Cracks in many soft solids behave very differently to the classical picture of fracture, where cracks are long and thin, with damage localised to a crack tip. In particular, small cracks in soft solids become highly rounded — almost circular — before they start to extend. However, despite being commonplace, this is still not well understood. We use a phase-separation technique in soft, stretched solids to controllably nucleate and grow small, nascent cracks. These give insight into the soft failure process. In particular, our results suggest fracture occurs in two regimes. When a crack is large, it obeys classical linear-elastic fracture mechanics, but when it is small it grows in a new, scale-free way at a constant driving stress.

# Past Industrial and Applied Mathematics Seminar

Transformation theory has long been known to be a mechanism for

the design of metamaterials. It gives rise to the required properties of the

material in order to direct waves in the manner desired. This talk will

focus on the mathematical theory underpinning the design of acoustic and

elastodynamic metamaterials based on transformation theory and aspects of

the experimental confirmation of these designs. In the acoustics context it

is well-known that the governing equations are transformation invariant and

therefore a whole range of microstructural options are available for design,

although designing materials that can harness incoming acoustic energy in

air is difficult due to the usual sharp impedance contrast between air and

the metamaterial in question. In the elastodynamic context matters become

even worse in the sense that the governing equations are not transformation

invariant and therefore we generally require a whole new class of materials.

In the acoustics context we will describe a new microstructure that consists

of rigid rods that is (i) closely impedance matched to air and (ii) slows

down sound in air. This is shown to be useful in a number of configurations

and in particular it can be employed to half the resonant frequency of the

standard quarter-wavelength resonator (or alternatively it can half the size

of the resonator for a specified resonant frequency) [1].

In the elastodynamics context we will show that although the equations are

not transformation invariant one can employ the theory of waves in

pre-stressed hyperelastic materials in order to create natural elastodynamic

metamaterials whose inhomogeneous anisotropic material properties are

generated naturally by an appropriate pre-stress. In particular it is shown

that a certain class of hyperelastic materials exhibit this so-called

“invariance property” permitting the creation of e.g. hyperelastic cloaks

[2,3] and invariant metamaterials. This has significant consequences for the

design of e.g. phononic media: it is a well-known and frequently exploited

fact that pre-stress and large deformation of hyperelastic materials

modifies the linear elastic wave speed in the deformed medium. In the

context of periodic materials this renders materials whose dynamic

properties are “tunable” under pre-stress and in particular this permits

tunable band gaps in periodic media [4]. However the invariant hyperelastic

materials described above can be employed in order to design a class of

phononic media whose band-gaps are invariant to deformation [5]. We also

describe the concept of an elastodynamic ground cloak created via pre-stress

[6].

[1] Rowley, W.D., Parnell, W.J., Abrahams, I.D., Voisey, S.R. and Etaix, N.

(2018) “Deepening subwavelength acoustic resonance via metamaterials with

universal broadband elliptical microstructure”. Applied Physics Letters 112,

251902.

[2] Parnell, W.J. (2012) “Nonlinear pre-stress for cloaking from antiplane

elastic waves”. Proc Roy Soc A 468 (2138) 563-580.

[3] Norris, A.N. and Parnell, W.J. (2012) “Hyperelastic cloaking theory:

transformation elasticity with pre-stressed solids”. Proc Roy Soc A 468

(2146) 2881-2903

[4] Bertoldi, K. and Boyce, M.C. (2008) “Mechanically triggered

transformations of phononic band gaps in periodic elastomeric structures”.

Phys Rev B 77, 052105.

[5] Zhang, P. and Parnell, W.J. (2017) “Soft phononic crystals with

deformation-independent band gaps” Proc Roy Soc A 473, 20160865.

[6] Zhang, P. and Parnell, W.J. (2018) “Hyperelastic antiplane ground

cloaking” J Acoust Soc America 143 (5)

This talk will concern the problem of inference when the posterior measure involves continuous models which require approximation before inference can be performed. Typically one cannot sample from the posterior distribution directly, but can at best only evaluate it, up to a normalizing constant. Therefore one must resort to computationally-intensive inference algorithms in order to construct estimators. These algorithms are typically of Monte Carlo type, and include for example Markov chain Monte Carlo, importance samplers, and sequential Monte Carlo samplers. The multilevel Monte Carlo method provides a way of optimally balancing discretization and sampling error on a hierarchy of approximation levels, such that cost is optimized. Recently this method has been applied to computationally intensive inference. This non-trivial task can be achieved in a variety of ways. This talk will review 3 primary strategies which have been successfully employed to achieve optimal (or canonical) convergence rates – in other words faster convergence than i.i.d. sampling at the finest discretization level. Some of the specific resulting algorithms, and applications, will also be presented.

A two-layer shear flow in the presence of surfactants is considered. The flow configuration comprises two superposed layers of viscous and immiscible fluids confined in a long horizontal channel, and characterised by different densities, viscosities and thicknesses. The surfactants can be insoluble, i.e. located at the interface between the two fluids only, or soluble in the lower fluid in the form of monomers (single molecules) or micelles (multi-molecule aggregates). A mathematical model is formulated, consisting of governing equations for the hydrodynamics and appropriate transport equations for the surfactant concentration at the interface, the concentration of monomers in the bulk fluid and the micelle concentration. A primary objective of this study is to investigate the effect of surfactants on the stability of the interface, and in particular surfactants in high concentrations and above the critical micelle concentration (CMC). Interfacial instabilities are induced due to the acting forces of gravity and inertia, as well as the action of Marangoni forces generated as a result of the dependence of surface tension on the interfacial surfactant concentration. The underlying physical mechanism responsible for the formation of interfacial waves will be discussed, together with the complex flow dynamics (typical nonlinear phenomena associated with interfacial flows include travelling waves, solitary pulses, quasi-periodic and chaotic dynamics).

