Past OCCAM Wednesday Morning Event

12 January 2011
10:10
Dr Oscar Lopez-Pamies
Abstract
<p>It is by now well established that loading conditions with sufficiently large triaxialities can induce the sudden appearance of internal cavities within elastomeric (and other soft) solids. The occurrence of such instabilities, commonly referred to as cavitation, can be attributed to the growth of pre-existing defects into finite sizes. </p> <p>&nbsp;</p> <p>In this talk, I will present a new theory to study the phenomenon of cavitation in soft solids that, contrary to existing approaches,</p> <p>simultaneously: (i) allows to consider general 3D loading conditions with arbitrary triaxiality, (ii)&nbsp; applies to large (including compressible and anisotropic) classes of nonlinear elastic solids, and</p> <p>(iii) incorporates direct information on the initial shape, spatial distribution, and mechanical properties of the underlying defects at which cavitation can initiate. The basic idea is to first cast cavitation in elastomeric solids as the homogenization problem of nonlinear elastic materials containing random distributions of zero-volume cavities, or defects. Then, by means of a novel iterated homogenization procedure, exact solutions are constructed for such a problem. These include solutions for the change in size of the underlying cavities as a function of the applied loading conditions, from which the onset of cavitation - corresponding to the event when the initially infinitesimal cavities suddenly grow into finite sizes - can be readily determined. In spite of the generality of the proposed approach, the relevant calculations amount to solving tractable Hamilton-Jacobi equations, in which the initial size of the cavities plays the role of "time" and the applied load plays the role of "space".</p> <p>An application of the theory to the case of Ne-Hookean solids containing a random isotropic distribution of vacuous defects will be presented.</p>
  • OCCAM Wednesday Morning Event
17 November 2010
10:15
Marcelo Epstein
Abstract
<p>Modern differential geometry is the art of the abstract that can be pictured. Continuum mechanics is the abstract description of concrete material phenomena. Their encounter, therefore, is as inevitable and as beautiful as the proverbial chance meeting of an umbrella and a sewing machine on a dissecting table. In this rather non-technical and lighthearted talk, some of the surprising connections between the two disciplines will be explored with a view at stimulating the interest of applied mathematicians.</p>
  • OCCAM Wednesday Morning Event
10 November 2010
10:10
Janet Elliott
Abstract
Cryopreservation (using temperatures down to that of liquid nitrogen at –196 °C) is the only way to preserve viability and function of mammalian cells for research and transplantation and is integral to the quickly evolving field of regenerative medicine. To cryopreserve tissues, cryoprotective agents (CPAs) must be loaded into the tissue. The loading is critical because of the high concentrations required and the toxicity of the CPAs. Our mathematical model of CPA transport in cartilage describes multi-component, multi-directional, non-dilute transport coupled to mechanics of elastic porous media in a shrinking and swelling domain. Parameters are obtained by fitting experimental data. We show that predictions agree with independent spatially and temporally resolved MRI experimental measurements. This research has contributed significantly to our interdisciplinary group’s ability to cryopreserve human articular cartilage.
  • OCCAM Wednesday Morning Event
3 November 2010
10:10
Abstract
We derive solutions of the Kirchhoff equations for a knot tied on an infinitely long elastic rod subjected to combined tension and twist. We consider the case of simple (trefoil) and double (cinquefoil) knots; other knot topologies can be investigated similarly. The rod model is based on Hookean elasticity but is geometrically non-linear. The problem is formulated as a non-linear self-contact problem with unknown contact regions. It is solved by means of matched asymptotic expansions in the limit of a loose knot. Without any a priori assumption, we derive the topology of the contact set, which consists of an interval of contact flanked by two isolated points of contacts. We study the influence of the applied twist on the equilibrium and find an instability for a threshold value of the twist.
  • OCCAM Wednesday Morning Event
20 October 2010
10:10
Dr Soumyendu Raha
Abstract
We shall discuss a simple low order numerical integration scheme for ODEs and DAEs. The scheme has a parameter that allows for regularization of Jacobian of stiff problems and for numerically elucidating multi-scale response, if any, in some problems.
  • OCCAM Wednesday Morning Event

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