Abstract

In this talk, we discuss two methods used in the modelling of materials with defects at different scales.   The first, known as the method of mesoscale asymptotic approximations [1],  is appropriate for the modelling of granular materials. In particular, we discuss the approximation of solutions to a particular class of transmission problems in solids containing clusters of many small inclusions, which can interact with each other [2]. The asymptotics are supplied with remainder estimates that are rigorously justified.    The second method concerns the dynamic failure of structured materials, commonly used in understanding the phenomena involved at multiple scales in the fracture of solids. The subject has been explored in the last 30 years for a variety of mass-spring systems [3].  Here, we take a different approach and consider faults propagating through periodic mass-beam systems [4, 5], which are more commonly found in civil engineering applications such as long rooftops, bridges and pipeline systems. We give a summary of analytical results, based on the Fourier transform and the Wiener-Hopf technique, concerning the dynamic behaviour of the structure during the failure.   Numerical simulations demonstrate the effectiveness of both approaches. It is envisaged that both techniques may find a new home in the modelling of phenomena associated with the behaviour of sea ice.   References:  [1] V. Maz’ya, A. Movchan and M. Nieves, (2013): Green’s Kernels and Meso-Scale Approximations in Perforated Domains, Lecture Notes in Mathematics 2077, Springer. [2] M.J. Nieves, (2017): Asymptotic analysis of solutions to transmission problems in solids with many inclusions, SIAM J . Appl. Math. 77 (4), 1417-1443. [3] L.I. Slepyan,  (2002): Models and Phenomena in Fracture Mechanics, Foundations of Engineering Mechanics, Springer. [4] M.J. Nieves, G.S. Mishuris, L.I. Slepyan, (2016): Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98, 699-713.  [5] M.J. Nieves, G.S. Mishuris, L.I. Slepyan, (2017): Transient wave in a transformable periodic flexural structure, Int. J. Solids Struct. 112, 185-208