The Society of Rheology: 70th Annual Meeting (Oct 1998) Paper SL7

MODELING OF MATERIALS SHOWING VISCOPLASTIC BEHAVIOR WITH NONLINEAR FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS

Session:	Rheology of Solids
Date:	Wednesday - October 7, 1998
Time:	10:35
Room:	San Carlos II

Speaker:	Alan D. Freed, Materials Division, NASA Lewis Research Center
Authors:	Alan D. Freed, Materials Division, NASA Lewis Research Center Kai Diethelm, Institut fur Mathematik, Universitat Hildesheim

Abstract

Internal state variable theory is used to demonstrate similarities and differences between viscoelastic and viscoplastic material models. The kinematics are for large deformations and are based on Biot's measures for stress and strain. Fractional-order, viscoelastic, material models have been around since the 1930's, and they constitute an important class of viscoelastic models. Cureously, the fractional calculus has not been applied to viscoplastic model development. This paper begins such a development process by examining the role that fractional-order evolution equations might play in viscoplastic material modeling.

To enable such a study to even exist in the first place requires a numerical technique capable of solving systems of nonlinear, fractional-order, differential equations, cast as initial-value problems. To complicate matters, this fractional system is coupled with a system of nonlinear, first-order, differential equations. There is no technique in existence today that can solve such an initial-value problem. Therefore, a second objective of this paper is to present the development of a numerical integrator that can handle systems of differential equations like these.

To investigate the applicability of fractional evolution in viscoplasticity, our numerical method is used to gain solutions to two types of boundary-value problems: the planar elongation of a solid in 1- or 2-D, and the tension\torsion\inflation of a thin-walled tube. The materials selected for modeling purposes include: PVC plastic, aluminum and copper. These model materials were selected because of the large amount of experimental data that exists for them in the literature.

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Developed by Albert Co, The Society of Rheology
Please e-mail suggestions and comments to albertco@umecheme.maine.edu.
Application Version: August 1998