ON WAVE PROPAGATION THROUGH REGULAR AND IRREGULAR MATERIAL FORMATIONS WITH PROPERTIES PERIODIC IN SPACE-TIME

Abstract eng:
Heterogeneous material structures assembled in space and time – dynamic materials (DM) – have shaped a novel material concept that finds applications to various problems, especially to material optimization in continuum dynamics. The specifics of DM resides in their universal feature to be thermodynamically open systems. To secure the temporal variability of material properties, a non-stop exchange of mass/momentum/energy should be maintained between the DM formation and the environment. Mathematically, the DM concept is expressed through linear hyperbolic equations with variable coefficients. When such coefficients (material properties) take at each point of space-time one out of two admissible systems of values, then we speak about “two materials” that may be distributed in space-time to form two types of layouts. One of them, termed regular, is characterized by the absence of collisions between the characteristics of the same family on the interfaces separating two different materials. Another type of a layout, termed irregular, allows for such collisions. This talk will offer examples illustrating both types of DM. The first (regular) case is given by a checkerboard structure assembled from two distinct materials and doubly periodic in space and time. When the structural parameters fall into certain continuous intervals, the waves traveling through this structure accumulate energy that is pumped into them by an external agent at the moments of temporal property switching. Another (irregular) example allows for a collision of characteristics, which may become necessary in the context of optimal design in dynamics. Physically speaking, the collision of characteristics means the rise of discontinuities in solutions of the dynamic equations. But when the equations are originally assumed linear (which is a part of the DM concept), then this property remains in effect only through a regular case. Contrary to that, in the irregular situation, one expects the appearance of non-linearity of some sort, generated by the physics of material collisions. Transformation of energy that accompanies such collisions may become irreversible, and, as a consequence, the problem may acquire features typical for a standard entropy situation. In this talk, there will be suggested a reasonable model that introduces a special dynamic scenario by assuming that the colliding elementary masses cling together to form clots – the moving finite masses. We require conservation of momentum through collisions to describe the subsequent motion of the clots. This requirement is implemented as a part of the concept of totally inelastic impact; through such collisions, the kinetic energy is lost (Carnot theorem), and the expected irreversibility follows. Technically, the original continuity equation becomes complemented with two additional ODEs governing the formation and the motion of clots; the extended system then exposes a substantial non-linearity that makes the optimization problem well-posed.

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