Filled elastomers: A theory of filler reinforcement based on hydrodynamic and interphasial effects

Abstract

Experimental evidence has by now established that i) the hydrodynamic effect and ii) the presence of stiff interphases (commonly referred to as bound rubber) "bonding" the underlying elastomer to the fillers are the dominant microscopic mechanisms typically responsible for the enhanced macroscopic stiffness of filled elastomers. Yet, because of the technical difficulties of dealing with these fine-scale effects within the realm of finite deformations, the theoretical reproduction of the macroscopic mechanical response of filled elastomers has remained an open problem. The object of this paper is to put forward a microscopic field theory with the capability to describe, explain, and predict the macroscopic response of filled elastomers under arbitrarily large nonlinear elastic deformations directly in terms of: i) the nonlinear elastic properties of the elastomeric matrix, ii) the concentration of filler particles, and iii) the thickness and stiffness of the surrounding interphases. Attention is restricted to the prominent case of isotropic incompressible elastomers filled with a random and isotropic distribution of comparatively rigid fillers. The central idea of the theory rests on the construction of a homogenization solution for the fundamental problem of a Gaussian elastomer filled with a dilute concentration of rigid spherical particles bonded through Gaussian interphases of constant thickness, and on the extension of this solution to non-Gaussian elastomers filled with finite concentrations of particles and interphases by means of a combination of iterative and variational techniques. For demonstration purposes, the theory is compared with full 3D finite-element simulations of the largedeformation response of Gaussian and non-Gaussian elastomers reinforced by isotropic distributions of rigid spherical particles bonded through interphases of various finite sizes and stiffnesses, as well as with experimental data available from the literature. Good agreement is found in all of these comparisons. The implications of this agreement are discussed.

Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects

“…28,29,30,31,32,33) are all positive and increasing in a convex manner together with an increasing α (w) . They are always lower than the input coefficient of variation and increase their magnitude together with an increase of E (w) .…”

“…RVEs have dimensions of [3δ, 3δ, 3δ], where δ is unit interval and they are composed of a polymeric matrix, a spherical carbon black C 60 particle and an interphase of constant thickness in between these two materials. Particle location is chosen in an accidental way and numerical algorithms presented in [31] or [32] are omitted here principally because this study focuses on an anisotropic representation of the homogenized composite and its impact on the effective tensor. A number of the particles representing an internal structure of the composite has been chosen as 27 because (1) it ensures the best balance between quality of its internal composition and mesh quality limited by the computational resources, (2) it allows adequate representation of internal structure in cubic composite and (3) it is very close to the number of monodisperse particle chosen in other studies [32].…”

Computational homogenization of the debonding of particle reinforced composites: The role of interphases in interfaces

“…For the most generality, we consider the particles to be polydisperse, represented through three families of particle sizes [27]. The particle locations are generated using a constrained adsorption algorithm [31,27,18]. The procedure we use to generate the polydisperse microstructures is as follows:…”

“…In this investigation, we set N p ¼ 10 such that the RUC contains a total of 80 particles [18]. Figs.…”

“…They extend the Mori-Tanaka homogenization scheme [17] to account for the debonding of composite materials under finite strains; however, they only consider hydrostatic loading in three-dimensions and do not account for the presence of interphases. More recently, Goudarzi et al [18] presented a theoretical framework capable of describing the influence of interphases on the macroscopic constitutive response of particle reinforced elastomers. They compare their formulation to both numerical and experimental results [6], and found excellent correlation with both.…”