Abstract
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.
Highlights
The Taylor theory and the Bishop-Hill theory for the plastic deformation of polycrystals have been outlined several times (Kocks, 1970; Gil Sevillano, Van Houtte and Aernoudt, 1980; Van Houtte and Wagner, 1985)
The identification of the active slip systems is accompanied by the calculation of the local plastic stress state, of the local rate of plastic work, of the slip rates on the individual slip systems and of the resulting rotation rate of the crystal orientation
The basic assumptions of the Taylor theory are: 1) All crystallites are subject to the same plastic strain (= the prescribed strain)
Summary
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. The concept of a "relaxed constraint" is formulated in a much more general way than has ever been done before It is shown why there are often multiple solutions for the slip rates, and that such difficulties can arise for the stress state as well.
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