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\DOI{10.5802/crmeca.349}
\datereceived{2026-01-16}
\daterevised{2026-02-09}
\dateaccepted{2026-02-10}
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\dateposted{2026-04-03}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{Revisiting intrinsic curve type strength  criteria for yield
design analyses}

\alttitle{Nouvelle approche de la courbe intrins\`{e}que dans le
contexte du calcul \`{a} la rupture}

\author{\firstname{Jean} \lastname{Salen\c{c}on}\CDRorcid{0000-0001-8472-5946}}
\address{Hong Kong Institute for Advanced Study, City University, Hong Kong}
\email{Jean.salencon@academie-sciences.fr}

\keywords{\kwd{Intrinsic curve}
\kwd{Yield criterion}
\kwd{Maximum resisting work}
\kwd{Yield design analyses}
\kwd{Vertical cut}}

\altkeywords{\kwd{Courbe intrins\`{e}que}
\kwd{Crit\`{e}re de r\'{e}sistance}
\kwd{Puissance r\'{e}sistante maximale}
\kwd{Calcul \`{a} la rupture}
\kwd{Tranch\'{e}e verticale}}

\begin{abstract} 
Intrinsic curve type strength criteria for isotropic materials rely on
the assumption that failure of a material element only depends on the
values of the major and minor principal stresses exerted on it. The
intrinsic curve is classically defined as the envelope of limit Mohr
circles corresponding to yielding. It can also be considered as the
envelope of a family of Coulomb's criteria depending on one scalar
parameter. This alternative definition is exploited for providing the
expressions for the maximum resisting work in the general case of a
material with an intrinsic curve type yield criterion, with or without
tension cutoff, both for strain rate tensors and velocity
discontinuities. The application of these results to yield design
analysis problems is discussed from various viewpoints, with one
additional contribution to determining the critical height of a
vertical cut.
\end{abstract} 

\begin{altabstract} 
Les crit\`{e}res de r\'{e}sistance de type courbe intrins\`{e}que pour
les mat\'{e}riaux isotropes reposent sur l'hypoth\`{e}se que la rupture
d'un \'{e}l\'{e}ment de mati\`{e}re est r\'{e}gie par les seules
valeurs des contraintes principales majeure et mineure qui lui sont
appliqu\'{e}es. La courbe intrins\`{e}que est classiquement d\'{e}finie
comme l'enveloppe des cercles de Mohr limites correspondant \`{a} la
rupture. Elle peut \'{e}galement \^{e}tre consid\'{e}r\'{e}e comme
l'enveloppe d'une famille de crit\`{e}res de Coulomb d\'{e}pendant d'un
param\`{e}tre scalaire. Cette d\'{e}finition alternative est
exploit\'{e}e pour obtenir les expressions de la puissance
r\'{e}sistante maximale dans le cas g\'{e}n\'{e}ral d'un mat\'{e}riau
avec crit\`{e}re de r\'{e}sistance de type courbe intrins\`{e}que,
\'{e}ventuellement tronqu\'{e} en traction, \`{a} la fois pour les
tenseurs de vitesse de d\'{e}formation et les discontinuit\'{e}s de
vitesse. L'application de ces r\'{e}sultats aux probl\`{e}mes de calcul
\`{a} la rupture est abord\'{e}e de diff\'{e}rents points de vue,
accompagn\'{e}e d'une contribution suppl\'{e}mentaire \`{a} la
d\'{e}termination de la hauteur critique d'une tranch\'{e}e verticale.
\end{altabstract} 

%\input{CR-pagedemetas}

\editornote{Article submitted by invitation}
\alteditornote{Article soumis sur invitation}

\maketitle

\end{noXML}

\section{Yield criteria of the intrinsic curve type}\label{sec1}
\renewcommand{\theequation}{\thesection.\arabic{equation}}

\subsection{Definition and general results}\label{sec1.1} 
After the experiments reported by Tresca in a series of Memoirs to the
French Academy of Sciences (1864--1870) (cf.~\cite{Salencon2021}), 
where the Tresca yield criterion was introduced together with a first
concept of a plastic flow rule, ``\textit{Mohr}~\cite{Mohr1900}
\textit{proposed a criterion of yielding to the effect that permanent
deformation occurred only when a stress circle touched a certain curve
in the} $(\sigma ,\tau )$ \textit{plane}''~\cite{Hill1950}
(Figure~\ref{fig1}). 

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}The \textit{H\"{u}llkurve},  (Hull curve;
Ref.~\cite{Mohr1900}). (By courtesy of  Professor Samuel Forest.)}
\end{figure}

The original French terminology  ``\textit{courbe intrins\`{e}que}'' is
referred to in Caquot and Kerisel's famous 
textbook~\cite{CaquotKerisel1949} as having been introduced more than
20 years earlier (than 1949) by one of the 
Authors\footnote{``\textit{C'est cette courbe \`{a} laquelle l'un de
nous a donn\'{e} il y a plus de 20 ans le nom de courbe intrins\`{e}que
du mat\'{e}riau dans les conditions des exp\'{e}riences''}. This is the
curve which, one of us, more than 20 years ago, named as the intrinsic
curve of the material within the considered experimental conditions. 
Viz.~\cite{Caquot1935}.}, and Figure~\ref{fig2} shows how the concept of
an \textit{intrinsic curve} appears in the lecture notes of the course
delivered by Caquot at the French \textit{\'{E}cole nationale
sup\'{e}rieure des mines}~\cite{Caquot1924}. 

