Comparison of time-domain finite element modelling of viscoelastic structures using an efficient fractional Voigt-Kelvin model of Prony series .

摘要

The thesis centers on time domain modelling of viscoelastic materials. Classical models are compared to models involving fractional derivatives, which are derivatives of an order between 0 and 1. Parameters for classical and fractional order models are found for two materials, polymethylmethacrylate and 3M ISD 112, an acrylic based material sold as a viscoelastic layer by 3M. In both cases, only Prony series with several parameters achieve a good representation over a large frequency range. In the case of 3M ISD 112, a fractional model with only two parameters gives a good representation over a frequency range of three decades, which is often sufficient.; An algorithm based on an approximated definition of the fractional derivative and a trapezoidal rule is described to solve constitutive equations with fractional derivatives. The algorithm is implemented in C and tested against a numerical Laplace inverse for the case of a material submitted to sinusoidal strains. The algorithm gives accurate results and does not require very small steps, which is usually the case for algorithms based on finite differences or Grünwald series.; The algorithm is adapted to the structure of a user subroutine of a commercial finite element package, Samcef, for a six component isotropic tensor. The model assumes a constant bulk modulus and has one fractional derivative of the deviatoric strain. The Jacobian of the constitutive equation with a fractional derivative is derived and implemented. The results from the subroutine are compared satisfactorily to results from the numerical Laplace inverse for a cubic element submitted to sinusoidal strains.; Finally, the different models are tested to represent the experimental behaviour of slewing beams made either of polymethylmethacrylate or steel covered by constrained viscoelastic layers. The classical models give generally a poor representation of the experimental behaviour, except for the Prony series. The fractional model give a representation as satisfactory as the ones obtained with the Prony series, but for a much higher CPU times due to the hereditary nature of the fractional derivative. It is therefore recommended to use Prony series models, unless the data to perform the parameter identification is limited. In that case, the fractional order model becomes interesting despite the higher demands on the CPU time.