Wave Propagation Analysis in the Homogenized Second-Gradient Medium: A Direct and Inverse Approach

In this work, we develop a method for homogenizing effective second-order gradient continuum models for 2D periodic composite materials. A constitutive law is formulated using a variational approach combined with the Hill macro-homogeneity condition for strain energy. Incorporating strain gradient effects enhances the constitutive law by linking the hyperstress tensor to the second-order gradient of displacement, capturing elastic size and microstructure effects beyond classical Cauchy elasticity. The effective strain gradient moduli are calculated for composites exhibiting strong internal length effects, validating the proposed approach by computing the strain energy at different scales. Additionally, we develop an inverse homogenization method to compute local mechanical properties (properties of the constituents) given known global properties (effective properties), showing good agreement with the literature data. This framework is extended to study wave propagation by analyzing longitudinal and shear waves in 2D composite materials. The effects of inclusion shape and volume percentage on wave propagation are examined, revealing that elliptic inclusions lead to a slight increase in both modes of propagation. Finally, we investigate the impact of property contrast between the inclusion and matrix, demonstrating its influence on wave dispersion.