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}The intrinsic curve as it appears  in
Caquot~\cite{Caquot1924}. (By courtesy of Professor Samuel Forest.)}
\end{figure}

As a matter of fact, 
the basic concept of an intrinsic curve originates
from experimental results obtained with granular materials and other
materials commonly used in civil engineering and considered as
isotropic, which indicate that the Mohr circles\footnote{The Mohr
circle with maximum diameter is classically called \textit{the} Mohr
circle.} corresponding to yielding admit an envelope in the $(\sigma
,\tau )$ plane of the Mohr stress representation.

Assuming this result to be valid for all types of experiments performed
on the material under consideration implies that its domain of
resistance is defined by a criterion that only involves the normal and
tangential components, $\sigma $ and $\tau $, of the stress vector
acting on any facet, whatever its orientation. 

From a mathematical viewpoint, the yield criterion is governed by the
values of the major and minor principal stresses, independently of the
value of the intermediate one. It follows that, the principal stresses
being ordered according to $\sigma _{3}\leq \sigma _{2}\leq \sigma
_{1}$ \footnote{Tensile stresses are counted positive.}, the yield
criterion can be written as a relationship between $\sigma _{1}$ and
$\sigma _{3}$ only 
{\begin{equation}\label{eq1.1}
(\sigma _{1}-\sigma _{3})-g(\sigma _{1}+\sigma _{3})\leq 0
\end{equation}}\unskip
or, equivalently, between $R=(\sigma _{1}-\sigma _{3})/2$ and $\Sigma
=(\sigma _{1}+\sigma _{3})/2$, which are, respectively, the radius of
\textit{the} Mohr circle and the abscissa of its centre in the plane of
the Mohr stress representation, in the form
{\begin{equation}\label{eq1.2}
R\leq R(\Sigma ), 
\end{equation}}\unskip
where $R(\Sigma )=g(2\Sigma )/2$ can be determined experimentally by
means of classical triaxial tests for instance.

Equality in~(\ref{eq1.2}) defines any \textit{limit Mohr circle}
through the positive scalar function $R$ that relates its radius to its
centre abscissa 
{\begin{equation}\label{eq1.3}
R=R(\Sigma ),
\end{equation}}\unskip
with $-\infty < \Sigma \leq \Sigma _{\mathrm{M}}$, where $\Sigma
_{\mathrm{M}}$ stands for the abscissa of the centre of the
\textit{ultimate} limit Mohr circle in the positive direction of the
$\sigma $ axis. It is worth noting that this ultimate limit Mohr circle
defines $T$, the maximum tensile normal stress that can be sustained by
the material (Figure~\ref{fig3}), in the form
{\begin{equation}\label{eq1.4}
T=\Sigma _{\mathrm{M}}+R(\Sigma _{\mathrm{M}})\geq 0. 
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Intrinsic curve and ultimate limit Mohr circle.}
\end{figure}

From its very definition, the intrinsic curve is the envelope of the
limit Mohr circles defined by~(\ref{eq1.3}); it consists of two arcs
that are symmetric about the $\sigma $ axis. In the particular case
when $R(\Sigma _{\mathrm{M}})=0$ these arcs meet on the $\sigma $ axis,
which corresponds to an intrinsic curve ``with a sharp summit'' in the
same way as the Coulomb yield criterion.

The existence of such a real non-degenerated envelope implies that
$R(\Sigma )$ must be a positive, decreasing function of $\Sigma $,
$R'(\Sigma )\leq 0$, with the additional  
condition~\cite[pp.~294--300]{Hill1950},
\cite[pp.~23--30]{HalphenSalencon1987}.
{\begin{equation}\label{eq1.5}
-R'(\Sigma )=
\left| \frac{\mathrm{d}R}{\mathrm{d}\Sigma }\right| < 1. 
\end{equation}}\unskip

The contact point of a limit Mohr circle with the intrinsic curve is
the point $P(\Sigma )$ such that (Figure~\ref{fig3})
{\begin{equation}\label{eq1.6}
\cos \gamma (\Sigma )=
-\mathrm{d}R/\mathrm{d}\Sigma =-R'(\Sigma ).
\end{equation}}\unskip

Assuming $R(\Sigma )$ to be a concave function of its argument, i.e.,
$R''(\Sigma )\leq 0$, implies the convexity of the domain delimited by
(both arcs of) the intrinsic curve \textit{and the ultimate limit Mohr
circle} in the $(\sigma ,\tau )$ plane, and vice versa. This domain
comes out as the \textit{domain of resistance} of the isotropic
material under concern in the $(\sigma ,\tau )$ plane of the Mohr
stress representation. It is usually modelled as unbounded for $\Sigma
\rightarrow -\infty $, as a description of the behaviour of the
material within the range of stress states that can practically be
exerted.

With the tangential and normal stresses corresponding to point
$P(\Sigma )$ being denoted by $\tau (\Sigma )$ and $\sigma (\Sigma )$
respectively, both arcs of the intrinsic curve are described in a
parametric form by~(\ref{eq1.6}) and
{\begin{equation}\label{eq1.7}
\begin{cases}
| \tau (\Sigma )| =
R(\Sigma )\sin \gamma (\Sigma )\\
\sigma (\Sigma )=
\Sigma +R(\Sigma )\cos \gamma (\Sigma )=
\Sigma -R(\Sigma )R'(\Sigma ).
\end{cases} 
\end{equation}}\unskip
 
At point $P(\Sigma )$, the tangent to the intrinsic curve is defined by
{\begin{equation}\label{eq1.8}
\frac{\mathrm{d}\tau }
{\mathrm{d}\sigma }
(\Sigma )=
-\cot \gamma (\Sigma ).
\end{equation}}\unskip

