Formulation and integration of the standard linear viscoelastic solid with fractional order rate laws

A physically sound three!dimensional anisotropic formulation of the standard linear viscoelastic solid with integer or fractional order rate laws for a _nite set of the pertinent internal variables is presented[ It is shown that the internal variables can be expressed in terms of the strain as convolution integrals with kernels of Mittag Le/er function type[ A time integration scheme\ based on the Generalized Midpoint rule together with the Gru nwald algorithm for numerical fractional di}erentiation\ for integration of the constitutive response is developed[ The predictive capability of the viscoelastic model for describing creep\ relaxation and damped dynamic responses is investigated both analytically and numerically[ The algorithm and the present general linear viscoelastic model are implemented into the general purpose _nite element code Abaqus[ The algorithm is then used together with an explicit di}erence scheme for integration of structural responses[ In numerical examples\ the quasi!static and damped responses of a viscoelastic ballast material that is subjected to loads simulating the overrolling of a train are investigated