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1. Static elastic cloaking, low-frequency elastic wave transparency and neutral inclusions

   https://doi.org/10.1098/rspa.2019.0725  

   Norris, Andrew N.; Parnell, William J. (August 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   New connections between static elastic cloaking, low-frequency elastic wave scattering and neutral inclusions (NIs) are established in the context of two-dimensional elasticity. A cylindrical core surrounded by a cylindrical shell is embedded in a uniform elastic matrix. Given the core and matrix properties, we answer the questions of how to select the shell material such that (i) it acts as a static elastic cloak, and (ii) it eliminates low-frequency scattering of incident elastic waves. It is shown that static cloaking (i) requires an anisotropic shell, whereas scattering reduction (ii) can be satisfied more simply with isotropic materials. Implicit solutions for the shell material are obtained by considering the core–shell composite cylinder as a neutral elastic inclusion. Two types of NI are distinguished, weak and strong with the former equivalent to low-frequency transparency and the classical Christensen and Lo generalized self-consistent result for in-plane shear from 1979. Our introduction of the strong NI is an important extension of this result in that we show that standard anisotropic shells can act as perfect static cloaks, contrasting previous work that has employed ‘unphysical’ materials. The relationships between low-frequency transparency, static cloaking and NIs provide the material designer with options for achieving elastic cloaking in the quasi-static limit. 

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2. Monitoring damage growth and topographical changes in plate structures using sideband peak count-index and topological acoustic sensing techniques

   https://doi.org/10.1016/j.ultras.2024.107354  

   Zhang, Guangdong; Deymier, Pierre A; Runge, Keith; Kundu, Tribikram (May 2024, Ultrasonics)

   Some topographies in plate structures can hide cracks and make it difficult to monitor damage growth. This is because topographical features convert homogeneous structures to heterogeneous one and complicate the wave propagation through such structures. At certain points destructive interference between incident, reflected and transmitted elastic waves can make those points insensitive to the damage growth when adopting acoustics based structural health monitoring (SHM) techniques. A newly developed nonlinear ultrasonic (NLU) technique called sideband peak count – index (or SPC-I) has shown its effectiveness and superiority compared to other techniques for nondestructive testing (NDT) and SHM applications and is adopted in this work for monitoring damage growth in plate structures with topographical features. The performance of SPC-I technique in heterogeneous specimens having different topographies is investigated using nonlocal peridynamics based peri-ultrasound modeling. Three types of topographies – “X” topography, “Y” topography and “XY” topography are investigated. It is observed that “X” and “XY” topographies can help to hide the crack growth, thus making cracks undetectable when the SPC-I based monitoring technique is adopted. In addition to the SPC-I technique, we also investigate the effectiveness of an emerging sensing technique based on topological acoustic sensing. This method monitors the changes in the geometric phase; a measure of the changes in the acoustic wave’s spatial behavior. The computed results show that changes in the geometric phase can be exploited to monitor the damage growth in plate structures for all three topographies considered here. The significant changes in geometric phase can be related to the crack growth even when these cracks remain hidden for some topographies during the SPC-I based single point inspection. Sensitivities of both the SPC-I and the topological acoustic sensing techniques are also investigated for sensing the topographical changes in the plate structures. 

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3. Safety assessemt based on physically-viable data-driven models

   https://doi.org/10.1109/CDC.2017.8264626  

   Ahmadi, Mohamadreza; Israel, Arie; Topcu, Ufuk (December 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC))

   We consider the problem of safety assessment of a dynamical system for which no model and just limited data on the states is available. That is, given samples of the state {x(t i )} N i=1 at time instances t 1 ≤ t 2 ≤ ··· ≤ t N and some other side information in terms of the regularity of the state evolutions, we are interested in checking whether x(T) ∉ Xu, where T > t N and Xu ⊂ R n (the unsafe set) are pre-specified. To this end, we use piecewise-polynomial approximations of the trajectories based on the data along with the regularity side information to formulate a data-driven differential inclusion model. For these classes of data-driven differential inclusions, we propose a safety assessment theorem based on barrier certificates. The barrier certificates are then found using polynomial optimization. The method is illustrated by two examples. 

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4. On the role of geometric phase in the dynamics of elastic waveguides

   https://doi.org/10.1098/rsta.2023.0357  

   Kumar, Mohit; Semperlotti, Fabio (September 2024, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue ‘Current developments in elastic and acoustic metamaterials science (Part 2)’. 

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5. Stabilization of the response of cyclically loaded lattice spring models with plasticity

   https://doi.org/10.1051/cocv/2020043  

   Gudoshnikov, Ivan; Makarenkov, Oleg (January 2021, ESAIM: Control, Optimisation and Calculus of Variations)

   null (Ed.)

   This paper develops an analytic framework to design both stress-controlled and displacement-controlled T -periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t ↦( e ( t ), p ( t )), where e i ( t ) is the elastic elongation and p i ( t ) is the relaxed length of spring i , defined on [ t 0 , ∞ ) by the initial condition ( e ( t 0 ), p ( t 0 )). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C ( t ) in a vector space E of dimension d , it becomes natural to expect (based on a result by Krejci) that the elastic component t ↦ e ( t ) always converges to a T -periodic function as t → ∞ . The achievement of this paper is in spotting a class of loadings where the Krejci’s limit doesn’t depend on the initial condition ( e ( t 0 ), p ( t 0 )) and so all the trajectories approach the same T -periodic regime. The proposed class of sweeping processes is the one for which the normals of any d different facets of the moving polyhedron C ( t ) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. We discover that the normals of any d different facets of the moving polyhedron C ( t ) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem. 

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