DEPARTMENT OF MATHEMATICAL SCIENCES
Analysis and PDE Seminar Series
Alexey Miroshnikov UCLA
On the Problem of Dynamic Cavitation in Nonlinear Elasticity
Co-author: Athanasios Tzavaras (KAUST)
ABSTRACT: In this work we study the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. J. Ball (1982) proposed to use continuum mechanics for modeling cavitation and used methods of the calculus of variations to construct radial cavitating solutions for equilibrium elasticity with polyconvex stored energy. In a subsequent important development, K.A. Pericak-Spector and S. Spector (1988) constructed a weak solution to a dynamic problem that correspond to a spherical cavity emerging at time t=0 from a homogeneously deformed state. Remarkably, the cavitating solution has lower mechanical energy, and thus provide a strikinexample of non-uniqueness of entropy weak solutions (in the sense of hyperbolic conservation laws) for polyconvex energies.
In our work, we established various properties of cavitating solutions. For the equations of radial elastodynamics 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.
Tuesday, September 19, 2017
Stratton Hall 304