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.