It follows that, at point $P(\Sigma )$, the intrinsic curve is tangent
to the Coulomb intrinsic curve defined by parameters $\phi (\Sigma
)=\rmpi /2-\gamma (\Sigma )> 0$ and $C(\Sigma )$, as indicated in 
Figure~\ref{fig4}, with 
{\begin{equation}\label{eq1.9}
\begin{cases} 
\sin \phi (\Sigma )=
\cos \gamma (\Sigma )=-R'(\Sigma )\\
\displaystyle
C(\Sigma )=
\frac{R(\Sigma )}{\cos \phi (\Sigma )}+
\Sigma \tan \phi (\Sigma ),
\end{cases} 
\end{equation}}\unskip
which corresponds to a cohesion pressure $H(\Sigma )$ 
{\begin{equation}\label{eq1.10}
H(\Sigma)=C(\Sigma)\cot \phi (\Sigma)=
\frac{R(\Sigma )}{\sin \phi (\Sigma )}+
\Sigma =
-\frac{R(\Sigma )}{R'(\Sigma )}+\Sigma 
\end{equation}}\unskip
and Coulomb's criterion being written as
{\begin{equation}\label{eq1.11}
| \tau (\Sigma )| 
-C(\Sigma )+\sigma (\Sigma )\tan \phi (\Sigma )\leq 0
\end{equation}}\unskip
or, equivalently, 
{\begin{equation}\label{eq1.12}
| \tau (\Sigma )| 
-(H(\Sigma )-\sigma (\Sigma ))
\tan \phi (\Sigma )\leq 0.
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Intrinsic curve and Coulomb intrinsic curve
tangent at point $P(\Sigma )$.}
\end{figure}

Positivity of $C(\Sigma )$ and $H(\Sigma )$ implies 
{\begin{equation}\label{eq1.13}
R(\Sigma )-\Sigma R'(\Sigma )> 0.
\end{equation}}\unskip

This condition being fulfilled over the range $-\infty < \Sigma \leq
\Sigma _{\mathrm{M}}$ ensures that the zero-stress state lies inside
the domain of resistance\footnote{Since $R(\Sigma )$ is a concave
function, it is sufficient that $R(\Sigma _{M})-\Sigma _{M}\,R'(\Sigma
_{M})> 0$.}.

\subsection{Parametric description}\label{sec1.2} 
The arcs of the intrinsic curve described in~(\ref{eq1.7}) constitute
the envelope of the family of Coulomb's intrinsic curves, the straight
lines defined by~(\ref{eq1.11}) or~(\ref{eq1.12}), depending on $\Sigma
$ as a parameter, under the condition that
{\begin{equation}\label{eq1.14}
\begin{array}{c}
R(\Sigma )=
(H(\Sigma )-\Sigma )
\sin \phi (\Sigma )
\text{ shall be a positive, 
decreasing,
concave function},\\
\text{with }R(\Sigma _{M})-\Sigma _{M}\,R'(\Sigma _{M})> 0.
\end{array}
\end{equation}}\unskip

Let $\phi _{-\infty }> 0$ denote the value of $\phi (\Sigma )$ when
$\Sigma \rightarrow -\infty $, describing the asymptotic direction of
the intrinsic curve. Hence, as a consequence of~(\ref{eq1.14}), the
parametric definition
{\begin{equation}\label{eq1.15}
\begin{cases} 
\Sigma \mapsto H(\Sigma )\\
\Sigma \mapsto \phi (\Sigma )
\end{cases} 
\end{equation}}\unskip
can be solved as a monotonous decreasing function
{\begin{equation}\label{eq1.16}
H=H(\phi )
\end{equation}}\unskip
over the range $0< \phi _{-\infty }\leq \phi \leq \phi _{\mathrm{M}}$.
It follows that the arcs of the intrinsic curve can also be defined as
the \textit{envelope of the straight lines}
{\begin{equation}\label{eq1.17}
0< \phi _{-\infty }\leq \phi \leq \phi _{\mathrm{M}},\qquad
| \tau | -(H(\phi )-\sigma )\tan \phi =0.
\end{equation}}\unskip

The contact point of the ultimate limit Mohr circle with the intrinsic
curve is $P(\Sigma _{M})$, as shown in Figure~\ref{fig3}, with centre
$\Sigma _{M}$ such that~(\ref{eq1.4}) can now be written as
{\begin{equation}\label{eq1.18}
T-\Sigma _{\mathrm{M}}=
R(\Sigma _{M})=
(H(\Sigma _{\mathrm{M}})-
\Sigma _{\mathrm{M}})\sin \phi (\Sigma _{\mathrm{M}}).
\end{equation}}\unskip

As a result, the domain of resistance in the $(\sigma ,\tau )$ plane is
defined by
{\begin{equation}\label{eq1.19}
\begin{cases} 
\underset{\phi _{-\infty }< \phi \leq \phi (\Sigma _{\mathrm{M}})}
{\mathrm{Max}}
\{|\tau | -(H(\phi )-\sigma )\tan \phi \}\leq 0\\
\underset{\phi (\Sigma _{\mathrm{M}})< \phi \leq \rmpi /2}
{\mathrm{Max}}
\{| \tau | -(T-\Sigma _{\mathrm{M}})\cos \phi \}\leq 0
\end{cases} 
\end{equation}}\unskip
which can be interpreted as follows:

\begin{itemize}
\item 
the first line in~(\ref{eq1.19}) describes the  domain delimited by
both arcs of the intrinsic curve as the intersection of the family of
Coulomb's domains depending on parameter $\phi $, with $\phi _{-\infty
}< \phi \leq \phi (\Sigma _{\mathrm{M}})$, these arcs being generated
as the envelope of the corresponding Coulomb intrinsic curves;
\item 
the second line, where $\phi (\Sigma _{\mathrm{M}})\leq \phi \leq \rmpi
/2$, describes the tangential generation of the circular cap of the
domain of resistance in the $(\sigma ,\tau )$ plane with the limitation
imposed by the maximum tensile normal stress.
\end{itemize}
This domain is obviously convex. 

