Table of Contents
Textures and Microstructures
Volume 8 –9

A Comprehensive Mathematical Formulation of an Extended Taylor–Bishop–Hill Model Featuring Relaxed Constraints, the Renouard–Wintenberger Theory and a Strain Rate Sensitivity Model

Department of Metallurgy and Materials Engineering, Katholieke Universiteit, Leuven, Belgium

Received 13 May 1987

Copyright © 1988 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The Taylor theory and the Bishop–Hill theory for the plastic deformation of polycrystals are expressed in a mathematical way which makes extensive use of vectors. These vectors represent either plastic strain rate tensors or deviatoric stress tensors, both in a unified five-dimensional stress-strain space. Such formulation permits a unified formulation of both theories, which can then easily be solved by means of linear programming. The computer implementation of this formalism (in Pascal) conserves this mathematical formalism to a high extent.

Relaxed constraints (or “mixed boundary conditions”) can very easily be incorporated in the method. The concept of a “relaxed constraint” is formulated in a much more general way than has ever been done before.

It is not only shown why there are often multiple solutions for the slip rates, but also that such difficulties can arise for the stress state as well. A few methods for making an appropriate choice among these equivalent solutions are explained, one based on the Renouard–Wintenberger theory that proposes a secondary energy criterion, and another that takes strain rate sensitivity effects into account.