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

A physically sound three-dimensional anisotropic formulation of the standar d linear viscoelastic solid with integer or fractional order rate laws for a finite set of the pertinent internal variables is presented. It is shown that the internal variables can be expressed in terms of the strain as conv olution integrals with kernels of Mittag-Leffler function type. A time inte gration scheme, based on the Generalized Midpoint rule together with the Gr unwald algorithm for numerical fractional differentiation, for integration of the constitutive response is developed. The predictive capability of the viscoelastic model for describing creep, relaxation and damped dynamic res ponses is investigated both analytically and numerically. The algorithm and the present general linear viscoelastic model are implemented into the gen eral purpose finite element code Abaqus. The algorithm is then used togethe r with an explicit difference scheme for integration of structural response s. In numerical examples, the quasi-static and damped responses of a viscoe lastic ballast material that is subjected to loads simulating the overrolli ng of a train are investigated. (C) 1999 Elsevier Science Ltd. All rights r eserved.