טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentKatz Shmuel
SubjectNon-Linear Dynamic Phenomena in Bistable Chains: the
Response to Impact, Solitary Waves, and Boomerons
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Josef Givli
Full Thesis textFull thesis text - English Version


Abstract

The design of architectured materials with bistable building blocks holds exciting possibilities. This new class of metamaterials exploits micro-level structural instabilities to obtain extraordinary physical and mechanical properties. Still, the dynamic behavior of these lattice structures is largely unexplored. Here, we study the dynamic response to impact of a 1-D bistable lattice, i.e. a FPU chain with springs having a non-convex double-well energy potential. In addition to metamaterials, this model-problem is prototypical to a large number of systems, such as unfolding/refolding of proteins, crack propagation, plasticity, and mechanisms underlying martensitic phase transformations. We show that, depending only on the stiffness-ratio associated with the two energy wells of the bistable springs, the system exhibits two fundamentally different responses to impact; (i) when the stiffness in the secondary well is smaller than that of the primary well, the impact energy is (almost entirely) trapped in the form of large undulations of the first few springs, and unique phenomena such as boomerons emerge. This is the first time a boomeron is observed in a free-standing lattice. (ii) On the other hand, when the stiffness in the secondary well is larger, the energy of the impact is (almost entirely) transferred along the chain in the form of a solitary wave that involves transition to the secondary energy-well and back. For the former, numerical simulations show that boomerons can emerge only on if negative spinodal region is present. Some parametric study, based on extensive numeric simulations, shows the broad existence of the phenomena and provides guideline for further research. For the latter we reveal, based on analytical treatment and extensive numerical simulations, a universal features of the solitary wave. Namely, the solitary wave solution is indifferent to the energy barrier separating the two equilibria of the double well potential, and that the shape of the wave can be described by means of merely two scalar properties of the potential of the springs, namely the ratio of stiffness in Phase II and Phase I, and the ratio between the Maxwell’s force and corresponding transition strain. Linear stability of the solitary-wave solution is studied using the Vakhitov-Kolokolov criterion applied to the approximate solutions. These results are validated by numerical simulations. We find that the solitary wave solution is stable provided that its velocity is higher than some critical value. It is shown that, the solitary waves are stable for almost the entire range of possible wave velocities. This is also manifested in the interaction between two solitary waves or between a solitary wave and a wall. Such interaction results in a minor change of height and shape of the solitary wave along with the formation of a trail of small undulations that follow the wave, as expected in a non-integrable system. Even after a significant number of interactions the changes in the wave height and shape are minor, suggesting that the bistable chain may be a useful platform for delivering information over long distances, even concurrently with additional information (other solitary waves) passing through the chain.