Model reduction by mean-field homogenization in viscoelastic composites. III. Dual theory

The mean-field homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2013 Int. J. Plasticity 42, 1–13 ( doi:10.1016/j.ijplas.2012.09.005 )) is revisited from the standpoint recently adopted in a companion paper (Idiart MI et al. 2020 Proc. R. Soc. A 20200407 ( doi:10.1098/rspa.2020.0407 )). It is shown that the scheme generates a reduced-order approximation wherein the microscopic kinetics of the composite are described in terms of a finite set of macroscopic forces identified with the phase averages and intraphase covariances of the various microscopic force fields, which can be evaluated by mean-field homogenization techniques. The approximation exhibits a two-potential structure with a convex complementary energy density but a non-convex force potential. The consequential properties of the approximation are exposed and their implications are discussed. The exposition is supplemented by proofs of equivalence between the present scheme and other candidate schemes proposed in the literature for composites with elementary local rheologies of Maxwellian type.

Impact and Bandgap Characteristics of Periodic Rods With Viscoelastic Inserts and Local Resonators

A comprehensive theoretical and experimental study is presented of the bandgap behavior of periodic viscoelastic material (VEM) composites subjected to impact loading. The composites under consideration consist of an assembly of aluminum sections integrated with periodic inserts which are arranged in one-dimensional configurations. The investigated inserts are manufactured either from VEM only or VEM with local resonators (LR). A finite element model (FEM) is developed to predict the dynamics of this class of VEM composites by integrating the dynamics of the solid aluminum sections with those of VEM using the Golla-Hughes-Mctavish (GHM) mini-oscillator approach. The integrated model enables, for the first time, the accurate predictions of the bandgap characteristics of periodic viscoelastic composites unlike previous studies where the viscoelastic damping is modeled using the complex modulus approach with storage modulus and loss factor are assumed constants and independent of the frequency or the unrealistic and physically inaccurate Kelvin–Voigt viscous-damping models. The predictions of the developed FEM are validated against the predictions of the commercial finite element package ansys. Furthermore, the FEM predictions are checked experimentally using prototypes of the VEM composites with VEM and VEM/LR inserts. Comparisons are also established against the behavior of plain aluminum rods in an attempt to demonstrate the effectiveness of the proposed class of composites in mitigation of the structural response under impact loading. Close agreements are demonstrated between the theoretical predictions and the obtained experimental results.