We present a variational coarse-graining framework for heterogeneous media in the spirit of FE² methods, that allows for a seamless transition from the traditional static scenario to dynamic loading conditions, while being applicable to general material behavior as well as to discrete or continuous representations of the material and its deformation, e.g., finite element discretizations or atomistic systems. The method automatically delivers the macroscopic equations of motion together with the generalization of Hill’s averaging relations to the dynamic setting. These include the expression of the macroscopic stresses and linear momentum as a function of the microscopic fields. We further demonstrate with a proof of concept example, that the proposed theoretical framework can be used to perform multiscale numerical simulations. The results are compared with standard single-scale finite element simulations, showcasing the capability of the method to capture the dispersive nature of the medium in the range of frequencies permitted by the multiscale strategy.

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finite element discretizations or atomistic systems macroscopic equations of motion dispersive nature of the medium dynamic loading the macroscopic stresses and linear momentum multiscale numerical simulations fesup locpost2sup method discrete or continuous representations of the material static scenario frequencies variational coarsegraining framework its deformation microscopic fields general material behavior generalization of hills averaging relations heterogeneous media

Chenchen Liu, Celia Reina,.Variational coarse-graining procedure for dynamic homogenization. 104 (0),187-206.

Chenchen Liu, Celia Reina,"Variational coarse-graining procedure for dynamic homogenization" 104

Chenchen Liu, Celia Reina,"Variational coarse-graining procedure for dynamic homogenization"