Dislocations, Plasticity and Metal Forming: Proceedings of PLASTICITY'03: The Tenth International Symposium on
Plasticity and its Current Applications, A.A. Khan, R. Kazmi and J. Zhou (eds.). Maryland: NEAT Press.
A UNIFIED MULTIAXIAL FORMULATION
TO DESCRIBE YIELDING, PLASTIC POTENTIAL,
AND LIMIT STATES OF ENGINEERING MATERIALS.
Michel Aubertin* and Li Li
Dept. CGM, Ecole Polytechnique
C.P. 6079, Station Centre-Ville, Montréal, Qc, Canada, H3C 3A7
* corresponding author: [email protected]
ABSTRACT- Most existing criteria developed to describe the yield condition, plastic
potential, and limit states of engineering materials present some apparently distinct
characteristics. In this short presentation, it is shown that a physically-based multiaxial
criterion (named MSDPu) can be used, for low and high porosity materials, to capture
their essential features in a three-dimensional stress space. Such unified representation is
very useful to reconcile many of the various approaches used for pressure dependent
criteria.
INTRODUCTION: Over the years, a large number of formulations have been
developed to describe the yield condition, plastic or viscoplastic potential, and limit
(failure) states of engineering materials such as metals, ceramics, concrete, powders,
rocks and soils. Although these formulations all aim at expressing the corresponding
criterion under a simplified mathematical form, appropriate to specific applications, a
diversity of backgrounds and assumptions have been used for their development.
Because of that, many different types of expressions exist, which can sometimes make it
difficult to compare the different approaches. It is shown in this paper that a large portion
of existing criteria, including simple ones such as von Mises and Drucker-Prager, and
more elaborate ones such as those of Schleicher, Gurson, Tvergaard, Lade-Duncan,
DiMaggio-Sandler, Ottosen, and Desai (to name only a few), can be represented by a
unique set of unified equations. This provides a generalized formulation with a criterion
applicable to a wide diversity of materials and loading states. This multiaxial criterion,
named MSDPu, is expressed from the complete stress tensor (or its main invariants). It
can be used for low porosity materials (like metals and hard rocks) and for high porosity
materials (like soils, fills, plasters, and powders). It includes a number of important
features such as a differential uniaxial resistance (tension vs compression), a friction
(pressure, mean stress) dependant surface, a Lode angle dependency in the octahedral
plane of stress space, a cap component to take into account volumetric compressibility
and yielding under hydrostatic compressive loads, and an explicit porosity-dependent
formulation for key model parameters (inspired by the theory of Continuum Damage
Mechanics). Although it has been developed for isotropic media, it can also be rendered
anisotropic for cases where stress orientation may become critical.
570
Dislocations, Plasticity and Metal Forming: Proceedings of PLASTICITY'03: The Tenth International Symposium on
Plasticity and its Current Applications, A.A. Khan, R. Kazmi and J. Zhou (eds.). Maryland: NEAT Press.
PROCEDURES, RESULTS AND DISCUSSION: Using common stress tensor
invariants, the unified multiaxial failure criterion MSDPu can be written as [Aubertin et
al. 2000]:
(1)
J21/2 - F0 Fπ = 0
The two functions F0 and Fπ can be defined as:
[
(
Fπ = b / [b
)
F0 = α 2 I12 − 2a1n I1 + a22n − a3n I1 − I cn
]
]
2 1/ 2
(2)
1/ 2
+ (1 − b 2 )sin 2 (45o − 1.5θ)
(3)
Here, I1 is the first invariant of the stress tensor σij (with compressive stresses taken as
positive); J2 is the second invariant of the deviatoric stress tensor Sij; parameter b is
related to the shape of the failure surface in the π plane (Aubertin et al. 2000); θ
represents the Lode angle defined in the octahedral (π) plane (-30° ≤ θ ≤ 30°):
2
)
(
θ = (1 / 3) sin −1 1.5 3 J 3 / J 23
(4)
where J3 is the third invariant of the deviatoric stress tensor Sij. Parameters α, a1n and a2n
are related to the porosity n and can be obtained from basic material properties: σcn
(positive) and σtn (negative), the (porosity dependent) uniaxial compressive and tensile
strength respectively, and φ, the friction angle on plane surfaces (φ ≅ φr or φb, the residual
or basic friction angle):
α = 2sinφ / 3 (3 − sinφ)
(5)
[
(
]
)(
a1n = (σ cn − | σ tn |) / 2 − σ c2n − (| σ tn | / b) 2 / 6α 2 (σ cn + | σ tn |)
[(
)
]
)
(6)
σ cn + (| σ tn | / b 2 ) / (3(σ cn + | σ tn |) ) − α 2 σ cn | σ tn |
a2n =
(7)
Other material parameters a3n and Icn are introduced to represent yielding and/or failure
under large hydrostatic compressive stresses; the surface closes on the positive side (as a
"cap") of I1 when the stress state exceeds Icn. The use of MacCauley brackets ( 〈 x 〉 = (x +
|x|)/2 ) in Eqn (2) implies that only a positive difference (I1 - Icn) is admissible.
