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

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.