663
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
On non-coaxial stress-dilatancy theories
Sur les théories de non co-axialité contrainte/dilatance
Anteneh Biru & Thomas Benz
Norwegian University of Science and Technology (NTNU)
ABSTRACT: The influence of non-coaxiality between principal stresses and principal strain increments on mechanical behavior of
soils has been investigated both experimentally and theoretically. In this paper, two non-coaxial stress-dilatancy theories for soils are
considered. The theoretical frameworks are investigated and inconsistencies are pointed out. Then a possible way of reconciling these
inconsistencies is proposed. Furthermore, a semi-empirical evolution equation is proposed for the degree of non-coaxiality.
RÉSUMÉ : L'influence, sur le comportement mécanique des sols, de la non co-axialité entre les contraintes principales et les
déformations principales, est l'objet d'études tant expérimentales que théoriques. Dans le présent article, deux théories de non co-
axialité contrainte/dilatance sont considérées. La structure théorique a été analysée, et certaines divergences ont pu être relevées entre
les deux théories. Une solution pour les concilier est alors proposée. De plus, une équation semi-empirique est proposée pour
exprimer le degré de non co-axialité.
KEYWORDS: non-coaxiality, non-coaxial plastic dissipation, stress-dilatancy
1 INTRODUCTION
Several stress-dilatancy formalisms assume coaxiality between
principal stress and principal plastic strain increments. The two
frequently applied stress-dilatancy formalisms are that follow
from Taylor’s (1948) work hypothesis and Rowe’s (1962)
stress-dilatancy theory. Both assume coaxiality. As shown in
Biru and Benz (2012) the two approaches can be seen from a
common point. In spite of this fact, the two approaches bear
differences.
The possible influence of non-coaxiality on stress-dilatancy
behavior of geomaterials has been first pointed out in de Jong
(1976). Gutierrez and Ishihara (2000) introduced non-coaxiality
into Taylor’s work hypothesis. Later, Gutierrez and Wang
(2009) introduced non-coaxiality to Rowe’s stress-dilatancy
theory. In this paper, the two non-coaxial approaches are
investigated. Differences in the two approaches are pointed out
and a possible way of reconciliation is proposed.
2 ON THE NON-COAXIAL TAYLOR AND ROWE
STRESS DILATNCY RELATIONSHIPS
This section focuses on the non-coaxial Taylor (Gutierrez and
Ishihara 2000) and non-coaxial Rowe (Gutierrez and Wang
2009) stress-dilatancy theories. The two approaches are
investigated and differences are pointed out.
2.1 Non-coaxiality for extended Taylor work hypothesis
The non-coaxial version of extended Taylor work hypothesis
(Gutierrez and Ishihara 2000), for triaxial compression, triaxial
extension and 2D plane strain deformation modes, is given by
s
m
p
p
p
p
M v
q
cv
q
p c q M p
e
e
e
D
= + =
,
(1)
where
p
M
is plastic dissipation and
c
D
is degree of (non-)
coaxiality ,
1 1
3 3
1
3
r
r
p
r r
s s
+
=
+
&
1
3
1 3
3
q
r r
s s
-
=
-
(2)
are mean stress and deviatoric stress respectively where
1
s
is
the major principal stress and
3
s
is the minor principal stress,
i
r
are such that
1 3
1
r r
for plane strain,
1
3
2 2
r r
for
triaxial extension and
1 3
2
2
r r
triaxial compression. The
corresponding work conjugate strain rate measures,
1 1
3 3
p
p
p
v
r
r
e e e
= +
&
1
3
1 3
2
p
p
p
q
r r
e e
e
-
=
+
,
(3)
are volumetric strain rate and deviatoric strain rate, respectively;
and
(
)
1 3
3 1
3 sin
3
sin
s
m
cv
cv
cv
r r
M
r r
j
j
=
- -
,
(4)
where
cv
j
is the friction angle at constant volumetric strain
.
The stress-dilatancy relationship obtained by rearranging Eq.
(1) is of the form
s
m
cv
M M c M
y
s
D
= -
,
(5)
where
p p
v q
M
y
e e
=
is dilatancy ratio and
M q p
s
=
is stress
ratio.
For plane strain condition, Eq. (5) simplifies to
ˆ sin
sin
sin
m
c
m
c
y j
j
D
= -
,
(6)
From this extension, the following points can be noted.
Firstly, in this modification the plastic dissipation remains
unaffected by non-coaxiality (See Figure 1
a
) but the stress-
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
1
On non-coaxial stress-dilatancy theories
Sur les théories de non co-axialité contrainte/dilatance
Biru Tsegaye A., Benz T.