The generalized MSDPu formulation given by Eqns (1) to (7) is an extension of a low
porosity failure criterion developed for rocks by the authors and their collaborators. It
has been extended to include the influence of initial porosity and of other defects, which
are represented by physically based parameters: a1n, a2n, a3n, and Icn. The physical
meaning and identification procedure for these parameters are given in a companion
paper, together with additional relationships for the porosity dependent functions used for
predictive applications.
In Table 1, it is shown graphically how the MSDPu criterion compares to some existing
criteria, commonly used from engineering materials. Other comparisons will also be
presented in the companion paper.
ACKNOWLEDGEMENTS: Part of this work has been financed through grants from
IRSST, NSERC, and participants to the Industrial Polytechnique-UQAT Chair
(http://www.polymtl.ca/enviro-geremi/).
571
Dislocations, Plasticity and Metal Forming: Proceedings of PLASTICITY'03: The Tenth International Symposium on
Plasticity and its Current Applications, A.A. Khan, R. Kazmi and J. Zhou (eds.). Maryland: NEAT Press.
Table 1: Comparison Between MSDPu and Some Existing Criteria
120
I
cn '
I
/I 1n
0
-100
m o d ifie d C a m - c la y
C a m - c la y
α CM
0
0
0
100
Drucker-Prager
M ises-Schlecher
M SDP u
Hoek-Brown
M ohr-C oulomb
I1
200
300
M SDPu
0 .1
H o e k - B ro w n
M o hr_ C o ulo mb
M SDPu
D ruc k e r- P rage r
M ise s- S c hle c her
40
= 2
C S L (w ith M = 1 )
1 /2
80
J2
J2
1 /2
0 .2
= 7
cn '
0 .2
0 .4
400
for I 1 = 0
M SDP u with I cn ' = 7
σx
0 .6
I 1 /I 1 n
0 .8
1
σx ( θ = 30°)
θ = 0°
M SDP u with
I cn ' = 2
θ = -30°
σy
σz
σz
σy
modified
C am-C lay
C am-C lay
0.8
fixed yield surface
yield caps and Sandler
DiMaggio
MSDPu
M SDPu
Gurson (1977)
1 /2
J2
1/2
12
J2
8
0.6
Tvergaard (1981)
Tvergaard and
et NNeedlemeant
eedleman 1984
Tvergaard
0.4
4
0.2
I1
I1
0
-10
-5
M SDP u
DiMaggio
C ap
and Sandler
0
5
10
σx ( θ = 30°)
15
20
25
θ = 0°
0
0
1
2
Tvergaard and
et NNeedlemeant
eedleman
(1984)
σx (θ = 30°)
Gurson
σz
4
θ = 0°
θ = -30°
θ = - 30°
σy
σy
3
Tvergaard (1981)
σz
M SDP u
REFERENCES:
Aubertin M., Li L. and Simon R. 2000, "A Multiaxial Stress Criterion for Short Term and
Long Term Strength of Isotropic Rock Media", Int. J. Rock Mech. Min. Sci. 37, 1169-1193.
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