The method of multiscale virtual power for the derivation of a second order mechanical model

• The Method of Multiscale Virtual Power is used to derive an RVE-based multiscale model of second-order continua.
• Body forces are considered in the formulation of the problem.
• The resulting model relies crucially on kinematical hypotheses and a Principle of Multiscale Virtual Power.
• Homogenisation formulae and the RVE equilibrium problems with kinematical constraints are consistently derived by means of variational arguments.
• Homogenised force- and stress-like quantities are identified as reactions to postulated RVE kinematical constraints, and characterised exclusively in terms of RVE boundary data.

A multi-scale model, based on the concept of Representative Volume Element (RVE), is proposed linking a classical continuum at RVE level to a macro-scale strain-gradient theory. The multi-scale model accounts for the effect of body forces and inertia phenomena occurring at the micro-scale. The Method of Multiscale Virtual Power recently proposed by the authors drives the construction of the model. In this context, the coupling between the macro- and micro-scale kinematical descriptors is defined by means of kinematical insertion and homogenisation operators, carefully postulated to ensure kinematical conservation in the scale transition. Micro-scale equilibrium equations as well as formulae for the homogenised (macro-scale) force- and stress-like quantities are naturally derived from the Principle of Multiscale Virtual Power – a variational extension of the Hill-Mandel Principle that enforces the balance of the virtual powers of both scales. As an additional contribution, further insight into the theory is gained with the enforcement of the RVE kinematical constraints by means of Lagrange multipliers. This approach unveils the reactive nature of homogenised force- and stress-like quantities and allows the characterisation of the homogenised stress-like quantities exclusively in terms of RVE boundary data in a straightforward manner.