Mathematical theory and analytical solutions for the wave catching-up phenomena in a nonlinearly elastic composite bar

Cracking induced by tensile wave at the free surface of an impacted target is an important issue in impact-resistant design. Here, we explore the use of material nonlinearity to undermine the strength of the tensile wave. More specifically, we consider waves in a two-material composite bar subjected to impact loading at one end. Multiple reflections cause a tensile wave being transmitted into the second material. The attention is on analytically and numerically studying the phenomenon that the tensile wave catches the first transmitted compressive wave. It turns out that, depending on the interval of the initial impact, catching-up phenomena can happen in two wave patterns. A general mathematical theory is provided to show the existence of these patterns together with some qualitative information. To gain more insights into such phenomena, asymptotic solutions are also constructed, which provide both qualitative and quantitative results on the requirement of the constitutive relation, the time and place at which the catching takes place, and how the initial impact, material and geometric parameters influence the solutions. Numerical simulations are also performed, confirming the validity of the analytical results. The analysis and results presented here could be useful for designing a composite structure that has a good impact-protection performance.

A N. Bourbaki Type General Theory and the Properties of Contracting Symbols and Corresponding Contracted Forms

A system of contracting symbols is introduced for a N. Bourbaki type general mathematical theory corresponding to a general classical mathematical theory .

Hele–Shaw flows with time-dependent free boundaries in which the fluid occupies a multiply-connected region

We consider the classical Hele-Shaw situation with two parallel planes separated by a narrow gap. A blob of Newtonian fluid is sandwiched between the planes, and we suppose its plan-view to occupy a bounded, multiply-connected domain; physically, we have a viscous fluid with the holes giving rise to the multiple connectivity occupied by relatively inviscid air. The relevant free boundary condition is taken to be one of constant pressure, but we allow different pressures to act within the different holes, and at the outer boundary. The motion is driven either by injection of further fluid into the blob at certain points, or by injection of air into the holes to change their area, or by a combination of these; suction, instead of injection, is also contemplated. A general mathematical theory of the above class of problems is developed, and applied to the particular situation that arises when fluid is injected into an initially empty gap bounded by two straight, semi-infinite barriers meeting at right-angles: injection into a quarterplane. For a range of positions of the injection point, air is trapped in the corner and, invoking images, the problem is equivalent to one involving a doubly-connected blob. When there is an air vent in the corner, so that the pressure is the same on the two free boundaries in these circumstances, the air hole rapidly disappears, as might be expected. If, however, there is no air vent and we suppose the air to be incompressible, so that the area of the region occupied by the air in the plan-view remains constant, we find there to be no solution within the framework of our model. Other scenarios within this same geometry, involving both suction and injection of fluid at the injection point, and air at the corner, are also examined.