# Phase transition processes in flexural structured systems with rotational inertia

WHTW01 - Factorisation of matrix functions: New techniques and applications

Failure and phase transition processes in mass-spring systems have been extensively studied in the literature, based on the approach developed in [1]. Only a few attempts at characterising these processes in flexural systems exist, see for instance [2, 3, 4, 5]. In comparison with mass-spring systems, flexural structures have a larger range  of applicability. They can describe phenomena in systems at various scales, including microlevel waves in materials and  dynamic processes in civil engineering assemblies such as bridges and buildings found in society. Flexural systems also provide a greater variety of modelling tools, related to loading configurations and physical parameters, that can be used to achieve a particular response.

Here we consider the role of rotational inertia in the process of phase transition in a one-dimensional flexural system, that may represent a simplified model of the  failure of a bridge exposed to hazardous vibrations. The phase transition process is assumed to occur with a uniform speed that is driven by feeding waves carrying energy produced by an applied oscillating moment and force. We show that the problem can be reduced to a functional equation via the Fourier transform which is solved using the Wiener-Hopf technique. From the solution we identify the dynamic behaviour of the system during the transition process. The minimum energy required to initiate the phase transition process with a given speed is determined and it is shown there exist parameter domains defined by the force and moment amplitudes where the phase transition can occur. The influence of the rotational inertia of the system on the wave radiation phenomenon connected with the phase transition is also discussed. All results are supplied with numerical illustrations confirming the analytical predictions.

Acknowledgement: M.J.N. and M.B. gratefully acknowledge the support of the EU H2020 grant MSCA -IF-2016-747334-CAT-FFLAP.

References
[1] Slepyan, L.I.: Models and Phenomena in Fracture Mechanics, Foundations of Engineering Mechanics, Springer, (2002).
[2] Brun, M., Movchan, A.B. and Slepyan, L.I.: Transition wave in a supported heavy beam, J. Mech. Phys. Solids 61, no. 10, pages 2067–2085, (2013).
[3] Brun, M., Giaccu, G.F., Movchan, A., B., and Slepyan, L. I.. Transition wave in the collapse of the San Saba Bridge. Front. Mater. 1:12, (2014). doi: 10.3389/fmats.2014.00012.
[4] Nieves, M.J., Mishuris, G.S., Slepyan, L.I.: Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98, pages 699–713, (2016).
[5] Garau, M., Nieves, M.J. and Jones, I.S. (2019): Alternating strain regimes for failure propagation in flexural systems, Q. J. Mech. Appl. Math., hbz008, https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1093%2Fqjmam%2Fhbz008&data=02%7C01%7C%7Cca24c94f14fb47b2a98908d6f19a9002%7Cd47b090e3f5a4ca084d09f89d269f175%7C0%7C0%7C636962043472937444&sdata=hFcD7qiLBQweKalUwfiI8DE4OoKVDBet7AwngVFgEf0%3D&reserved=0.

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