Issue 29

G. Maurelli et alii, Frattura ed Integrità Strutturale, 29 (2014) 351-363; DOI: 10.3221/IGF-ESIS.29.31 352 key issue when modeling and designing rubber devices such as base isolators. The adopted formulation resembles the one presented in [4] in that an additive decomposition of the stress field is considered that finds its motivation if the adoption of a Maxwell-type phenomenological model. However, a weak enforcement of the stress tensor symmetry is considered in [4] as opposed to a strong approach adopted herein. The Arnold-Winther finite element is used as to the discretization of the stress field [5]. To the author's knowledge, it has never been used for the analysis of viscoelastic structures whereas a few applications in a purely elastic framework are available in the literature [6]. Optimal design with respect to eigenvalues is a topic that has received much attention in the last decades and [7, 8] and [9] may be cited among the most interesting contributions in this respect. However, optimal design of nonclassically damped viscoelastic structures is far from being a mature topic and further research seems to be in order. To gain insight into such a complex problem, a viscoelastic thin-beam truly-mixed formulation is introduced that has the merit to allow a deep understanding of the spectral properties of the discretized structure so as to end up with convincing optimal topologies. The first three eigenvalues shall be object of optimization and a strict relation between the optimal density distribution and the relevant eigenmode shapes clearly determined. Though preliminary, not many such results are available in the literature and should open the way to more complex results concerning two-dimensional systems. F ORMULATION AND DISCRETIZATION ( CONTINUUM CASE ) oal of this section is the definition of a truly-mixed variational formulation for two-dimensional viscoelastic continua. Based on a parallel phenomenological model, compatibility, equilibrium and viscoelastic laws are written in strong form so as to allow the definition of a truly mixed variational formulation of Hellinger-Reissner type. Eventually, the Arnold-Winther finite-element is introduced along with some technical details concerning its implementation. Figure 1 : Standard solid phenomenological model. Strong form Reference is made to Fig. 1 for the standard viscoelastic solid model that is the basis for the continuum and thin-beam models to be developed hereafter. As usual when adopting Hellinger-Reissner variational principles, compliance tensors relating strains to stresses are introduced that allow one to write 0 0 0 0 1 1 ( ) ( ) E V E A A v A v                (1) where 0 E A and 0 V A are the elastic and viscous compliance tensors of the viscoelastic component, 1 E A is the elastic compliance tensor that is in parallel to the viscoelastic one and v is the velocity field. One should notice that a stress- velocity formulation is being used that presents several advantages over more classical stress-displacement approaches, including the ease with which dynamic effects may be considered in the analysis. Therefore, compatibility relations are written in terms of strain velocities as G