In the Haigh--Westergaard stress space (cf.~\cite{Mase1970}), with
coordinates the non-ordered principal stresses $\sigma _{i},\sigma
_{j},\sigma _{k}$, the domain of resistance characterised
by~(\ref{eq1.1}) and~(\ref{eq1.4}) is defined by 
{\begin{equation}\label{eq1.20}
f(\underline{\underline{\sigma }})=
\mathrm{Max}\{(\sigma _{i}-\sigma _{j})-
g(\sigma _{i}+\sigma _{j}),
\sigma _{i}-T\mid i,j=1,2,3\}\leq 0. 
\end{equation}}\unskip

Consistently with~(\ref{eq1.19}), it is delimited by the envelope of
the corresponding Coulomb's domains depending on parameter $\phi $,
with $\phi _{-\infty }< \phi \leq \phi (\Sigma _{\mathrm{M}})$, and the
limitation set on the principal stresses by the maximum tensile normal
stress $T=\Sigma _{\mathrm{M}}+R(\Sigma _{\mathrm{M}})$. Its definition
can be written as
{\begin{equation}\label{eq1.21}
\begin{cases} 
T=\Sigma _{\mathrm{M}}+R(\Sigma _{\mathrm{M}}),\\
\phi _{-\infty }< \phi \leq \phi (\Sigma _{\mathrm{M}}),\\
\mathrm{Max}\{\sigma _{i}(1+\sin \phi )-
\sigma _{j}(1-\sin \phi )-
2H(\phi )\sin \phi \mid i,j=1,2,3\}\leq 0\\
\mathrm{Max}\{\sigma _{i}-T\mid i=1,2,3\}\leq 0.
\end{cases} 
\end{equation}}\unskip
Incidentally, it is worth noting that, from this definition, any Mohr
circle such that
{\begin{equation}\label{eq1.22}
\sigma _{1}=T,\quad 
T-2g(\Sigma _{M})< \sigma _{3}< \sigma _{1}
\end{equation}}\unskip
comes out as a \textit{limit} Mohr circle, which is physically
consistent.

\section{Intrinsic curve type strength criteria: maximum resisting
work}\label{sec2}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
Except for the case of Tresca's criterion, intrinsic curve type
strength criteria are not associated with any concept of a constitutive
law, or plastic flow rule, regarding yielding when the criterion is
saturated. Consistently, they are only considered in yield design
analyses, where the mathematical compatibility between the equilibrium
equations for the structure under concern and the resistance of its
constituent material shall be checked, within the framework of the
theory of Yield design~\cite{Salencon1983,Salencon2013}, through the
implementation of static approaches and, more usually, kinematic
approaches. The latter ones call for the definition of the concept of
\textit{maximum resisting work}, whose mathematical expression is only
derived from the yield criterion itself.

\subsection{Definition}\label{sec2.1}
For a material whose domain of resistance is defined by a convex
function $f$ of the stress tensor $\underline{\underline{\sigma }}$ in
the form
{\begin{equation}\label{eq2.1}
f(\underline{\underline{\sigma }})\leq 0,
\end{equation}}\unskip
given a strain rate tensor $\underline{\underline{d}}$, it is recalled
that the \textit{maximum resisting work} that can be developed in
$\underline{\underline{d}}$ by a stress tensor
$\underline{\underline{\sigma }}$ abiding by~(\ref{eq2.1}) is the
volume density $\rmpi (\underline{\underline{d}})$ defined through
{\begin{equation}\label{eq2.2}
\rmpi (\underline{\underline{d}})=
\mathrm{Sup}\{\,\underline{\underline{\sigma }}\colon 
\underline{\underline{d}}\mid 
f(\underline{\underline{\sigma }})\leq 0\},
\end{equation}}\unskip
where the notation `` : '' stands for the doubly contracted product of
two 2nd Rank tensors i.e., $\underline{\underline{\sigma }}\colon
\underline{\underline{d}}=\sigma _{ij}d_{ji}$ (with summation on
repeated indices).

In the same way, in the case of a discontinuity $\underline{V}$, of the
velocity field $\underline{U}$, across a jump surface with outward
normal $\underline{n}$, the \textit{maximum resisting work} that can be
developed by a stress tensor $\underline{\underline{\sigma }}$ abiding
by~(\ref{eq2.1}) is the surface density $\rmpi
(\underline{V},\underline{n})$ defined through
{\begin{equation}\label{eq2.3}
\rmpi (\underline{V},\underline{n})=
\mathrm{Sup}\{\,\underline{V}\cdot
\underline{\underline{\sigma }}\cdot 
\underline{n}\mid 
f(\underline{\underline{\sigma }})\leq 0\}.
\end{equation}}\unskip
From a mathematical viewpoint, within the context of convex analysis,
function $\rmpi (\underline{\underline{d}})$ is the \textit{support
function} of the convex domain defined by
$f(\underline{\underline{\sigma }})\leq 0$. It is obtained as the value
of the product $\underline{\underline{\sigma }}\colon
\underline{\underline{d}}$ at any point $\underline{\underline{\sigma
}}*(\underline{\underline{d}})$ on the convex yield surface defined by
$f(\underline{\underline{\sigma }})=0$, where
$\underline{\underline{d}}$ is oriented along an outward normal:
{\begin{equation}\label{eq2.4}
\rmpi (\underline{\underline{d}})=
\underline{\underline{\sigma }}*
(\underline{\underline{d}})\colon 
\underline{\underline{d}}.
\end{equation}}\unskip

As a result, $\rmpi (\underline{\underline{d}})$ is finite, except when
$\underline{\underline{d}}$ is oriented in a direction along which the
domain $f(\underline{\underline{\sigma }})\leq 0$ is not bounded, in
which case $\rmpi (\underline{\underline{d}})$ is infinite.

