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Implementation of material laws with fractional derivatives in the finite element methods. (Implementation von Stoffgesetzen mit fraktionalen Ableitungen in die finite Elemente Methode.) (German) Zbl 1205.74022

Summary: The material behavior of polymers and of rubber-like materials can often be classified as linear viscoelastic. Typical phenomena of viscoelastic material are the appearance of creep and relaxation processes. Considering periodic loads, a part of the deformation energy is stored in the material, whereas the rest is dissipated. This phenomenon is called material damping. When describing viscoelastic materials, often rheological models consisting of springs and dashpots are used. They are known to have poor curve-fitting properties over a broad range of time or frequency. In this paper, a generalized rheological element will be deduced using fractional derivatives and the properties of some “fractional” rheological models are discussed. Free decay experiments are carried out in order to perform a parameter identification of a “fractional” 3-parameter model in the frequency domain. The 3d implementation of this model into an FE code is demonstrated and one of the experiments is simulated in order to verify the implementation.

MSC:

74D05 Linear constitutive equations for materials with memory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
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