I am an applied mathematician, interested in mathematical modelling and applying computational and analytical tools to solve problems in the natural sciences. My work is focussed on the continuum scale; in particular problems relating to mechanical biology and physiology, growth and pattern formation, morphoelasticity, and elastic mechanisms in nature. Earlier work involved ferrohydrodynamics, minimal surfaces, and soap films.
Growth is the process by which a material gains mass. It is ubiquitous in nature, common in some form to all living entities, and found in many industrial applications and physical phenomena as well. It is the process that links the graceful neck of the giraffe to the powerful horns of the ram to the formation of a planet as a dust cloud collapses under gravity. In biological tissues, such as arteries, skin tissues, airways, and plants, growth is a very complex process, which can be analysed from multiple perspectives (biological, chemical, evolutionary, ...) as well as on multiple length and time scales.
My interest is on the mechanics of growth processes. In particular, I am interested in explaining the form of biological structures through the mechanical forces underlying their morphogenesis.
Surface growth, or accretion, refers to the deposition of new material on the surface of a body. A striking example of surface growth found in nature is the morphogenesis of seashells. Seashells have intrigued scientists for centuries, and many mathematical descriptions of the shapes of shells have been provided over the years. However, the developmental mechanisms underlying shell formation are largely not understood. We have developed a general model for shell growth based entirely on the local geometry and mechanics of the aperture, where shell growth occurs, and the mollusc, the creature that lives in and builds the shell.
On the left is an example of a Giant Clam shell simulated through our model. The ribbed shell opening is incrementally determined by finding the shape of an elastic beam on an evolving foundation; the shape evolution is then coupled through our growth framework to a kinematics description to produce the logarithmic shell coiling.
Elastic filaments, structures characterised by a length much greater than the cross-sectional width, are very common - for instance bacterial fibers, roots, and spine - and can have complicated 3D geometry. Understanding the mechanical behavior of these structures in the context of growth poses a significant challenge. We have developed a general framework to study such structures, which we term morphoelastic rods. Our approach combines the typical Kirchhoff equations for elastic rods with the decomposition of gradient tensor from general finite elasticity theory.
As an example, a growing elastic ring, constrained by its own closed geometry, will buckle at a critical value; the buckled shape and the evolution as it continues to grow is seen below.
I am exploring the stability of knotted elastic rods. In particular, we have found theoretically for the first time a stable elastic trefoil knot with no points of self-contact (below). Aside from increasing the understanding of the rich solution structure of the Kirchhoff equations, in recent years the study of elastic knots has found relevance in other areas as well, particularly in the micro biological world, where knots can be found in long polymers, proteins, and DNA plasmids.
Within the theory of finite elasticity are the most sophisticated tools for analysing the mechanical behaviour of tissues. Our modelling philosophy is to work within simplified geometries – such as cylindrical and spherical – where analytical progress may be made, and to incorporate significant mechanical complexity. For instance, we explore the effects of anisotropy, growth, residual stress, multiple material layers, and varying constitutive laws on material response.
Cylindrical tubes are particularly pervasive biological structures, forming the basic geometry of arteries, airways, digestive tract, and many others. These structures are often highlighted by a complex make-up, including anisotropy, differential growth, and multiple layers. The basic mechanical behaviour -- for instance response to forces such as external pressure or muscle contraction -- plays a vital role in the physiological properties and is often a component in degradation and disease.
I have developed a model to investigate the effect of growth on the circumferential buckling instability of growing bilayer elastic tubes. We demonstrated the important effect of growth on stability properties in airway remodelling in asthma, an often fatal phenomenon for which previous mechanical investigations did not consider large deformations or differential growth effects.
We are currently using non-linear elasticity to explore muscle contraction in arteries and the projection of the chameleon tongue.
A vertical soap film draining under gravity is a classic experiment, explored since the time of Newton. In our lab at the University of Delaware, we created magnetic soap-films by adding magnetic nano-particles to a standard soap-film recipe. These films, when placed vertically under a strong bar magnet, could be made to drain upwards against gravity with a strong enough magnet.
We developed a 2D model for a draining magnetic film using lubrication theory. We have captured well the basic phenomenon of reverse draining; we have also explored the effect of a disjoining pressure due to molecular forces. On the left is a regular draining film, on the right a magnetic film undergoes reverse draining (the black signals the thinnest, drained region).