\subsection{Maximum resisting work for a material with a Coulomb yield
criterion with tension cutoff}\label{sec2.2}

\vspace*{-1pt}

Coulomb criterion with tension cutoff is the particular case of a
criterion evoked in Section~\ref{sec1}, when the classical Coulomb's
criterion is capped by an ultimate limit Mohr circle with 
\vspace*{-1pt}
{\begin{equation}\label{eq2.5}
\sigma _{1}=
T\geq 0,\Sigma _{\mathrm{M}}=
(T-C\cos \phi )/(1-\sin \phi ). 
\end{equation}}\unskip
In terms of non-ordered principal stresses, this criterion can be
written as 
\vspace*{-1pt}
{\begin{equation}\label{eq2.6}
f(\underline{\underline{\sigma }})=
\mathrm{Max}
\{\sigma _{i}(1+\sin \phi )-
\sigma _{j}(1-\sin \phi )-
2C\cos \phi ,
\sigma _{i}-T\mid i,j=1,2,3\}\leq 0.
\end{equation}}\unskip

Referring to Section~\ref{sec1}, the corresponding domain of resistance
in the $(\sigma ,\tau )$ plane can also be described
through~(\ref{eq1.19}) as
\vspace*{-1pt}
{\begin{equation}\label{eq2.7}
\begin{cases} 
\mathrm{Max}\{| \tau | -C+\sigma \tan \phi \}\leq 0\\
\underset{\phi < \alpha \leq \rmpi /2}{\mathrm{Max}}
\{| \tau | -(T-\Sigma _{\mathrm{M}})\cos \alpha \}\leq 0.
\end{cases} 
\end{equation}}\unskip

The expressions of the maximum resisting work for this criterion can be
found in Salen\c{c}on~\cite{Salencon2013} in the forms
\vspace*{-1pt}
{\begin{equation}\label{eq2.8}
\left\{\begin{array}{rcl}
\underset{\phi ,H,T}{\rmpi}
(\underline{\underline{d}})&=&
+\infty \quad\text{if } \mathrm{tr}\,
\underline{\underline{d}}< 
(| d_{1}| +| d_{2}| +| d_{3}| )\sin \phi \\
\underset{\phi ,H,T}{\rmpi }
(\underline{\underline{d}})&=&
C(| d_{1}| +
| d_{2}| +
| d_{3}| -
\mathrm{tr}\,
\underline{\underline{d}})\tan (\rmpi /4+\phi /2)
\vspace*{4pt}\\
&&+\, \displaystyle
\frac{T}{1-\sin \phi }
(\mathrm{tr}\,
\underline{\underline{d}}-
(| d_{1}| +| d_{2}| +
| d_{3}| )\sin \phi )
\quad\text{if tr}\;
\underline{\underline{d}}\geq 
(| d_{1}| +| d_{2}| +| d_{3}| )\sin \phi 
\end{array} \right.
\end{equation}}\unskip
and
\vspace*{-1pt}
{\begin{equation}\label{eq2.9}
\left\{\begin{array}{rcl}
\underset{\phi ,H,T}{\rmpi }
(\underline{V},\underline{n}) &=& +\infty 
\quad\text{if }\underline{V}\cdot
\underline{n}< | \underline{V}| \sin \phi \\
\underset{\phi ,H,T}{\rmpi }
(\underline{V},\underline{n}) &=&
C(| \underline{V}| -\underline{V}\cdot
\underline{n})\tan (\rmpi /4+\phi /2)
\vspace*{4pt}\\
 &&+\, \displaystyle
\frac{T}{1-\sin \phi }
(\underline{V}\cdot
\underline{n}-
| \underline{V}| 
\sin \phi )
\quad\text{if }\;
\underline{V}\cdot
\underline{n}\geq 
| \underline{V}| \sin \phi.
\end{array} \right.
\end{equation}}\unskip

It is worth noting that Equations~(\ref{eq2.9}) illustrate the result
that, from  definition~(\ref{eq2.3}) and the characteristic property of
\textit{the} Mohr circle, $\underset{\phi ,H,T}{\rmpi}
(\underline{V},\underline{n})$ is the \textit{support function} of the
convex domain delimited, in the Mohr stress representation plane
$(\sigma ,\tau )$, by \textit{the Coulomb intrinsic curve and the
ultimate limit Mohr circle}.

\vspace*{-2pt}

\subsection{Maximum resisting work in the case of an intrinsic curve
type yield criterion}\label{sec2.3}

\vspace*{-1pt}

As recalled in Section~\ref{sec2.1}, for any given value of
$\underline{\underline{d}}$, determining the maximum resisting work
$\underset{\mathrm{IC}}{\rmpi}(\underline{\underline{d}})$ amounts to
looking for a point $\underline{\underline{\sigma
}}*(\underline{\underline{d}})$ on the boundary of the domain of
resistance defined  by~(\ref{eq1.21}), where
$\underline{\underline{d}}$ is oriented along an outward normal. If
such a point exists, then 
\vspace*{-1pt}
{\begin{equation}\label{eq2.10}
\underset{\mathrm{IC}}{\rmpi }
(\underline{\underline{d}})=
\underline{\underline{\sigma }}*
(\underline{\underline{d}})\colon
\underline{\underline{d}}. 
\end{equation}}\unskip

