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Applied Math and Comp Sci Colloquium

Friday, April 20, 2018 - 2:00pm

Rohan Abeyaratne



University of Pennsylvania


In this talk I will illustrate and discuss certain issues pertaining to the continuum mechanical formulation of problems that involve accretion on a rigid impermeable surface. One characteristic of the problem we consider is that accretion takes place on the interior surface that separates the body from its support (rather than on its exterior surface).  As a result, each new layer  formed must push away the layers formed previously. This leads to a build-up of stress in the growing body.  Simultaneously, ablation takes place at the outer surface where material is removed from the body. 

The issues that I will address include (a) choosing a reference configuration that allows one to cope with the continually evolving material structure; (b) the nonstandard boundary conditions that couple diffusion to growth; and (c) the driving force associated with growth. I will illustrate the theory by a problem involving growth on a spherical bead.  I will show in particular that  the build-up of stress at the inner surface slows down accretion, while the increase in strain energy at the outer surface promotes ablation.  Eventually, the system may reach a point where internal accretion is balanced by external ablation, a regime referred to as ``treadmilling''. We derive conditions under which such a state exists, and show that when it does exist, it is unique.  Finally (d)  we will use mixture theory to derive a (more detailed) coupled model of the solid that accounts for the affect of diffusion on the solid.  

Parts of this work were carried out jointly with Rami Abi-Akl (MIT), Tal Cohen (MIT) and Giuseppe Tomassetti (University of Rome).