MiroshnikovProf. Alexey Miroshnikov
Visiting Assistant Professor of Department of Mathematics - University of Massachusetts Amherst
On the properties of weak solutions describing dynamic cavitation in nonlinear elasticity

Thursday, Nov 13, 2014, 12:00pm to 1:00pm | Room 1-390

In this work we study the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. Cavitating solutions were introduced by J.M. Ball [1982, Phil. Trans. R. Soc. Lond. A] in elastostatics and by K.A. Pericak-Spector and S. Spector [1988, Arch. Rational Mech. Anal.] in elastodynamics. They turn out to decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (in the sense of hyperbolic conservation laws) for polyconvex energies. In our work we established various further properties of cavitating solutions. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions d = 2,3 we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress-free cavities the critical stretching associated with dynamically cavitating solutions coincides with the critical stretching in the bifurcation diagram of equilibrium elasticity.

Dr. Miroshnikov has been a visiting assistant professor of mathematics at the University of Massachusetts Amherst since 2012. In 2006, he received Michael Brin Fellowship from the University of Maryland, College Park (UMD) and graduated with a PhD in Mathematics from UMD in 2012. In addition, in 2008-2010, while a PhD student at UMD, he held Marie Curie Early Stage Research position at the Institute of Applied and Computational Mathematics at FORTH in Crete, Greece. Prior to this, Dr. Miroshnikov received an MSc in Mathematics from Virginia Tech in 2004. He also obtained BSc in Mechanical Engineering from Moscow State Technological University "Stankin" in 2000 and obtained MSc in Computer Science from the same university in 2002.