Variational Linear Comparison estimates for the macroscopic response and field statistics of elasto-viscoplastic composites

Abstract

This work provides a generalization of the Variational Linear Comparison method (Ponte Castañeda, 1991) for elasto-viscoplastic (EVP) composites. To this end, use is made of Rayleigh’s least dissipation variational principle, leading to the characterization of the time-incremental macroscopic response of the composites in terms of suitably defined macroscopic Rayleigh potentials. By combining this variational framework with linearization schemes based on the notion of ‘comparison’ linear viscoelastic (LVE) composites, estimates are obtained for the macroscopic constitutive relation of the EVP composite in terms of the constitutive response of the comparison LVE composite, where the viscosities of the phases correspond to the secant viscosities of the phases of the EVP composite, evaluated at the instantaneous values of the second moments of the stress or strain-rate fields in the phases of the LVE comparison composite. The methodology is illustrated for particulate microstructures by application of the estimates of Willis (1977) for the comparison LVE composite, for which the use of time-differential operators can be exploited to generate evolution equations not only for the macroscopic fields but also for the field statistics — including the covariance of the fluctuations (which are not normally available from the correspondence principle). After testing the methodology for LVE composites, results are obtained for the class of EVP composites with particulate microstructures and general compressible, isotropic behavior for the phases. The new method recovers earlier results for the incompressible case, but it is also computationally robust when the phases are compressible (where other approximate methods have significant issues, especially when the phases are not well-ordered or when cyclic loading is applied), and it is the only method to date to be able to recover exactly, by construction, the limiting cases of purely elastic, purely viscoplastic and LVE behaviors.