Injection of a gas into a gas/liquid foam is known to give rise to instability phenomena on a variety of time and length scales. Macroscopically, one observes a propagating gas-filled structure that can display properties of liquid finger propagation as well as of fracture in solids. Using a discrete model, which incorporates the underlying film instability as well as viscous resistance from the moving liquid structures, we describe brittle cleavage phenomena in line with experimental observations. We find that the dimensions of the foam sample significantly affect the speed of the cracks as well as the pressure necessary to sustain them: cracks in wider samples travel faster at a given driving stress, but are able to avoid arrest and maintain propagation at a lower pressure (the velocity gap becomes smaller). The system thus becomes a study case for stress concentration and the transition between discrete and continuum systems in dynamical fracture; taking into account the finite dimensions of the system improves agreement with experiment.

Many fluid flows in natural systems are highly complex, with an often beguilingly intricate and confusing detailed structure. Yet, as with many systems, a good deal of insight can be gained by testing the consequences of simple mathematical models that capture the essential physics. We’ll tour two such problems. In the summer melt seasons in Greenland, lakes form on the surface of the ice which have been observed to rapidly drain. The propagation of the meltwater in the subsurface couples the elastic deformation of the ice and, crucially, the flow of water within the deformable subglacial till. In this case the poroelastic deformation of the till plays a subtle, but crucial, role in routing the surface meltwater which spreads indefinitely, and has implications for how we think about large-scale motion in groundwater aquifers or geological carbon storage. In contrast, when magma erupts onto the Earth’s surface it flows before rapidly cooling and crystallising. Using analogies from the kitchen we construct, and experimentally test, a simple model of what sets the ultimate extent of magmatic intrusions on Earth and, as it turns out, on Venus. The results are delicious! In both these cases, we see how a simplified mathematical analysis provides insight into large scale phenomena.

Blisters form when a thin surface layer of a solid body separates/delaminates from the underlying bulk material over a finite, bounded region. It is ubiquitous in a range of industrial applications, e.g. blister test is applied to assess the strength of adhesion between thin elastic films and their solid substrates, and during natural processes, such as formation and spreading of laccoliths or retinal detachment.

We study a special case of blistering, in which a thin elastic membrane is adhered to the substrate by a thin layer of viscous fluid. In this scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface in a radial geometry. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. We find that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction, particularly in the case of rectangular blisters. Furthermore, changes to the geometry of the channel cross-section in the latter case lead to a rich variety of possible interfacial patterns. Our experiments provide a link between studies of airway reopening, Saffman-Taylor fingering and printer’s instability.

Accurate simulation of electromagnetic scattering by ice crystals in clouds is an important problem in atmospheric physics, with single scattering results feeding directly into the radiative transfer models used to predict long-term climate behaviour. The problem is challenging for numerical simulation methods because the ice crystals in a given cloud can be extremely varied in size and shape, sometimes exhibiting fractal-like geometrical characteristics and sometimes being many hundreds or thousands of wavelengths in diameter. In this talk I will focus on the latter "high-frequency" issue, describing a hybrid numerical-asymptotic boundary element method for the simplified problem of acoustic scattering by penetrable convex polygons, where high frequency asymptotic information is used to build a numerical approximation space capable of achieving fixed accuracy of approximation with frequency-independent computational cost.

Motion at the microscale is a fascinating field visualizing non-equilibrium behaviour of matter. Evolution has optimized the ability of microscale swimming on different length scales from tedpoles, to sperm and bacteria. A constant metabolic energy input is required to achieve active propulsion which means these systems are obeying the laws imposed by a low Reynolds number. Several strategies - including topography1 , chemotaxis or rheotaxis2 - have been used to reliably determine the path of active particles. Curiously, many of these strategies can be recognized as analogues of approaches employed by nature and are found in biological microswimmers.

Bacteria attached to the metal caps of Janus particles3

However, natural microswimmers are not limited to being exemplary systems on the way to artificial micromotion. They certainly enable us to observe how nature overcame problems such as a lack of inertia, but natural microswimmers also offer the possibility to couple them to artificial microobjects to create biohybrid systems. Our group currently explores different coupling strategies and to create so called ‘BacteriaBots’.4

1 J. Simmchen, J. Katuri, W. E. Uspal, M. N. Popescu, M. Tasinkevych, and S. Sánchez, Nat. Commun., 2016, 7, 10598.

2 J. Katuri, W. E. Uspal, J. Simmchen, A. Miguel-López and S. Sánchez, Sci. Adv., , DOI:10.1126/sciadv.aao1755.

3 M. M. Stanton, J. Simmchen, X. Ma, A. Miguel-Lopez, S. Sánchez, Adv. Mater. Interfaces, DOI:10.1002/admi.201500505.

4 J. Bastos-Arrieta, A. Revilla-Guarinos, W. E. Uspal and J. Simmchen, Front. Robot. AI, 2018, 5, 97.

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of partial differential equations, harmonic, stochastic and statistical analysis, and optimisation. Starting with a discussion on the intrinsic structure of images and their mathematical representation, in this talk we will learn about variational models for image analysis and their connection to partial differential equations, and go all the way to the challenges of their mathematical analysis as well as the hurdles for solving these - typically non-smooth - models computationally. The talk is furnished with applications of the introduced models to image de-noising, motion estimation and segmentation, as well as their use in biomedical image reconstruction such as it appears in magnetic resonance imaging.