Anisotropic stress-strain relations and complex moduli of a viscoelastic composite with aligned spheroidal inclusions

Abstract

By means of a micromechanical theory this study seeks to uncover the influence of the inclusion shape on the stress-strain behavior and complex moduli of a class of composites containing aligned spheroidal inclusions. Before we set out the analysis it is shown first that, by combining two Maxwell or two Voigt constituents, the composite as whole is generally not of the Maxwell or the Voigt type; however, under the conditions that the Poisson's ratio of both phases remains constant and the ratios of their shear modulus to shear viscosity are equal, a transversely isotropic Maxwell or Voigt composite can be constructed. The strain-rate sensitivity of the stress-strain behavior is then examined for a system containing elastic inclusions and a viscoelastic matrix, at various inclusion shapes. It is found that the stress-strain curves are strongly dependent upon the applied strain rate in most cases, and that the precise increase of flow stress with increasing strain rate is intimately related to the inclusion shape and loading mode. Except for the axial tension with continuous fibers and the transverse tension, shear, and biaxial plane-strain with aligned discs, most stress-strain curves exhibit a saturation stress under a constant strain-rate loading. The real and imaginary parts of the five complex moduli are subsequently examined for their dependence upon the inclusion shape and concentration, and loading frequency. It is observed that the real parts of the moduli tend to increase with increasing frequency, and eventually approach their corresponding elastic moduli. The imaginary parts of the moduli may actually increase with increasing amount of elastic inclusions, albeit droping to zero again as the entire composite turns into inclusions. Their dependence on the frequency starts out at zero initially and again returns to zero as the frequency approaches infinity, but in the intermediate range multiple maxima are experienced with all inclusion shapes.