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Understanding dynamic crack growth in structured systems with the Wiener-Hopf technique: Lecture 1

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WHT - Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications

Crack propagation is a process accompanied by multiple phenomena at different scales. In particular, when a crack grows, microstructural  vibrations are released, emanating from the crack tip. Continuous   models   of  dynamic   cracks   are  well  known   to  omit  information   concerning   these microstructural processes [1]. On the other hand, tracing these vibrations on the microscale is possible if one considers a crack propagating in a structured system, such as a lattice [2, 3].  These models have a particular relevance in the design of metamaterials,  whose microstructure  can be tailored to control dynamic effects for a variety of practical purposes [4]. Similar approaches have been recently paving new pathways to understanding failure processes in civil engineering systems [5, 6].  

In this lecture, we aim to demonstrate the importance of the Wiener-Hopf technique in the analysis and solution  of problems  concerning  waves and crack propagation  in discrete periodic  media. We begin with the model of a lattice system containing  a crack and show how this can be reduced to a scalar Wiener-Hopf  equation  through  the Fourier  transform.  From  this functional  equation  we identify  all possible  dynamic  processes  complementing   the  crack  growth.  We  determine  the  solution  to  the problem  and  how  this  is  used  to  predict  crack  growth  regimes  in  numerical  simulations.  Other applications of the adopted method, including the analysis of the progressive collapse of large-scale structures, are discussed.

References [1] Marder, M. and Gross, S. (1995): Origin of crack tip instabilities, J. Mech. Phys. Solids 43, no. 1, 1- 48.   [2] Slepyan, L.I. (2001): Feeding and dissipative  waves in fracture and phase transition  I. Some 1D structures and a square-cell lattice, J. Mech. Phys. Solids 49, 469-511.   [3] Slepyan, L.I. (2002): Models and Phenomena  in Fracture Mechanics, Foundations  of Engineering Mechanics, Springer.   [4] Mishuris, G.S., Movchan, A.B. and Slepyan, L.I., (2007): Waves and fracture in an inhomogeneous lattice structure, Wave Random Complex 17, no. 4, 409-428.   [5] Brun, M., Giaccu, G.F., Movchan, A.B., and Slepyan, L. I., (2014): Transition wave in the collapse of the San Saba Bridge, Front. Mater. 1:12. doi: 10.3389/fmats.2014.00012.   [6] Nieves, M.J., Mishuris,  G.S., Slepyan,  L.I., (2016): Analysis  of dynamic  damage propagation  in discrete beam structures, Int. J. Solids Struct. 97-98, 699-713.

This talk is part of the Isaac Newton Institute Seminar Series series.

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