An RVE-based multiscale theory of solids with micro-scale inertia and body force effects

•An RVE-based multiscale theory of solids with inertia and body forces is developed.•The theory is derived variationally based on an extended Hill–Mandel Principle.•Volume averages of inertia and body forces are only relevant to the macroscale.•Fluctuations of inertia and body forces are only relevant to the microscale.

A multiscale theory of solids based on the concept of representative volume element (RVE) and accounting for micro-scale inertia and body forces is proposed. A simple extension of the classical Hill–Mandel Principle together with suitable kinematical constraints on the micro-scale displacements provide the variational framework within which the theory is devised. In this context, the micro-scale equilibrium equation and the homogenisation relations among the relevant macro- and micro-scale quantities are rigorously derived by means of straightforward variational arguments. In particular, it is shown that only the fluctuations of micro-scale inertia and body forces about their RVE volume averages may affect the micro-scale equilibrium problem and the resulting homogenised stress. The volume average themselves are mechanically relevant only to the macro-scale.