667
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
On the geometry of plastic potential surfaces and isochoric stress paths
Sur la géométrie des surfaces potentielles plastiques et des chemins de contraintes isochores
Anteneh Biru, Thomas Benz & Steinar Nordal
Norwegian university of Science and Technology (NTNU)
ABSTRACT: In this paper, isochoric stress paths are studied considering an elastic perfectly plastic model and a simple hardening
plasticity model. Both associated and non-associated flow rules are considered. The effect of geometry of plastic potential functions
on the evolution of stress paths is illustrated. Plastic potential functions of the Mohr-Coulomb type and the Drucker-Prager type are
considered. Finally, relevant conclusions are given.
RÉSUMÉ : Dans cet article, des chemins de contraintes isochores sont étudiés, en considérant deux modèles: un modèle élastique
parfaitement plastique, et un modèle à simple durcissement plastique. Les lois d'écoulement, tant associées que non-associées, sont
considérées. L'effet de la géométrie des fonctions potentielles plastiques, sur l'évolution des chemins de contraintes, a été illustré. Les
fonctions potentielles plastiques de types Mohr-Coulomb et Drucker-Prager sont considérées. Enfin, des conclusions sont données sur
l'ensemble de ces sujets.
KEYWORDS: geometric non-coaxiality, deviatoric non-associativity, deviatoric plane plots,
1 INTRODUCTION
Several soil models of varying degree of complexity, many of
them within the elastoplastic framework, have been developed.
Often, limited types of deformation modes are plotted for
illustration of model responses. In this paper, two simple
elastoplastic models are considered and various isochoric stress
paths are plotted in a deviatoric plane such that the mobilization
of different stress paths relative to the strain increment direction
is investigated.
The mobilization of stress relative to (plastic) strain increment
is studied in non-coaxiality theories. Findings indicate that (
e.g
.,
Roscoe 1970, Thornton and Zhang 2006, Arthur
et al.
1986):
non-coaxiality vanishes with plastic shear strain
although contradicting reports exist during post
bifurcation deformation states (
e.g
., Vardoulakis and
Georgopoulos 2004, Gutierrez and Vardoulakis 2007).
for an isotropic state, plastic strain increments and stress
paths show reasonable coaxiality.
Similar observations can be inferred from a limited number of
true triaxial tests in literature. For example, for isotropic state
and proportional loading, the tests by Yamada and Ishihara
(1981), and Jafarzadeh
et al.
(2008) show that
at lower mobilizations, for radial proportional loading in
a deviatoric plane, stress paths are also radial. At higher
mobilization strain increment vectors show some
deviation from the radial direction but are reasonably
close (see for example,
Figure 1
).
the total strain rate direction and the plastic strain rate
directions are nearly the same.
stresses and strain increments are hence reasonably
coaxial.
Note: In the following sections stresses and friction angles are
effective.
0°
-15°
-30°
-15°
0°
15°
15°
0°
-15°
-30°
15°
Direction of strain
increment
Failure stress
Figure 1: Deviatoric plots of effective stress path and orientation of
plastic deviatoric strain increment (according to Yamada and Ishihara
1979).
2 ELASTO PLASTIC FRAMEWORK
Generally, in plasticity theory strain rate is additively
decomposed into elastic and plastic such that the stress
increment and the strain increment are related as
ep
=
σ C ε
,
(1)
where
ep
C
is the elastoplastic tangent stiffness tensor and may
be decomposed as
ep
e
p
= +
C C C
,
(2)
where
e
C
is the elastic stiffness tensor and
p
C
is the stiffness
degradation due to plasticity.
Furthermore, it is assumed that a stress state obeys a certain
(yield) function and plastic strains are oriented normal to a
plastic potential function. The plastic flow is distinguished
associated if the potential function is the same as the yield
function. Elastoplastic constitutive models for soil usually
consider plastic potential functions different from the yield
function. The plastic flow rule is then called non-associated
flow rule. As will be shown in the following, geometric
properties of the various surfaces affect the response of these
models.
