  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.
