Abstract
In this paper, we introduce a formulation of fractional constitutive equations for finite element analysis using the reformulated infinite state representation of fractional derivatives. Thereby, the fractional constitutive law is approximated by a high-dimensional set of ordinary differential and algebraic equations describing the relation of internal and external system states. The method is deduced for a three-dimensional linear viscoelastic continuum, for which the hydrostatic and deviatoric stress-strain relations are represented by a fractional Zener model. One- and two-dimensional finite elements are considered as benchmark problems with known closed form solutions in order to evaluate the performance of the scheme.
Highlights
Constitutive Laws Using theThe mechanical behavior of viscoelastic materials is characterized by a combined viscous and elastic material response to external loads and displacements depending on time or frequency
In order to simulate the mechanical behavior of arbitrary viscoelastic structures described by fractional constitutive laws, an associated finite element formulation is required, which has to be combined with a suitable solver for fractional-order differential equations
In order to describe the viscoelastic material behavior of an arbitrary structure by a fractional constitutive law, the above models have to be implemented in the finite element method
Summary
The mechanical behavior of viscoelastic materials is characterized by a combined viscous and elastic material response to external loads and displacements depending on time or frequency. In order to simulate the mechanical behavior of arbitrary viscoelastic structures described by fractional constitutive laws, an associated finite element formulation is required, which has to be combined with a suitable solver for fractional-order differential equations. The authors in [33] derive the three-dimensional generalization of a fractional viscoelastic constitutive law for the isotropic case and introduce a finite element approach solving fractional-order equations of order 2 using the Grünwald-Letnikov scheme. A novel improved infinite state scheme was presented in [41], but is not yet accessible for a finite element formulation of fractional constitutive laws. It is the aim of this paper to fill that gap and combine the scheme in [41] with the finite element method (FEM).
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