Implementation of fractional order viscoelastic models to finite element method

---

Tezin Türü: Yüksek Lisans

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Bilimleri Enstitüsü, Türkiye

Tezin Onay Tarihi: 2019

Tezin Dili: İngilizce

Öğrenci: PARNIAN HESAMMOKRI

Danışman: Ergin Tönük

Özet:

In the latest decades, fractional calculus has been commonly used to define the behavior of viscoelastic materials. Real viscoelastic materials such as rubbers, polymers, soft biological tissues, asphalt mixtures, soils, etc. represent power law creep and relaxation behaviors. In Scientific literature relaxation and creep of this type of material has been modelled, primarily through single and/or linear combinations of exponential functions, in an effort to capture the contributions of both solid and fluid phases. This strategy does not allow experimental findings to fit correctly. In this study, isotropic 3-D constitutive equations are evaluated using the fractional calculus by means of the concept of fading memory for a single spring pot, the fractional Kelvin-Voigt model, and the fractional standard linear solid model to reproduce the actual behavior of these materials. Using the UMAT subroutine in ABAQUS / Standard, a finite element code is developed for each model. To reach the strain and stress history of all fractional models, the Boltzmann superposition concept and the fractional derivatives evaluated by Grünwald-Letnikov were used. Relaxation and creep responses have been obtained for each fractional model and these computational results are compared to analytical results to demonstrate the correctness of the finite element codes. Access to the history of strain and stress at each Gauss point of each component is essential for the implementation of the model in a constructive way which is one of the most important aims of this study which has been reached by developing the finite element code using the Jacobian matrix not the strain energy density function which utilized widely in literature. These codes can describe the transition of the viscoelastic models’ behavior smoothly from rubbery to glassy just by changing the fractional coefficients. It has been shown that using this technique the process of extracting material parameters can be much easier as less coefficients are required compared to other techniques in constitutive models. This study demonstrates that 3D fractional viscoelastic models can be readily and effectively implemented in finite element software.