Issue 29

G. Maurelli et alii, Frattura ed Integrità Strutturale, 29 (2014) 351-363; DOI: 10.3221/IGF-ESIS.29.31 351 Focussed on: Computational Mechanics and Mechanics of Materials in Italy Mixed methods for viscoelastodynamics and topology optimization Giacomo Maurelli, Nadia Maini, Paolo Venini Department of Civil Engineering and Architecture, University of Pavia, Via Ferrata 3, 27100 Pavia, Italy paolo.venini@unipv.it A BSTRACT . A truly-mixed approach for the analysis of viscoelastic structures and continua is presented. An additive decomposition of the stress state into a viscoelastic part and a purely elastic one is introduced along with an Hellinger-Reissner variational principle wherein the stress represents the main variable of the formulation whereas the kinematic descriptor (that in the case at hand is the velocity field) acts as Lagrange multiplier. The resulting problem is a Differential Algebraic Equation (DAE) because of the need to introduce static Lagrange multipliers to comply with the Cauchy boundary condition on the stress. The associated eigenvalue problem is known in the literature as constrained eigenvalue problem and poses several difficulties for its solution that are addressed in the paper. The second part of the paper proposes a topology optimization approach for the rationale design of viscoelastic structures and continua. Details concerning density interpolation, compliance problems and eigenvalue-based objectives are given. Worked numerical examples are presented concerning both the dynamic analysis of viscoelastic structures and their topology optimization. K EYWORDS . Viscoelasticity; Mixed finite elements; Topology optimization. I NTRODUCTION iscoelasticity is a constitutive feature of materials that finds applications in several areas such as structural engineering (concrete), biomedics (soft tissues) and also synthetic materials and polymers fabrication [1]. From a modeling viewpoint, the most classic approach to viscoelasticity is based on hereditary integral formulations that rely on the interpretation of viscoelastic materials as materials with memory: the current stress is based on a time integral of the strain history. Given such a complex constitutive response, the design of viscoelastic structures is quite a challenging task that calls for a robust numerical model and a rationale design approach. Within this scenario, object of this paper is the proposal of a truly-mixed finite element method for the analysis of viscoelastic structures coupled to a topology optimization approach with the ultimate goal of the development of an integrated analysis-design tool for viscoelastic systems. For this paper's sake, eigenvalue-based objectives shall be pursued within a few numerical applications that are based on the purely elastic cases discussed in [2]. As to the mixed finite-element formulation, the truly-mixed setting in a general Hellinger-Reissner framework has been selected mainly because of the need to pursue an accurate approximation of the stress field, feature that is not shared by any standard displacement-based approach. Furthermore, such a formulation is well known to pass the inf-sup condition [3] even in the limiting case of incompressible materials with no further tricks or enrichments and this could represent a V