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
1
On the geometry of plastic potential surfaces and isochoric stress paths
Sur la géométrie des surfaces potentielles plastiques et des chemins de contraintes isochores
Biru Tsegaye A., Benz T.,Nordal S.
Norwegian university of Science and Technology (NTNU)
ABSTRACT: In this paper, isochoric stress paths are studied considering an elastic perfectly plastic model and a simple hardening
plasticity model. Both associated and non-associated flow rules are considered. The effect of geometry of plastic potential functions
on the evolution of stress paths is illustrated. Plastic potential functions of the Mohr-Coulomb type and the Drucker-Prager type are
considered. Finally, relevant conclusions are given.
RÉSUMÉ : Dans cet article, des chemins de contraintes isochores sont étudiés, en considérant deux modèles: un modèle élastique
parfaitement plastique, et un modèle à simple durcissement plastique. Les lois d'écoulement, tant associées que non-associées, sont
considérées. L'effet de la géométrie des fonctions potentielles plastiques, sur l'évolution des chemins de contraintes, a été illustré. Les
fonctions potentielles plastiques de types Mohr-Coulomb et Drucker-Prager sont considérées. Enfin, des conclusions sont données sur
l'ensemble de ces sujets.
KEYWORDS: Mohr-Coulomb yield surface, deviatoric plane plots, deviatoric non-associativity,
1
INTRODUCTION
Several soil models of varying degree of complexity, many of
them within the elastoplastic framework, have been developed.
Often, limited types of deformation modes are plotted for
illustration of model responses. In this paper, two simple
elastoplastic models are considered and various isochoric stress
paths are plotted in a deviatoric plane such that the mobilization
of different stress paths relative to the strain increment direction
is investigated.
The mobilization of stress relative to (plastic) strain increment
is studied in non-coaxiality theories. Findings indicate that (
e.g
.,
Roscoe 1970, Thornton and Zhang 2006, Arthur
et al.
1986):
non-coaxiality vanishes with plastic shear strain
although contradicting reports exist during post
bifurcation deformation states (
e.g
., Vardoulakis and
Georgopoulos 2004, Gutierrez and Vardoulakis 2007).
for an isotropic state, plastic strain increments and stress
paths show reasonable coaxiality.
Additionally useful observations can be inferred from a limited
number of true triaxial tests in literature. For example, for
isotropic state and proportional loading, the tests by Yamada
and Ishihara (1981), and Jafarzadeh
et al.
(2008) show that
at lower mobilizations, for radial proportional loading in
a deviatoric plane, stress paths are also radial. At higher
mobilization strain increment vectors show some
deviation from the radial direction but are reasonably
close (see for example, Figure 1).
the strain rate direction and the plastic strain rate
directions are nearly the same.
stresses and strain increments direction are hence
reasonably coinciding.
Note: In the following sections stresses and friction angles are
effective.
0°
-15°
-30°
-15°
0°
15°
15°
0°
-15°
-30°
15°
Direction of strain
increment
Failure stre s
Figure 1: Plots of effective stress path and orientation of deviatoric
strain increment in the deviatoric plane (according to Yamada and
Ishihara 1979),
Q
= Lode angle.
2
ELASTO PLASTIC FRAMEWORK
Generally, in plasticity theory strain rate is additively
decomposed into elastic and plastic such that the stress
increment and the strain increment are related as
EP
=
σ C ε
,
(1)
where
EP
C
is the elastoplastic tangent stiffness tensor and may
be d compo ed as
EP
E
P
= +
C C C
,
(2)
where
E
C
is the elastic stiffness tensor and
P
C
is the stiffness
deg adation du to plastic ty.
Furthermore, it is assumed that a stress state obeys a certain
(yield) function and plastic strains are oriented normal to a
plastic potential function. The plastic flow is distinguished
associated if the potential function is the same as the yi ld
function. Elastoplastic constitutive models for soil usually
consider plastic potential functions diff rent from the yield
function. The plastic flow rule is then called non-associated
flow rule. As will be shown in the following, ge metric
properties of the various surfaces affect the response f these
models.
