Elastomers filled with liquid inclusions: Theory, numerical implementation, and some basic results

Experimental and theoretical results of late have pointed to elastomers filled with various types of liquid inclusions as a promising new class of materials with unprecedented properties. Motivated by these findings, the first of two objectives of this paper is to formulate the homogenization problem that describes the macroscopic mechanical response of elastomers filled with liquid inclusions under finite quasistatic deformations. The focus is on the non-dissipative case when the elastomer is a hyperelastic solid, the liquid making up the inclusions is a hyperelastic fluid, the interfaces separating the solid elastomer from the liquid inclusions feature their own hyperelastic behavior (which includes surface tension as a special case), and the inclusions are initially spherical in shape. The macroscopic behavior of such filled elastomers turns out to be that of a hyperelastic solid, albeit one that depends directly on the size of the inclusions and the constitutive behavior of the interfaces. It is hence characterized by an effective stored-energy function W ¯ ( F ¯ ) of the macroscopic deformation gradient F ¯ . Strikingly, in spite of the fact that there are local residual stresses within the inclusions (due to the presence of initial interfacial forces), the resulting macroscopic behavior is free of residual stresses, that is, ∂ W ¯ ( I ) / ∂ F ¯ = 0 . What is more, in spite of the fact that the local moduli of elasticity in the bulk and the interfaces in the small-deformation limit do not possess minor symmetries (due to the presence of residual stresses and initial interfacial forces), the resulting effective modulus of elasticity does possess the standard minor symmetries, that is, L ¯ i j k l = L ¯ j i k l = L ¯ i j l k , where L ¯ i j k l ≔ ∂ 2 W ¯ ( I ) / ∂ F ¯ i j ∂ F ¯ k l . The second objective of this paper is to implement and deploy a finite-element scheme to numerically generate solutions for the macroscopic response of a basic class of elastomers filled with liquid inclusions, that of isotropic suspensions of incompressible liquid inclusions of monodisperse size embedded in incompressible Neo-Hookean elastomers wherein the interfaces feature a constant surface tension. With guidance from the numerical solutions, the last part of this paper is devoted to proposing a simple explicit approximation for the effective stored-energy function W ¯ ( F ¯ ) .