FE Implementation of Viscoelastic Constitutive Stress-Strain Relations Involving Fractional Time Derivatives

Viscoelastic material behavior implies the capability to store a portion of its deformation energy whereas the remaining portion is dissipated, so called material damping. The damping properties of a structure may be modeled locally or globally using differential operators or hereditary integral viscoelastic constitutive equations. Rheological damping models consisting of springs and dashpots result in constitutive stress-strain relations of differential operator type. They are known to have deficiencies when being applied to a broad range of time or frequencies. These drawbacks can only be minimized by using a large number of material parameters. Improved adaptivity with respect to measured constitutive behavior is obtained by the differential operator concept including fractional derivatives, where the theory of fractional derivatives can be considered as an extension of derivatives of integer order. This generalization to any real-order derivative results in non-local operators. Material models involving fractional time derivatives provide good curve-fitting properties, require only few parameters and lead to causal behavior. In addition, the concept of fractional derivatives in conjunction with viscoelastic constitutive equations is physically justified. The implementation of fractional constitutive equations based on the Grünwaldian formulation into an elastic FE code is demonstrated. Parameter identifications for the fractional 3-parameter model in the time domain as well as in the frequency domain are carried out. The identified material model is used to perform an FE analysis of a viscoelastic structure.

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