Scale-dependent homogenization of elastic-viscoelastic random composites

Microstructural randomness is one of the most basic characteristics of nearly all solid materials and plays a key role in the prediction of their macroscopic properties. Evidently, any material that displays heterogeneity on a micro scale has properties depending on the scale of approximating continuum. The key issue, which commonly arises when dealing with structure-property relations of such materials, is the validity of separation of scales of the continuum mechanical model d < L_{RVE} < L macro. Here L_{RVE} refers to the size of so-called Representative Volume Element (RVE), while d is the microstructural scale (microconstituents’ size) and Lmacro is the macroscale. In the randomly structured media, the RVE is considered to contain a sufficiently large number of micro constituents (e.g. inclusions) so that the volume can be regarded as statistically homogeneous and structurally typical of the mixture. This research focuses on the investigation of the scale-dependent homogenization from a Statistical Volume Element (SVE) (i.e., mesoscale level) to a Representative Volume Element (RVE) (i.e., macroscale level) for linear viscoelastic random composites with perfectly bonded microconstituents. The theoretical framework employed is based on the Hill-Mandel homogenization condition, with adoption to constitutive relations including time derivative for viscoelastic materials. Requiring the material statistics to be spatially homogeneous and ergodic, the mesoscale bounds are obtained by taking the ensemble average of two stochastic initial-boundary value problems set up, respectively, under uniform kinematic and traction boundary conditions. Convergence of mesoscale responses as the scale approaches that of RVE is proved using the extended minimum theorem in viscoelasticity and also numerically verified in both time and frequency domains through computational mechanics of planar random microstructures. The frequency-dependent scaling to RVE is further described through a complex-valued scaling function, which generalizes the concept originally developed for linear elastic random composites. This scaling function is shown to apply for all different phase combinations and essentially, uniquely characterizes the geometric effects of the microstructure in the homogenization trend from SVE to RVE. The transition between elasticity and viscoelasticity in elastic-viscoelastic random composites is also investigated by simulating the microstructures responses at various volume fractions of the viscoelastic phase. Systems with a significantly high contrast between the elastic and viscoelastic phase are being considered. For the random checkerboard, a sharp transition in the microstructures response occurs at the volume fraction 0.4, with the response transitioning from a time-independent type to time-dependent type. This critical volume fraction is consistent with the probability threshold of site percolation and the transition indicates a shift of dominance of one phase over the other. Fractional calculus models are known to be robust descriptors of the behavior of real viscoelastic polymers. By allowing the derivative in viscoelastic constitutive relations to be of the fractional type, the response of a viscoelastic material can be determined using a much smaller number of empirical parameters than is the case with conventional calculus. Compared to a dashpot in classical viscoelasticity, the fractional calculus element spring-pot has the property of a continuously varying constitutive equation from ideal solid state to ideal fluid state by changing the fractional derivative order from 0 to 1. Responses of elastic-viscoelastic composites are investigated under the four-parameter factional models over the full range of volume fractions of the viscoelastic phase. It is found that if the viscoelastic phase is of Zener model (standard solid model of integer order), the best fit of the effective elastic-viscoelastic composites is also of integer order and independent of the volume fraction of viscoelastic phases. This suggests that the microstructural randomness of a composite material is not the cause of the fractional order viscoelasticity, rather it is due to the nature of the constituent phase(s). In other words, in viscoelastic composites, if the component phase is described by classical viscoelasticity of integer order, the responses of the composite are of the same type