Spatial and Temporal Multiscale Modelling Strategies for the Elastodynamic Response of Periodic Composites

The distinctive multiscale properties of composite materials render them increasingly popular in engineering applications for exploiting customizable characteristics to achieve superior material properties. This thesis aims to address challenges in modelling their microstructural features under dynamic excitations, using widely adopted multiscale methods to predict their mechanical behaviour. This thesis introduces novel techniques to address three prominent challenges: achieving accurate micro-macro averaging, attaining effective scale transitions, and selecting appropriate scale parameters in dynamics. In order to improve the accuracy of micro-macro averaging, a method where the standard Hill-Mandel Principle is extended with time averaging is developed to obtain homogenised material properties from a transient dynamic numerical model when subjected to transient dynamic loads. Thus, in addition to a Representative Volume Element (RVE) to carry out the averaging in space, a sufficiently large time window is required to carry out the time averaging. The proposed space-time averaging approach is used to predict the dynamic RVE size by increasing static RVE sizes as well as time averaging windows, thereby capturing dynamic microstructural effects in the homogenized macroscopic response. A further concern addressed in this thesis is to enhance the accuracy of dynamic scale transitions in computational homogenisation. By decoupling microstructures from its associated macrostructure in both space and time, a novel scale transition approach is presented to incorporate the simultaneous separation of length and time scales. This technique exploits the benefits of separations of length and time scales by increasing the RVE size as well as the time period on the microscale, ultimately leading to more accurate macroscopic response of a material. The approach to increasing the RVE size for transient dynamic problems allows better understanding of material scale parameters in analytical homogenisation. The issue of quantifying length scale parameters in dynamics is addressed in this work in a gradient elasticity framework. A procedure is presented to select the dynamic length scale parameter linked to the dynamic RVE size through curve fitting optimisation. This provides a suitable value for the dynamic length scale parameter, which ensures an accurate material response without having deviations caused by the RVE size. The efficacy and efficiency of the proposed methods in addressing the challenges are demonstrated through several numerical examples on a one-dimensional periodic laminate bar under various boundary conditions, material contrasts, and loading scenarios.