Joint Materials/Solid Mechanics Seminar Series
Institute of Continuous Media Mechanics of the Russian Academy of Sciences
Abstract: Statistical theory of the evolution of typical mesoscopic defects (microcracks, microshears) allowed us to establish new type of critical phenomena in solid with defects -- structural- scaling transitions, to develop thermodynamics of solid with defects and to propose the phenomenology based on generalization of the Ginzburg-Landau theory and constitutive equations. The key results of the statistical theory and statistically based phenomenology are the establishment of two order parameters responsible for the structure evolution -- the defect density tensor (deformation induced by defects) and the structural scaling parameter, which reflects the scaling transition in the course of nucleation and growth of defects, and the generation of characteristic collective modes of defects responsible for the relaxation (shear transformation zones -STZ) and failure (damage transformation zone -DTZ). These modes have the nature of the solitary wave and blow-up dissipative structure and provide the mechanisms of plastic relaxation and damage- failure transition and can be excited in the resonance regime in dynamically loaded and shocked materials. Dynamic, shock wave experiments and structural study supported the linkage of the evolution of these modes with material responses in the large range of strain rates and loading intensity and allowed us to propose the interpretation of the following effects: (i) Nonlinear crack dynamics and the transition from the steady- state to the branching regime of crack propagation and the onset of the fragmentation dynamics. The existence of two critical velocities, three characteristic regimes of crack dynamics and the role of mentioned collective modes of defects were studied experimentally in the preloaded PMMA plate using the REMIX high speed framing of crack propagation, scaling analysis of stress phase portraits in the process zone of dynamically propagated cracks, correlation analysis of failure surface roughness using high resolution interferometer-profiler NEW VIEW. The linkage of different types of fracture (fragmentation) statistics with above regimes was shown: stress (stress intensity factor) controlled statistics, the statistics (close to the Poisson statistics) related to the dynamics of blow-up collective modes and the intermediate statistics (similar to the Weibull statistics).
(ii) The mechanisms of failure wave generation and propagation that has the nature of delayed failure with the delay time, which corresponds to the time of the excitation of blow-up collective modes (DTZ) in the microshear ensemble. The comparative analysis of the conditions of failure wave generation showed that the excitation of failure wave has the "resonance" dynamics in the case of the formation of DTZ as the self-similar blow-up distribution of microshear ensemble due to the pass of shock wave. Experimental study of mechanism of failure wave generation and propagation was studied for the Taylor test in the fused quartz rod using high-speed framing and supported the "delayed" mechanism of failure wave generation and propagation.
(iii) The fourth power universality of the steady-state plastic front as the consequence of the subjection of the relaxation kinetics to the dynamics of the STZ was confirmed both theoretically and experimentally. Experimental study of the plastic wave structure was carried out in the plate impact test for the copper and the NEW VIEW scaling analysis of the STZ distribution in the recovered specimen in the cross-section of wave propagation. High correlated distribution of STZ was established that supported scaling law at the steady-state plastic front.
Center for Fluid Mechanics Seminar Series
Abstract: Turbulent, stably-stratified flows occur in most of the ocean below the mixed layer, in the atmosphere above the tropopause, and in the stable atmospheric boundary layer. As a consequence, understanding the fluid dynamics in this flow regime is important for predicting environmental phenomena such as weather, climate, and the food chain in the ocean, and technological processes such as sensor performance in the ocean and atmosphere and turbulence signatures left by submerged vehicles or structures. In this seminar, results from very high resolution direct numerical simulations of strongly stratified decaying turbulence will be presented. The simulations span a wide range of Reynolds and Froude numbers, and are run on numerical grids having up to 500 million grid points. The computational results will be presented in the context of theoretical predictions and scaling arguments.
Abstract: Suppose we have well-posedness for some PDE both in R^d and in a simply connected bounded domain U. Imagine we take the initial data for the whole space and restrict it to U, modifying it slightly to satisfy any required boundary conditions. How much does the resulting solution in the whole spacediffer from the solution on U over time? Or, what is perhaps slightly easier to answer, if we scale U by r, does the solution on rU approach the solution in R^d as r goes to infinity in some appropriate sense?
We address this last question for weak solutions to the Navier-Stokes and Euler equations in two dimensions, and show that strong convergence occurs in some norms and weak convergence in other norms of interest. Critical to this is showing that the initial velocity can be modified to match the appropriate boundary conditions while changing its norms by a vanishingly small amount as r goes to infinity. A secondary goal is to establish, for initial velocity that decays at infinity as slowly as possible, the same existence, uniqueness, and regularity results as for classical (that is, finite energy) solutions.
The original motivation for this work was to establish the existence and uniqueness of statistical solutions to the Navier-Stokes and Euler equations in the entire plane with infinite energy and, for the Euler equations, unbounded initial vorticities, by taking advantage of the well developed theory of statistical solutions in a bounded domain of the plane. Time allowing, we will discuss this application. (This is work in progress.)
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