On problems with discontinuous motions in solid mechanics: shear bands and cavitation
Talk by Prof. Dr. Athanasios Tzavaras
Date: 04.12.14 Time: 12.15 - 13.15 Room: ETH HG G 19.2
Various problems in solid mechanics concern phenomena where discontinuous motions emerge. Such phenomena lie outside the traditional premiss of continuum modeling and their study raises questions at the interface of discrete and continuum modeling and poses challenges to the mathematical theory. In this talk we will review recent work on two specific examples: In the first part we will present results on nonuniqueness on the system of radial elasticity and the notion of singular induced from continuum solutions, which is a strengthening of the notion of weak solutions attempting to account for the surface energy required to open a cavity. The second part will consider the onset of localization and the formation of shear bands in high strain-rate plasticity of metals. Shear strain localization is tyically associated with ill-posedness of an underlying initial value problem, what has coined the term Hadamard-instability for its description in the mechanics literature. It should however be noted that while Hadamard instability indicates the catastrophic growth of oscillations around a mean state, it does not by itself explain the formation of coherent structures typically observed in localization. For a simple model capturing the essence of the mechanism of localization in metals we will construct self-similar solutions that describe the self-organization into a localized solution starting from well prepared data.