Dynamical Homogenization of Microstructured Materials in the Context of Strain Gradient Effective Continua

This paper introduces a dynamical homogenization methodology for microstructured materials, modeled as strain gradient effective continua, to address scale-dependent dynamic behaviors that classical continuum mechanics cannot capture. The approach derives effective macroscopic properties, including kinetic energy and constitutive laws, by employing a polynomial expansion of the microscopic displacement field completed with periodic fluctuations. The strain gradient framework incorporates the gradient of the strain tensor to account for microstructural effects, introducing internal length scales that reflect the impact of the material's inner architecture. Numerical evaluations for periodic lattice structures, such as the tetrachiral and square lattices, illustrate the model's ability to predict their dynamic responses, highlighting the influence of unit cell topology and intrinsic length scales. This methodology provides a powerful tool in engineering applications involving advanced materials like composites and metamaterials, enabling improved design and performance analysis.