Spatial and temporal averaging in the homogenisation of the elastodynamic response of periodic laminates

This paper suggests a methodology to obtain homogenised material properties from a transient dynamic numerical model. The standard Hill–Mandel Principle, based on spatial averages, is extended with time averaging. Thus, in addition to a sufficiently large 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 space–time averaging procedure is validated for a periodic laminate bar subjected to a variety of boundary conditions, impedance contrasts and loading conditions. The homogenised results converge to the analytical solutions and confirm that having a higher impedance contrast between laminate components requires not only larger RVE sizes but also longer time averaging windows. The most efficient macroscopic approximation is obtained by a balanced increase in RVE size and time averaging window.