Browsing by Author "Ortner, Christoph"
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- ItemAn approach to nonlinear viscoelasticity via metric gradient flows(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Ortner, Christoph; Şengül, YaseminWe formulate quasistatic nonlinear finite-strain viscoelasticity of rate-type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame-indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous in some case in a specific case.
- ItemMini-Workshop: Mathematical Models, Analysis, and Numerical Methods for Dynamic Fracture(Zürich : EMS Publ. House, 2011) Larsen, Christopher J.; Ortner, ChristophThe mathematical foundation of fracture mechanics has seen considerable advances in the last fifteen years. While this progress has been substantial, it has been largely limited to quasi-static evolutions based on global energy minimization, which is known to produce non-physical results. What is missing is a generally accepted mathematical theory of dynamic crack growth, which accounts for material inertia. Such a theory would not only be able to describe the most physically realistic setting, but it would also provide a trusted starting point to resolve pressing questions about quasistatic evolutions, e.g., a rigorous justification of the quasi-static setting as an asymptotic limit of inertial dynamics. This workshop brought together researchers in mathematical analysis, mechanics, applied mathematics, and numerical analysis and laid the groundwork for progress on these questions.