Stability analysis of laminated beam systems with delay using lyapunov functional

This work is concerned with systems of laminated beams model subject to linear and nonlinear delay feedback. In a dynamic laminated beam, time delay manifests in the form of lags in restoring the desired system stability after perturbations. Four prevalent categories of time delay are considered. For laminated beams with relatively high adhesive stiffness, a constant delay feedback is considered for systems made up of individual beams with same elasticity, and neutral delay otherwise. In systems where delay is significantly due to adhesive softening, distributed delay is considered. Lastly, in structures where the mechanism of dissipating energy is nonlinear, a corresponding nonlinear delay effect is investigated. The mechanism of stabilization mainly relies on the intrinsic structural damping, unlike in previous works where researchers introduced additional dampings such as boundary feedback and material damping. The objective of this work is to establish the asymptotic behavior of a vibrating Timoshenko laminated beam using structural or utmost a single frictional damping in presence of different forms of time delay. The energy method for partial differential equations is the main tool used to establish wellposedness results and asymptotic behavior. The existence and uniqueness of the solution is proved using the linear semi group theory, whereas for energy decay properties, the multiplier technique involving constructing a suitable Lyapunov functional equivalent to the energy is utilized. With appropriate assumptions on the delay weight and wave speeds, it is established that the energy of the system at least decays exponentially due to structural damping. Furthermore, a single additional frictional damping guarantees polynomial decay despite the presence of constant or distributed delay feedback. For nonlinear structural damping, with help of some convexity arguments, general decay result is achieved. In summary, depending on the damping mechanism(s), exponential, polynomial, or general decay results of a laminated beam system subject to different forms of delay feedback are established

Frictional sliding without geometrical reflection symmetry

The dynamics of frictional interfaces play an important role in many physical systems spanning a broad range of scales. It is well-known that frictional interfaces separating two dissimilar materials couple interfacial slip and normal stress variations, a coupling that has major implications on their stability, failure mechanism and rupture directionality. In contrast, interfaces separating identical materials are traditionally assumed not to feature such a coupling due to symmetry considerations. We show, combining theory and experiments, that interfaces which separate bodies made of macroscopically identical materials, but lack geometrical reflection symmetry, generically feature such a coupling. We discuss two applications of this novel feature. First, we show that it accounts for a distinct, and previously unexplained, experimentally observed weakening effect in frictional cracks. Second, we demonstrate that it can destabilize frictional sliding which is otherwise stable. The emerging framework is expected to find applications in a broad range of systems.Comment: 14 pages, 5 figures + Supplementary Material. Minor change in the title, extended analysis in the second par

Spatiotemporal dynamics of frictional systems: The interplay of interfacial friction and bulk elasticity

Frictional interfaces are abundant in natural and engineering systems, and predicting their behavior still poses challenges of prime scientific and technological importance. At the heart of these challenges lies the inherent coupling between the interfacial constitutive relation -- the macroscopic friction law -- and the bulk elasticity of the bodies that form the frictional interface. In this feature paper, we discuss the generic properties of the macroscopic friction law and the many ways in which its coupling to bulk elasticity gives rise to rich spatiotemporal frictional dynamics. We first present the widely used rate-and-state friction constitutive framework, discuss its power and limitations, and propose extensions that are supported by experimental data. We then discuss how bulk elasticity couples different parts of the interface, and how the range and nature of this interaction are affected by the system's geometry. Finally, in light of the coupling between interfacial and bulk physics, we discuss basic phenomena in spatially-extended frictional systems, including the stability of homogeneous sliding, the onset of sliding motion and a wide variety of propagating frictional modes (e.g. rupture fronts, healing fronts and slip pulses). Overall, the results presented and discussed in this feature paper highlight the inseparable roles played by interfacial and bulk physics in spatially-extended frictional systems.Comment: An invited feature paper (24 pages including Appendices, 8 figures

Analysis and design of large space structures with nonlinear joints

Issued as Final report, Project no. E-25-62

On the speed of fast and slow rupture fronts along frictional interfaces

The transition from stick to slip at a dry frictional interface occurs through the breaking of the junctions between the two contacting surfaces. Typically, interactions between the junctions through the bulk lead to rupture fronts propagating from weak and/or highly stressed regions, whose junctions break first. Experiments find rupture fronts ranging from quasi-static fronts with speeds proportional to external loading rates, via fronts much slower than the Rayleigh wave speed, and fronts that propagate near the Rayleigh wave speed, to fronts that travel faster than the shear wave speed. The mechanisms behind and selection between these fronts are still imperfectly understood. Here we perform simulations in an elastic 2D spring--block model where the frictional interaction between each interfacial block and the substrate arises from a set of junctions modeled explicitly. We find that a proportionality between material slip speed and rupture front speed, previously reported for slow fronts, actually holds across the full range of front speeds we observe. We revisit a mechanism for slow slip in the model and demonstrate that fast slip and fast fronts have a different, inertial origin. We highlight the long transients in front speed even in homogeneous interfaces, and we study how both the local shear to normal stress ratio and the local strength are involved in the selection of front type and front speed. Lastly, we introduce an experimentally accessible integrated measure of block slip history, the Gini coefficient, and demonstrate that in the model it is a good predictor of the history-dependent local static friction coefficient of the interface. These results will contribute both to building a physically-based classification of the various types of fronts and to identifying the important mechanisms involved in the selection of their propagation speed.Comment: 29 pages, 21 figure