Dynamic mechanical behavior of filled polymers. I. Theoretical developments

Abstract

A constitutive theory is developed for modeling the mechanical response of dynamically loaded filled-polymer composites. The basis for this work is Mori and Tanaka’s effective medium theory. Expressions derived by Weng and co-workers are used for the elastic stiffness tensor of the composite. The filler is a low volume concentration of randomly positioned elastic ellipsoidal particles. Random and aligned orientations of the ellipsoids are considered. The viscoelastic stress–strain behavior of the polymer matrix is modeled using the Boltzmann superposition principle with a Prony series representation for the stress relaxation functions. We argue that for rubbery polymers it is reasonable to express the composite stress relaxation functions as series expansions about the ratio of the polymer shear relaxation function to its bulk modulus. The smallness of this ratio allows accurate results to be obtained when truncating the expansion at first order. Inverse Laplace transformations required by the theory can then be done analytically. The result of these manipulations is again a Prony series for the stress relaxation functions of the composite, but with series coefficients that are now functions of the filler concentration, ellipsoidal aspect ratio, and moduli. In the limit of low filler concentration the theory reduces to known results derived in classical suspension theory.