Solution of nonlinear variational problems for generalized fractional viscoelastic models

Generalized fractional viscoelastic models (shortly GFV models) are considered based on the Caputo derivative of fractional order. Constitutive relations of GFV models are given by Volterra hereditary integrals with creep functions described by Prony series replaced with a Mittag-Leffler function. In a quasi-static situation, the time-dependent equilibrium problem is set in the variational formulation. For linear as well as nonlinear material responses, a solution to the GFV model is provided by an analytical formula of convolution with a solution of the corresponding elastic problem. The numerical example is presented in 1D for a creep test under isotropic expansion, which is analyzed with respect to fractional derivative order and non-linearity.