Time domain analytical bounds to the homogenized viscous kernels of linear viscoelastic composites

Abstract

New extremum principles in linear viscoelasticity are derived from general stationarity ones proposed in Carini and Mattei (2015), exploiting suitable selections of the admissible fields in the associated convolutive functionals. These new extremum principles have therefore a restricted validity. Analytical bounds to the homogenized viscous kernels of linear viscoelastic composites are derived in the time domain. In the restricted case of macroscopically isotropic composite materials, six new bounds are obtained from the new extremum principles. These bounds can be derived exploiting the choice of Representative Volume Elements (RVEs) loaded in a purely deviatoric way only. Two strict lower bounds to the homogenized viscous kernels, of the Reuss type, are also derived. One of these was already proposed in Huet (1995), and is valid for generic linear viscoelastic composites under general stress and strain states. The other Reuss-type strict lower bound is new, but has the same limited validity as the first six ones. The new upper bounds for isotropic composites, obtained both in terms of viscous kernels and of their time rates, as well as the strict lower bounds, are extensions to viscoelasticity of the Voigt and Reuss bounds for linear elastic composites. The performance of the obtained bounds is checked by comparison with numerical solutions. It is worth remarking that the use of the new extremum theorems for the purpose of deriving bounds is not possible in a general RVE stress case, i.e., deviatoric plus volumetric. As a consequence, the non-strict bounds holding for the case of deviatoric RVE loading are not valid for the volumetric case. The reason for this difference is not yet fully understood.