Since $\underline{\underline{\sigma }}*(\underline{\underline{d}})$
lies on the boundary of the domain of resistance, it determines the
Coulomb yield criterion with tension cutoff defined by $\phi $ and
$H(\phi )$, that is tangent to the domain of resistance at this point.
As a result, $\underset{\mathrm{IC}}{\rmpi }
(\underline{\underline{d}})$, the value of the maximum resisting work
for strain rate tensor $\underline{\underline{d}}$, is equal to the
value of $\underset{\phi ,H}{\rmpi } (\underline{\underline{d}})$ for the
tangent Coulomb criterion. Then,  from~(\ref{eq2.8}), we derive that
the corresponding expression of $\underset{\mathrm{IC}}{\rmpi
}(\underline{\underline{d}})$ is governed by the value of
$\mathrm{tr}\,\underline{\underline{d}}/\sum _{i}| d_{i}| $ in the
following way:
\vspace*{-1pt}
{\begin{equation}\label{eq2.11}
\begin{cases}
\displaystyle
\underset{\mathrm{IC}}{\rmpi }
(\underline{\underline{d}})=
+\infty 
\quad\text{if }\phi =
\sin ^{-1}\left(\mathrm{tr}\,\underline{\underline{d}}/
\sum\limits _{i}| d_{i}| \right)< \phi _{-\infty }\\
\displaystyle
\underset{\mathrm{IC}}{\rmpi }
(\underline{\underline{d}})=
\underset{\phi ,H(\phi )}{\rmpi }
(\underline{\underline{d}})=
H(\phi )\mathrm{tr}\,\underline{\underline{d}}
\quad\text{if }\phi _{-\infty }\leq \phi =
\sin ^{-1}\left(\mathrm{tr}\,\underline{\underline{d}}/
\sum\limits _{i}| d_{i}| \right)\leq \phi _{\mathrm{M}}.
\end{cases} 
\end{equation}}\unskip

In the same way, for a velocity jump $\underline{V}$,
$\underset{\mathrm{IC}}{\rmpi } (\underline{V},\underline{n})$ is
governed by the value of $\underline{V}\cdot\underline{n}/|
\underline{V}| $:
{\begin{equation}\label{eq2.12}
\begin{cases}
\underset{\mathrm{IC}}{\rmpi }
(\underline{V},\underline{n})=
+\infty \quad\text{if }\phi =
\sin ^{-1}
(\underline{V}\cdot\underline{n}/| \underline{V}| )<
\phi _{-\infty }\\
\underset{\mathrm{IC}}{\rmpi }
(\underline{V},\underline{n})=
\underset{\phi ,H(\phi )}{\rmpi }
(\underline{V},\underline{n})=
H(\phi )\underline{V}\cdot
\underline{n}\quad\text{if }\phi _{-\infty }\leq 
\phi =\sin ^{-1}
(\underline{V}\cdot\underline{n}/
| \underline{V}| )\leq \phi _{\mathrm{M}}.
\end{cases} 
\end{equation}}\unskip

When $\underline{\underline{d}}$ is oriented along an outward normal to
the ultimate limit Mohr circle, i.e., when 
{\begin{equation}\label{eq2.13}
\phi _{\mathrm{M}}\leq 
\sin ^{-1}
\left(\mathrm{tr}\,\underline{\underline{d}}/
\sum _{i}| d_{i}| \right)\leq \rmpi /2, 
\end{equation}}\unskip
the expressions of $\underset{\mathrm{IC}}{\rmpi
}(\underline{\underline{d}})$ are given 
by~(\ref{eq2.8})--(\ref{eq2.9}) with $\phi =\phi _{\mathrm{M}}$.

