Theory of differential inclusions and its application in mechanics

The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load change is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems

Underwater contact adhesion and microarchitecture in polyelectrolyte complexes actuated by solvent exchange

Polyelectrolyte complexation is critical to the formation and properties of many biological and polymeric materials, and is typically initiated by aqueous mixing followed by fluid-fluid phase separation, such as coacervation. Yet little to nothing is known about how coacervates evolve into intricate solid microarchitectures. Inspired by the chemical features of the cement proteins of the sandcastle worm, here we report a versatile and strong wet-contact microporous adhesive resulting from polyelectrolyte complexation triggered by solvent exchange. After premixing a catechol-functionalized weak polyanion with a polycation in dimethyl sulphoxide (DMSO), the solution was applied underwater to various substrates whereupon electrostatic complexation, phase inversion, and rapid setting were simultaneously actuated by water-DMSO solvent exchange. Spatial and temporal coordination of complexation, inversion and setting fostered rapid (???25 s) and robust underwater contact adhesion (Wad ??? 2 J m-2) of complexed catecholic polyelectrolytes to all tested surfaces including plastics, glasses, metals and biological materials.close