Industrial and Applied Mathematics Seminar

We develop a comprehensive geometrically-exact theory for an end-loaded elastic rod constrained to deform on a cylindrical surface. By viewing the rod-cylinder system as a special case of an elastic braid, we are able to obtain all forces and moments imparted by the deforming rod to the cylinder as well as all contact reactions. This framework allows us to give a complete treatment of static friction consistent with force and moment balance. In addition to the commonly considered model of hard frictionless contact, we analyse two friction models in which the rod, possibly with intrinsic curvature, experiences either lateral or tangential friction. As applications of the theory we study buckling of the constrained rod under compressive and torsional loads, finding critical loads to depend on Coulomb-like friction parameters, as well as the tendency of the rod to lift off the cylinder under further loading. The cylinder can also have arbitrary orientation relative to the direction of gravity. The cases of a horizontal and vertical cylinder, with gravity having only a lateral or axial component, are amenable to exact analysis, while numerical results map out the transition in buckling mechanism between the two extremes. Weight has a stabilising effect for near-horizontal cylinders, while for near-vertical cylinders it introduces the possibility of buckling purely due to self-weight. Our results are relevant for many engineering and medical applications in which a slender structure winds inside or outside a cylindrical boundary.

Traditionally, stochastic processes are modelled one of two ways: a continuum Fokker-Planck approach, where a PDE is solved to determine the time evolution of the probability density, or a Langevin approach, where the SDE describing the system is sampled, and multiple simulations are used to collect statistics. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also have much to offer classical stochastic processes (and statistical physics).  

In this talk I will introduce the formalism at a physicist’s level of rigour, and focus on determining the dominant contribution to the path integral when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian dynamics in an effective potential [1,2,3]. I will then discuss some applications, including reaction pathways conditioned on finite time [2]. We demonstrate that the most probable pathway at a finite time may be very different from the usual minimum energy path used to calculate the average reaction rate. If time permits, I will also discuss the extremely nonlinear crystal dislocation response to applied stress [4].  

[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)     

[2] Fitzgerald, Steve, et al. J. Chem. Phys. 158.12 (2023).

[3] Honour, Tom and Fitzgerald, Steve. in press J. Phys. A (2024)

[4] Fitzgerald, Steve. Sci. Rep. 6 (1) 39708 (2016)

What do fiber polymers and ice sheets have in common? They both flow with a directionally dependent - anisotropic - viscosity. This behaviour occurs in other geophysical flows, such as the Earth's mantle, where a material's microstructure affects its large-scale flow. In ice, the alignment of crystal orientations can cause the viscosity to vary by an order of magnitude, consequently having a strong impact on the flow of ice sheets and glaciers. However, the effect of anisotropy on large-scale flow is not well understood, due to a lack of understanding of a) the best physical approximations to model crystal orientations, and b) how crystal orientations affect rheology. In this work, we aim to address both these questions by linking rheology to crystal orientation predictions, and testing a range of models against observations from the Greenland ice sheet. The results show assuming all grains experience approximately the same stress provides realistic predictions, and we suggest a set of equations and parameters which can be used in large-scale models of ice sheets. 

Mathematical modelling can support decontamination processes in a variety of ways.  In this talk, we focus on the contamination step: understanding how much of a chemical spill has seeped into the Earth or a building material, and how far it has travelled, are essential for making good decisions about how to clean it up.  

We consider an infiltration problem in which a chemical is poured on an initially unsaturated porous medium, and seeps into it via capillary action. Capillarity-driven flow through partially-saturated porous media is often modelled using Richards’ equation, which is a simplification of the Buckingham-Darcy equation in the limit where the infiltrating phase is much more viscous than the receding phase.  In this talk, I will explore the limitations of Richards equation, and discuss some scenarios in which predictions for small-but-finite viscosity ratios are very different to the Richards simplification.

applied mathematics.

In geophysical and astrophysical flows, we are often interested in understanding the impact of internal waves on the non-wavelike flow. For example, oceanic internal waves generated at the surface and the seafloor transfer energy from the large scale flow to dissipative scales, thereby influencing the global ocean state. A primary challenge in the study of wave-flow interactions is how to separate these processes – since waves and non-wavelike flows can vary on similar spatial and temporal scales in the Eulerian frame. However, in a Lagrangian flow-following frame, temporal filtering offers a convenient way to isolate waves. Here, I will discuss a recently developed method for evolving Lagrangian mean fields alongside the governing equations in a numerical simulation, and extend this theory to allow effective filtering of waves from non-wavelike processes.

Even though most of us tie our shoelaces "wrongly," knots in ropes and filaments have been used as functional structures for millennia, from sailing and climbing to dewing and surgery. However, knowledge of the mechanics of physical knots is largely empirical, and there is much need for physics-based predictive models. Tight knots exhibit highly nonlinear and coupled behavior due to their intricate 3D geometry, large deformations, self-contact, friction, and even elasto-plasticity. Additionally, tight knots do not show separation of the relevant length scales, preventing the use of centerline-based rod models. In this talk, I will present an overview of recent work from our research group, combining precision experiments, Finite Element simulations, and theoretical analyses. First, we study the mechanics of two elastic fibers in frictional contact. Second, we explore several different knotted structures, including the overhand, figure-8, clove-hitch, and bowline knots. These knots serve various functions in practical settings, from shoelaces to climbing and sailing. Lastly, we focus on surgical knots, with a particularly high risk of failure in clinical settings, including complications such as massive bleeding or the unraveling of high-tension closures. Our research reveals a striking and robust power law, with a general exponent, between the mechanical strength of surgical knots, the applied pre-tension, and the number of throws, providing new insights into their operational and safety limits. These findings could have potential applications in the training of surgeons and enhanced control of robotic-assisted surgical devices.