Klas Adolfsson

Fractional Derivative Viscoelasticity at Large Deformations

A time domain viscoelastic model for large three-dimensional responses underisothermal conditions is presented. Internal variables with fractional orderevolution equations are used to model the time dependent part of the response. By using fractional order rate laws, the characteristics of the timedependency of many polymeric materials can be described using relatively fewparameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does nothold (e.g., suspensions, vibration isolators and rubber bushings). A numerical algorithm for the constitutive response is developed and implemented into a finite element code forstructural dynamics. The algorithm calculates the fractional derivatives by means of the Grünwald–Lubich approach.Analytical and numerical calculations of the constitutive response in the nonlinearregime are presented and compared. The dynamicstructural response of a viscoelastic bar as well as the quasi-static response of athick walled tube are computed, including both geometrically and materiallynonlinear effects. Moreover, it isshown that by applying relatively small load magnitudes, the responses ofthe linear viscoelastic model are recovered.

Space-time discretization of an integro-differential equation modeling quasi-static fractional-order viscoelasticity

We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind, with a weakly singular kernel in the time variable, and which also involves partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used at each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, which can be used as the basis for an adaptive strategy.

Discretization of integro-differential equations modeling dynamic fractional order viscoelasticity

We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integro-differential equation with a weakly singular convolution kernel. We discretize in the spatial variable by a standard Galerkin finite element method. We prove stability and regularity estimates which show how the convolution term introduces dissipation into the equation of motion. These are then used to prove a priori error estimates. A numerical experiment is included.

ADAPTIVE DISCRETIZATION OF AN INTEGRO-DIFFERENTIAL EQUATION MODELING QUASI-STATIC FRACTIONAL ORDER VISCOELASTICITY

Abstract We study a quasi-static model for viscoelastic materials based on a constitutive equation of fractional order. In the quasi-static case this results in a Volterra integral equation of the second kind with a weakly singular kernel in the time variable involving also partial derivatives of second order in the spatial variables. We discretize by means of a discontinuous Galerkin finite element method in time and a standard continuous Galerkin finite element method in space. To overcome the problem of the growing amount of data that has to be stored and used in each time step, we introduce sparse quadrature in the convolution integral. We prove a priori and a posteriori error estimates, and develop an adaptive strategy based on the a posteriori error estimate.

A LARGE DEFORMATION VISCOELASTIC MODEL WITH FRACTIONAL ORDER RATE LAWS

A time domain viscoelastic model for large three dimensional responses under isothermal conditions is presented. Internal variables with fractional orders evolution equations are used to model the time dependent part of the response. By use fractional orders rate laws, the characteristics of the time dependency of many polymeric materials can be described using relatively few parameters. Moreover, here we take into account that polymeric materials are often used in applications where the small deformations approximation does not hold (e.g, suspensions, vibration isolators and flexible joints). A numerical algorithm for the constitutive response is developed and implemented into a finite element code for structural dynamics. The algorithm calculates the fractional derivatives by means of the Griinwald-Lubich approach. Analytical and numerical comparisons of the constitutive response in the nonlinear regime are presented. The dynamic structural response of a viscoelastic bar is computed, including both geometrically and materially nonlinear effects. Moreover it is shown that by applying relatively small load magnitudes, the responses of the linear viscoelastic model are recovered.