Homogenization of elastomers filled with liquid inclusions: The small-deformation limit

Abstract: This paper presents the derivation of the homogenized equations that describe the macroscopic mechanical response of elastomers filled with liquid inclusions in the setting of small quasistatic deformations. The derivation is carried out for materials with periodic microstructure by means of a two-scale asymptotic analysis. The focus is on the non-dissipative case when the elastomer is an elastic solid, the liquid making up the inclusions is an elastic fluid, the interfaces separating the solid elastomer from the liquid inclusions are elastic interfaces featuring an initial surface tension, and the inclusions are initially $n$-spherical ($n=2,3$) in shape. Remarkably, in spite of the presence of local residual stresses within the inclusions due to an initial surface tension at the interfaces, the macroscopic response of such filled elastomers turns out to be that of a linear elastic solid that is free of residual stresses and hence one that is simply characterized by an effective modulus of elasticity $\bar{\textbf{L}}$. What is more, in spite of the fact that the local moduli of elasticity in the bulk and the interfaces do not possess minor symmetries (due to the presence of residual stresses and the initial surface tension at the interfaces), the resulting effective modulus of elasticity $\bar{\textbf{L}}$ does possess the standard minor symmetries of a conventional linear elastic solid, that is, $\bar{L}{ijkl}=\bar{L}{jikl}=\bar{L}_{ijlk}$. As a first application, numerical results are worked out and analyzed for the effective modulus of elasticity of isotropic suspensions of incompressible liquid $2$-spherical inclusions of monodisperse size embedded in an isotropic incompressible elastomer.