An extended multiscale principle of virtual velocities approach for evolving microstructure

Abstract

A hierarchical multiscale approach is presented for modeling microstructure evolution in heterogeneous materials. Preservation of momentum across each scale transition is incorporated through the application of the principle of virtual velocities at the fine scale giving rise to the appropriate continuum momentum balance equations at the coarse scale. In addition to satisfying momentum balance and invariance of momentum among scales, invariance of elastic free energy, stored free energy, and dissipation between two scales of observation is regarded as crucial to the physics of each scale transition. The preservation of this energy partitioning scheme is obtained through construction of constitutive relations within the framework of internal state variable theory. Internal state variables that are directly computed from the fine scale response are introduced to augment the state equations and describe the inelastic energy storage and dissipation within the fine scale. By virtue of a second gradient kinematic decomposition, the framework naturally gives rise to couple stresses.