two fast variables and one slow variable," Ph.D. dissertation, Cornell University, 05 2010.
[6] C. K. R. T. Jones, Geometric singular perturbation theory.