Norw gian Univ rsity of Science and Techn logy (NTNU)
ABSTRACT: The influence of non-coaxiality between principal stresses and principal strain increments on mechanical behavior of
soils has been investigated both experimentally and theoretically. In this paper, two non-coaxial stress-dilatancy theories for soils are
considered. The theoretical frameworks are investigated and inconsistencies are pointed out. Then a possible way of reconciling these
inconsistencies is proposed. Furthermore, a semi-empirical evolution equation is proposed for the degree of non-coaxiality.
RÉSUMÉ : L'influence, sur le comportement mécanique des sols, de la non co-axialité entre les contraintes principales et les
déformations principales, est l'objet d'études tant expérimentales que théoriques. Dans le présent article, deux théories de non co-
axialité contrainte/dilatance sont considérées. La structure théorique a été analysée, et certaines divergences ont pu être relevées entre
les deux théories. Une solution pour les concilier est alors proposée. De plus, une équation semi-empirique est proposée pour
exprimer le degré de non co-axialité.
KEYWORDS: non-co xiality, non-coaxial plastic dissipation, stress-dilatancy
1
INTRODUCTION
Several stress-dilatancy formalisms assume coaxiality between
principal stress and principal plastic strain increments. The two
frequently applied stress-dilatancy formalisms are that follow
from Taylor’s (1948) work hypothesis and Rowe’s (1962)
stress-dilatancy theory. Both assume coaxiality. As shown in
Biru and Benz (2013) the two approaches can be seen from a
common point. In spite of this fact, the two approaches bear
differences.
The possible influence of non-coaxiality on stress-dilatancy
behavior of geomaterials has been first pointed out in de Jong
(1976). Gutierrez and Ishihara (2000) introduced non-coaxiality
into Taylor’s work hypothesis. Later, Gutierrez and Wang
(2009) introduced non-coaxiality to Rowe’s stress-dilatancy
theory. In this paper, the two approaches of incorporating effect
of degree non-coaxiality into plastic dissipation and hence into
stress-dilatancy formalisms are investigated. Differences in the
two approaches are pointed out and a possible way of
reconciliation is proposed.
2
ON THE NON-COAXIAL TAYLOR AND ROWE
STRESS DILATNCY RELATIONSHIPS
This section focuses on the non-coaxial Taylor (Gutierrez and
Ishihara 2000) and non-coaxial Rowe (Gutierrez and Wang
2009) stress-dilatancy theories. The two approaches are
investigated and differences are pointed out.
2.1 Non-coaxiality for extended Taylor work hypothesis
The non-coaxial version of extended Taylor work hypothesis
(Gutierrez and Ishihara 2000), for triaxial compression, triaxial
extension and 2D plane strain deformation modes, is given by
s
m
p
p
p
p
M v
q
cv
q
p c q M p
e
e
e
D
= + =
,
(1)
where
p
M
is plastic dissipation and
c
D
is degree of (non-)
coaxiality,
1 1
3 3
1
3
r
r
p
r r
s s
+
=
+
&
1
3
1 3
3
q
r r
s s
-
=
-
(2)
are mean stress and deviatoric stress respectively where
1
s
is
the major principal stress and
3
s
is the minor principal stress,
i
r
are such that
1 3
1
r r
for plane strain,
1
3
2 2
r r
for
triaxial extension and
1 3
2
2
r r
triaxial compression. The
corresponding work conjugate strain rate measures,
1 1
3 3
p
p
p
v
r
r
e e e
= +
&
1
3
1 3
2
p
p
q
r r
e e
e
-
=
+
,
(3)
are volumetric strain rate and deviatoric strain rate, respectively;
and
(
)
1 3
3 1
3 sin
3
sin
s
m
cv
c
cv
r r
M
r r
j
j
=
- -
,
(4)
where
cv
j
is the friction angle at constant volumetric strain
.
The stress-dilatancy relationship obtained by rearranging Eq.
(1) is of the form
s
m
cv
M M c M
y
s
D
= -
,
(5)
where
p p
v q
M
y
e e
=
is dilatancy ratio and
M q p
s
=
is stress
ratio.
For plane strain condition, Eq. (5) simplifies to
ˆ sin
sin
sin
m
c
m
c
y j
j
D
= -
,
(6)
From this extension, the following points can be noted.
Firstly, in this modification the plastic dissipation remains
unaffected by non-coaxiality (See Figure 1
a
) but the stress-
