1. Introduction

Rankine’s law is formulated in terms of main constraints. This formulation assumes the isotropy of the material (see [1] and [2]) Indeed, this condition is necessary to ensure that the radial return method preserves the main directions. Its interest lies in the fact that it simplifies the writing of equations and therefore allows very efficient (because they are almost analytical) resolution methods.

The elastic behavior is purely linear.

The load surface is characterized by three planes in the space of the main stresses \({\mathrm{\sigma }}_{1}\ge {\mathrm{\sigma }}_{2}\ge {\mathrm{\sigma }}_{3}\). Each of these planes is characterized by an equation of the type:

(1.1)\[ {R} _ {i=\ mathrm {1,2,3}}}\ left ({\ mathrm {\ sigma}} ^ {\ text {+}}\ right) = {\ mathrm {\ sigma}}} _ {i}} _ {i}} _ {i}} ^ {\ text {+}}} - {\ mathrm {\ sigma}} _ {t} =0\]

Where \({\mathrm{\sigma }}_{t}\) is the only material data and characterizes the material’s tensile limit. The law is associated.

Notes

\({\sigma }_{1}\mathrm{\ge }{\sigma }_{2}\mathrm{\ge }{\sigma }_{3}\)	Key Constraints (in this order)
\(E\)	Young’s module
\(\nu\)	Poisson’s ratio
\(K\mathrm{=}\frac{E}{3(1\mathrm{-}2\nu )}\)	Elastic Compression Modulus
\(G\mathrm{=}\frac{E}{2(1+\nu )}\)	Elastic shear modulus
\({\mathrm{\sigma }}_{t}\)	Material traction limit
\(p=\frac{{I}_{1}}{3}=\frac{\mathit{trace}\left(\sigma \right)}{3}\)	Average Stress
\(p<0\)	Sign convention for compressive stress
\({\mathrm{\sigma }}^{e}\)	Elastic stress prediction tensor
\(\mathrm{\epsilon }={\mathrm{\epsilon }}^{e}+{\mathrm{\epsilon }}^{p}\)	Total, elastic, and plastic deformation increments tensors
\({\mathrm{\epsilon }}_{v}^{p}=\mathit{trace}\left({\mathrm{\epsilon }}^{p}\right)\)	Volume plastic deformation increment
\({\stackrel{~}{\mathrm{\epsilon }}}^{p}={\mathrm{\epsilon }}^{p}-\frac{{\mathrm{\epsilon }}_{v}^{p}}{3}1\)	Increment in deviatoric plastic deformation
\({\mathrm{\epsilon }}^{p}=\Vert {\stackrel{~}{\mathrm{\epsilon }}}^{p}\Vert =\sqrt{\frac{3}{2}{\stackrel{~}{\mathrm{\epsilon }}}^{p}\mathrm{:}{\stackrel{~}{\mathrm{\epsilon }}}^{p}}\)	Norm of the deviatoric plastic deformation increment, or equivalent deformation increment