Finally, with $T=\Sigma _{\mathrm{M}}+g(\Sigma _{\mathrm{M}})\geq 0$,
we get 
{\begin{equation}\label{eq2.14}
\left\{\begin{array}{rcl}
\displaystyle
\underset{\mathrm{IC},T}{\rmpi }
(\underline{\underline{d}})&=&
\underset{\phi ,H,T}{\rmpi }
(\underline{\underline{d}})+
\infty \quad\text{if }\phi =
\sin ^{-1}
\left(\mathrm{tr}\,\underline{\underline{d}}/
\sum \limits_{i}| d_{i}| \right)< \phi _{-\infty }\\
\displaystyle
\underset{\mathrm{IC},T}{\rmpi }
(\underline{\underline{d}}) &=&
\underset{\phi ,H(\phi )}{\rmpi }
(\underline{\underline{d}})=
H(\phi )\mathrm{tr}\,\underline{\underline{d}}
\quad\text{if }
\phi _{-\infty }\leq \phi =
\sin ^{-1}
\left(\mathrm{tr}\,\underline{\underline{d}}/
\sum\limits _{i}| d_{i}| \right)\leq 
\phi _{\mathrm{M}}\\
\underset{\mathrm{IC},T}{\rmpi }
(\underline{\underline{d}})&=&
\underset{\phi _{\mathrm{M}},H_{\mathrm{M}},T}{\rmpi }
(\underline{\underline{d}})=
C_{\mathrm{M}}
(| d_{1}| +
| d_{2}| +
| d_{3}| -
\mathrm{tr}\,\underline{\underline{d}})
\tan (\rmpi /4+\phi _{\mathrm{M}}/2)
\vspace*{4pt}\\
&&\displaystyle
+\,
\frac{T}{1-\sin \phi _{\mathrm{M}}}
(\mathrm{tr}\,\underline{\underline{d}}-
(| d_{1}| +
| d_{2}| +
| d_{3}| )
\sin \phi _{\mathrm{M}})
\quad\text{if }\mathrm{tr}\,
\underline{\underline{d}}\geq
(| d_{1}| +
| d_{2}| +
| d_{3}| )
\sin \phi _{\mathrm{M}}
\end{array} \right.
\end{equation}}\unskip
and for a velocity jump, 
{\begin{equation}\label{eq2.15}
\left\{\begin{array}{rcl}
\underset{\mathrm{IC},T}{\rmpi }
(\underline{V},\underline{n})&=&
+\infty \quad\text{if }\phi =
\sin ^{-1}
(\underline{V}\cdot
\underline{n}/| \underline{V}| )<
\phi _{-\infty }\\
\underset{\mathrm{IC},T}{\rmpi }
(\underline{V},\underline{n})&=&
\underset{\phi ,H(\phi )}{\rmpi }
(\underline{V},\underline{n})=
H(\phi )\underline{V}\cdot
\underline{n}\quad\text{if }\phi _{-\infty }\leq 
\phi =
\sin ^{-1}
(\underline{V}\cdot
\underline{n}/| \underline{V}| )\leq 
\phi _{\mathrm{M}},\\
\underset{\mathrm{IC},T}{\rmpi }
(\underline{V},\underline{n})&=&
\underset{\phi _{\mathrm{M}},H_{\mathrm{M}},T}{\rmpi }
(\underline{V},\underline{n})=
C_{\mathrm{M}}
(| \underline{V}| -
\underline{V}\cdot
\underline{n})\tan 
(\rmpi /4+\phi _{\mathrm{M}}/2)
\vspace*{4pt}\\
&&\displaystyle
+\,
\frac{T}{1-\sin \phi _{\mathrm{M}}}
(\,\underline{V}\cdot
\underline{n}-
| \underline{V}| \sin \phi _{\mathrm{M}})
\quad\text{if }\,\underline{V}\cdot
\underline{n}\geq
| \underline{V}| \sin \phi _{\mathrm{M}}.
\end{array} \right.
\end{equation}}\unskip
Consistently with the final remark in  Section~\ref{sec2.2}, 
Equation~(\ref{eq2.15}) shows that $\underset{\mathrm{IC}}{\rmpi }
(\underline{V},\underline{n})$ is the  \textit{support function of the
domain delimited by the intrinsic curve and the ultimate limit Mohr
circle in the} $(\sigma ,\tau )$ \textit{plane} 
(Figure~\ref{fig5})\footnote{To obtain a more comprehensive perspective
on this result,  refer to~\cite[pp.~53--55]{Salencon1983}.}.

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Maximum resisting work for a velocity  jump.}
\end{figure}

\section{Applications to Yield design analysis}\label{sec3}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\setcounter{equation}{0}
Yield design analyses for a material with an intrinsic curve criterion
proceed from the implementation of the interior and exterior approaches
of the theory. 

Without any consideration for a tension cutoff, i.e., assuming that the
intrinsic curve exhibits a sharp summit, an abundant literature has
been devoted to plane limit equilibrium solutions, which are based upon
K\"{o}tter's equations~\cite{Kotter1903} for the stress field in the
case of a material with a Coulomb or Tresca yield criterion.

These equations were completed by Mandel~\cite{Mandel1943} in the
general case. They can be used to build up complete statical solutions
to be implemented within the interior approach framework, or partial
statical solutions, without any conclusive relevance in that case. They
can also be the basis for building up ``incomplete solutions'',
following Bishop's terminology~\cite{Bishop1953}, where a plane
velocity field is associated through the (outwards) normality rule with
a partial limit statically admissible stress field. It may be worth
recalling that these solutions fall within the exterior approach
framework of the yield design theory: as first established by Bishop,
the result of the corresponding exterior approach comes out directly
from the partial stress field, without it being necessary to compute
the maximum resisting work developed in the velocity field.

Within the framework of the Haar--Karman 
hypothesis~\cite{HaarvonKarman1909},  similar solutions have been proposed
for axially symmetrical problems, which refer to the counterparts of
K\"{o}tter's equations in the general case of a non-homogeneous
material, with an intrinsic curve type yield criterion with a sharp 
summit~\cite{Berezantzev1952,Salencon1973,Salencon1977}.

The fact that the solutions here above only concern  intrinsic curve
type yield criteria with a sharp summit, essentially (if not uniquely)
Coulomb or Tresca yield criteria, enhances the importance of the
results displayed in  Equations~(\ref{eq2.14}) and~(\ref{eq2.15}). They
open the way to pure kinematical exterior approaches based upon the
design of virtual velocity fields in the general case of a domain of
resistance described  by~(\ref{eq1.19}). Such virtual velocity fields
$\underline{U}$ can be piecewise continuous and continuously
differentiable (strain rate $\underline{\underline{d}}$) with velocity
discontinuities $\underline{V}$ across jump surfaces with normal
$\underline{n}$.

Since the exterior approach of the yield design theory states that the
work by external forces exerted on the considered structure shall not
be superior to the maximum resisting rate of work in any virtual
velocity field, it follows that, for the exterior approach to yield
significant results, only virtual velocity fields that comply with the
condition that $\underset{\mathrm{IC}}{\rmpi
}(\underline{\underline{d}})$ and $\underset{\mathrm{IC}}{\rmpi
}(\underline{V},\underline{n})$, $\underset{\mathrm{IC},T}{\rmpi
}(\underline{\underline{d}})$ and $\underset{\mathrm{IC},T}{\rmpi
}(\underline{V},\underline{n})$ remain finite, shall be considered.
This condition defines \textit{relevant virtual velocity fields} for
the problem, without any reference to any constitutive law whatsoever.
It acts both as a constraint and a guide in devising relevant virtual
velocity fields, for example, when considering rigid body velocity
fields with velocity jump lines.

