Fractional derivative viscoelasticity at large deformations

A time domain viscoelastic model for large three-dimensional responses under isothermal conditions is presented. Internal variables with fractional order evolution equations are used to model the time dependent part of the response. By using fractional order rate laws, the characteristics of the time dependency of many polymeric materials can be described using relatively few parameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does not hold (e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive response is developed and implemented into a finite element code for structural dynamics. The algorithm calculates the fractional derivatives by means of the Grunwald-Lubich approach. Analytical and numerical calculations of the constitutive response in the nonlinear regime are presented and compared. The dynamic structural response of a viscoelastic bar as well as the quasi-static response of a thick walled tube are computed, including both geometrically and materially nonlinear effects. Moreover, it is shown that by applying relatively small load magnitudes, the responses of the linear viscoelastic model are recovered.