Willis coupling in micromorphic elasticity

Acoustic and elastic metamaterials (AEMM) are artificial materials whose performance is engineered to exceed naturally occurring materials and conventional composites, often leveraging complex microstructural geometry and strong dispersion. Generalized continuum (GC) theories, such as micromorphic elasticity, are extensions of classical elasticity that capture the effects of microstructure via inclusion of internal kinematic variables and/or higher-order gradients of displacement, and have received renewed interest as potential reduced-order models for AEMM. An alternative AEMM modeling framework is the elastodynamic homogenization theory of Willis, which yields an effective medium whose constitutive relations contain stress-acceleration and momentum-strain-rate couplings. Despite the recent popularity of the GC and Willis methods, a comprehensive description of their relationship remains lacking. In the present work, we derive an explicit connection between the Willis and GC models. Specifically, we augment the variational formulation of Mindlin [Arch. Ration. Mech. Anal., 16, 51–78 (1964)] to account for microscale asymmetry of mass density and demonstrate that the resulting boundary value problem may be expressed in the form of a Willis material. This approach helps to elucidate the physical origins of Willis coupling and may offer insights about how other emergent material properties arise from microstructural constituents. [Work supported by ONR.]