Vol. 8, No. 2, 2020

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Modeling the linear dynamics of continuous viscoelastic systems on their infinite-dimensional central subspace

Angelo Luongo and Francesco D’Annibale

Vol. 8 (2020), No. 2, 127–151
Abstract

A metamodel of linear viscoelastic continuum is formulated. Internal variables, of arbitrary number, are introduced to describe the viscous part of the strain, and a wide class of constitutive laws, suggested by rheological models, is considered. The spectral properties of the system are discussed. Based on the separation of the eigenvalues occurring when the viscous moduli are small, the system is reduced to its infinite-dimensional central subspace, on which the steady dynamics takes place. Both the center manifold method and the multiple scales method are used to build the reduced model, which is formulated in terms of the only observable variables. Examples relevant to one-, two-, and three-dimensional continua are worked out to illustrate the theory, in conjunction with the standard three-parameter model and the five-parameter model.

Keywords
continuous viscoelastic metamodel, internal variables, linear dynamics, center manifold, multiple scales method, viscoelastic beam on viscoelastic Winkler soil, viscoelastic plate, viscoelastic Cauchy continuum
Mathematical Subject Classification 2010
Primary: 74D05, 74H10, 74H40, 74H45, 74Q10
Milestones
Received: 21 October 2019
Revised: 9 January 2020
Accepted: 18 February 2020
Published: 19 May 2020

Communicated by Francesco dell'Isola
Authors
Angelo Luongo
Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale
Il Centro Internazionale di Ricerca per la Matematica e Meccanica dei Sistemi Complessi
Università degli Studi dell’Aquila
L’Aquila
Italy
Francesco D’Annibale
Dipartimento di Ingegneria Civile, Edile-Architettura e Ambientale
Il Centro Internazionale di Ricerca per la Matematica e Meccanica dei Sistemi Complessi
Università degli Studi dell’Aquila
L’Aquila
Italy