Tresca and Coulomb criteria with zero-tension cutoff, which means that
$T$ is set to zero  in~(\ref{eq2.5}), were first considered by Drucker
and Prager~\cite{DruckerPrager1952}, Drucker~\cite{Drucker1953} for the
stability analysis of a vertical cut.  Chatzigogos, Pecker and
Salen\c{c}on~\cite{SalenconPecker1995,Chatzigogosetal2007}  also
referred to a Tresca criterion with zero-tension cutoff for the
determination of the ultimate bearing capacity of shallow foundations
under inclined eccentric loads. From a general viewpoint, these
criteria which include cohesionless Coulomb's criterion, illustrate the
concept of strength criteria for which the zero-stress state lies on
the boundary of the domain of resistance, as considered by Fr\'{e}mond
and Fria\`{a}~\cite{FremondFriaa1978}.  For such criteria, it turns out
that, when implementing the exterior approach of the Yield design
theory, the minimization process often results in ``vanishing'' virtual
velocity fields, such as those implemented in 
Drucker~\cite{Drucker1953} or Salen\c{c}on~\cite{Salencon1974}.

Recently, in an attempt to assess the sensitivity of yield design
analyses to the tensile resistance of the constituent material, the
problem of the critical height of a vertical cut was revisited within
the framework of Tresca's and Coulomb's criteria with a non-zero
tension cutoff~\cite{Salencon2024}, implementing kinematical exterior
approaches with the same virtual collapse mechanism as first devised by
Drucker and Prager  (Figure~\ref{fig6}), with
$\underset{\mathrm{IC},T}{\rmpi }(\underline{\underline{d}})$ and
$\underset{\mathrm{IC},T}{\rmpi} (\underline{V},\underline{n})$ expressed
by~(\ref{eq2.14}) and~(\ref{eq2.15}). With $\varepsilon =(e/h)$ as a
non-dimensional geometrical parameter, the description of this virtual
collapse mechanism can be briefly recalled as follows. Zone  \xboxed{3}
remaining motionless, the virtual velocity field $\underline{U}$ in
zone  \xboxed{1} defined as $\Omega A'B'B$ consists of an anticlockwise
rigid body rotational motion, with angular velocity $\omega $ about
point $\Omega $. This implies that zone  \xboxed{1} separates from zone
\xboxed{3} with a velocity discontinuity $\underline{V}(y)$ along the
$\xi $ axis when crossing $\Omega B$, whose magnitude is
{\begin{equation}\label{eq2.16}
V(y)=\omega [h(1-\varepsilon \tan (\rmpi /4+\phi /2))+y]. 
\end{equation}}\unskip

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Drucker--Prager's virtual collapse mechanism.}
\end{figure}

The velocity field is continuous across $\Omega A'$ and across $\Omega
A$. Complying with these boundary conditions, the velocity field in
zone \xboxed{2}, delimited by $\Omega AA'$, is defined as follows:
referring to the $\alpha $ and $\beta $ lines, $\underline{U}$ is
constant along any $\beta $ line and normal to $\Omega A'$ with
magnitude $\omega \,x^{\alpha }$, where $x^{\alpha }$ denotes the
abscissa of the considered $\beta $ line along $\Omega \alpha $. 

Using properly chosen non-dimensional factors, it turned out that the
concept of a vanishing virtual collapse mechanism corresponds to the
fact that the value $\varepsilon _{\mathrm{m}}=(e/h)_{\mathrm{m}}$ that
defines the optimum virtual mechanism, tends to zero with $T$, as shown
in  Figure~\ref{fig7}. 

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig7}$(e/h)_{\mathrm{m}}\tan (\rmpi /4+\phi /2)$ as a
function of $T\tan (\rmpi /4+\phi /2)/2C$.}
\vspace*{10pt}
\end{figure}

In addition, for any value of $T$, an upper bound to the critical
height of the cut was obtained in the form
{\begin{equation}\label{eq3.1}
h_{\mathrm{cr}}\leq 
\frac{C}{\gamma }\tan 
(\rmpi /4+\phi /2)F
\left(\frac{T}{2C}\tan (\rmpi /4+\phi /2)\right)
\end{equation}}\unskip
where function $F$ is depicted in  Figure~\ref{fig8}.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig8}$(\gamma \,h_{\mathrm{cr}}/C) \tan (\rmpi /4-\phi
/2)$ as a function $F$ of $T\tan (\rmpi /4+\phi /2)/2C$.}
\end{figure}

Based upon this result and taking advantage  of~(\ref{eq1.16})
and~(\ref{eq1.19}) as a definition of an intrinsic curve type domain of
resistance with tension cutoff, we derive that, in the case when the
vertical cut is characterized by such a yield criterion, an upper bound
to its critical height can be straightforwardly obtained in the form
{\begin{equation}\label{eq3.2}
h_{\mathrm{cr}}\leq 
\underset{\phi _{\infty \leq \phi \leq {\phi _{\mathrm{M}}}}}
{\mathrm{Inf}}
\left[\frac{H(\phi )\sin \phi }{\gamma (1-\sin \phi )}F
\left(\frac{T(1+\sin \phi )}{2H(\phi )\sin \phi }\right)\right].
\end{equation}}\unskip

In essence, this is the determination of the Coulomb criterion with
maximum normal  stress $T$, which most accurately fits the intrinsic
curve as an upper bound, for the problem under consideration.

\section*{Acknowledgments}
The author is grateful to the reviewers for their valuable comments and
suggestions.

\section*{Declaration of interests}
The author does not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and has
declared no affiliations other than his research organization.

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