When a liquid drop is deposited over a solid surface whose temperature is sufficiently above the boiling point of the liquid, the drop does not experience nucleate boiling but rather levitates over a thin layer of its own vapor. This is known as the Leidenfrost effect. Whilst highly undesirable in certain cooling applications, because of a drastic decrease of the energy transferred between the solid and the evaporating liquid due to poor heat conductivity of the vapor, this effect can be of great interest in many other processes profiting from this absence of contact with the surface that considerably reduces the friction and confers an extreme mobility on the drop. During this presentation, I hope to provide a good vision of some of the knowledge on this subject through some recent studies that we have done. First, I will present a simple fitting-parameter-free theory of the Leidenfrost effect, successfully validated with experiments, covering the full range of stable shapes, i.e., from small quasi-spherical droplets to larger puddles floating on a pocketlike vapor film. Then, I will discuss the end of life of these drops that appear either to explode or to take-off. Finally, I will show that the Leidenfrost effect can also be observed over hot baths of non-volatile liquids. The understanding of the latter situation, compare to the classical Leidenfrost effect on solid substrate, provides new insights on the phenomenon, whether it concerns levitation or its threshold.

# Past Industrial and Applied Mathematics Seminar

Tendons are vital connective tissues that anchor muscle to bone to allow the transfer of forces to the skeleton. They exhibit highly non-linear viscoelastic mechanical behaviour that arises due to their complex, hierarchical microstructure, which consists of fibrous subunits made of the protein collagen. Collagen molecules aggregate to form fibrils with diameters of tens to hundreds of nanometres, which in turn assemble into larger fibres called fascicles with diameters of tens to hundreds of microns. In this talk, I will discuss the relationship between the three-dimensional organisation of the fibrils and fascicles and the macroscale mechanical behaviour of the tendon. In particular, I will show that very simple constitutive behaviour at the microscale can give rise to highly non-linear behaviour at the macroscale when combined with geometrical effects.

**Likely instabilities in stochastic hyperelastic solids **

*L. Angela Mihai *

*School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK *

*E-mail: MihaiLA@cardiff.ac.u*k

Nonlinear elasticity has been an active topic of fundamental and applied research for several decades. However, despite numerous developments and considerable attention it has received, there are important issues that remain unresolved, and many aspects still elude us. In particular, the quantification of uncertainties in material parameters and responses resulting from incomplete information remain largely unexplored. Nowadays, it is becoming increasingly apparent that deterministic approaches, which are based on average data values, can greatly underestimate, or overestimate, mechanical properties of many materials. Thus, stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions. In this talk, I will consider stochastic hyperelastic material models described by a strain-energy density where the parameters are characterised by probability distributions. These models, which are constructed through a Bayesian identification procedure, rely on the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest. Similar modelling approaches can be developed for other mechanical systems. To demonstrate the effect of probabilistic model parameters on large strain elastic responses, specific case studies include the classic problem of the Rivlin cube, the radial oscillatory motion of cylindrical and spherical shells, and the cavitation and finite amplitude oscillations of spheres.

Chemical reaction network (CRN) theory focusses on making claims about dynamical behaviours of reaction networks which are, as far as possible, dependent on the network structure but independent of model details such as functions chosen and parameter values. The claims are generally about the existence, nature and stability of limit sets, and the possibility of bifurcations, in models of CRNs with particular structural features. The methodologies developed can often be applied to large classes of models occurring in biology and engineering, including models whose origins are not chemical in nature. Many results have a natural algorithmic formulation. Apart from the potential for application, the results are often pleasing mathematically for their power and generality.

This talk will concern some recent themes in CRN theory, particularly focussed on how the presence or absence of particular subnetworks ("motifs") influences allowed dynamical behaviours in ODE models of a CRN. A number of recent results take the form: "a CRN containing no subnetworks satisfying condition X cannot display behaviour of type Y"; but also, in the opposite direction, "if a CRN contains a subnetwork satisfying condition X, then some model of this CRN from class C admits behaviour of type Y". The proofs of such results draw on a variety of techniques from analysis, algebra, combinatorics, and convex geometry. I'll describe some of these results, outline their proofs, and sketch some current challenges in this area.

## Further Information:

Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, CA 92093-0411

**Abstract **

The dynamical fields that emanate from self-similarly expanding ellipsoidal regions undergoing phase change (change in density, i.e., volume collapse, and change in moduli) under pre-stress, constitute the dynamic generalization of the seminal Eshelby inhomogeneity problem (as an equivalent inclusion problem), and they consist of pressure, shear, and M waves emitted by the surface of the expanding ellipsoid and yielding Rayleigh waves in the crack limit. They may constitute the model of Deep Focus Earthquakes (DFEs) occurring under very high pressures and due to phase change. Two fundamental theorems of physics govern the phenomenon, the Cauchy-Kowalewskaya theorem, which based on dimensional analysis and analytic properties alone, dictates that there is zero particle velocity in the interior, and Noether’s theorem that extremizes (minimizes for stability) the energy spent to move the boundary so that it does not become a sink (or source) of energy, and determines the self-similar shape (axes expansion speeds). The expression from Noether’s theorem indicates that the expanding region can be planar, thus breaking the symmetry of the input and the phenomenon manifests itself as a newly discovered one of a “dynamic collapse/ cavitation instability”, where very large strain energy condensed in the very thin region can escape out. In the presence of shear, the flattened very thin ellipsoid (or band) will be oriented in space so that the energy due to phase change under pre-stress is able to escape out at minimum loss condensed in the core of dislocations gliding out on the planes where the maximum configurational force (Peach-Koehler) is applied on them. Phase change occurring planarly produces in a flattened expanding ellipdoid a new defect present in the DFEs. The radiation patterns are obtained in terms of the equivalent to the phase change six eigenstrain components, which also contain effects due to planarity through the Dynamic Eshelby Tensor for the flattened ellipsoid. Some models in the literature of DFEs are evaluated and excluded on the basis of not having the energy to move the boundary of phase discontinuity. Noether’s theorem is valid in anisotropy and nonlinear elasticity, and the phenomenon is independent of scales, valid from the nano to the very large ones, and applicable in general to other dynamic phenomena of stress induced martensitic transformations, shear banding, and amorphization.

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.

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.