Variational multiscale methods to embed the macromechanical continuum formulation with fine-scale strain gradient theories

A variational basis is presented to link ne-scale theories of material behaviour with the classical, macromechanical continuum theory. The approach is based on the weak form of the linear momentum balance equations, and a separation of the weighting function and displacement elds into coarse and ne-scale components. Coarse and ne-scale weak forms are de ned. The latter is used to introduce a strain gradient theory that operates at ner scales of deformation. Attention is focused upon applications requiring the enhanced physical accuracy of the ne-scale strain gradient theory, without the computational cost of discretization that spans the range from coarse to ne scales. A variationally consistent method is developed to embed the ne-scale strain gradient theory in the macromechanical formulation. The embedding is achieved by eliminating the ne-scale displacement eld from the problem. Two examples demonstrate the numerical e ciency of the method, while retaining physical and mathematical properties of the ne-scale strain gradient theory