Thermalizing and Damping in Structural Dynamics

Abstract

Structural damping, that is the presence of a velocity dependent dissipative term in the equation of motion, is rationalized as a thermalization process between a structure (here a beam) and an outside bath (understood in a broad sense as a system property). This is achieved via the introduction of the kinetic temperature of structures and formalized by means of an extended Lagrangian formulation of a structure in contact with an outside bath at a given temperature. Using the Nos e-Hoover thermostat, the heat exchange rate between structure and bath is identified as a mass damping coefficient, which evolves in time in function of the kinetic energy/temperature history exhibited by the structure. By way of application to a simple beam structure subjected to eigen-vibrations and dynamic buckling, commonality and differences of the Nos e-Hoover beam theory with constant mass damping models are shown, which permit a handshake between classical damping models and statistical mechanics-based thermalization models. The solid foundation of these thermalization models in statistical physics provides new insights into stability and instability for engineering structures. Specifically, since two systems are considered in (thermodynamic) equilibrium when they have the same temperature, we show in the case of dynamic buckling that a persistent steady-state difference in kinetic temperature between structure and bath is but indicative of the instability of the system. This shows that the kinetic temperature can serve as a structural order parameter to identify and comprehend failure of structures, possibly well beyond the elastic stability considered here.

Mode Coarsening or Fracture: Energy Transfer Mechanisms in Dynamic Buckling of Rods

“…Noteworthy is the fact that the asymptotic value of the structure's kinetic temperature is well above the bath's temperature T 0 → 0, prescribed in the simulations. This observation is readily attributed to the instability in dynamic buckling [23]: evoking the zeroth law (two systems are in equilibrium when they have the same temperature), it becomes apparent that the rod is not in thermodynamic equilibrium with the bath when the structure exhibits instabilities.…”

CASCO: a simulator of load paths in 2D frames during progressive collapse

“…These would differ from existing network models, in that they should capture some peculiar topological features that emerged from the simulations in this manuscript, such as the coexistence of different types of beams and links, and the possibility for their number to decrease as well as increase during failure propagation. Such conceptual models would help making structural collapse amenable to analysis using the tools of statistical mechanics [19,28], for a more fundamental understanding of the phenomenon. In this sense, CASCO is a first effort to develop a minimal model of structural collapse that can help understand the main drivers of progressive collapse, akin to how the Ising [25] and Fibre Bundle models [13] have scaffolded the rationalisation of phase transitions and fracture in materials.…”

Viscoelastic Characteristics of Mechanically Assembled Three-Dimensional Structures Formed by Compressive Buckling

“…One major challenge in the design of 3D MEMS arises from the effects of viscoelasticity in the constituent materials. Such effects lead to dissipation of energy, changes in stress distributions, displacements, and frequencies during vibration [41][42][43][44][45]. For example, increases in the ambient temperature and/or humidity can increase the viscosity [46][47][48], which, in turn, can reduce the efficiency of 3D MEMS devices.…”