Determination of an image point on a surface based on a $$\pi $$ π plane-based algorithm

Abstract

Constitutive models of particulate materials often rely on distances between the current stress state in stress space and various surfaces. Examples of these surfaces include the bounding surface and the dilatancy surface. This paper proposes a rigorous method for determination of distance to a surface in stress space. It starts by examining operations on stress variables defined in the $$\pi $$ plane. Algorithms for determination of an image point on a surface are then presented as a function of the location of the current stress state with respect to the surface. For points within the surface, the bisection method is used; otherwise, the secant method is used. The paper shows that implementation of the proposed algorithm locates the image point on a surface in stress space with accuracy and rigor, providing an accurate measure of the distance to the surface that can be used in hardening or flow rules.